Calculus Illustrated. Volume 4: Calculus in Higher Dimensions [4]

This is the fourth volume of the series Calculus Illustrated, a textbook for undergraduate students. Mathematical think

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Table of contents :
Preface
Functions in multidimensional spaces
Multiple variables, multiple dimensions
Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,...
Geometry of distances
Where vectors come from
Vectors in Rn
Algebra of vectors
Convex, affine, and linear combinations of vectors
The magnitude of a vector
Parametric curves
The angles between vectors; the dot product
Projections and decompositions of vectors
Sequences and topology in Rn
The coordinatewise treatment of sequences
Partitions of the Euclidean space
Discrete forms
Parametric curves
Parametric curves
Limits
Continuity
Location - velocity - acceleration
The change and the rate of change: the difference and the difference quotient
The instantaneous rate of change: derivative
Computing derivatives
Properties of difference quotients and derivatives
Compositions and the Chain Rule
What the derivative says about the difference quotient: the Mean Value Theorem
Sums and integrals
The Fundamental Theorem of Calculus
Algebraic properties of sums and integrals
The rate of change of the rate of change
Reversing differentiation: antiderivatives
The speed
The curvature
The arc-length parametrization
Re-parametrization
Lengths of curves
Arc-length integrals: weight
The helix
Functions of several variables
Overview of functions
Linear functions: lines in R2 and planes in R3
An example of a non-linear function
Graphs
Limits
Continuity
The difference and the partial difference quotients
The average and the instantaneous rates of change
Linear approximations and differentiability
Partial differentiation and optimization
The second difference quotient with respect to a repeated variable
The second difference and the difference quotient with respect to mixed variables
The second partial derivatives
The gradient
Overview of differentiation
Gradients vs. vector fields
The change and the rate of change of a function of several variables
The gradient
Algebraic properties of the difference quotients and the gradients
Compositions and the Chain Rule
Differentiation under multiplication and division
The gradient is perpendicular to the level curves
Monotonicity of functions of several variables
Differentiation and anti-differentiation
When is anti-differentiation possible?
When is a vector field a gradient?
The integral
Volumes and the Riemann sums
Properties of the Riemann sums
The Riemann integral over rectangles
The weight as the 3d Riemann sum
The weight as the 3d Riemann integral
Lengths, areas, volumes, and beyond
Outside the sandbox
Triple integrals
The n-dimensional case
The center of mass
Vector fields
What are vector fields?
Motion under forces: a discrete model
The algebra and geometry of vector fields
Summation along a curve: flow and work
Line integrals: work
Sums along closed curves reveal exactness
Path-independence of integrals
How a ball is spun by the stream
The Fundamental Theorem of Discrete Calculus of degree 2
Green's Theorem: the Fundamental Theorem of Calculus for vector fields in dimension 2
Exercises
Exercises: Basic calculus
Exercises: Algebra and geometry
Exercises: Parametric curves
Exercises: Functions of several variables
Exercises: Integrals
Exercises: Vector fields
Examples
Index
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Calculus Illustrated. Volume 4: Calculus in Higher Dimensions [4]

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✶

✷

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚♦ t❤❡ st✉❞❡♥t ▼❛t❤❡♠❛t✐❝s ✐s ❛ s❝✐❡♥❝❡✳ ❏✉st ❛s t❤❡ r❡st ♦❢ t❤❡ s❝✐❡♥t✐sts✱ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛r❡ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❯♥✐✈❡rs❡ ♦♣❡r❛t❡s ❛♥❞ ❞✐s❝♦✈❡r ✐ts ❧❛✇s✳

❲❤❡♥ s✉❝❝❡ss❢✉❧✱ t❤❡② ✇r✐t❡ t❤❡s❡ ❧❛✇s ❛s s❤♦rt st❛t❡♠❡♥ts

❝❛❧❧❡❞ ✏t❤❡♦r❡♠s✑✳ ■♥ ♦r❞❡r t♦ ♣r❡s❡♥t t❤❡s❡ ❧❛✇s ❝♦♥❝❧✉s✐✈❡❧② ❛♥❞ ♣r❡❝✐s❡❧②✱ ❛ ❞✐❝t✐♦♥❛r② ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ✐s ❛❧s♦ ❞❡✈❡❧♦♣❡❞❀ ✐ts ❡♥tr✐❡s ❛r❡ ❝❛❧❧❡❞ ✏❞❡✜♥✐t✐♦♥s✑✳ ❚❤❡s❡ t✇♦ ♠❛❦❡ ✉♣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛rt ♦❢ ❛♥② ♠❛t❤❡♠❛t✐❝s ❜♦♦❦✳ ❚❤✐s ✐s ❤♦✇ ❞❡✜♥✐t✐♦♥s✱ t❤❡♦r❡♠s✱ ❛♥❞ s♦♠❡ ♦t❤❡r ✐t❡♠s ❛r❡ ✉s❡❞ ❛s ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ t❤❡ s❝✐❡♥t✐✜❝ t❤❡♦r② ✇❡ ♣r❡s❡♥t ✐♥ t❤✐s t❡①t✳ ❊✈❡r② ♥❡✇ ❝♦♥❝❡♣t ✐s ✐♥tr♦❞✉❝❡❞ ✇✐t❤ ✉t♠♦st s♣❡❝✐✜❝✐t②✳

❉❡✜♥✐t✐♦♥ ✵✳✵✳✶✿ sq✉❛r❡ r♦♦t ❙✉♣♣♦s❡

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ x✱ s✉❝❤ t❤❛t x2 = a✳

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r

❚❤❡ t❡r♠ ❜❡✐♥❣ ✐♥tr♦❞✉❝❡❞ ✐s ❣✐✈❡♥ ✐♥ ✐t❛❧✐❝s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❛r❡ t❤❡♥ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ◆❡✇ s②♠❜♦❧✐s♠ ♠❛② ❛❧s♦ ❜❡ ✐♥tr♦❞✉❝❡❞✳

❙q✉❛r❡ r♦♦t √

a

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥♦t❛t✐♦♥ ✐s ❢r❡❡❧② ✉s❡❞ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❲❡ ♠❛② ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ✐♥st❛♥❝❡ ♦❢ ❛ ♥❡✇ ❝♦♥❝❡♣t ❡✐t❤❡r ❜❡❢♦r❡ ♦r ❛❢t❡r ✐t ✐s ❡①♣❧✐❝✐t❧② ❞❡✜♥❡❞✳

❊①❛♠♣❧❡ ✵✳✵✳✷✿ ❧❡♥❣t❤ ♦❢ ❞✐❛❣♦♥❛❧ 1 × 1 sq✉❛r❡❄ ❚❤❡ sq✉❛r❡ ✐s ♠❛❞❡ ♦❢ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ ▲❡t✬s ❝❛❧❧ ✐t a✳ ❚❤❡♥✱ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ t❤❡ sq✉❛r❡ ♦❢

❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛

❞✐❛❣♦♥❛❧ ✐s t❤❡✐r s❤❛r❡❞ ❤②♣♦t❡♥✉s❡✳ a ✐s 12 + 12 = 2✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡✿

a2 = 2 . ❲❡ ✐♠♠❡❞✐❛t❡❧② s❡❡ t❤❡ ♥❡❡❞ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✦ ❚❤❡ ❧❡♥❣t❤ ✐s✱ t❤❡r❡❢♦r❡✱

a=

√

2✳

❨♦✉ ❝❛♥ s❦✐♣ s♦♠❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ✇✐t❤♦✉t ✈✐♦❧❛t✐♥❣ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s✱ ❛t ②♦✉r ♦✇♥ r✐s❦✳ ❆❧❧ ♥❡✇ ♠❛t❡r✐❛❧ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ❢❡✇ ❧✐tt❧❡ t❛s❦s✱ ♦r q✉❡st✐♦♥s✱ ❧✐❦❡ t❤✐s✳

❊①❡r❝✐s❡ ✵✳✵✳✸ ❋✐♥❞ t❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s

1✳

❚❤❡ ❡①❡r❝✐s❡s ❛r❡ t♦ ❜❡ ❛tt❡♠♣t❡❞ ✭♦r ❛t ❧❡❛st ❝♦♥s✐❞❡r❡❞✮ ✐♠♠❡❞✐❛t❡❧②✳ ▼♦st ♦❢ t❤❡ ✐♥✲t❡①t ❡①❡r❝✐s❡s ❛r❡ ♥♦t ❡❧❛❜♦r❛t❡✳

❚❤❡② ❛r❡♥✬t✱ ❤♦✇❡✈❡r✱ ❡♥t✐r❡❧② r♦✉t✐♥❡ ❛s t❤❡② r❡q✉✐r❡

✉♥❞❡rst❛♥❞✐♥❣ ♦❢✱ ❛t ❧❡❛st✱ t❤❡ ❝♦♥❝❡♣ts t❤❛t ❤❛✈❡ ❥✉st ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✳ ❆❞❞✐t✐♦♥❛❧ ❡①❡r❝✐s❡ s❡ts ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❛s ✇❡❧❧ ❛s ❛t t❤❡ ❜♦♦❦✬s ✇❡❜s✐t❡✿ ❝❛❧❝✉❧✉s✶✷✸✳❝♦♠✳ ❉♦ ♥♦t st❛rt ②♦✉r st✉❞② ✇✐t❤ t❤❡ ❡①❡r❝✐s❡s✦ ❑❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❡①❡r❝✐s❡s ❛r❡ ♠❡❛♥t t♦ t❡st ✕ ✐♥❞✐r❡❝t❧② ❛♥❞ ✐♠♣❡r❢❡❝t❧② ✕ ❤♦✇ ✇❡❧❧ t❤❡ ❝♦♥❝❡♣ts ❤❛✈❡ ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡r❡ ❛r❡ s♦♠❡t✐♠❡s ✇♦r❞s ♦❢ ❝❛✉t✐♦♥ ❛❜♦✉t ❝♦♠♠♦♥ ♠✐st❛❦❡s ♠❛❞❡ ❜② t❤❡ st✉❞❡♥ts✳

❚♦ t❤❡ st✉❞❡♥t✳✳✳

✸

❲❛r♥✐♥❣✦ 2 √ (−1) = 1✱ 1✱ 1 = 1✳

■♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♦♥❡ sq✉❛r❡ r♦♦t ♦❢

t❤❡r❡ ✐s ♦♥❧②

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝ts ❛❜♦✉t t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ❛r❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡r✳

❚❤❡♦r❡♠ ✵✳✵✳✹✿ Pr♦❞✉❝t ♦❢ ❘♦♦ts ❋♦r ❛♥② t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs

a

b✱

❛♥❞

√

a·

√

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t②✿

b=

√

a·b

❚❤❡ t❤❡♦r❡♠s ❛r❡ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡♦r❡♠s ♠❛② ❝♦♥t❛✐♥ ❢♦r♠✉❧❛s❀ ❛ t❤❡♦r❡♠ s✉♣♣❧✐❡s ❧✐♠✐t❛t✐♦♥s ♦♥ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐t ❝♦♥t❛✐♥s✳

❋✉rt❤❡r♠♦r❡✱ ❡✈❡r② ❢♦r♠✉❧❛ ✐s ❛ ♣❛rt ♦❢ ❛ t❤❡♦r❡♠✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❢♦r♠❡r ✇✐t❤♦✉t

❦♥♦✇✐♥❣ t❤❡ ❧❛tt❡r ✐s ♣❡r✐❧♦✉s✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ♠❡♠♦r✐③❡ ❞❡✜♥✐t✐♦♥s ♦r t❤❡♦r❡♠s ✭❛♥❞ ❢♦r♠✉❧❛s✮✱ ✐♥✐t✐❛❧❧②✳ ❲✐t❤ ❡♥♦✉❣❤ t✐♠❡ s♣❡♥t ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✱ t❤❡ ♠❛✐♥ ♦♥❡s ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❢❛♠✐❧✐❛r ❛s t❤❡② ❝♦♥t✐♥✉❡ t♦ r❡❛♣♣❡❛r ✐♥ t❤❡ t❡①t✳ ❲❛t❝❤ ❢♦r ✇♦r❞s ✏✐♠♣♦rt❛♥t✑✱ ✏❝r✉❝✐❛❧✑✱ ❡t❝✳ ❚❤♦s❡ ♥❡✇ ❝♦♥❝❡♣ts t❤❛t ❞♦ ♥♦t r❡❛♣♣❡❛r ✐♥ t❤✐s t❡①t ❛r❡ ❧✐❦❡❧② t♦ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥❡①t ♠❛t❤❡♠❛t✐❝s ❜♦♦❦ t❤❛t ②♦✉ r❡❛❞✳ ❨♦✉ ♥❡❡❞ t♦✱ ❤♦✇❡✈❡r✱ ❜❡ ❛✇❛r❡ ♦❢ ❛❧❧ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❛♥❞ ❜❡ ❛❜❧❡ t♦ ✜♥❞ t❤❡ r✐❣❤t ♦♥❡ ✇❤❡♥ ♥❡❝❡ss❛r②✳ ❖❢t❡♥✱ ❜✉t ♥♦t ❛❧✇❛②s✱ ❛ t❤❡♦r❡♠ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ t❤♦r♦✉❣❤ ❛r❣✉♠❡♥t ❛s ❛ ❥✉st✐✜❝❛t✐♦♥✳

Pr♦♦❢✳ ❙✉♣♣♦s❡

A=

√

a

❛♥❞

B=

√

b✳

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

a = A2

❛♥❞

b = B2 .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

❍❡♥❝❡✱

√

a · b = A2 · B 2 = A · A · B · B = (A · B) · (A · B) = (AB)2 . ab = A · B ✱

❛❣❛✐♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✳

❙♦♠❡ ♣r♦♦❢s ❝❛♥ ❜❡ s❦✐♣♣❡❞ ❛t ✜rst r❡❛❞✐♥❣✳ ■ts ❤✐❣❤❧② ❞❡t❛✐❧❡❞ ❡①♣♦s✐t✐♦♥ ♠❛❦❡s t❤❡ ❜♦♦❦ ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❢♦r s❡❧❢✲st✉❞②✳ ■❢ t❤✐s ✐s ②♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ♠② s✉❣❣❡st✐♦♥s✳ ❲❤✐❧❡ r❡❛❞✐♥❣ t❤❡ ❜♦♦❦✱ tr② t♦ ♠❛❦❡ s✉r❡ t❤❛t ②♦✉ ✉♥❞❡rst❛♥❞ ♥❡✇ ❝♦♥❝❡♣ts ❛♥❞ ✐❞❡❛s✳ ❤♦✇❡✈❡r✱ t❤❛t s♦♠❡ ❛r❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛t ♦t❤❡rs❀ t❤❡② ❛r❡ ♠❛r❦❡❞ ❛❝❝♦r❞✐♥❣❧②✳

❑❡❡♣ ✐♥ ♠✐♥❞✱

❈♦♠❡ ❜❛❝❦ ✭♦r ❥✉♠♣

❢♦r✇❛r❞✮ ❛s ♥❡❡❞❡❞✳ ❈♦♥t❡♠♣❧❛t❡✳ ❋✐♥❞ ♦t❤❡r s♦✉r❝❡s ✐❢ ♥❡❝❡ss❛r②✳ ❨♦✉ s❤♦✉❧❞ ♥♦t t✉r♥ t♦ t❤❡ ❡①❡r❝✐s❡ s❡ts ✉♥t✐❧ ②♦✉ ❤❛✈❡ ❜❡❝♦♠❡ ❝♦♠❢♦rt❛❜❧❡ ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✳ ❲❤❛t t♦ ❞♦ ❛❜♦✉t ❡①❡r❝✐s❡s ✇❤❡♥ s♦❧✉t✐♦♥s ❛r❡♥✬t ♣r♦✈✐❞❡❞❄ ❋✐rst✱ ✉s❡ t❤❡ ❡①❛♠♣❧❡s✳ ▼❛♥② ♦❢ t❤❡♠ ❝♦♥t❛✐♥ ❛ ♣r♦❜❧❡♠ ✕ ✇✐t❤ ❛ s♦❧✉t✐♦♥✳ ❚r② t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✕ ❜❡❢♦r❡ ♦r ❛❢t❡r r❡❛❞✐♥❣ t❤❡ s♦❧✉t✐♦♥✳ ❨♦✉ ❝❛♥ ❛❧s♦ ✜♥❞ ❡①❡r❝✐s❡s ♦♥❧✐♥❡ ♦r ♠❛❦❡ ✉♣ ②♦✉r ♦✇♥ ♣r♦❜❧❡♠s ❛♥❞ s♦❧✈❡ t❤❡♠✦ ■ str♦♥❣❧② s✉❣❣❡st t❤❛t ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② ✇r✐tt❡♥✳ ❨♦✉ s❤♦✉❧❞ ✇r✐t❡ ✐♥ ❝♦♠♣❧❡t❡ s❡♥t❡♥❝❡s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿

❲r♦♥❣✿

1+1 2

❘✐❣❤t✿

1+1 =2

✹

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚❤❡ ❧❛tt❡r r❡❛❞s ✏♦♥❡ ❛❞❞❡❞ t♦ ♦♥❡ ✐s t✇♦✑✱ ✇❤✐❧❡ t❤❡ ❢♦r♠❡r ❝❛♥♥♦t ❜❡ r❡❛❞✳ ❨♦✉ s❤♦✉❧❞ ❛❧s♦ ❥✉st✐❢② ❛❧❧ ②♦✉r st❡♣s ❛♥❞ ❝♦♥❝❧✉s✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿ ❲r♦♥❣✿

2x = 4 x=2

❘✐❣❤t✿

2x = 4 ; x = 2.

t❤❡r❡❢♦r❡✱

❚❤❡ st❛♥❞❛r❞s ♦❢ t❤♦r♦✉❣❤♥❡ss ❛r❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ❜♦♦❦✳ ◆❡①t✱ ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② r❡❛❞✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s❡❧❢✲❝r✐t✐❝✐s♠✿ ▲♦♦❦ ❢♦r ❡rr♦rs ❛♥❞ ✇❡❛❦ s♣♦ts✳ ■t s❤♦✉❧❞ ❜❡ r❡✲r❡❛❞ ❛♥❞ t❤❡♥ r❡✇r✐tt❡♥✳ ❖♥❝❡ ②♦✉ ❛r❡ ❝♦♥✈✐♥❝❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❝♦rr❡❝t ❛♥❞ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✐s s♦❧✐❞✱ ②♦✉ ♠❛② s❤♦✇ ✐t t♦ ❛ ❦♥♦✇❧❡❞❣❡❛❜❧❡ ♣❡rs♦♥ ❢♦r ❛ ♦♥❝❡✲♦✈❡r✳ ◆❡①t✱ ②♦✉ ♠❛② t✉r♥ t♦ ♠♦❞❡❧✐♥❣ ♣r♦❥❡❝ts✳ ❙♣r❡❛❞s❤❡❡ts ✭▼✐❝r♦s♦❢t ❊①❝❡❧ ♦r s✐♠✐❧❛r✮ ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ ✉s❡❞ ❢♦r ❣r❛♣❤✐♥❣ ❛♥❞ ♠♦❞❡❧✐♥❣✳ ❖♥❡ ❝❛♥ ❛❝❤✐❡✈❡ ❛s ❣♦♦❞ r❡s✉❧ts ✇✐t❤ ♣❛❝❦❛❣❡s s♣❡❝✐✜❝❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r t❤❡s❡ ♣✉r♣♦s❡s✱ ❜✉t s♣r❡❛❞s❤❡❡ts ♣r♦✈✐❞❡ ❛ t♦♦❧ ✇✐t❤ ❛ ✇✐❞❡r s❝♦♣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✳ ♦♣t✐♦♥✳ ●♦♦❞ ❧✉❝❦✦ ❆✉❣✉st ✽✱ ✷✵✷✵

Pr♦❣r❛♠♠✐♥❣ ✐s ❛♥♦t❤❡r

❚♦ t❤❡ t❡❛❝❤❡r

✺

❚♦ t❤❡ t❡❛❝❤❡r ❚❤❡ ❜✉❧❦ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡ ❜♦♦❦ ❝♦♠❡s ❢r♦♠ ♠② ❧❡❝t✉r❡ ♥♦t❡s✳ ❚❤❡r❡ ✐s ❧✐tt❧❡ ❡♠♣❤❛s✐s ♦♥ ❝❧♦s❡❞✲❢♦r♠ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥s✳ ■ ❞♦ t❤✐♥❦ t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ✐♥t❡❣r❛t❡❞ ❜② ❤❛♥❞ ✭♦r ❞✐✛❡r❡♥t✐❛t❡❞✱ ♦r ❛♣♣❧✐❡❞ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛✱ ❡t❝✳✮ ❝❛♥♥♦t ♣♦ss✐❜❧② ✉♥❞❡rst❛♥❞ ✐♥t❡❣r❛t✐♦♥ ✭♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ♦r q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ❛ ❧❛r❣❡ ♣r♦♣♦rt✐♦♥ ♦❢ t✐♠❡ ❛♥❞ ❡✛♦rt ❝❛♥ ❛♥❞ s❤♦✉❧❞ ❜❡ ❞✐r❡❝t❡❞ t♦✇❛r❞✿

•

✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ❛♥❞

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♠♦❞❡❧✐♥❣ ✐♥ r❡❛❧✐st✐❝ s❡tt✐♥❣s✳

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✐t r❡q✉✐r❡s ♠♦r❡ ❛❜str❛❝t✐♦♥ r❛t❤❡r t❤❛♥ ❧❡ss✳ ❱✐s✉❛❧✐③❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ t♦♦❧ ✉s❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤✐s ❝❤❛❧❧❡♥❣❡✳ ■❧❧✉str❛t✐♦♥s ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❡✈❡r② ❝♦♥❝❡♣t✱ ❜✐❣ ♦r s♠❛❧❧✳ ❚❤❡ ♣✐❝t✉r❡s t❤❛t ❝♦♠❡ ♦✉t ❛r❡ s♦♠❡t✐♠❡s ✈❡r② ♣r❡❝✐s❡ ❜✉t s♦♠❡t✐♠❡s s❡r✈❡ ❛s ♠❡r❡ ♠❡t❛♣❤♦rs ❢♦r t❤❡ ❝♦♥❝❡♣ts t❤❡② ✐❧❧✉str❛t❡✳ ❚❤❡ ❤♦♣❡ ✐s t❤❛t t❤❡② ✇✐❧❧ s❡r✈❡ ❛s ✈✐s✉❛❧ ✏❛♥❝❤♦rs✑ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✇♦r❞s ❛♥❞ ❢♦r♠✉❧❛s✳ ■t ✐s ✉♥❧✐❦❡❧② t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ♣❧♦tt❡❞ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ❤❛♥❞ ❝❛♥ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ♦r ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ♣❧♦t ♠♦r❡ t❤❛♥ ❥✉st ❛ ❢❡✇ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ ❝✉r✈❡s✱ s✉r❢❛❝❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❡t❝✳❄

❙♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ❣r❛♣❤✐❝ ❝❛❧❝✉❧❛t♦rs ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥ ♣✉r♣♦s❡s

❜❡❝❛✉s❡ t❤❡② r❡♣r❡s❡♥t t❤❡ s❤♦rt❡st st❡♣ ❛✇❛② ❢r♦♠ ♣❡♥ ❛♥❞ ♣❛♣❡r✳

■♥❞❡❡❞✱ t❤❡ ❞❛t❛ ✐s ♣❧♦tt❡❞ ✐♥ t❤❡

s✐♠♣❧❡st ♠❛♥♥❡r ♣♦ss✐❜❧❡✿ ♦♥❡ ❝❡❧❧ ✲ ♦♥❡ ♥✉♠❜❡r ✲ ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✳ ❋♦r ♠♦r❡ ❛❞✈❛♥❝❡❞ t❛s❦s s✉❝❤ ❛s ♠♦❞❡❧✐♥❣✱ s♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ♦t❤❡r s♦❢t✇❛r❡ ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ♦♣t✐♦♥s ❢♦r t❤❡✐r ✇✐❞❡ ❛✈❛✐❧❛❜✐❧✐t② ❛♥❞✱ ❛❜♦✈❡ ❛❧❧✱ t❤❡✐r s✐♠♣❧✐❝✐t②✳ ◆✐♥❡ ♦✉t ♦❢ t❡♥✱ t❤❡ s♣r❡❛❞s❤❡❡t s❤♦✇♥ ✇❛s ✐♥✐t✐❛❧❧② ❝r❡❛t❡❞ ❢r♦♠ s❝r❛t❝❤ ✐♥ ❢r♦♥t ♦❢ t❤❡ st✉❞❡♥ts ✇❤♦ ✇❡r❡ ❧❛t❡r ❛❜❧❡ t♦ ❢♦❧❧♦✇ ♠② ❢♦♦tst❡♣s ❛♥❞ ❝r❡❛t❡ t❤❡✐r ♦✇♥✳ ❆❜♦✉t t❤❡ t❡sts✳ ❚❤❡ ❜♦♦❦ ✐s♥✬t ❞❡s✐❣♥❡❞ t♦ ♣r❡♣❛r❡ t❤❡ st✉❞❡♥t ❢♦r s♦♠❡ ♣r❡❡①✐st✐♥❣ ❡①❛♠❀ ♦♥ t❤❡ ❝♦♥tr❛r②✱ ❛ss✐❣♥♠❡♥ts s❤♦✉❧❞ ❜❡ ❜❛s❡❞ ♦♥ ✇❤❛t ❤❛s ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡ st✉❞❡♥ts✬ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♥❡❡❞s t♦ ❜❡ t❡st❡❞ ❜✉t✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥❞✐r❡❝t❧②✳ ❚❤❡r❡❢♦r❡✱ ❛ ❝❡rt❛✐♥ s❤❛r❡ ♦❢ r♦✉t✐♥❡✱ ♠❡❝❤❛♥✐❝❛❧ ♣r♦❜❧❡♠s ✐s ✐♥❡✈✐t❛❜❧❡✳ ◆♦♥❡t❤❡❧❡ss✱ ♥♦ t♦♣✐❝ ❞❡s❡r✈❡s ♠♦r❡ ❛tt❡♥t✐♦♥ ❥✉st ❜❡❝❛✉s❡ ✐t✬s ❧✐❦❡❧② t♦ ❜❡ ♦♥ t❤❡ t❡st✳ ■❢ ❛t ❛❧❧ ♣♦ss✐❜❧❡✱ ❞♦♥✬t ♠❛❦❡ t❤❡ st✉❞❡♥ts ♠❡♠♦r✐③❡ ❢♦r♠✉❧❛s✳ ■♥ t❤❡ ♦r❞❡r ♦❢ t♦♣✐❝s✱ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s t❡①t❜♦♦❦ ✐s t❤❛t s❡q✉❡♥❝❡s ❝♦♠❡ ❜❡❢♦r❡ ❡✈❡r②t❤✐♥❣ ❡❧s❡✳ ❚❤❡ r❡❛s♦♥s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

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❙❡q✉❡♥❝❡s ❛r❡ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s✳

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▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ❛r❡ s✐♠♣❧❡r t❤❛♥ ❧✐♠✐ts ♦❢ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥s ✭✐♥❝❧✉❞✐♥❣ t❤❡ ♦♥❡s ❛t ✐♥✜♥✐t②✮✳

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❚❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♠❛❦❡ ♠♦r❡ s❡♥s❡ t♦ ❛ st✉❞❡♥t ✇✐t❤

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❆ q✉✐❝❦ tr❛♥s✐t✐♦♥ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ♦❢t❡♥ ❧❡❛❞s t♦ ❝♦♥❢✉s✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✳

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❙❡q✉❡♥❝❡s ❛r❡ ♥❡❡❞❡❞ ❢♦r ♠♦❞❡❧✐♥❣✱ ✇❤✐❝❤ s❤♦✉❧❞ st❛rt ❛s ❡❛r❧② ❛s ♣♦ss✐❜❧❡✳

❛ s♦❧✐❞ ❜❛❝❦❣r♦✉♥❞ ✐♥ s❡q✉❡♥❝❡s✳

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s

✻

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ■t✬s ♥♦ s❡❝r❡t t❤❛t ❛ ✈❛st ♠❛❥♦r✐t② ♦❢ ❝❛❧❝✉❧✉s st✉❞❡♥ts ✇✐❧❧ ♥❡✈❡r ✉s❡ ✇❤❛t t❤❡② ❤❛✈❡ ❧❡❛r♥❡❞✳ P♦♦r ❝❛r❡❡r ❝❤♦✐❝❡s ❛s✐❞❡✱ ❛ ❢♦r♠❡r ❝❛❧❝✉❧✉s st✉❞❡♥t ✐s ♦❢t❡♥ ✉♥❛❜❧❡ t♦ r❡❝♦❣♥✐③❡ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ✐s s✉♣♣♦s❡❞ t♦ s✉rr♦✉♥❞ ❤✐♠✳ ❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✳ ❋r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✱ ✐ts ♣❡❝✉❧✐❛r ❝❤❛❧❧❡♥❣❡ ❤❛s ❜❡❡♥ t♦ st✉❞② ❛♥❞

❝♦♥t✐♥✉♦✉s ❝❤❛♥❣❡✿ ❝✉r✈❡s ❛♥❞ ♠♦t✐♦♥ ❛❧♦♥❣ ❝✉r✈❡s✳ ❢♦r♠✉❧❛s✳ ❙❦✐❧❧❢✉❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ t❤♦s❡ ❢♦r♠✉❧❛s ✐s ✇❤❛t

♠❡❛s✉r❡

❚❤❡s❡ ❝✉r✈❡s ❛♥❞ t❤✐s ♠♦t✐♦♥ ❛r❡ r❡♣r❡s❡♥t❡❞

❜②

s♦❧✈❡s ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s✳ ❋♦r ♦✈❡r ✸✵✵ ②❡❛rs✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳

❚❤❡ s✉❝❝❡ss❡s ❛r❡ ✇❡❧❧✲❦♥♦✇♥✿

♣r♦❥❡❝t✐❧❡ ♠♦t✐♦♥✱ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✱ ✢♦✇ ♦❢ ❧✐q✉✐❞s✱ ❤❡❛t tr❛♥s❢❡r✱ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✱ ❡t❝✳ ❚❡❛❝❤✐♥❣ ❝❛❧❝✉❧✉s ❢♦❧❧♦✇s t❤✐s ❛♣♣r♦❛❝❤✿ ❆♥ ♦✈❡r✇❤❡❧♠✐♥❣ ♠❛❥♦r✐t② ♦❢ ✇❤❛t t❤❡ st✉❞❡♥t ❞♦❡s ✐s ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❇✉t t❤✐s ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ♣r♦❜❧❡♠s t❤❡ st✉❞❡♥t ❢❛❝❡s ✇❡r❡ ✭♦r ❝♦✉❧❞ ❤❛✈❡ ❜❡❡♥✮ s♦❧✈❡❞ ✐♥ t❤❡ ✶✽t❤ ♦r ✶✾t❤ ❝❡♥t✉r✐❡s✦ ❚❤✐s ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ s✐♥❝❡ t❤❡♥❄ ❚❤❡ ❝♦♠♣✉t❡rs ❤❛✈❡ ❛♣♣❡❛r❡❞✱ ♦❢ ❝♦✉rs❡✱ ❛♥❞ ❝♦♠♣✉t❡rs ❞♦♥✬t ♠❛♥✐♣✉❧❛t❡ ❢♦r♠✉❧❛s✳

❚❤❡② ❞♦♥✬t ❤❡❧♣ ✇✐t❤ s♦❧✈✐♥❣ ✕ ✐♥ t❤❡ tr❛❞✐t✐♦♥❛❧ s❡♥s❡ ♦❢

t❤❡ ✇♦r❞ ✕ t❤♦s❡ ♣r♦❜❧❡♠s ❢r♦♠ t❤❡ ♣❛st ❝❡♥t✉r✐❡s✳

✐♥❝r❡♠❡♥t❛❧

■♥st❡❛❞ ♦❢

❝♦♥t✐♥✉♦✉s✱

❝♦♠♣✉t❡rs ❡①❝❡❧ ❛t ❤❛♥❞❧✐♥❣

♣r♦❝❡ss❡s✱ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❢♦r♠✉❧❛s t❤❡② ❛r❡ ❣r❡❛t ❛t ♠❛♥❛❣✐♥❣ ❞✐s❝r❡t❡ ✭❞✐❣✐t❛❧✮ ❞❛t❛✳ ❚♦ ✉t✐❧✐③❡

t❤❡s❡ ❛❞✈❛♥t❛❣❡s✱ s❝✐❡♥t✐sts ✏❞✐s❝r❡t✐③❡✑ t❤❡ r❡s✉❧ts ♦❢ ❝❛❧❝✉❧✉s ❛♥❞ ❝r❡❛t❡ ❛❧❣♦r✐t❤♠s t❤❛t ♠❛♥✐♣✉❧❛t❡ t❤❡ ❞✐❣✐t❛❧ ❞❛t❛✳

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❛♣♣r♦①✐♠❛t❡ ❜✉t t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ✐s ✉♥❧✐♠✐t❡❞✳

❙✐♥❝❡ t❤❡ ✷✵t❤ ❝❡♥t✉r②✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✿ ❛❡r♦❞②♥❛♠✐❝s ✭❛✐r♣❧❛♥❡ ❛♥❞ ❝❛r ❞❡s✐❣♥✮✱ s♦✉♥❞ ❛♥❞ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ s♣❛❝❡ ❡①♣❧♦r❛t✐♦♥✱ str✉❝t✉r❡ ♦❢ t❤❡ ❛t♦♠ ❛♥❞ t❤❡ ✉♥✐✈❡rs❡✱ ❡t❝✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ❛❧s♦ ❝✐r❝✉✐t♦✉s✿ ❊✈❡r② ❝♦♥❝❡♣t ✐♥ ❝❛❧❝✉❧✉s

st❛rts

✕ ♦❢t❡♥ ✐♠♣❧✐❝✐t❧② ✕ ❛s ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥

♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥♦♥✦

❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✱

❜♦t❤

✐♥❝r❡♠❡♥t❛❧ ❛♥❞ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ❢♦r♠❡r ♣❛rt ✕ t❤❡ s♦✲❝❛❧❧❡❞ ❞✐s❝r❡t❡

❝❛❧❝✉❧✉s ✕ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ st✉❞② ♦❢ ✐♥❝r❡♠❡♥t❛❧ ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥❞✐✈✐s✐❜❧❡

❜② t❤❡✐r

✈❡r② ♥❛t✉r❡✿ ♣❡♦♣❧❡✱ ❛♥✐♠❛❧s✱ ❛♥❞ ♦t❤❡r ♦r❣❛♥✐s♠s✱ ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ❧♦❝❛t✐♦♥s ♦❢ s♣❛❝❡✱ ♣❛rt✐❝❧❡s✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❞✐❣✐t❛❧ ✐♠❛❣❡s ❛♥❞ ♦t❤❡r ♠❛♥✲♠❛❞❡ ❞❛t❛✱ ❡t❝✳ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♠❛❝❤✐♥❡r② ❝❛❧❧❡❞ ✏❧✐♠✐ts✑✱ ✇❡ ✐♥✈❛r✐❛❜❧② ❝❤♦♦s❡ t♦ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢ ❝❛❧❝✉❧✉s✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ✇❡ ❢❛❝❡ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡

❡✐t❤❡r ❜② t❤❡✐r ♥❛t✉r❡ ♦r ❜② ❛ss✉♠♣t✐♦♥✿ t✐♠❡✱

s♣❛❝❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❡t❝✳ ❈❛❧❝✉❧✉s ♣r♦❞✉❝❡s ❞❡✜♥✐t✐✈❡ r❡s✉❧ts ❛♥❞ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝② ✕ ❜✉t ♦♥❧② ❢♦r ♣r♦❜❧❡♠s ❛♠❡♥❛❜❧❡ t♦ ✐ts ♠❡t❤♦❞s✦ ■♥ t❤❡ ❝❧❛ssr♦♦♠✱ t❤❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧✐✜❡❞ ✉♥t✐❧ t❤❡② ❜❡❝♦♠❡ ♠❛♥❛❣❡❛❜❧❡❀ ♦t❤❡r✇✐s❡✱ ✇❡ ❝✐r❝❧❡ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ✐♥ s❡❛r❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲✐t❤✐♥ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s ❝♦✉rs❡✱ t❤❡ st✉❞❡♥t s✐♠♣❧② ♥❡✈❡r ❣❡ts t♦ ❝♦♠♣❧❡t❡ t❤❡ ✏❝✐r❝❧❡✑✦

▲❛t❡r ♦♥✱ t❤❡

❣r❛❞✉❛t❡ ✐s ❧✐❦❡❧② t♦ t❤✐♥❦ ♦❢ ❝❛❧❝✉❧✉s ♦♥❧② ✇❤❡♥ ❤❡ s❡❡s ❢♦r♠✉❧❛s ❛♥❞ r❛r❡❧② ✇❤❡♥ ❤❡ s❡❡s ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ■♥ t❤✐s ❜♦♦❦✱ ❡✈❡r② ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s ✐s ✜rst ✐♥tr♦❞✉❝❡❞ ✐♥ ✐ts ❞✐s❝r❡t❡✱ ✏♣r❡✲❧✐♠✐t✑✱ ✐♥❝❛r♥❛t✐♦♥ ✕ ❡❧s❡✇❤❡r❡ t②♣✐❝❛❧❧② ❤✐❞❞❡♥ ✐♥s✐❞❡ ♣r♦♦❢s ✕ ❛♥❞ t❤❡♥ ✉s❡❞ ❢♦r ♠♦❞❡❧✐♥❣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✇❡❧❧ ❜❡❢♦r❡ ✐ts ❝♦♥t✐♥✉♦✉s ❝♦✉♥t❡r♣❛rt ❡♠❡r❣❡s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠❡r ❛r❡ ❞✐s❝♦✈❡r❡❞ ✜rst ❛♥❞ t❤❡♥ t❤❡ ♠❛t❝❤✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❢♦✉♥❞ ❜② ♠❛❦✐♥❣ t❤❡ ✐♥❝r❡♠❡♥t s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✱ ❛t t❤❡ ❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s

∆x→0

−−−−−−−−−−→

❧✐♠✐t ✿

❝♦♥t✐♥✉♦✉s ❝❛❧❝✉❧✉s

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✼

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞ ❚❤✐s ❜♦♦❦ ✐s ❛ ♣❛rt ♦❢ t❤❡ s❡r✐❡s ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❚❤❡ s❡r✐❡s ❝♦✈❡rs t❤❡ st❛♥❞❛r❞ ♠❛t❡r✐❛❧ ♦❢ t❤❡ ✉♥❞❡r✲ ❣r❛❞✉❛t❡ ❝❛❧❝✉❧✉s ✇✐t❤ ❛ s✉❜st❛♥t✐❛❧ r❡✈✐❡✇ ♦❢ ♣r❡❝❛❧❝✉❧✉s ❛♥❞ ❛ ♣r❡✈✐❡✇ ♦❢ ❡❧❡♠❡♥t❛r② ♦r❞✐♥❛r② ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❇❡❧♦✇ ✐s t❤❡ ❧✐st ♦❢ t❤❡ ❜♦♦❦s ♦❢ t❤❡ s❡r✐❡s✱ t❤❡✐r ❝❤❛♣t❡rs✱ ❛♥❞ t❤❡ ✇❛② t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✭♣❛r❡♥t❤❡t✐❝❛❧❧②✮ r❡❢❡r❡♥❝❡s t❤❡♠✳ 

✶ P❈✲✶ ✶ P❈✲✷ ✶ P❈✲✸ ✶ P❈✲✹ ✶ P❈✲✺



✷ ❉❈✲✶ ✷ ❉❈✲✷ ✷ ❉❈✲✸ ✷ ❉❈✲✹ ✷ ❉❈✲✺ ✷ ❉❈✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✹✿ ❈❛❧❝✉❧✉s ✐♥ ❍✐❣❤❡r ❉✐♠❡♥s✐♦♥s

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❚❤❡ ❣r❛❞✐❡♥t ❚❤❡ ✐♥t❡❣r❛❧ ❱❡❝t♦r ✜❡❧❞s 

✺ ❉❊✲✶ ✺ ❉❊✲✷ ✺ ❉❊✲✸ ✺ ❉❊✲✹ ✺ ❉❊✲✺ ✺ ❉❊✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✸✿ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s

❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ■♥t❡❣r❛t✐♦♥ ❲❤❛t ✇❡ ❝❛♥ s♦ ✇✐t❤ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❙❡r✐❡s 

✹ ❍❉✲✶ ✹ ❍❉✲✷ ✹ ❍❉✲✸ ✹ ❍❉✲✹ ✹ ❍❉✲✺ ✹ ❍❉✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✷✿ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s 

✸ ■❈✲✶ ✸ ■❈✲✷ ✸ ■❈✲✸ ✸ ■❈✲✹ ✸ ■❈✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✶✿ Pr❡❝❛❧❝✉❧✉s

❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ❈❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✺✿ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❱❡❝t♦r ✈❛r✐❛❜❧❡s ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ❙②st❡♠s ♦❢ ❖❉❊s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❊❛❝❤ ✈♦❧✉♠❡ ❝❛♥ ❜❡ r❡❛❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✽

❆ ♣♦ss✐❜❧❡ s❡q✉❡♥❝❡ ♦❢ ❝❤❛♣t❡rs ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳ ❆♥ ❛rr♦✇ ❢r♦♠ ❆ t♦ ❇ ♠❡❛♥s t❤❛t ❝❤❛♣t❡r ❇ s❤♦✉❧❞♥✬t ❜❡ r❡❛❞ ❜❡❢♦r❡ ❝❤❛♣t❡r ❆✳

❆❜♦✉t t❤❡ ❛✉t❤♦r

❆❜♦✉t t❤❡ ❛✉t❤♦r P❡t❡r ❙❛✈❡❧✐❡✈ ✐s ❛ ♣r♦❢❡ss♦r ♦❢ ♠❛t❤❡♠❛t✐❝s ❛t ▼❛rs❤❛❧❧ ❯♥✐✈❡rs✐t②✱ ❍✉♥t✲ ✐♥❣t♦♥✱ ❲❡st ❱✐r❣✐♥✐❛✱ ❯❙❆✳ ❆❢t❡r ❛ P❤✳❉✳ ❢r♦♠ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛✲❈❤❛♠♣❛✐❣♥✱ ❤❡ ❞❡✈♦t❡❞ t❤❡ ♥❡①t ✷✵ ②❡❛rs t♦ t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✳ P❡t❡r ✐s t❤❡ ❛✉t❤♦r ♦❢ ❛ ❣r❛❞✉❛t❡ t❡①t❜♦♦❦ ❚♦♣♦❧♦❣② ■❧❧✉str❛t❡❞ ♣✉❜❧✐s❤❡❞ ✐♥ ✷✵✶✻✳ ❍❡ ❤❛s ❛❧s♦ ❜❡❡♥ ✐♥✈♦❧✈❡❞ ✐♥ r❡s❡❛r❝❤ ✐♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣② ❛♥❞ s❡✈❡r❛❧ ♦t❤❡r ✜❡❧❞s✳ ❍✐s ♥♦♥✲❛❝❛❞❡♠✐❝ ♣r♦❥❡❝ts ❤❛✈❡ ❜❡❡♥✿ ❞✐❣✐t❛❧ ✐♠❛❣❡ ❛♥❛❧②s✐s✱ ❛✉t♦♠❛t❡❞ ✜♥❣❡r♣r✐♥t ✐❞❡♥t✐✜❝❛t✐♦♥✱ ❛♥❞ ✐♠❛❣❡ ♠❛t❝❤✐♥❣ ❢♦r ♠✐s✲ s✐❧❡ ♥❛✈✐❣❛t✐♦♥✴❣✉✐❞❛♥❝❡✳

✾

❈♦♥t❡♥ts Pr❡❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

 ❈❤❛♣t❡r ✶✿ ❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✶ ▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s 1✱ 2✱ 3✱✳✳✳ ✶✳✸ ●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❱❡❝t♦rs ✐♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✶ Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ▲✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❈♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✳ ✳ ✳ ✳ ✷✳✻ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✳ ✷✳✶✶ ❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✹ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✺ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✻ ❚❤❡ s♣❡❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✼ ❚❤❡ ❝✉r✈❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✽ ❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✾ ❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✵ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✶ ❆r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧s✿ ✇❡✐❣❤t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✷ ❚❤❡ ❤❡❧✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✶✸✸ ✳ ✶✸✸ ✳ ✶✹✵ ✳ ✶✹✼ ✳ ✶✺✹ ✳ ✶✺✽ ✳ ✶✻✶ ✳ ✶✻✻ ✳ ✶✻✾ ✳ ✶✼✺ ✳ ✶✼✼ ✳ ✶✽✷ ✳ ✶✽✼ ✳ ✶✾✷ ✳ ✶✾✺ ✳ ✶✾✽ ✳ ✷✵✶ ✳ ✷✵✺ ✳ ✷✶✵ ✳ ✷✶✺ ✳ ✷✶✾ ✳ ✷✷✷ ✳ ✷✷✹

 ❈❤❛♣t❡r ✷✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

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✶✸ ✶✽ ✸✶ ✸✽ ✹✽ ✺✻ ✻✹ ✼✶ ✼✾ ✾✶ ✶✵✶ ✶✵✻ ✶✶✷ ✶✶✽ ✶✷✹

 ❈❤❛♣t❡r ✸✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✽

✸✳✶ ❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✽ ✸✳✷ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥ R2 ❛♥❞ ♣❧❛♥❡s ✐♥ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✸ ✸✳✸ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✽

❈♦♥t❡♥ts

✶✶

✸✳✹ ●r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ▲✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❈♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s ✸✳✶✸ ❚❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 

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✷✹✶ ✷✹✽ ✷✺✼ ✷✻✹ ✷✼✷ ✷✼✺ ✷✽✶ ✷✽✼ ✷✾✶ ✷✾✺

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✷✾✼ ✷✾✾ ✸✵✻ ✸✵✽ ✸✶✹ ✸✷✵ ✸✷✺ ✸✸✵ ✸✸✻ ✸✸✾ ✸✹✷ ✸✹✻

❈❤❛♣t❡r ✺✿ ❚❤❡ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺✷ ✺✳✶ ❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✳ ✳ ✳ ✳ ✺✳✷ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✳ ✳ ✳ ✳ ✺✳✸ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦✈❡r r❡❝t❛♥❣❧❡s ✳ ✺✳✹ ❚❤❡ ✇❡✐❣❤t ❛s t❤❡ ✸❞ ❘✐❡♠❛♥♥ s✉♠ ✳ ✳ ✺✳✺ ❚❤❡ ✇❡✐❣❤t ❛s t❤❡ ✸❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✺✳✻ ▲❡♥❣t❤s✱ ❛r❡❛s✱ ✈♦❧✉♠❡s✱ ❛♥❞ ❜❡②♦♥❞ ✳ ✺✳✼ ❖✉ts✐❞❡ t❤❡ s❛♥❞❜♦① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✽ ❚r✐♣❧❡ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✾ ❚❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✵ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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❈❤❛♣t❡r ✹✿ ❚❤❡ ❣r❛❞✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✼ ✹✳✶ ❖✈❡r✈✐❡✇ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✹✳✹ ❚❤❡ ❣r❛❞✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ✳ ✹✳✻ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✽ ❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✵ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✶ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✷ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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✸✺✷ ✸✻✵ ✸✻✹ ✸✼✷ ✸✼✻ ✸✽✶ ✸✽✺ ✸✾✵ ✸✾✸ ✸✾✺

❈❤❛♣t❡r ✻✿ ❱❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵✸ ✻✳✶ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹ ❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✵ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2 ✳

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✹✵✸ ✹✵✽ ✹✶✸ ✹✶✽ ✹✷✹ ✹✸✶ ✹✸✽ ✹✹✺ ✹✺✷ ✹✺✾

❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻✼ ✶ ❊①❡r❝✐s❡s✿ ✷ ❊①❡r❝✐s❡s✿ ✸ ❊①❡r❝✐s❡s✿ ✹ ❊①❡r❝✐s❡s✿ ✺ ❊①❡r❝✐s❡s✿

❇❛s✐❝ ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ■♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✹✻✼ ✹✼✻ ✹✼✽ ✹✽✸ ✹✾✵

❈♦♥t❡♥ts

✶✷

✻ ❊①❡r❝✐s❡s✿ ❱❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾✸ ✼ ❊①❛♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾✻

■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵✼

❈❤❛♣t❡r ✶✿ s♣❛❝❡s

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧

❈♦♥t❡♥ts

✶✳✶ ▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s 1✱ 2✱ 3✱✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❱❡❝t♦rs ✐♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✶ Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸ ✶✽ ✸✶ ✸✽ ✹✽ ✺✻ ✻✹ ✼✶ ✼✾ ✾✶ ✶✵✶ ✶✵✻ ✶✶✷ ✶✶✽ ✶✷✹

✶✳✶✳ ▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s

❲❤② ❞♦ ✇❡ ♥❡❡❞ t♦ st✉❞② ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s❄ ❚❤❡s❡ ❛r❡ t❤❡ ♠❛✐♥ s♦✉r❝❡s ♦❢ s♣❛❝❡s ♦❢ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s ✿ ✶✳ ❚❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡✱ ❞✐♠❡♥s✐♦♥ 3✳ ✷✳ ▼✉❧t✐♣❧❡ s♣❛❝❡s ♦❢ s✐♥❣❧❡ ❞✐♠❡♥s✐♦♥ ✐♥t❡r❝♦♥♥❡❝t❡❞ ✈✐❛ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥s✿ ❚❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝✲ t✐♦♥s ❧✐❡ ✐♥ ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s✳ ✸✳ ▼✉❧t✐♣❧❡ q✉❛♥t✐t✐❡s✱ ❤♦♠♦❣❡♥❡♦✉s ✭s✉❝❤ ❛s st♦❝❦ ❛♥❞ ❝♦♠♠♦❞✐t② ♣r✐❝❡s✮ ❛♥❞ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ✭s✉❝❤ ❛s ♦t❤❡r ❞❛t❛✮✿ ❚❤❡② ❛r❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♣♦✐♥ts ✐♥ ❛❜str❛❝t s♣❛❝❡s✳ ❚❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ r❡♣r❡s❡♥ts ❛ s✐❣♥✐✜❝❛♥t ❝❤❛❧❧❡♥❣❡ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣❧❛♥❡✳ ❋✉rt❤❡r♠♦r❡✱ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t✐♠❡ ✇✐❧❧ ♠❛❦❡ ✐t 4✲❞✐♠❡♥s✐♦♥❛❧✳ ❋✉rt❤❡r♠♦r❡✱ ♣❧❛♥♥✐♥❣ ❛ ✢✐❣❤t ♦❢ ❛ ♣❧❛♥❡ ✇♦✉❧❞ r❡q✉✐r❡ 3 s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s✱ ❜✉t t❤❡ ♥✉♠❜❡r ✐♥❝r❡❛s❡s t♦ 6 ✐❢ ✇❡ ❛r❡ t♦ ❝♦♥s✐❞❡r t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✐♥ t❤❡ ❛✐r✿ t❤❡ r♦❧❧✱ t❤❡ ♣✐t❝❤✱ ❛♥❞ t❤❡ ②❛✇✳ ◆❡①t✱ ❧❡t✬s ♥♦t✐❝❡ t❤❛t ❡✈❡♥ ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ ♦♥❧② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡ ❣r❛♣❤ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❧✐❡s ✐♥ t❤❡ xy ✲♣❧❛♥❡✱ ❛ s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2✳ ❲❤❛t ✐❢ t❤❡r❡ ❛r❡ t✇♦ s✉❝❤ ❢✉♥❝t✐♦♥s❄

✶✳✶✳

▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s

❊①❛♠♣❧❡ ✶✳✶✳✶✿ r♦❛❞ tr✐♣

▲❡t✬s ✐♠❛❣✐♥❡ ❛ ❝❛r ❞r✐✈❡♥ t❤r♦✉❣❤ ❛ ♠♦✉♥t❛✐♥ t❡rr❛✐♥✳ ■ts ❧♦❝❛t✐♦♥ ❛♥❞ ✐ts s♣❡❡❞✱ ❛s s❡❡♥ ♦♥ t❤❡ ♠❛♣✱ ❛r❡ ❦♥♦✇♥✳ ❚❤❡ ❣r❛❞❡ ♦❢ t❤❡ r♦❛❞ ✐s ❛❧s♦ ❦♥♦✇♥✳ ❍♦✇ ❢❛st ✐s t❤❡ ❝❛r ❝❧✐♠❜✐♥❣❄

❚❤❡ ✜rst ✈❛r✐❛❜❧❡ ✐s t✐♠❡✱ t✳ ❲❡ ❛❧s♦ ❤❛✈❡ t✇♦ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s✿ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ x ❛♥❞ t❤❡ ❡❧❡✈❛t✐♦♥ ✭t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✮ z ✳ ❚❤❡♥ z ❞❡♣❡♥❞s ♦♥ x✱ ❛♥❞ x ❞❡♣❡♥❞s ♦♥ t✳ ❚❤❡r❡❢♦r❡✱ z ❞❡♣❡♥❞s ♦♥ t ✈✐❛ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✿

P❧♦tt✐♥❣ ❜♦t❤ ❢✉♥❝t✐♦♥s t♦❣❡t❤❡r r❡q✉✐r❡s ❛ 3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ❲❡ ❝❛♥ t❛❦❡ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥s✿ • ❚❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ x = 2t − 1✳ • ❚❤❡ ❡❧❡✈❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ z = 3x + 7✳ • ❚❤❡♥ ❡❧❡✈❛t✐♦♥ ✐s✱ t♦♦✱ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ z = 3(2t − 1) + 7✳ ❲❡ ❝❛♥ ♥♦✇ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ❞✐r❡❝t❧②✿ dz d = ((2t)2 ) = 8t , dt dt

♦r ✈✐❛ t❤❡ ❈❤❛✐♥ ❘✉❧❡✿ dz dz dx d 2 d = · = (x ) · (2t) = 2x · 2 = 2(2t) · 2 = 8t . dt dx dt dx dt ❊①❛♠♣❧❡ ✶✳✶✳✷✿ ❛♥♦t❤❡r r♦❛❞ tr✐♣

◆♦✇✱ t❤❡ r♦❛❞ ✐♥ t❤✐s ❡①❛♠♣❧❡ ✐s str❛✐❣❤t✦ ▼♦r❡ r❡❛❧✐st✐❝❛❧❧②✱ ✐t s❤♦✉❧❞ ❤❛✈❡ t✉r♥s ❛♥❞ ❝✉r✈❡s✳✳✳

✶✹

✶✳✶✳

✶✺

▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s

❋♦rt✉♥❛t❡❧②✱ ♦✉r ♠❛♣s ❤❛s ❡❧❡✈❛t✐♦♥ ✐♥❢♦r♠❛t✐♦♥✿ t❤❡ ❝✉r✈❡s ✐♥❞✐❝❛t❡ t❤❛t ❤♦✇ ❤✐❣❤ ✐s ❡✈❡r② ❧♦❝❛t✐♦♥ ♦♥ t❤❡ r♦❛❞ ❛♥❞ ❛r♦✉♥❞✳ ❚❤❡ ✜rst ✈❛r✐❛❜❧❡ ✐s t✐♠❡✱ t✱ ❛❣❛✐♥✳ ❲❡ ❛❧s♦ ❤❛✈❡ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s✿ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ ♣♦✐♥ts (x, y) ♦♥ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❡❧❡✈❛t✐♦♥ ✭t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✮ z ✳ ❚❤❡♥ z ❞❡♣❡♥❞s ♦♥ (x, y) ❛♥❞ (x, y) ❞❡♣❡♥❞s ♦♥ t✳ ❚❤❡r❡❢♦r❡✱ z ❞❡♣❡♥❞s ♦♥ t ✈✐❛ t❤❡ ❝♦♠♣♦s✐t✐♦♥✳ ❲❡ ❝❛♥ t❛❦❡ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥s✿ • ❚❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ x = 2t ❛♥❞ y = sin t❀ • ❚❤❡ ❡❧❡✈❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ z = x2 + y 2 ✳ • ❚❤❡♥✱ t❤❡ ❡❧❡✈❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ z = (2t)2 + (sin t)2 ✳ ❲❡ ❝❛♥ ♥♦✇ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ❞✐r❡❝t❧②✿ dz d = ((2t)2 + (sin t)2 ) = 8t + 2 sin t cos t . dt dt

■s t❤❡r❡ t❤❡ ❈❤❛✐♥ ❘✉❧❡❄ ■❢ t❤❡r❡ ✐s✱ t❤❡ ❛❜♦✈❡ ❞❡r✐✈❛t✐✈❡ ✐s ♠❛❞❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦✉r✳ ❋✐rst✱ ♦✉r ♠♦t✐♦♥ ✐s r❡❝♦r❞❡❞ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❝♦♥s✐sts ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts✿ dx = 2 ❛♥❞ dt

dy = cos t . dt ❚❤❡ t❡rr❛✐♥ ♠❛♣✬s st❡❡♣♥❡ss ✐s ❢♦✉♥❞ ✐♥ t❤❡ t✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥✱ x ❛♥❞ y ✱ ❛s t❤❡ t✇♦ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s

♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

∂z = 2x ❛♥❞ ∂x

∂z = 2y . ∂y

❊①❛♠♣❧❡ ✶✳✶✳✸✿ ❤✐❦✐♥❣

▲❡t✬s ♥♦✇ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❝♦♠♣❧❡① tr✐♣✳ P❧❛♥♥✐♥❣ ❛ ❤✐❦❡✱ ✇❡ ❝r❡❛t❡ ❛ ♣❧❛❝❡s ❛r❡ ♣✉t ♦♥ ❛ s✐♠♣❧❡ ♠❛♣ ♦❢ t❤❡ ❛r❡❛✿

❚❤✐s ✐s ❛ ♣❛r❛♠❡tr✐❝

❝✉r✈❡

tr✐♣ ♣❧❛♥

✿ ❚❤❡ t✐♠❡s ❛♥❞ t❤❡

✿ x = f (t), y = g(t) ,

✇✐t❤ x ❛♥❞ y ♣r♦✈✐❞✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ②♦✉r ❧♦❝❛t✐♦♥✳ ❈♦♥✈❡rs❡❧②✱ ♠♦t✐♦♥ ✐♥ t✐♠❡ ✐s ❛ ❣♦✲t♦ ♠❡t❛♣❤♦r ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✦ ❲❡ t❤❡♥ ❜r✐♥❣ t❤❡

t❡rr❛✐♥ ♠❛♣

♦❢ t❤❡ ❛r❡❛✳ ❚❤❡ ❞❛t❛ ✐s ❝♦❧♦r❡❞ ❛❝❝♦r❞✐♥❣❧②✿

✶✳✶✳ ▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s

✶✻

❙✉❝❤ ❛ t♦♣♦❣r❛♣❤✐❝ ♠❛♣ ❤❛s t❤❡ ❝♦❧♦rs ✐♥❞✐❝❛t✐♥❣ t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ ❛❝t✉❛❧ t❡rr❛✐♥✿

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✿

z = f (x, y) . ❈♦♥✈❡rs❡❧②✱ ❛ t❡rr❛✐♥ ♠❛♣ ✐s ❛ ❣♦✲t♦ ♠❡t❛♣❤♦r ❢♦r ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✦

◆♦✇✱ ❜❛❝❦ t♦ t❤❡ s❛♠❡ q✉❡st✐♦♥✿ ❍♦✇ ❢❛st ✇✐❧❧ ✇❡ ❜❡ ❝❧✐♠❜✐♥❣❄ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ r❡q✉✐r❡❞ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

✶✳✶✳ ▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s✱ ♠✉❧t✐♣❧❡ ❞✐♠❡♥s✐♦♥s

✶✼

❲❡ ❢❛❝❡ ♥❡✇ ❦✐♥❞s ♦❢ ❢✉♥❝t✐♦♥s✿

−→

tr✐♣ ♠❛♣

t

❇♦t❤ ❢✉♥❝t✐♦♥s ❞❡❛❧ ✇✐t❤

3

−→

(x, y)

z

t❡rr❛✐♥ ♠❛♣

✈❛r✐❛❜❧❡s ❛t t❤❡ s❛♠❡ t✐♠❡✱ ✇✐t❤ ❛ t♦t❛❧ ♦❢

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡r❡ ❛r❡ ♠❛♥② ❢✉♥❝t✐♦♥s ♦❢ t❤❡

2

♦r

3

4✦

✈❛r✐❛❜❧❡s ♦❢ ❧♦❝❛t✐♦♥✿ t❤❡ t❡♠♣❡r❛t✉r❡ ❛♥❞

t❤❡ ♣r❡ss✉r❡ ♦❢ t❤❡ ❛✐r ♦r ✇❛t❡r✱ t❤❡ ❤✉♠✐❞✐t②✱ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❝❤❡♠✐❝❛❧✱ ❡t❝✳

❚❤❡ ♦❜s❡r✈❛t✐♦♥s ❛❜♦✉t t❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ❛r❡ st✐❧❧ ❛♣♣❧✐❝❛❜❧❡✿

•

■❢ ✇❡ ❞♦✉❜❧❡ ♦✉r ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ✭✇✐t❤ t❤❡ s❛♠❡ t❡rr❛✐♥✮✱ t❤❡ ❝❧✐♠❜ ✇✐❧❧ ❜❡ t✇✐❝❡ ❛s ❢❛st✳

•

■❢ ✇❡ ❞♦✉❜❧❡ st❡❡♣♥❡ss ♦❢ t❤❡ t❡rr❛✐♥ ✭✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ s♣❡❡❞✮✱ t❤❡ ❝❧✐♠❜ ✇✐❧❧ ❜❡ t✇✐❝❡ ❛s ❢❛st✳

■t ❢♦❧❧♦✇s t❤❛t t❤❡ s♣❡❡❞ ♦❢ t❤❡ ❝❧✐♠❜ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ❜♦t❤ ♦✉r ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ❛♥❞ t❤❡ st❡❡♣♥❡ss ♦❢ t❤❡ t❡rr❛✐♥✳ ❚❤❛t✬s t❤❡ ❈❤❛✐♥ ❘✉❧❡✳

❲❤❛t ✐s ✐t ✐♥ t❤✐s ♥❡✇ s❡tt✐♥❣❄ ❇♦t❤ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ ❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦

x ❛♥❞ y

✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ✐♥✈♦❧✈❡❞✳

❲❡ ✇✐❧❧ s❤♦✇ t❤❛t ✐t ✐s t❤❡ s✉♠ ♦❢ t❤♦s❡✿

∆z ∆z ∆x ∆z ∆y = + ∆t ∆x ∆t ∆y ∆t

♦r

dz ∂z dx ∂z dy = + . dt ∂x dt ∂y dt

❚❤✐s ♥✉♠❜❡r ✐s t❤❡♥ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡s❡ t✇♦ ✈❡❝t♦rs ✿

•

t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ tr✐♣✱ ✐✳❡✳✱ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t②✿

 •

∆x ∆y , ∆t ∆t



♦r



dx dy , dt dt



.

❚❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ t❡rr❛✐♥ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s✿



∆z ∆z , ∆x ∆x



♦r



∂z ∂z , ∂x ∂x



.

❊①❛♠♣❧❡ ✶✳✶✳✹✿ ❝♦sts ❛♥❞ ♣r✐❝❡s ❲❡ s❛✇ ✐♥ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✹✮ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❛❜str❛❝t s♣❛❝❡✿

t❤❡ s♣❛❝❡ ♦❢ ♣r✐❝❡s✳

❆t ✐ts

s✐♠♣❧❡st✱ t❤❡ ❜❛❦❡r ❞♦❡s t✇♦ t❤✐♥❣s✿ ✶✳ ❍❡ ✇❛t❝❤❡s t❤❡ ♣r✐❝❡s ♦❢ t❤❡ t✇♦ ✐♥❣r❡❞✐❡♥ts ♦❢ ❤✐s ❜r❡❛❞✿ s✉❣❛r ❛♥❞ ✇❤❡❛t✳ ✷✳ ❍❡ ❞❡❝✐❞❡s✱ ❜❛s❡❞ ♦♥ t❤❡s❡ t✇♦ ♥✉♠❜❡rs✱ ✇❤❛t t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❜r❡❛❞ s❤♦✉❧❞ ❜❡✳ ❚❤❡ s♣❛❝❡✬s ❞✐♠❡♥s✐♦♥ ✇❛s

2✱ ✇✐t❤ ♦♥❧② t❤❡ t✇♦ ♣r✐❝❡s ♦❢ t❤❡ t✇♦ ✐♥❣r❡❞✐❡♥ts ♦❢ ❜r❡❛❞✳

❚❤❡ ❞❡♣❡♥❞❡♥❝❡

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

✐s ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ ❝♦sts

t

7→

(x, y)

1✱ 2✱ 3✱✳✳✳

7 →

✶✽

z

♣r✐❝❡

▼✉❧t✐♣❧❡ ✈❛r✐❛❜❧❡s ❧❡❛❞ t♦ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❛❜str❛❝t s♣❛❝❡s✱ s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♣r✐❝❡ ♦❢ ❛ ❝❛r ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ♣r✐❝❡s ♦❢ 1000 ♦❢ ✐ts ♣❛rts✿

t 7→ (x1 , x2 , ..., x1000 ) 7→ z ❲❡ ❝❛♥ ❞❡✈❡❧♦♣ ❛❧❣❡❜r❛✱ ❣❡♦♠❡tr②✱ ❛♥❞ ❝❛❧❝✉❧✉s t❤❛t ✇✐❧❧ ❜❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❲❡ r❡♣❧❛❝❡ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✐♥ ❛ s♣❛❝❡ ♦❢ ❛ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ P ❜❡❧♦✇ ✐s s✉❝❤ ❛ ✈❛r✐❛❜❧❡✱ ✐✳❡✳✱ ❛ ❧♦❝❛t✐♦♥ ✐♥ ❛ 1000✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✿

t 7→ P 7→ z ■♥✐t✐❛❧❧② ❤♦✇❡✈❡r ✇❡ ✇✐❧❧ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ ❞✐♠❡♥s✐♦♥s t❤❛t ✇❡ ❝❛♥ ✈✐s✉❛❧✐③❡✦

❈❖◆❱❊◆❚■❖◆ ❲❡ ✇✐❧❧ ✉s❡ ✉♣♣❡r ❛♥❞ ✈❡❝t♦rs✿

❝❛s❡

❧❡tt❡rs ❢♦r t❤❡ ❡♥t✐t✐❡s t❤❛t ❛r❡ ✭♦r ♠❛② ❜❡✮ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧✱ s✉❝❤ ❛s ♣♦✐♥ts

A, B, C, ... P, Q, ... , ❛♥❞

❧♦✇❡r ❝❛s❡

❧❡tt❡rs ❢♦r ♥✉♠❜❡rs✿

a, b, c, ... , x, y, z, .. .

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

❲❡ st❛rt ✇✐t❤ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 1✳ ■t ✐s ❛ ❧✐♥❡ ✇✐t❤ ❛ ❝❡rt❛✐♥ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❢❡❛t✉r❡s ✕ t❤❡ ♦r✐❣✐♥✱ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✱ ❛♥❞ t❤❡ ✉♥✐t ✕ ❛❞❞❡❞✿

❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✭✐✳❡✳✱ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✮✿ ❛ ❧♦❝❛t✐♦♥ P ←→ ❛ r❡❛❧ ♥✉♠❜❡r x.

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

✶✾

❲❡ ❝❛♥ ❤❛✈❡ s✉❝❤ ✏❈❛rt❡s✐❛♥ ❧✐♥❡s✑ ❛s ♠❛♥② ❛s ✇❡ ❧✐❦❡ ❛♥❞ ✇❡ ❝❛♥ ❛rr❛♥❣❡ t❤❡♠ ✐♥ ❛♥② ✇❛② ✇❡ ❧✐❦❡✳ ❚❤❡♥ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ ❛t

90

2

✐s ♠❛❞❡ ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ❞✐♠❡♥s✐♦♥

❞❡❣r❡❡s ✭♦❢ r♦t❛t✐♦♥✮ ❢r♦♠ ♣♦s✐t✐✈❡

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ❞✐♠❡♥s✐♦♥

x

t♦ ♣♦s✐t✐✈❡

1

❛❧✐❣♥❡❞

y✿

3✳

❚❤❡r❡ ✐s ♠✉❝❤ ♠♦r❡ ❣♦✐♥❣ ♦♥ ✐♥ ✏s♣❛❝❡✑ t❤❛♥ ♦♥ ❛ ♣❧❛♥❡✿

❚❤❛t ✐s ✇❤② ✇❡✬❧❧ ♥❡❡❞ t❤r❡❡ ♥✉♠❜❡rs t♦ r❡♣r❡s❡♥t t❤❡ ❧♦❝❛t✐♦♥s✳

3 ✐s ♠❛❞❡ ♦❢ t❤r❡❡ ❝♦♣✐❡s ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ❞✐♠❡♥s✐♦♥ 1✳ ❏✉st 2 ❛❜♦✈❡✱ t❤❡s❡ ❝♦♣✐❡s ❞♦♥✬t ❤❛✈❡ t♦ ❜❡ ✐❞❡♥t✐❝❛❧❀ t❤❡✐r ✉♥✐ts ♠✐❣❤t ❜❡ ✉♥r❡❧❛t❡❞✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥

❚❤❡ s②st❡♠ ✐s ❜✉✐❧t ✐♥ s❡✈❡r❛❧ st❛❣❡s✿ ✶✳ ❚❤r❡❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ❝❤♦s❡♥✿ t❤❡

x✲❛①✐s✱

t❤❡

y ✲❛①✐s✱

❛♥❞ t❤❡

90✲❞❡❣r❡❡ t✉r♥ ❢r♦♠ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ x t♦ y t♦ z t♦ x✳

✷✳ ❚❤❡ t✇♦ ❛①❡s ❛r❡ ♣✉t t♦❣❡t❤❡r ❛t t❤❡✐r ♦r✐❣✐♥s s♦ t❤❛t ✐t ✐s ❛ ♦❢ ♦♥❡ ❛①✐s t♦ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♥❡①t ✕ ❢r♦♠ ✸✳ ❯s❡ t❤❡ ♠❛r❦s ♦♥ t❤❡ ❛①✐s t♦ ❞r❛✇ ❣r✐❞s ♦♥ t❤❡ ♣❧❛♥❡s✳

z ✲❛①✐s✳

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

✷✵

✹✳ ❲❡ r❡♣❡❛t t❤❡s❡ t❤r❡❡ ❣r✐❞s ✐♥ ♣❛r❛❧❧❡❧ t♦ ❝r❡❛t❡ t❤r❡❛❞s ✐♥ s♣❛❝❡✳

❚❤❡ ❧❛st st❡♣ ✐s s❤♦✇♥ ❜❡❧♦✇✿

❚❤❡ s❡❝♦♥❞ r❡q✉✐r❡♠❡♥t ✐s ❝❛❧❧❡❞ t❤❡ ❘✐❣❤t

❍❛♥❞ ❘✉❧❡

✿

■t r❡❛❞s✿ ◮ ■❢ ✇❡ ❝✉r❧ ♦✉r ✜♥❣❡rs ❢r♦♠ t❤❡ x✲❛①✐s t♦ t❤❡ y ✲❛①✐s✱ ♦✉r t❤✉♠❜ ✇✐❧❧ ♣♦✐♥t ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ z ✲❛①✐s✳

❲❡ ❝❛♥ ❛❧s♦ ✉♥❞❡rst❛♥❞ t❤✐s ✐❞❡❛ ✐❢ ✇❡ ✐♠❛❣✐♥❡ t✉r♥✐♥❣ ❛ s❝r❡✇❞r✐✈❡r ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ❛♥❞ s❡❡✐♥❣ ✇❤✐❝❤ ✇❛② t❤❡ s❝r❡✇ ❣♦❡s✳ ❚❤❡ ❛①❡s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✱ ❜✉t t❤❡r❡ ✐s ♠♦r❡✦ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ x✲ ❛♥❞ y ✲❛①✐s ❜❡✐♥❣ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ z ✲❛①✐s✱ ❛❧❧ ❧✐♥❡s ✐♥ t❤❡ xy ✲♣❧❛♥❡ ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐t✿

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

✷✶

1✱ 2✱ 3✱✳✳✳

❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ r❡♠❛✐♥s t❤❡ s❛♠❡❀ ✐t ✐s t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ❛ ❧♦❝❛t✐♦♥ P ←→

❛ tr✐♣❧❡

♦❢ r❡❛❧ ♥✉♠❜❡rs (x, y, z)

❲❛r♥✐♥❣✦ ❚❤❡ t❤r❡❡ ✈❛r✐❛❜❧❡s ♦r q✉❛♥t✐t✐❡s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ t❤r❡❡ ❛①❡s ♠❛② ❜❡ ✉♥r❡❧❛t❡❞✳

❚❤❡♥ ♦✉r ✈✐s✉❛❧✐③❛✲

t✐♦♥ ✇✐❧❧ r❡♠❛✐♥ ✈❛❧✐❞ ✇✐t❤ r❡❝t❛♥❣❧❡s ✐♥st❡❛❞ ♦❢ sq✉❛r❡s✱ ❛♥❞ ❜♦①❡s ✐♥st❡❛❞ ♦❢ ❝✉❜❡s✳

❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ s②st❡♠ ✐s ❜✉✐❧t ❢r♦♠ t❤r❡❡ ❝♦♣✐❡s ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡✿ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ yz ✲♣❧❛♥❡✱ ❛♥❞ t❤❡ zx✲♣❧❛♥❡✳ ❚❤❡② ❛r❡ ❛rr❛♥❣❡❞ ❛t 90 ❞❡❣r❡❡s ❛s ✇❛❧❧s ♦❢ ❛ r♦♦♠✿

❚❤❡s❡ ♣❧❛♥❡s ❛r❡ ❝❛❧❧❡❞ t❤❡

✳

❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s

❚❤✐s ✐s ❤♦✇ t❤❡ s②st❡♠ ✇♦r❦s✿ ❋✐rst✱ s✉♣♣♦s❡ P ✐s ❛ ❧♦❝❛t✐♦♥ ✐♥ t❤✐s s♣❛❝❡✳ ❲❡ ✜♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❜♦① ✇✐t❤ ♦♥❡ ❝♦r♥❡r ❛t O ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ❛t P ✳ ❲❡ ✜♥❞ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ t❤r❡❡ ♣❧❛♥❡s t♦ t❤❛t ❧♦❝❛t✐♦♥ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ t❤❡ r❡s✉❧t ✐s t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡s ♦❢ P ✱ s♦♠❡ ♥✉♠❜❡rs x✱ y ✱ ❛♥❞ z ✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ yz ✲♣❧❛♥❡ ✐s ♠❡❛s✉r❡❞ ❛❧♦♥❣ t❤❡ x✲❛①✐s✱ ❡t❝✳ ❲❡ ✉s❡ t❤❡ ♥❡❛r❡st ♠❛r❦ t♦ s✐♠♣❧✐❢② t❤❡ t❛s❦✿

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ x, y, z ❛r❡

1✱ 2✱ 3✱✳✳✳

✷✷

✳ ■❢ ✇❡ ♥❡❡❞ t♦ ❜✉✐❧❞ ❛ ❜♦① ✇✐t❤ t❤❡s❡ ❞✐♠❡♥s✐♦♥s✿

♥✉♠❜❡rs

• ❋✐rst✱ ✇❡ ♠❡❛s✉r❡ x ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ yz ✲♣❧❛♥❡ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ • ❙❡❝♦♥❞✱ ✇❡ ♠❡❛s✉r❡ y ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ xz ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✳ • ❚❤✐r❞✱ ✇❡ ♠❡❛s✉r❡ z ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ xy ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ z ✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t❤r❡❡ ♣❧❛♥❡s ✕ ❛s ✐❢ t❤❡s❡ ✇❡r❡ t❤❡ t✇♦ ✇❛❧❧s ❛♥❞ t❤❡ ✢♦♦r ✐♥ ❛ r♦♦♠ ✕ ✐s ❛ P = (x, y, z) ✐♥ t❤❡ s♣❛❝❡✳ ❲❡ ✉s❡ t❤❡ ♥❡❛r❡st ♠❛r❦s t♦ s✐♠♣❧✐❢② t❤❡ t❛s❦✿

❧♦❝❛t✐♦♥

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

❚❤✐s

3✲❞✐♠❡♥s✐♦♥❛❧

1✱ 2✱ 3✱✳✳✳

✷✸

❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❝❛❧❧❡❞ t❤❡ ❈❛rt❡s✐❛♥ s♣❛❝❡ ♦r t❤❡

3✲s♣❛❝❡✳

❖♥❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ✐♥ ♣❧❛❝❡✱ ✐t ✐s ❛❝❝❡♣t❛❜❧❡ t♦ t❤✐♥❦ ♦❢ ❧♦❝❛t✐♦♥ ❛s tr✐♣❧❡s ♦❢ ♥✉♠❜❡rs ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ✇r✐t❡✿

P = (x, y, z) . ❈♦♥s✐❞❡r ♠♦r❡ ♦❢ t❤❡ ♣❧❛♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✿

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t t❤❡s❡ ♣❧❛♥❡s✿

x = k, y = k, ❢♦r s♦♠❡ r❡❛❧

♦r

z = k,

k✳

❲❡ ❝❛♥ ✉s❡ t❤✐s ✐❞❡❛ t♦ r❡✈❡❛❧ t❤❡ ✐♥t❡r♥❛❧ str✉❝t✉r❡ ♦❢ t❤❡ s♣❛❝❡✳

❚❤❡♦r❡♠ ✶✳✷✳✶✿ P❧❛♥❡s P❛r❛❧❧❡❧ t♦ ❈♦♦r❞✐♥❛t❡ P❧❛♥❡s ✶✳ ■❢

L

✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

z ✲❝♦♦r❞✐♥❛t❡✳

xy ✲♣❧❛♥❡✱

t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥

❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥

L

L

❤❛✈❡ t❤❡ s❛♠❡

♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts

z ✲❝♦♦r❞✐♥❛t❡✱ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✳ yz ✲♣❧❛♥❡✱ t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥ L ❤❛✈❡ t❤❡ s❛♠❡ x✲❝♦♦r❞✐♥❛t❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥ L ♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ✇✐t❤ t❤❡ s❛♠❡ x✲❝♦♦r❞✐♥❛t❡✱ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳

✇✐t❤ t❤❡ s❛♠❡ ✷✳ ■❢

L

✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

✸✳ ■❢

L

✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

y ✲❝♦♦r❞✐♥❛t❡✳

❛♥❛❧②t✐❝ ❣❡♦♠❡tr②

◆♦✇ t❤❛t ❡✈❡r②t❤✐♥❣ ✐s ❞✐♥❛t❡s✳

zx✲♣❧❛♥❡✱

t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥

❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥

✇✐t❤ t❤❡ s❛♠❡

❲❡ t✉r♥ t♦

✷✹

1✱ 2✱ 3✱✳✳✳

y ✲❝♦♦r❞✐♥❛t❡✱ L

L

L

❤❛✈❡ t❤❡ s❛♠❡

♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts

✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

zx✲♣❧❛♥❡✳

♦❢ t❤❡ 3✲s♣❛❝❡✳ ✱ ✇❡ ❝❛♥ s♦❧✈❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ❜② ❛❧❣❡❜r❛✐❝❛❧❧② ♠❛♥✐♣✉❧❛t✐♥❣ ❝♦♦r✲

♣r❡✲♠❡❛s✉r❡❞

❚❤❡ ✜rst ❣❡♦♠❡tr✐❝ t❛s❦ ✐s ✜♥❞✐♥❣ t❤❡ ❞✐st❛♥❝❡ ✿ ❲❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥s P ❛♥❞ Q ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s (x, y, z) ❛♥❞ (x′ , y ′ , z ′ )❄ ❋♦r ❞✐♠❡♥s✐♦♥ 2✱ ✇❡ ✉s❡❞ t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❲❡ ❢♦✉♥❞ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❛s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✕ ✇✐t❤ ✐ts s✐❞❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✕ t❤❛t ❤❛s t❤❡s❡ ♣♦✐♥ts ❛t t❤❡ ♦♣♣♦s✐t❡ ❝♦r♥❡rs✿

❋✐rst✱ ✇❡ ♥❡❡❞ t♦ r❡❛❧✐③❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐ts❡❧❢ ✐s 1✲❞✐♠❡♥s✐♦♥❛❧✦ ■♥❞❡❡❞✱ ❛♥② t✇♦ ♣♦✐♥ts✱ ✐♥ ❛♥② s♣❛❝❡ ✕ 1✲✱ 2✲✱ 3✲✱ ♦r n✲❞✐♠❡♥s✐♦♥❛❧ ✕ ❝❛♥ ❜❡ ❝♦♥♥❡❝t❡❞ ❜② ❛ ❧✐♥❡✱ ❛♥❞ ❛❧♦♥❣ t❤❛t ❧✐♥❡ ✕ ❛ 1✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✕ ✇❡ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡✿

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❥✉st ❛ ♠❡❛♥s t♦ ❛♥ ❡♥❞✳ ❙♦✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐♥ s♣❛❝❡ ❛s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ❜♦① ✕ ✇✐t❤ ✐ts ❡❞❣❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ ❛♥❞ s✐❞❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ✕ t❤❛t ❤❛s t❤❡s❡ ♣♦✐♥ts ❛t t❤❡ ♦♣♣♦s✐t❡ ❝♦r♥❡rs✿

❲❡ ♥♦✇ ✉t✐❧✐③❡ t❤❡s❡ t✇♦ ❢❛❝ts✿ ✶✳ ❊✈❡r② ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ♦❢ t❤❡ 3✲s♣❛❝❡ ❤❛s ✐ts ♦✇♥✱ 2✲❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ✷✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✳ ❚❤✐s ✐s t❤❡ ♦✉t❧✐♥❡ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥✿

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

✷✺

❚❤❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐s t♦ ❜❡ ❛♣♣❧✐❡❞ ✇✐t❤✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ❛♥❞ t❤❡♥ ✇✐t❤✐♥ ❛ ❝❡rt❛✐♥ ✈❡rt✐❝❛❧ ♣❧❛♥❡✳ ❚❤❡ ❢♦r♠✉❧❛ ✐s✱ ❛s ✇❡ ❛♥t✐❝✐♣❛t❡❞✱ s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞✐♠❡♥s✐♦♥s✿ ❚❤❡♦r❡♠ ✶✳✷✳✷✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❋♦r ❉✐♠❡♥s✐♦♥

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ✇✐t❤ ❝♦♦r❞✐♥❛t❡s P ✐s d(P, Q) =

3

= (x, y, z)

❛♥❞ Q = (x′, y′, z ′)

p (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2

Pr♦♦❢✳

❚❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts

P = (x, y, z)

❛♥❞

Q = (x′ , y ′ , z ′ )

1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ s❡♣❛r❛t❡❧② ❢♦r x ❛♥❞ x′ ♦♥ t❤❡ x✲❛①✐s ✐s |x − x′ |✳ y ❛♥❞ y ′ ♦♥ t❤❡ y ✲❛①✐s ✐s |y − y ′ |✳ z ❛♥❞ z ′ ♦♥ t❤❡ z ✲❛①✐s ✐s |z − z ′ |✳

t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ ✶✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✷✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✸✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥

✐s t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤✐s ✏❜♦①✑✳ ❲❡ ✉s❡

❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❛①❡s✱ ❛s ❢♦❧❧♦✇s✿

❚❤❡s❡ ❛r❡ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❜♦①✳

◆❡①t ✇❡ ✉s❡ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ t✇✐❝❡✳ ❲❡ ✜rst ✜♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❜♦① ❛♥❞ t❤❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠❛✐♥ ❞✐❛❣♦♥❛❧✿ P❚ ✶✿ P❚ ✷✿

d(P, A) = |x − x′ |,

d(A, B) = |y − y ′ | =⇒ d(P, B)2 = (x − x′ )2 + (y − y ′ )2 d(P, B)2 = (x − x′ )2 + (y − y ′ )2 , d(B, Q) = |z − z ′ | =⇒ d(P, Q)2 = d(P, B)2 + d(B, Q)2 = (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2

❊①❡r❝✐s❡ ✶✳✷✳✸

Pr♦✈❡ t❤❛t ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ t❤❡ tr✐❛♥❣❧❡ ✐s ✐♥❞❡❡❞ ❛ r✐❣❤t tr✐❛♥❣❧❡✳

❘❡❧❛t✐♦♥s ❛r❡ ✉s❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❜❡❢♦r❡ ❜✉t ✇✐t❤ ♠♦r❡ ✈❛r✐❛❜❧❡s✳

♥✉♠❜❡rs

(x, y, z)

❆ r❡❧❛t✐♦♥ ♣r♦❝❡ss❡s ❛ tr✐♣❧❡ ♦❢

❛s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s ❛♥ ♦✉t♣✉t✱ ✇❤✐❝❤ ✐s✿ ❨❡s ♦r ◆♦✳ ■❢ ✇❡ ❛r❡ t♦ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢

1✱ 2✱ 3✱✳✳✳

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

✷✻

❛ r❡❧❛t✐♦♥✱ t❤✐s ♦✉t♣✉t ❜❡❝♦♠❡s✿ ❛ ♣♦✐♥t ♦r ♥♦ ♣♦✐♥t✳ ❋♦r ❡①❛♠♣❧❡✿ ♦✉t❝♦♠❡✿ P❧♦t ♣♦✐♥t ♣❧❛♥❡✿ 3

R

tr✐♣❧❡✿

r❡❧❛t✐♦♥✿

→ (x, y, z) →

❚❘❯❊

→

x + y + z = 2?

(x, y, z).

ր

❋❆▲❙❊

ց

❉♦♥✬t ♣❧♦t ❛♥②t❤✐♥❣✳

❲❡ ❝❛♥ ❞♦ ✐t ❜② ❤❛♥❞✿

❲❡ ❝❛♥ ✉s❡✱ ❛s ❜❡❢♦r❡✱ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ ✿

{(x, y, z) :

x, y, z} .

❛ ❝♦♥❞✐t✐♦♥ ♦♥

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛❜♦✈❡ r❡❧❛t✐♦♥ ✐s ❛ s✉❜s❡t ♦❢

R3

❣✐✈❡♥ ❜②✿

{(x, y, z) : x + y + z = 2} . ❲❤❛t ❛❜♦✉t ❞✐♠❡♥s✐♦♥

4

❛♥❞ ❤✐❣❤❡r❄

❲❡ ❝❛♥♥♦t ✉s❡ ♦✉r ♣❤②s✐❝❛❧ s♣❛❝❡ ❛s ❛ r❡❢❡r❡♥❝❡ ❛♥②♠♦r❡✦ ❲❡ ❝❛♥✬t ✉s❡ ✐t ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥ ❡✐t❤❡r✳ ❚❤❡ s♣❛❝❡ ✐s ❛❜str❛❝t✳ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡ r❡♠❛✐♥s t❤❡ s❛♠❡❀ ✐t ✐s t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡✿

❛ ❧♦❝❛t✐♦♥

P ←→

❛ str✐♥❣ ♦❢

n

r❡❛❧ ♥✉♠❜❡rs

(x1 , x2 , x3 , ..., xn )

❯s✐♥❣ t❤❡ s❛♠❡ ❧❡tt❡r ✇✐t❤ s✉❜s❝r✐♣ts ✐s ♣r❡❢❡r❛❜❧❡ ❡✈❡♥ ❢♦r ❞✐♠❡♥s✐♦♥ ❛♥❞ ✈❛r✐❛❜❧❡s ❛r❡ ❡❛s✐❡r t♦ ❞❡t❡❝t ❛♥❞ ✉t✐❧✐③❡✳ ❍♦✇❡✈❡r✱ ✉s✐♥❣ ❥✉st

P

3 ❛s t❤❡ s②♠♠❡tr✐❡s ❜❡t✇❡❡♥ t❤❡ ❛①❡s

✐s ♦❢t❡♥ ❡✈❡♥ ❜❡tt❡r✦

❇❡❝❛✉s❡ ♦❢ t❤❡ ❞✐✣❝✉❧t② ♦r ❡✈❡♥ ✐♠♣♦ss✐❜✐❧✐t② ♦❢ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡s❡ ✏❧♦❝❛t✐♦♥s✑ ✐♥ ❞✐♠❡♥s✐♦♥

4✱

t❤✐s

❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡❝♦♠❡s ♠✉❝❤ ♠♦r❡ t❤❛♥ ❥✉st ❛ ✇❛② t♦ ❣♦ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ✇❤❡♥❡✈❡r ❝♦♥✈❡♥✐❡♥t✳ ❚❤✐s t✐♠❡✱ ✇❡ ❥✉st s❛② ✏■t✬s t❤❡ s❛♠❡ t❤✐♥❣✑✳

❊①❛♠♣❧❡ ✶✳✷✳✹✿ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ✈❛r✐❛❜❧❡s ❚❤✐s ♠❛② ❜❡ t❤❡ ❞❛t❛ ❝♦♥t✐♥✉♦✉s❧② ❝♦❧❧❡❝t❡❞ ❜② ❛ ✇❡❛t❤❡r ❝❡♥t❡r ✿

1

2

3

4

5

t❡♠♣❡r❛t✉r❡

♣r❡ss✉r❡

♣r❡❝✐♣✐t❛t✐♦♥

❤✉♠✐❞✐t②

s✉♥❧✐❣❤t

... ...

❚❤❡② ❛r❡ ❛❧❧ ♠❡❛s✉r❡❞ ✐♥ ❞✐✛❡r❡♥t ✉♥✐ts ❛♥❞ ❝❛♥♥♦t ❜❡ s❡❡♥ ❛s ❛♥ ❛♥❛❧♦❣ ♦❢ ♦✉r ♣❤②s✐❝❛❧ s♣❛❝❡✳

❍♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤✐s

n✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡❄

▲❡t✬s ✜rst r❡❛❧✐③❡ t❤❛t✱ ✐♥ ❛ s❡♥s❡✱ ✇❡ ❤❛✈❡ ❢❛✐❧❡❞ ❡✈❡♥ ✇✐t❤ t❤❡ t❤r❡❡ ✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✦ ❲❡ ❤❛✈❡ ❤❛❞ t♦ sq✉❡❡③❡ t❤❡s❡ t❤r❡❡ ❞✐♠❡♥s✐♦♥s ♦♥ ❛ t✇♦ ✲❞✐♠❡♥s✐♦♥❛❧ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❲✐t❤♦✉t t❤❡ ♥✉♠❜❡rs t❡❧❧✐♥❣ ✉s ✇❤❛t t♦ ❡①♣❡❝t✱ ✇❡ ✇♦✉❧❞♥✬t ❜❡ ❛❜❧❡ t♦ t❡❧❧ t❤❡ ❞✐♠❡♥s✐♦♥ ✭t♦♣ r♦✇✮✿

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

✷✼

❆t ❜❡st✱ ✇❡ ❛r❡ s❡❡✐♥❣ t❤❡ s❤❛❞♦✇s ♦❢ t❤❡ ❧✐♥❡s ✭❜♦tt♦♠ r♦✇✮✳ ❚❤❡s❡ ❛r❡ t❤❡ s♣❛❝❡s ✇❡ ✇✐❧❧ st✉❞② ❛♥❞ t❤❡ ♥♦t❛t✐♦♥s ❢♦r t❤❡♠✿

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s • • • • • • •

R✱ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs ✭❧✐♥❡✮ R2 ✱ ❛❧❧ ♣❛✐rs ♦❢ r❡❛❧ ♥✉♠❜❡rs ✭♣❧❛♥❡✮ R3 ✱ ❛❧❧ tr✐♣❧❡s ♦❢ r❡❛❧ ♥✉♠❜❡rs ✭s♣❛❝❡✮ R4 ✱ ❛❧❧ q✉❛❞r✉♣❧❡s ♦❢ r❡❛❧ ♥✉♠❜❡rs

✳✳✳

Rn ✱ ❛❧❧ str✐♥❣s ♦❢ n r❡❛❧ ♥✉♠❜❡rs

✳✳✳

❊❛❝❤ ♦❢ t❤❡♠ ✐s s✉♣♣❧✐❡❞ ✇✐t❤ ✐ts ♦✇♥ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②✳ ❲❡ ❝❛♥ ❜✉✐❧❞ t❤❡s❡ ❜② ❝♦♥s❡❝✉t✐✈❡❧② ❛❞❞✐♥❣ ♦♥❡ ❞✐♠❡♥s✐♦♥ ❛t ❛ t✐♠❡✳ • ■❢ R ✐s ❣✐✈❡♥✱ ✇❡ tr❡❛t ✐t ❛s t❤❡ x✲❛①✐s ❛♥❞ t❤❡♥ ❛❞❞ ❛♥♦t❤❡r ❛①✐s✱ t❤❡ y ✲❛①✐s✱ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✜rst✳ • ❚❤❡ r❡s✉❧t ✐s R2 ✱ ✇❤✐❝❤ ✇❡ tr❡❛t ❛s t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ t❤❡♥ ❛❞❞ ❛♥♦t❤❡r ❛①✐s✱ t❤❡ z ✲❛①✐s✱ ♣❡r♣❡♥❞✐❝✉❧❛r

t♦ t❤❡ ✜rst t✇♦✳

• ❚❤❡ r❡s✉❧t ✐s R3 ✱ ✇❤✐❝❤ ✇❡ tr❡❛t ❛s t❤❡ xyz ✲s♣❛❝❡ ❛♥❞ t❤❡♥ ❛❞❞ ❛♥♦t❤❡r ❛①✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✜rst

t❤r❡❡❀ ❛♥❞ s♦ ♦♥✳

❍❡r❡ ✐s t❤❡ s✉♠♠❛r②✿

❲✐t❤ ♦✉r 1✲❞✐♠❡♥s✐♦♥❛❧ ✜♥❣❡r✱ ✇❡ ♣✉♥❝t✉r❡ t❤❡ s♣❛❝❡✳ ■♥ R4 ✱ t❤❡ s❛♠❡ t❤✐♥❣ ❤❛♣♣❡♥s❀ t❤❡ ✜♥❣❡r ❞✐s❛♣♣❡❛rs✿

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

❚❤❡ ❢♦r♠✉❧❛ t❤❛t r❡♣r❡s❡♥ts t❤❡ ❧✐♥❡ ✐♥ t❤❡ ✜rst r♦✇ ✐s✿ y = 0 ♦r x2 = 0 .

❚❤❡ ❢♦r♠✉❧❛ t❤❛t r❡♣r❡s❡♥ts t❤❡ ♣❧❛♥❡ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇ ✐s✿ z = 0 ♦r x3 = 0 .

❚❤✐s s♣❛❝❡ ✐s ❛❜str❛❝t ❜✉t ✐s st✐❧❧ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❧♦✇❡r✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s✿ ✶✳ ❋♦✉r ❝♦♣✐❡s ♦❢ R✿ t❤❡ ❢♦✉r ❝♦♦r❞✐♥❛t❡ ❛①❡s✳ ✷✳ ❙✐① ❝♦♣✐❡s ♦❢ R2 ✿ t❤❡ s✐① ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✱ ❡❛❝❤ s♣❛♥♥❡❞ ♦♥ ❛ ♣❛✐r ♦❢ t❤♦s❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳ ✸✳ ❋♦✉r ❝♦♣✐❡s ♦❢ R3 ✿ ❢♦✉r s♣❛❝❡s✱ ❡❛❝❤ ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ❢r❛♠❡ ♦❢ t❤r❡❡ ♦❢ t❤♦s❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✳

❊①❡r❝✐s❡ ✶✳✷✳✺

❲❤❛t ✐s t❤❡ ❢♦r♠✉❧❛ t❤❛t r❡♣r❡s❡♥ts ❡❛❝❤ ♦❢ t❤❡s❡ s♣❛❝❡s❄ ❊①❡r❝✐s❡ ✶✳✷✳✻

❍♦✇ ♠❛♥② ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ❛r❡ t❤❡r❡ ✐♥ R5 ❄ Rn ❄ ❍♦✇ ♠❛♥② ❝♦♦r❞✐♥❛t❡ s♣❛❝❡s❄ ❙♦✱ t❤❡s❡ s♣❛❝❡s ❛r❡♥✬t ✉♥r❡❧❛t❡❞✦ ■♥ ♦r❞❡r t♦ r❡✈❡❛❧ t❤❡ ✐♥t❡r♥❛❧ str✉❝t✉r❡ ♦❢ ❛ s♣❛❝❡s✱ ✇❡ ❧♦♦❦ ❢♦r ❧♦✇❡r✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s ✐♥ ✐t✳ ❚❤❡ ♣❧❛♥❡ ✐s ❛ st❛❝❦

✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ❥✉st ❛ ❝♦♣② ♦❢ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✿

♦❢ ❧✐♥❡s

✷✽

✶✳✷✳ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

❚❤❡s❡ ❧✐♥❡s ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s ❢♦r ❡❛❝❤ r❡❛❧

a

♦r

1✱ 2✱ 3✱✳✳✳

✷✾

b✿

x = a, y = b ❚❤❡② ❛r❡ ❝♦♣✐❡s ♦❢

R

❛♥❞ ✇✐❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②✳

■♥ ❢❛❝t✱ t❤❡② ❝❛♥ ❤❛✈❡ t❤❡✐r ♦✇♥

❝♦♦r❞✐♥❛t❡ s②st❡♠s✿

◆♦✇✱ ♦♥❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡

3✲s♣❛❝❡

❛s ❛ st❛❝❦ ♦❢ ♣❧❛♥❡s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ❥✉st ❛ ❝♦♣② ♦❢ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡

♣❧❛♥❡s✿

❚❤❡② ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s ❢♦r ❛❧❧ r❡❛❧

a, b, c✿ x = a, y = b, z = c

❚❤❡s❡ ❛r❡ ❝♦♣✐❡s ♦❢ ■❢ ❛

2✲❞✐♠❡♥s✐♦♥❛❧

R2 ✳

♣❡rs♦♥ ❝❛♥ r❡❝♦❣♥✐③❡ ✕ t❤✐♥❦✐♥❣ ♠❛t❤❡♠❛t✐❝❛❧❧② ✕ t❤❛t t❤❡

❝♦♣✐❡s ♦❢ ❤✐s ♦✇♥ s♣❛❝❡✱ ✇❡ ❝❛♥ s❡❡ ♦✉r ♣❤②s✐❝❛❧ ❙♦✱

R4

✐s ❛ ✏st❛❝❦✑ ♦❢

R3 s✳

3✲s♣❛❝❡

❛s ❥✉st ❛ s✐♥❣❧❡ ✏❧❛②❡r✑

3✲s♣❛❝❡ 4 ✐♥ R ✳

✐s ♠❛❞❡ ♦❢ ❧❛②❡rs ♦❢

❍♦✇ t❤❡② ✜t t♦❣❡t❤❡r ✐s ❤❛r❞ t♦ ✈✐s✉❛❧✐③❡✱ ❜✉t t❤❡② ❛r❡ st✐❧❧ ❝♦♣✐❡s ♦❢

R3

❣✐✈❡♥

❜② ❡q✉❛t✐♦♥s✿

x1 = a1 , x2 = a2 , x3 = a3 , x4 = a4 ❊①❡r❝✐s❡ ✶✳✷✳✼

❲❤❛t ✐s ❛ ❧✐♥❡ ❛ st❛❝❦ ♦❢ ❄

❙♦✱ ✇❡ ❝❛♥ s❡❡ ♠❛♥② ❝♦♣✐❡s ♦❢

Rm

✐♥

Rn ✱

✇✐t❤

n > m✳

❇❡②♦♥❞ ❛ ❝❡rt❛✐♥ ♣♦✐♥t✱ t❤❡ ❝❤❛♥❝❡ t♦ ✈✐s✉❛❧✐③❡ t❤❡ s♣❛❝❡ ✐s ❣♦♥❡✳ ❲❡✱ ❤♦✇❡✈❡r✱ ❛r❡ st✐❧❧ ❛❜❧❡ t♦ ✈✐s✉❛❧✐③❡ t❤❡ s♣❛❝❡ ♦♥❡ ❡❧❡♠❡♥t ❛t ❛ t✐♠❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ♣♦✐♥t ✐♥ t❤❡ ✇✐t❤

n

t❡r♠s✿

n✲❞✐♠❡♥s✐♦♥❛❧

k 1 2 3 4 5 ... n xk x1 x2 x3 x4 x5 ... xn

s♣❛❝❡ ✐s ♥♦t❤✐♥❣ ❜✉t ❛ s❡q✉❡♥❝❡

✶✳✷✳

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥s

1✱ 2✱ 3✱✳✳✳

✸✵

■t✬s ❥✉st ❛ ❢✉♥❝t✐♦♥✱ ✇✐t❤ t❤❡ ✐♥♣✉ts ✐♥ t❤❡ ✜rst r♦✇ ❛♥❞ t❤❡ ♦✉t♣✉ts ✐♥ t❤❡ s❡❝♦♥❞✳ ❲❡ ✈✐s✉❛❧✐③❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡✐r ❣r❛♣❤s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s str✐♥❣ ♦❢ 6 ♥✉♠❜❡rs ✐s ❛ ♣♦✐♥t ✐♥ t❤❡ 6✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✿

❲❛r♥✐♥❣✦ ❚❤❡ ❝✉r✈❡ t❤❛t ②♦✉ s❡❡ ✐s ✐♥❝✐❞❡♥t❛❧ ❜❡❝❛✉s❡ t❤❡ r♦✇s ♦❢ t❤❡ t❛❜❧❡ ❝❛♥ ❜❡ r❡✲❛rr❛♥❣❡❞✳

◆❡①t✱ t❤❡r❡ ✐s ♥♦ ♣♦✐♥t ✐♥ st✉❞②✐♥❣ s♣❛❝❡s ✇✐t❤♦✉t st✉❞②✐♥❣ ❢✉♥❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡♠✳ ▲❡t✬s r❡✈✐❡✇ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳ ❲❡ ♣❧❛❝❡ t❤❡♠ ✐♥ ❛ t❛❜❧❡ ✇✐t❤ t✇♦ ❛①❡s r❡♣r❡s❡♥t✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❝♦❞♦♠❛✐♥✿

❲❡ ❛❧✇❛②s st❛rt ❛t t❤❡ ✈❡r② ✜rst ❝❡❧❧✳ Pr❡✈✐♦✉s❧② ✇❡ ♠❛❞❡ ❛ st❡♣ ✐♥ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥ ❛♥❞ ❡①♣❧♦r❡❞ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ t❤✐s t❛❜❧❡✳ ❲❡ ❛❧s♦ ♠♦✈❡❞ t♦ t❤❡ r✐❣❤t✳ ❲✐t❤ ❛❧❧ t❤✐s ❝♦♠♣❧❡①✐t②✱ ✇❡ s❤♦✉❧❞♥✬t ♦✈❡r❧♦♦❦ t❤❡ ❣❡♥❡r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ ❢✉♥❝t✐♦♥s✳ ❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t ♦❢ ✇❤❛t❡✈❡r ♥❛t✉r❡✿ ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t X

7→

f

7→

Y

✶✳✸✳ ●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

✸✶

▲❡t✬s t❛❦❡ ♦♥❡ ❢r♦♠ t❤❡ ❧❡❢t✲♠♦st ❝♦❧✉♠♥ ❛♥❞ ♦♥❡ ❢r♦♠ t❤❡ ❜♦tt♦♠ r♦✇✿ ❢✉♥❝t✐♦♥ ♦❢

♣❛r❛♠❡tr✐❝ ✐♥♣✉t

t 7→ R

❝✉r✈❡

F

✐♥♣✉t

♦✉t♣✉t

X 7→ Rm

7→ X Rm

♥✉♠❜❡r

♣♦✐♥t

t✇♦ ✈❛r✐❛❜❧❡s

f

♣♦✐♥t

t✐♠❡

♣r✐❝❡s ♦❢ ♣❛rts

t✐♠❡

♣r✐❝❡s ♦❢ st♦❝❦s

♣r✐❝❡s ♦❢ ♣❛rts ♣r✐❝❡s ♦❢ st♦❝❦s

♦✉t♣✉t

7→ z R ♥✉♠❜❡r ♣r✐❝❡ ♦❢ ❝❛r ✈❛❧✉❡ ♦❢ ♣♦rt❢♦❧✐♦

❚❤❡② ❝❛♥ ❜❡ ❧✐♥❦❡❞ ✉♣ ❛♥❞ ♣r♦❞✉❝❡ ❛ ❝♦♠♣♦s✐t✐♦♥✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳ ❆❜♦✈❡ ✐s ❛ ✈✐❡✇ ♦❢ ✏❣❡♥❡r✐❝✑ ❢✉♥❝t✐♦♥s✳ ■♥ t❤❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ ❝♦♥t❡①t✱ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ t❛❜❧❡ ❛r❡ s✐♠♣❧❡r ❛♥❞ s♦ ❛r❡ t❤❡✐r ✈✐s✉❛❧✐③❛t✐♦♥s✿

❲❡ t✉r♥ t♦ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② ♦❢

n✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡s✳

✶✳✸✳ ●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

❚❤❡ ❛①❡s ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠

R3

❢♦r ♦✉r ♣❤②s✐❝❛❧ s♣❛❝❡ r❡❢❡r t♦ t❤❡ s❛♠❡✿ ❞✐st❛♥❝❡s t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ Rn r❡❢❡r

♣❧❛♥❡s✳ ❚❤❡② ❛r❡ ✭♦r s❤♦✉❧❞ ❜❡✮ ♠❡❛s✉r❡❞ ✐♥ t❤❡ s❛♠❡ ✉♥✐t✳ ❊✈❡♥ t❤♦✉❣❤✱ ✐♥ ❣❡♥❡r❛❧✱ t❤❡ ❛①❡s ♦❢

t♦ ✉♥r❡❧❛t❡❞ q✉❛♥t✐t✐❡s✱ t❤❡② ♠❛② ❜❡ ♠❡❛s✉r❡❞ ✐♥ t❤❡ s❛♠❡ ✉♥✐t✱ s✉❝❤ ❛s t❤❡ ♣r✐❝❡s ♦❢ n ❝♦♠♠♦❞✐t✐❡s ❜❡✐♥❣ n tr❛❞❡❞✳ ❲❤❡♥ t❤✐s ✐s t❤❡ ❝❛s❡✱ ❞♦✐♥❣ ❣❡♦♠❡tr② ✐♥ R ❜❛s❡❞ ❡♥t✐r❡❧② ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts ✐s ♣♦ss✐❜❧❡✳ ❆ ❈❛rt❡s✐❛♥ s②st❡♠ ❤❛s ❡✈❡r②t❤✐♥❣ ✐♥ t❤❡ s♣❛❝❡ ♣r❡✲♠❡❛s✉r❡❞✳

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

✸✷

■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❝♦♠♣✉t❡ ✭r❛t❤❡r t❤❛♥ ♠❡❛s✉r❡✮ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥s ❜❡❝❛✉s❡ t❤❡ ❞✐st❛♥❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❧♦❝❛t✐♦♥s✳

❚❤❡♦r❡♠ ✶✳✸✳✶✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✶

❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ♣♦✐♥t r❡s♣❡❝t✐✈❡❧② ✐s

P

t♦ ♣♦✐♥t

✐♥

Q

R

❣✐✈❡♥ ❜② r❡❛❧ ♥✉♠❜❡rs

x

❛♥❞

x′

d(P, Q) = |x − x′ | ❍❡r❡✱ t❤❡ ❣❡♦♠❡tr② ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ ❞✐st❛♥❝❡s r❡❧✐❡s ♦♥ t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❛❧ ♥✉♠❜❡rs ✭t❤❡ s✉❜tr❛❝t✐♦♥✮✳ ◆♦✇ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 2✳ ❚❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥s P ❛♥❞ Q ✐♥ ′ ′ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s (x, y) ❛♥❞ (x , y ) ✐s ❢♦✉♥❞ ❜② ✉s✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ❢♦r ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❛①❡s ✐♥ ♦r❞❡r t♦ ✜♥❞ ✶✳ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥

x

❛♥❞

x′ ✱

✇❤✐❝❤ ✐s

✷✳ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥

y

❛♥❞

y′✱

✇❤✐❝❤ ✐s

|x − x′ | = |x′ − x|✱

|y − y ′ | = |y ′ − y|✱

❚❤❡♥ t❤❡ t✇♦ ♥✉♠❜❡rs ❛r❡ ♣✉t t♦❣❡t❤❡r ❜② t❤❡

❛♥❞ r❡s♣❡❝t✐✈❡❧②✳

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤✐s s✐♠♣❧✐✜❝❛t✐♦♥✿

|x − x′ |2 = (x − x′ )2 , |y − y ′ |2 = (y − y ′ )2 . ❚❤❡♦r❡♠ ✶✳✸✳✷✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✷

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts P ❛♥❞ Q ✐♥ R2 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s (x, y) ❛♥❞ (x′, y′) r❡s♣❡❝t✐✈❡❧② ✐s d(P, Q) =

❚❤❡ t✇♦ ❡①❝❡♣t✐♦♥❛❧ ❝❛s❡s ✇❤❡♥

P

❛♥❞

Q

p

(x − x′ )2 + (y − y ′ )2

❧✐❡ ♦♥ t❤❡ s❛♠❡ ✈❡rt✐❝❛❧ ♦r t❤❡ s❛♠❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ✭❛♥❞ t❤❡

tr✐❛♥❣❧❡ ✏❞❡❣❡♥❡r❛t❡s✑ ✐♥t♦ ❛ s❡❣♠❡♥t✮ ❛r❡ tr❡❛t❡❞ s❡♣❛r❛t❡❧②✳ ◆♦✇ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥

❉✐st❛♥❝❡ ❋♦r♠✉❧❛✳

3✳

❲❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡r❡ ✇✐❧❧ ❜❡ ❛♥♦t❤❡r t❡r♠ ✐♥ t❤❡ s✉♠ ♦❢ t❤❡

❚❤❡♦r❡♠ ✶✳✸✳✸✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✸

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts (x′ , y ′ , z ′ ) r❡s♣❡❝t✐✈❡❧② ✐s d(P, Q) =

P

❛♥❞

Q

✐♥

R3

✇✐t❤ ❝♦♦r❞✐♥❛t❡s

p (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2

(x, y, z)

❛♥❞

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

✸✸

❆ ♣❛tt❡r♥ st❛rts t♦ ❛♣♣❡❛r✿

◮

❚❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ✐s t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦❢ t❤❡ ❞✐st❛♥❝❡s ❛❧♦♥❣ ❡❛❝❤ ♦❢ t❤❡

❝♦♦r❞✐♥❛t❡s✳ ❚❤✐♥❦✐♥❣ ❜② ❛♥❛❧♦❣②✱ ✇❡ ❝♦♥t✐♥✉❡ ♦♥ t♦ ✐♥❝❧✉❞❡ t❤❡ ❝❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥

❞✐♠❡♥s✐♦♥

♣♦✐♥ts

❝♦♦r❞✐♥❛t❡s

1

P Q P Q P Q P Q ...

x x′ (x, y) (x′ , y ′ ) (x, y, z) (x′ , y ′ , z ′ ) (x1 , x2 , x3 , x4 ) (x′1 , x′2 , x′3 , x′4 ) ...

2 3 4 ...

❚❤❡r❡ ❛r❡

n

t❡r♠s ✐♥ ❞✐♠❡♥s✐♦♥

4✿

❞✐st❛♥❝❡

d(P, Q)2 = (x − x′ )2 d(P, Q)2 = (x − x′ )2 + (y − y ′ )2 d(P, Q)2 = (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2 d(P, Q)2 = (x1 − x′1 )2 + (x2 − x′2 )2 + (x3 − x′3 )2 + (x4 − x′4 )2 ...

n✿

❞✐♠❡♥s✐♦♥

♣♦✐♥ts

❝♦♦r❞✐♥❛t❡s

n

P Q

(x1 , x2 , ..., xn ) (x′1 , x′2 , ..., x′n ) d(P, Q)2 = (x1 − x′1 )2 + (x2 − x′2 )2 + ... + (xn − x′n )2

❊①❛♠♣❧❡ ✶✳✸✳✹✿ ❣❡♦♠❡tr② ♦❢

❚❤❡ ❢♦r♠✉❧❛ ❢♦r

n = 1, 2, 3

R3

✐♥

❞✐st❛♥❝❡

R4

✐s ❥✉st✐✜❡❞ ❜② ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡✳ ❲❤❛t ❛❜♦✉t n = 4 R3 t❤❛t ♠❛❦❡ ✉♣ R4 ✳ ❖♥❡ ♦❢ t❤❡♠ ✐s ❣✐✈❡♥ ❜② x4 = a4

❛♥❞ ❛❜♦✈❡❄ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ❝♦♣✐❡s ♦❢ ❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡r

a4 ✳

■❢ ✇❡ t❛❦❡ ❛♥② t✇♦ ♣♦✐♥ts

P, Q

✇✐t❤✐♥ ✐t✱ t❤❡ ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿

p d(P, Q) = p(x1 − x′1 )2 + (x2 − x′2 )2 + (x3 − x′3 )2 + (a4 − a4 )2 = (x1 − x′1 )2 + (x2 − x′2 )2 + (x3 − x′3 )2 .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐st❛♥❝❡ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ❢♦r ❞✐♠❡♥s✐♦♥ 3 ♦❢ s✉❝❤ ❛ ❝♦♣② ♦❢ R ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✏♦r✐❣✐♥❛❧✑✦

3✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❣❡♦♠❡tr②

❊①❡r❝✐s❡ ✶✳✸✳✺

❙❤♦✇ t❤❛t t❤❡ ❣❡♦♠❡tr② ♦❢

❛♥② ♣❧❛♥❡ ✐♥ R3 ❛♥❞ R4 ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ R2✳

❈❛♥ ✇❡ ❥✉st✐❢② t❤✐s ❢♦r♠✉❧❛ ✇✐t❤ ♠♦r❡ t❤❛♥ ❥✉st ✏■t✬s ❛ ♣❛tt❡r♥✑❄ ❨❡s✱ ✇❡ ♣r♦❣r❡ss ❢r♦♠ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ n n+1 ❣❡♦♠❡tr② ♦❢ X = R t♦ t❤❛t ♦❢ Y = R ✱ ❡✈❡r② t✐♠❡✳ ❙✉♣♣♦s❡ t❤❡ ❞✐st❛♥❝❡s ✐♥

X = Rn

❛r❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✳ n+1 ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ r❡st ✕ t♦ ❝r❡❛t❡ Y = R ✿

❚❤❡♥✱ ✇❡ ❛❞❞ ❛♥ ❡①tr❛ ❛①✐s ✕

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

❚❤❡♥✱ t❤❡

✸✹

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠

✐s ❛♣♣❧✐❡❞ ✭t❤❡ ❣r❡❡♥ tr✐❛♥❣❧❡✮✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❥✉st ❛s ✐♥ t❤❡ ❝❛s❡

n=3

♣r❡s❡♥t❡❞ ❛❜♦✈❡✿

d(P, Q)2 = d(P, R)2 + d(R, Q)2 = (x1 − x′1 )2 + (x2 − x′2 )2 + ... + (xn − x′n )2 + (xn+1 − x′n+1 )2 . ❚❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s t♦ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥

n = 1, 2, 3✳

n✳

■t ♠❛t❝❤❡s t❤❡ ♠❡❛s✉r❡❞ ❞✐st❛♥❝❡ ✐♥ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡✿

❲❡ ❝❛♥✬t s❛② t❤❡ s❛♠❡ ❛❜♦✉t t❤❡ s♣❛❝❡s ♦❢ ❞✐♠❡♥s✐♦♥s

❢♦r♠✉❧❛✱ t❤❡r❡❢♦r❡✱ ✐s s❡❡♥ ❛s t❤❡

❞❡✜♥✐t✐♦♥

n > 3✳

❚❤❡② ❛r❡ ❛❜str❛❝t s♣❛❝❡s✳ ❚❤❡

♦❢ t❤❡ ❞✐st❛♥❝❡ ❢♦r t❤❡s❡ s♣❛❝❡s✿

❉❡✜♥✐t✐♦♥ ✶✳✸✳✻✿ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ❚❤❡

❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts P

❛♥❞

Q ✐♥ Rn

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ sq✉❛r❡

r♦♦t ♦❢ t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦❢ t❤❡ ❞✐st❛♥❝❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s✿

P = (x1 , x2 , ..., xn ) Q = (x′1 , x′2 , ..., x′n ) p d(P, Q) = (x1 − x′1 )2 + (x2 − x′2 )2 + ... + (xn − x′n )2

❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝✳ ❚❤❡ s♣❛❝❡ Rn ❡q✉✐♣♣❡❞ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ✐s ❝❛❧❧❡❞ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳ ❲❡ r❡❢❡r t♦ t❤❡ ❢♦r♠✉❧❛ ❛s t❤❡

✇✐t❤

◆♦✇✱ t❤❡ ❢♦r♠✉❧❛ ✐s s♦♠❡✇❤❛t ❝♦♠♣❧✐❝❛t❡❞✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛ ❢❡✇ s✐♠♣❧❡ r✉❧❡s ❛♣♣❧② ❡q✉❛❧❧② t♦ ❛❧❧ ❞✐♠❡♥s✐♦♥s✱ ✇✐t❤♦✉t r❡❢❡r❡♥❝❡ t♦ t❤❡ ❢♦r♠✉❧❛s❄ ❲❡ ❢♦r♠✉❧❛t❡ t❤r❡❡ ✈❡r② s✐♠♣❧❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐st❛♥❝❡s✳

❋✐rst✱ t❤❡ ❞✐st❛♥❝❡s ❝❛♥✬t ❜❡ ♥❡❣❛t✐✈❡ ❛♥❞✱

♠♦r❡♦✈❡r✱ ❢♦r t❤❡ ❞✐st❛♥❝❡ t♦ ❜❡ ③❡r♦✱ t❤❡ t✇♦ ♣♦✐♥ts ❤❛✈❡ t♦ ❜❡ t❤❡ s❛♠❡✳ ❙❡❝♦♥❞✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠

Q

✐s t❤❡ s❛♠❡ ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠

Q

t♦

P✳

P

t♦

❆♥❞ s♦ ♦♥✳

❚❤❡♦r❡♠ ✶✳✸✳✼✿ ❆①✐♦♠s ♦❢ ▼❡tr✐❝ ❙♣❛❝❡ ❙✉♣♣♦s❡ P, Q, S ❛r❡ ♣♦✐♥ts ✐♥ R3 ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❛r❡ s❛t✐s✜❡❞✿ • P♦s✐t✐✈✐t②✿ d(P, Q) ≥ 0❀ ❛♥❞ d(P, Q) = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ P = Q✳ • ❙②♠♠❡tr②✿ d(P, Q) = d(Q, P )✳ • ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✿ d(P, Q) + d(Q, S) ≥ d(P, S)✳

Pr♦♦❢✳ ❙✉♣♣♦s❡

d(P, Q) = 0✳

❚❤❡♥

0 = d(P, Q)2 = (x1 − x′1 )2 + (x2 − x′2 )2 + ... + (xn − x′n )2 . ❙✐♥❝❡ ♥♦♥❡ ♦❢ t❤❡ t❡r♠s ✐s ♥❡❣❛t✐✈❡✱ ❛❧❧ ❤❛✈❡ t♦ ❜❡ ③❡r♦✿

(x1 − x′1 )2 = 0, (x2 − x′2 )2 = 0, ..., (xn − x′n )2 = 0 . ❚❤❡r❡❢♦r❡✱

x1 = x′1 , x2 = x′2 , ..., xn = x′n . ■t ❢♦❧❧♦✇s t❤❛t

P = Q✳

❊①❡r❝✐s❡ ✶✳✸✳✽ Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳

❚❤❡s❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ❤❛✈❡ ❜❡❡♥ ❥✉st✐✜❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❢❛♠✐❧✐❛r ❣❡♦♠❡tr② ♦❢ t❤❡ ✏♣❤②s✐❝❛❧ s♣❛❝❡✑

R3 ✳

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

❍♦✇❡✈❡r✱ t❤❡② ❛❧s♦ s❡r✈❡ ❛s ❛

✸✺

st❛rt✐♥❣ ♣♦✐♥t ❢♦r ❢✉rt❤❡r ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❇❡❧♦✇✱ ✇❡ ✇✐❧❧ ❞❡✜♥❡

t❤❡ ♥❡✇ ❣❡♦♠❡tr② ♦❢ t❤❡ ❛❜str❛❝t s♣❛❝❡

Rn

❛♥❞ ❞❡♠♦♥str❛t❡ t❤❛t t❤❡s❡ ✏❛①✐♦♠s✑ ❛r❡ st✐❧❧ s❛t✐s✜❡❞✳

❊①❡r❝✐s❡ ✶✳✸✳✾ ❚❤❡ ❞✐st❛♥❝❡ ✐s ❛ ❢✉♥❝t✐♦♥✳ ❊①♣❧❛✐♥✳

❲❡ ❦♥♦✇ t❤❡ ❧❛st ♣r♦♣❡rt② ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✿

❲❡ ❝❛♥ ❥✉st✐❢② ✐t ❢♦r ❞✐♠❡♥s✐♦♥

n≥4

❜② r❡❢❡rr✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿ ❆♥② t❤r❡❡ ♣♦✐♥ts ❧✐❡ ✇✐t❤✐♥ ❛ s✐♥❣❧❡

♣❧❛♥❡✳ ❚❤✐s ❢❛❝t ❜r✐♥❣s ✉s ❜❛❝❦ t♦ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✳✳✳ ✐❢ t❤❛t✬s ✇❤❛t ✇❡ ✇❛♥t✳

❊①❛♠♣❧❡ ✶✳✸✳✶✵✿ ❝✐t② ❜❧♦❝❦s ❚❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r t❤❡ ♣❧❛♥❡ ❣✐✈❡s ✉s t❤❡ ❞✐st❛♥❝❡ ♠❡❛s✉r❡❞

❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛s ✐❢ ✇❡ ❛r❡

✇❛❧❦✐♥❣ t❤r♦✉❣❤ ❛ ✜❡❧❞✳ ❇✉t ✇❤❛t ✐❢ ✇❡ ❛r❡ ✇❛❧❦✐♥❣ t❤r♦✉❣❤ ❛ ❝✐t②❄ ❲❡ t❤❡♥ ❝❛♥♥♦t ❣♦ ❞✐❛❣♦♥❛❧❧② ❛s ✇❡ ❤❛✈❡ t♦ ❢♦❧❧♦✇ t❤❡ ❣r✐❞ ♦❢ str❡❡ts✳ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❧♦❝❛t✐♦♥s

P = (x, y)

❚❤✐s ❢❛❝t ❞✐❝t❛t❡s ❤♦✇ ✇❡ ♠❡❛s✉r❡ ❞✐st❛♥❝❡s✳ ❛♥❞

Q = (u, v)✱

✇❡ ♠❡❛s✉r❡

❚♦ ✜♥❞ t❤❡

❛❧♦♥❣ t❤❡ ❣r✐❞ ♦♥❧②✿

❚❤❡ ❢♦r♠✉❧❛ ✐s✱ t❤❡r❡❢♦r❡✿

■t ✐s ❝❛❧❧❡❞ t❤❡ ✐s

2

t❛①✐❝❛❜ ♠❡tr✐❝✳

dT (P, Q) = |x − u| + |y − v| ■t ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ❛s t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛

1

sq✉❛r❡

✉♥✐ts ❧♦♥❣ ✉♥❞❡r t❤✐s ❣❡♦♠❡tr②✳

❊①❡r❝✐s❡ ✶✳✸✳✶✶ Pr♦✈❡ t❤❛t t❤❡ t❛①✐❝❛❜ ♠❡tr✐❝ s❛t✐s✜❡s t❤❡ t❤r❡❡ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ t❤❡♦r❡♠✳

■❢ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡ ❝❛♥ ❜❡ r❡❛s♦♥❛❜❧② tr❡❛t❡❞ ✇✐t❤ ♥♦♥✲❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡s✱ t❤❡ ✐❞❡❛ ✐s ❡✈❡♥ ♠♦r❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ ■❢ ✇❡ ❛r❡ ❣✐✈❡♥ s♣❛❝❡ ♦❢ ❧♦❝❛t✐♦♥s ♦r ✏st❛t❡s✑✱

Rn ✱

✐t ✐s ♦✉r ❝❤♦✐❝❡ t♦ ♣✐❝❦ ❛♥ ❛♣♣r♦♣r✐❛t❡ ✇❛② t♦ ❝♦♠♣✉t❡

❞✐st❛♥❝❡s ❢r♦♠ ❝♦♦r❞✐♥❛t❡s✿

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✷✿ t❤r❡❡ ♠❡tr✐❝s ❙✉♣♣♦s❡ ♣♦✐♥ts r❡s♣❡❝t✐✈❡❧②✳

P

❛♥❞

Q ✐♥ Rn ❤❛✈❡ ❝♦♦r❞✐♥❛t❡s (x1 , x2 , ..., xn ) ❛♥❞ (x′1 , x′2 , ..., x′n )

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

✸✻ ✶✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥

♠❡tr✐❝✱ ♦r t❤❡ L2 ✲♠❡tr✐❝✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ v u n uX d2 (P, Q) = t (xk − x′ )2 k

k=1

✷✳ ❚❤❡ t❛①✐❝❛❜

♠❡tr✐❝✱ ♦r t❤❡ L1 ✲♠❡tr✐❝✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ d1 (P, Q) =

n X k=1

✸✳ ❚❤❡ ♠❛①

|xk − x′k |

♠❡tr✐❝✱ ♦r t❤❡ L∞ ✲♠❡tr✐❝✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ d∞ (P, Q) = max |xk − x′k | k=1,...,n

❚❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ✭n = 40✮✿

❋♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝✱ ✇❡ ❝♦♠♣✉t❡ ❢♦r ❡❛❝❤ r♦✇✿ ❂❆❇❙✭❘❈❬✲✷❪✲❘❈❬✲✶❪✮

❲❡ ♣❧♦t ✐t✱ t❤❡♥ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛✿

❂❙❯▼✭❘❬✶❪❈✿❘❬✹✵❪❈✮

❋♦r t❤❡ t❛①✐❝❛❜ ♠❡tr✐❝✱ ✇❡ ❝♦♠♣✉t❡ ❢♦r ❡❛❝❤ r♦✇✿ ❂❘❈❬✲✶❪✂ ✷

❲❡ t❤❡♥ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛✿

❂❙◗❘❚✭❙❯▼✭❘❬✶❪❈✿❘❬✹✵❪❈✮✮

❚❤❡ ❢♦r♠✉❧❛ ❢♦r ♠❛① ♠❡tr✐❝ ✐s s✐♠♣❧②✿ ❂▼❆❳✭❘❬✶❪❈❬✲✷❪✿❘❬✹✵❪❈❬✲✷❪✮

✶✳✸✳

●❡♦♠❡tr② ♦❢ ❞✐st❛♥❝❡s

✸✼

❊①❡r❝✐s❡ ✶✳✸✳✶✸

Pr♦✈❡ t❤❡ ❆①✐♦♠s ♦❢ ▼❡tr✐❝ ❙♣❛❝❡ ❢♦r t❤❡s❡ ❢♦r♠✉❧❛s✳

❲❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ t❤❡ ✏♣❤②s✐❝❛❧ s♣❛❝❡✑ ✭n ✐♠♣❧✐❡❞✳ ❋♦r t❤❡ ✏❛❜str❛❝t s♣❛❝❡s✑ ✭n

= 1, 2, 3✮ ❛s ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠s✱ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ✐s = 1, 2, ...✮✱ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ✐s t❤❡ ❞❡❢❛✉❧t ❝❤♦✐❝❡❀ ❤♦✇❡✈❡r✱ t❤❡r❡

❛r❡ ♠❛♥② ❡①❛♠♣❧❡s ✇❤❡♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ✭❛❦❛ t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛✮ ❞♦♥✬t ❛♣♣❧②✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✹✿ ❛ttr✐❜✉t❡s

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ♣r✐❝❡s ♦❢ ✇❤❡❛t ❛♥❞ s✉❣❛r ❛❣❛✐♥✳ ❚❤❡ s♣❛❝❡ ♦❢ ♣r✐❝❡s ✐s t❤❡ s❛♠❡✱

R2 ✳

❍♦✇❡✈❡r✱

♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♣r✐❝❡s ✇✐t❤ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ❧❡❛❞s t♦ ✉♥❞❡✲ s✐r❛❜❧❡ ❡✛❡❝ts✳ ❋♦r ❡①❛♠♣❧❡✱ s✉❝❤ ❛ tr✐✈✐❛❧ st❡♣ ❛s ❝❤❛♥❣✐♥❣ t❤❡ ❧❛tt❡r ❢r♦♠ ✏♣❡r t♦♥✑ t♦ ✏♣❡r ❦✐❧♦❣r❛♠✑ ✇✐❧❧ ❝❤❛♥❣❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ■t ✐s ❛s ✐❢ t❤❡ s♣❛❝❡ ✐s str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣♦✐♥t

P

t❤❛t ✉s❡❞ t♦ ❜❡ ❝❧♦s❡r t♦ ♣♦✐♥t

A t❤❛♥ t♦ B ♠✐❣❤t ♥♦✇ s❛t✐s❢② t❤❡ ♦♣♣♦s✐t❡ ❝♦♥❞✐t✐♦♥✳

❋✉rt❤❡r♠♦r❡✱ t❤❡ t✇♦ ✭♦r ♠♦r❡✮ ♠❡❛s✉r❡♠❡♥ts ♦r ♦t❤❡r ❛ttr✐❜✉t❡s ♠✐❣❤t ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤

✉♥✐ts✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❝♦♠♣❛r❡ t✇♦ ❜✉✐❧t ❜❛s❡❞ t♦ t❤❡ t✇♦ ♠❛✐♥ ♠❡❛s✉r❡♠❡♥ts✿ ✇❡✐❣❤t ❛♥❞ ❤❡✐❣❤t✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✐❢ ✇❡ s✉❜st✐t✉t❡

♦t❤❡r✳ ■♥ s♦♠❡ ♦❜✈✐♦✉s ❝❛s❡s✱ t❤❡② ✇✐❧❧ ❡✈❡♥ ❤❛✈❡ ❞✐✛❡r❡♥t ♣❡rs♦♥s

s✉❝❤ ♥✉♠❜❡rs ✐♥t♦ ♦✉r ❢♦r♠✉❧❛✱ ✇❡ ✇✐❧❧ ❜❡ ❛❞❞✐♥❣ ♣♦✉♥❞s t♦ ❢❡❡t✦

❙♦♠❡ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♦❢ ❣❡♦♠❡tr② ✜♥❞ t❤❡✐r ❛♥❛❧♦❣s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r✿

• •

❝✐r❝❧❡ ♦♥ t❤❡ ♣❧❛♥❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ❛ ❣✐✈❡♥ ❞✐st❛♥❝❡ ❛✇❛② ❢r♦♠ ✐ts ❝❡♥t❡r✳ ❆ s♣❤❡r❡ ✐♥ t❤❡ s♣❛❝❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ❛ ❣✐✈❡♥ ❞✐st❛♥❝❡ ❛✇❛② ❢r♦♠ ✐ts ❝❡♥t❡r✳

❆

❲❤❛t ❛❜♦✉t ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s❄ ❚❤❡ ♣❛tt❡r♥ ✐s ❝❧❡❛r✿

•

❆

❤②♣❡rs♣❤❡r❡ ✐♥ Rn ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ❛ ❣✐✈❡♥ ❞✐st❛♥❝❡ ❛✇❛② ❢r♦♠ ✐ts ❝❡♥t❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❡❛❝❤ ♣♦✐♥t

P

♦♥ t❤❡ ❤②♣❡rs♣❤❡r❡ s❛t✐s✜❡s✿

d(P, Q) = R , ✇❤❡r❡

Q

✐s ✐ts ❝❡♥t❡r ❛♥❞

R

✐s ✐ts r❛❞✐✉s✳

❊①❛♠♣❧❡ ✶✳✸✳✶✺✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇✱ t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ✐s

• •

♣r♦♣♦rt✐♦♥❛❧ t♦ ❡✐t❤❡r ♦❢ t❤❡✐r ♠❛ss❡s✱ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ❝❡♥t❡rs✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢♦r❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛✿

F =G

mM , r2

✇❤❡r❡✿

• • • • •

F ✐s t❤❡ ❢♦r❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝ts❀ G ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t❀ m ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♦❜❥❡❝t❀ M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t❀ r ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢

F

♦♥

m

❛♥❞

M

♦❢ t❤❡ ♠❛ss ♦❢ t❤❡ t✇♦✳

✐s ✈❡r② s✐♠♣❧❡ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ ♠❛ss❡s ♦❢

♣❧❛♥❡ts ❛r❡ r❡♠❛✐♥ t❤❡ s❛♠❡✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✿

F (r) = G ▼♦r❡ ♣r❡❝✐s❡❧②✱

r

❞❡♣❡♥❞s ♦♥ t❤❡

❧♦❝❛t✐♦♥ P

mM . r2

♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t ✐♥ t❤❡

r = d(O, P ) ,

3✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✿

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✸✽

✐❢✱ ❢♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✜rst ♦❜❥❡❝t ✐s ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✳ ◆♦t❡ t❤❛t t❤✐s ❢♦r❝❡ ✐s ❝♦♥st❛♥t ❛❧♦♥❣ ❛♥② ♦❢ t❤❡ s♣❤❡r❡s ❝❡♥t❡r❡❞ ❛t O✳ ◆♦✇✱ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❧❛✇ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ P ✿ F (P ) =

GmM d(O, P )2

❋✉rt❤❡r♠♦r❡✱ ✐❢ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ t❤r❡❡ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s x, y, z ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ P ✱ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❧❛✇ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ F (x, y, z) =

GmM GmM GmM = p . 2 = 2 2 d(O, P ) x + y2 + z2 x2 + y 2 + z 2

■❢ ✇❡ ✐❣♥♦r❡ t❤❡ t❤✐r❞ ✈❛r✐❛❜❧❡ ✭z = 0✮✱ ✇❡ ❝❛♥ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

❇✉t ✇❤❛t ❛❜♦✉t t❤❡

❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ❢♦r❝❡❄ ❚❤✐s q✉❡st✐♦♥ ✐s ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✸✳✶✻ ❱✐s✉❛❧✐③❡ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❝❛s❡ ♦❢ 3 ❞✐♠❡♥s✐♦♥s✳

✶✳✹✳ ❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

❞✐r❡❝t✐♦♥s

❲❡ ✐♥tr♦❞✉❝❡❞ ✈❡❝t♦rs ✐♥ ♣r❡✈✐♦✉s❧② t♦ ♣r♦♣❡r❧② ❤❛♥❞❧❡ t❤❡ ❣❡♦♠❡tr✐❝ ✐ss✉❡ ♦❢ ❛♥❞ ❛♥❣❧❡s ❜❡t✇❡❡♥ ❞✐r❡❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✈❡❝t♦rs ❛❧s♦ ❛♣♣❡❛r ❢r❡q✉❡♥t❧② ✐♥ ♦✉r st✉❞② ♦❢ t❤❡ ♥❛t✉r❛❧ ✇♦r❧❞✳

❉❡✜♥✐t✐♦♥ ✶✳✹✳✶✿ ❞✐s♣❧❛❝❡♠❡♥t ❲❤❡♥ t❤❡ ♣♦✐♥ts ✐♥ Rn ❛r❡ ❝❛❧❧❡❞ ❧♦❝❛t✐♦♥s ♦r ♣♦s✐t✐♦♥s✱ t❤❡ ✈❡❝t♦rs ❛r❡ ❝❛❧❧❡❞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ P ❛♥❞ Q ❛r❡ t✇♦ ❧♦❝❛t✐♦♥s✱ t❤❡♥ t❤❡ ✈❡❝t♦r P Q ✐s t❤❡ ❢r♦♠ P t♦ Q✳

❞✐s♣❧❛❝❡♠❡♥ts ❞✐s♣❧❛❝❡♠❡♥t

❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❛♥② s♣❛❝❡ Rn ❜✉t ✇❡ ✇✐❧❧ st❛rt ✇✐t❤ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡ ❞❡✈♦✐❞ ♦❢ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❋r♦♠ t❤✐s ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❛

✈❡❝t♦r ✐s ❛ ♣❛✐r✱ P Q✱ ♦❢ ❧♦❝❛t✐♦♥s P ❛♥❞ Q✳

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✸✾

❲❛r♥✐♥❣✦ ✏❱❡❝t♦r✑ ✐s ♥♦t s②♥♦♥②♠♦✉s ✇✐t❤ ✏s❡❣♠❡♥t✑❀ ✐t✬s ♥♦t ❡✈❡♥ ❛ s❡t✳

❚❤❡ s❡❣♠❡♥t t❤❛t ②♦✉ s❡❡ ✐s ❥✉st ❛

✈✐s✉❛❧✐③❛t✐♦♥✳

❲❡ s❛✇ ✈❡❝t♦rs ✐♥ ❛❝t✐♦♥ ♣r❡✈✐♦✉s❧②✱ ❜✉t t❤❡ ❣♦❛❧ ✇❛s ❧✐♠✐t❡❞ t♦ ✉s✐♥❣ ✈❡❝t♦rs t♦ ✉♥❞❡rst❛♥❞ ❞✐r❡❝t✐♦♥s ❛♥❞ ❛♥❣❧❡s ❜❡t✇❡❡♥ t❤❡♠✳ ❖✉r ✐♥t❡r❡st ❤❡r❡ ✐s t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ✈❡❝t♦rs✳

♦r❞❡r❡❞

■❢ ❛ ✈❡❝t♦r ✐s ❛♥ ♣❛✐r✱ t❤✐s ♠❡❛♥s t❤❛t P Q 6= QP ✳ ❇✉t ✐s t❤❡r❡ ❛ r❡❧❛t✐♦♥❄ ❚❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ P t♦ Q ✐s t❤❡ ♦♣♣♦s✐t❡ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ Q t♦ P ✿ QP = −P Q

❚❤❡ ❧♦❝❛t✐♦♥s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♣♦✐♥ts ❛♥❞ ✈❡❝t♦rs ❛r❡ s✉❜❥❡❝t t♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s t❤❛t ❝♦♥♥❡❝t t❤❡♠✿ P + PQ = Q

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❛❞❞ ❛ ✈❡❝t♦r t♦ ❛ ♣♦✐♥t t❤❛t ✐s ✐ts ✐♥✐t✐❛❧ ♣♦✐♥t ❛♥❞ t❤❡ r❡s✉❧t ✐s ✐ts t❡r♠✐♥❛❧ ♣♦✐♥t✳

■t ❢♦❧❧♦✇s✿ PQ = Q − P .

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ✈❡❝t♦r ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ✐ts t❡r♠✐♥❛❧ ❛♥❞ ✐ts ✐♥✐t✐❛❧ ♣♦✐♥ts✳ ■t ❢♦❧❧♦✇s t❤❛t QP = P − Q = −(Q − P ) = −P Q .

❲❡ ❛r❡ ❜❛❝❦ t♦ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✳

❉❡✜♥✐t✐♦♥ ✶✳✹✳✷✿ ❛✣♥❡ s♣❛❝❡

❚❤❡ ❛✣♥❡ s♣❛❝❡ ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn ✐s t❤❡ s❡t ♦❢ ♦r❞❡r❡❞ ♣❛✐rs P Q ♦❢ ♣♦✐♥ts P ❛♥❞ Q ✐♥ t❤✐s s♣❛❝❡✳ ❚❤❡s❡ ♣❛✐rs ❛r❡ ❝❛❧❧❡❞ ✈❡❝t♦rs✳ ❲❡ ♥♦✇ r❡✈✐❡✇ t❤❡ ❛❧❣❡❜r❛✳ ❋✐rst✱ ❞✐♠❡♥s✐♦♥ 1✳

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✵

❞✐r❡❝t❡❞ s❡❣♠❡♥ts

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ✇❡ ❝❛♥ st✐❧❧✱ ❡✈❡♥ ✇✐t❤♦✉t ❛ ❈❛rt❡s✐❛♥ s②st❡♠✱ t❤✐♥❦ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❚❤❡

❛❞❞✐t✐♦♥

✳

♦❢ t✇♦ ✈❡❝t♦rs ✐s ❡①❡❝✉t❡❞ ❜② ❛tt❛❝❤✐♥❣ t❤❡ ❤❡❛❞ ♦❢ t❤❡ s❡❝♦♥❞ ✈❡❝t♦r t♦ t❤❡ t❛✐❧ ♦❢ t❤❡ ✜rst✱

❛s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡ ♥❡❣❛t✐✈❡ ♥✉♠❜❡r ✭r❡❞✮ ✐s ❛ s❡❣♠❡♥t ❞✐r❡❝t❡❞ ❜❛❝❦✇❛r❞s s♦ t❤❛t ✐ts t❛✐❧ ✐s ♦♥ ✐ts ❧❡❢t✳ ◆♦✇ ❞✐♠❡♥s✐♦♥

2✳

❊①❛♠♣❧❡ ✶✳✹✳✸✿ ❝♦♥s❡❝✉t✐✈❡ ❞✐s♣❧❛❝❡♠❡♥ts ❲❡ ♠♦✈❡ ♣♦✐♥t t♦ ♣♦✐♥t t❤r♦✉❣❤ t❤❡ s♣❛❝❡✿

❚❤✐s ✐s ❤♦✇ ✇❡ ❝❛♥ ✉♥❞❡rst❛♥❞ ❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs ❛s ❞✐s♣❧❛❝❡♠❡♥ts✿ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥

❞✐s♣❧❛❝❡♠❡♥t

P Q R S

PQ QR RS ST

t❡r♠✐♥❛❧ ❧♦❝❛t✐♦♥

P + PQ Q + QR R + RS S + ST

=Q =R =S =T

= P + (P Q + QR) = P + (P Q + QR + RS) = P + (P Q + QR + RS + ST )

❚❤❡ r✐❣❤t ❝♦❧✉♠♥ s❤♦✇s ❤♦✇ ❛❞❞✐♥❣ ✈❡❝t♦r t♦ ♣♦✐♥t ❝❛♥ ❜❡ s❡❡♥ ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤✿ ❲❡ ❛❞❞ ✈❡❝t♦r t♦ ✈❡❝t♦r ✜rst✳

❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢

m

m X

st❡♣s✱ ✇❡ ❤❛✈❡ t❤❡s❡ t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s✿

Xk Xk+1

= X0 X1 + ...

+Xm−1 Xm

= X0 Xm

k=0

m X k=0

❙✐♥❝❡ ♠♦✈✐♥❣ ❢r♦♠

P

(Xk+1 − Xk ) = (X1 − X0 ) + ... +(Xm − Xm−1 ) = Xm − X0

t♦

Q

❛♥❞ t❤❡♥ ❢r♦♠

Q

t♦

R

❛♠♦✉♥ts t♦ ♠♦✈✐♥❣ ❢r♦♠

P

t♦

R✱

t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s✱

❛❣❛✐♥✱ ❛ ✏❤❡❛❞✲t♦✲t❛✐❧✑ ❛❧✐❣♥♠❡♥t ♦❢ ✈❡❝t♦rs✿

P R = P Q + QR . ❲❛r♥✐♥❣✦ ❲❡ ✉s❡ t❤❡ s❛♠❡ s②♠❜♦❧ ✏ +✑ ❛s ❢♦r ❛❞❞✐t✐♦♥ ♦❢ ♥✉♠❜❡rs✳

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✶

❍♦✇❡✈❡r✱ ✐♥ t❤❡ ♣❤②s✐❝❛❧ ✇♦r❧❞✱ t❤❡r❡ ❛r❡ ♦t❤❡r ✏♠❡t❛♣❤♦rs✑ ❢♦r ✈❡❝t♦rs ❜❡s✐❞❡s t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✳ ❊①❛♠♣❧❡ ✶✳✹✳✹✿ ✈❡❧♦❝✐t② ♦❢ str❡❛♠

❲❡ ❧♦♦❦ ❛t t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ ♣❛rt✐❝❧❡s ✐♥ ❛ str❡❛♠ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✳ ❚❤❡♥ t❤❡② ♠❛② ❜❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ s♣❡❡❞ ♦❢ r♦✇✐♥❣ ♦❢ t❤❡ ❜♦❛t✿

s❛♠❡

❲❡ ❛r❡ t♦ ❛❞❞ t❤❡s❡ t✇♦ ✈❡❝t♦rs ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ ❜✉t t❤❡②✱ ✐♥ ❝♦♥tr❛st t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✱ st❛rt ❛t t❤❡

♣♦✐♥t✦

❊①❡r❝✐s❡ ✶✳✹✳✺

❲✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s ❛s s❤♦✇♥✱ ✇❤❛t ✐s t❤❡ ❜❡st str❛t❡❣② t♦ ❝r♦ss t❤❡ ❝❛♥❛❧❄

❊①❛♠♣❧❡ ✶✳✹✳✻✿ ❢♦r❝❡s

▲❡t✬s ❛❧s♦ ❧♦♦❦ ❛t

❢♦r❝❡s ❛s ✈❡❝t♦rs

✳ ❋♦r ❡①❛♠♣❧❡✱ s♣r✐♥❣s ❛tt❛❝❤❡❞ t♦ ❛♥ ♦❜❥❡❝t ✇✐❧❧ ♣✉❧❧ ✐t ✐♥ t❤❡✐r

r❡s♣❡❝t✐✈❡ ❞✐r❡❝t✐♦♥s✿

❲❡ ❛❞❞ t❤❡s❡ ✈❡❝t♦rs t♦ ✜♥❞ t❤❡ ❝♦♠❜✐♥❡❞ ❢♦r❝❡ ❛s ✐❢ ♣r♦❞✉❝❡❞ ❜② ❛ s✐♥❣❧❡ s♣r✐♥❣✳

❚❤❡ ❢♦r❝❡s ❛r❡

✈❡❝t♦rs t❤❛t st❛rt ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✹✳✼✿ ❞✐s♣❧❛❝❡♠❡♥ts

❲❡ ❝❛♥ ✐♥t❡r♣r❡t t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✱ t♦♦✱ ❛s ✈❡❝t♦rs ❛❧✐❣♥❡❞ t♦ t❤❡✐r st❛rt✐♥❣ ♣♦✐♥ts✳ ❛r❡ ❝r♦ss✐♥❣ ❛ r✐✈❡r

3

■♠❛❣✐♥❡ ✇❡

♠✐❧❡s ✇✐❞❡ ❛♥❞ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❝✉rr❡♥t ✭✇✐t❤ ♥♦ r♦✇✐♥❣✮ t❛❦❡s ✉s

2

♠✐❧❡s

❞♦✇♥str❡❛♠✳ ❚❤r❡❡ ❞✐✛❡r❡♥t ✇❛②s t❤✐s ❝❛♥ ❤❛♣♣❡♥✿ ✶✳ ❛ ❢r❡❡✲✢♦✇ tr✐♣ ✷✳ ❛ ✇❛❧❦

2

3

♠✐❧❡s ♥♦rt❤ ❢♦❧❧♦✇❡❞ ❜② ❛ ✇❛❧❦

2

♠✐❧❡s ❡❛st ♦✈❡r ❛ ❜r✐❞❣❡❀ ♦r

♠✐❧❡s ❡❛st ♦✈❡r ❛♥♦t❤❡r ❜r✐❞❣❡ ❢♦❧❧♦✇❡❞ ❜② ❛ ❢r❡❡✲✢♦✇ tr✐♣

✸✳ ❛ r♦✇✐♥❣ tr✐♣ ❛❧♦♥❣ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ♦♥❡ s✐❞❡ ❣♦✐♥❣ ♠✐❧❡s ❡❛st✳ ❚❤❡ t❤r❡❡ ♦✉t❝♦♠❡s ❛r❡ t❤❡ s❛♠❡✿

3 3

♠✐❧❡s ♥♦rt❤❀ ❜✉t ❛❧s♦ ♠✐❧❡s ♥♦rt❤ ❛♥❞ ❛♥♦t❤❡r

2

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✷

❚❤❡ s✉♠ ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs ✐s t❤❡ s❛♠❡ ✐♥ ❛♥② ♦r❞❡r✿ 3 ♠✐❧❡s ♥♦rt❤ ❛♥❞ 2 ♠✐❧❡s ❡❛st

❙♦✱ t♦ ❛❞❞ t✇♦ ✈❡❝t♦rs✱ ✇❡ ❢♦❧❧♦✇ ❡✐t❤❡r ✶✳ ❚❤❡ ❤❡❛❞✲t♦✲t❛✐❧✿ t❤❡ tr✐❛♥❣❧❡ ❝♦♥str✉❝t✐♦♥✳ ✷✳ ❚❤❡ t❛✐❧✲✇✐t❤✲t❛✐❧✿ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❝♦♥str✉❝t✐♦♥✳ ❚❤❡② ❤❛✈❡ t♦ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✦ ❚❤❡② ❞♦✱ ❛s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❝♦♣②

✶✳ ❋♦r t❤❡ ❢♦r♠❡r✱ ✇❡ ♠❛❦❡ ❛ B ′ ♦❢ B ✱ ❛tt❛❝❤ ✐t t♦ t❤❡ ❡♥❞ ♦❢ A✱ ❛♥❞ t❤❡♥ ❝r❡❛t❡ ❛ ♥❡✇ ✈❡❝t♦r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t t❤❛t ♦❢ A ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥t t❤❛t ♦❢ B ′ ✳ ✷✳ ❋♦r t❤❡ ❧❛tt❡r✱ ✇❡ ♠❛❦❡ ❛ ❝♦♣② B ′ ♦❢ B ✱ ❛tt❛❝❤ ✐t t♦ t❤❡ ❡♥❞ ♦❢ A✱ ❛❧s♦ ♠❛❦❡ ❛ ❝♦♣② A′ ♦❢ A✱ ❛tt❛❝❤ ✐t t♦ t❤❡ ❡♥❞ ♦❢ B ✳ ❚❤❡♥ t❤❡ A+B A B ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣♦✐♥t ✐s t❤❡ ✈❡❝t♦r ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣♦✐♥t t❤❛t ✐s t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ✇✐t❤ s✐❞❡s A ❛♥❞ B ✳

s✉♠

♦❢ t✇♦ ✈❡❝t♦rs ❛♥❞

❊①❡r❝✐s❡ ✶✳✹✳✽

Pr♦✈❡ t❤❛t t❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡ ❛❝❝♦r❞✐♥❣ t♦ ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✳ ■t ✐s t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥✳ ❲❡ t❤✐♥❦ ❛❜♦✉t ✈❡❝t♦rs ❛s ❧✐♥❡ s❡❣♠❡♥ts ✐♥ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳ ❆s s✉❝❤✱ ✐t ❤❛s ❛ ❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ ❧❡♥❣t❤✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ t❤❡ ❧❡♥❣t❤ t♦ ❜❡ 0❀ t❤❛t✬s t❤❡ ✳ ■ts ❞✐r❡❝t✐♦♥ ✐s ✉♥❞❡✜♥❡❞✳

s✉❜tr❛❝t✐♦♥

③❡r♦ ✈❡❝t♦r

❖♥❝❡ ✇❡ ❦♥♦✇ ❛❞❞✐t✐♦♥✱ ✐s ✐ts ✐♥✈❡rs❡ ♦♣❡r❛t✐♦♥✳ ■♥❞❡❡❞✱ ❣✐✈❡♥ ✈❡❝t♦rs A ❛♥❞ B ✱ ✜♥❞✐♥❣ t❤❡ ✈❡❝t♦r C s✉❝❤ t❤❛t B + C = A ❛♠♦✉♥ts t♦ s♦❧✈✐♥❣ ❛♥ ❡q✉❛t✐♦♥✱ ❥✉st ❛s ✇✐t❤ ♥✉♠❜❡rs✿ A − B = C =⇒ B + C = A .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❞♦ ■ ❛❞❞ t♦ B t♦ ❣❡t A❄ ❆♥ ❡①❛♠✐♥❛t✐♦♥ r❡✈❡❛❧s t❤❡ ❛♥s✇❡r✿

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✸

❙♦✱ ✇❡ ❝♦♥str✉❝t t❤❡ ✈❡❝t♦r ❢r♦♠ t❤❡ ❡♥❞ ♦❢ s❛♠❡ st❛rt✐♥❣ ♣♦✐♥t ❛s

B

t♦ t❤❡ ❡♥❞ ♦❢

A✳

❖♥❡ ♠♦r❡ st❡♣✿ ♠❛❦❡ ❛ ❝♦♣② ♦❢

C

✇✐t❤ t❤❡

A✳

■❢ ✇❡ ✇❛♥t t♦ ❣♦ ❢❛st❡r✱ ✇❡ r♦✇ t✇✐❝❡ ❛s ❤❛r❞❀ t❤❡ ✈❡❝t♦r ❤❛s t♦ ❜❡ str❡t❝❤❡❞✦

❖r✱ ♦♥❡ ❝❛♥ ❛tt❛❝❤ t✇♦

s♣r✐♥❣s ✐♥ ❛ ❝♦♥s❡❝✉t✐✈❡ ♠❛♥♥❡r t♦ ❞♦✉❜❧❡ t❤❡ ❢♦r❝❡✱ ♦r ❝✉t ❛♥② ♣♦rt✐♦♥ ♦❢ t❤❡ s♣r✐♥❣ t♦ r❡❞✉❝❡ t❤❡ ❢♦r❝❡ ♣r♦♣♦rt✐♦♥❛❧❧②✳ ❆ ❢♦r❝❡ ♠✐❣❤t ❦❡❡♣ ✐ts ❞✐r❡❝t✐♦♥ ❜✉t ❝❤❛♥❣❡ ✐ts ♠❛❣♥✐t✉❞❡✦ ■t ♠✐❣❤t ❛❧s♦ ❝❤❛♥❣❡ t❤❡ ❞✐r❡❝t✐♦♥ t♦ t❤❡ ♦♣♣♦s✐t❡✳ ❚❤❡r❡ ✐s t❤❡♥ ❛♥♦t❤❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ♦♥ ✈❡❝t♦rs✳ ❚❤✐s ✐s ❞✐♠❡♥s✐♦♥

❚❤✐s ✐s ❞✐♠❡♥s✐♦♥

1✿

2✿

❆s ②♦✉ ❝❛♥ s❡❡✱ ❡✈❡r② ♣♦✐♥t ❤❛s ❛ s♣❡❝✐❛❧ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ ✐t✳ ❋♦r ❡✈❡r② ♣♦✐♥t

❲❡ s❛② t❤❛t

0

s❡r✈❡ ❛s t❤❡

❚❤✉s✱ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t

A✱

s❛♠❡ ❛s t❤❛t ♦❢

•

♦♣♣♦s✐t❡ t♦ t❤❛t ♦❢

•

③❡r♦ ✇❤❡♥

A

✐❞❡♥t✐t②

c = 0✳

③❡r♦ ✈❡❝t♦r

✐s✿

✳

c·A

✇❤❡♥

A

t❤❡

0 = PP .

♦❢ ❛ ✈❡❝t♦r

A

✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ✇❤✐❝❤ ✐s

•

P✱

c > 0✱

✇❤❡♥

c < 0✱

❛♥❞

❛♥❞ ❛ r❡❛❧ ♥✉♠❜❡r

c

✐s t❤❡ ✈❡❝t♦r ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣♦✐♥t ❛s

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✹ ❲❛r♥✐♥❣✦ ❲❡ ✉s❡ t❤❡ s❛♠❡ s②♠❜♦❧ ✏ ·✑ ❛s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♥✉♠❜❡rs✳

❊①❛♠♣❧❡ ✶✳✹✳✾✿ ✈❡❧♦❝✐t② ♦❢ ✇✐♥❞

❱❡❧♦❝✐t✐❡s ❛♣♣❡❛r ❛s t❤❡ ✇✐♥❞ s♣❡❡❞ ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✿

■❢ t❤❡ ✈❡❧♦❝✐t✐❡s ❛r❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥ts✱ ✇❡ ❝❛♥ ✜♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦❢ t❤❡ ♣❛rt✐❝❧❡s ♦❢ t❤❡ ❛✐r✳ ❲❡ ❝❛♥ ❛❧s♦ ♣❧♦t ❛ ✇❤♦❧❡ tr✐♣ ♦❢ ♦♥❡ s✉❝❤ ♣❛rt✐❝❧❡✱ ❥✉st ❛s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ❲✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s Vk , k = 1, 2, ..., m✱ r❡s♣❡❝t✐✈❡❧②✱ t❛❦❡s t❤❡ ❢♦r♠✿ m X k=0

Vk · ∆t = Xm − X0 .

❊①❡r❝✐s❡ ✶✳✹✳✶✵

P❧♦t ❛ ❢❡✇ ♠♦r❡ ♣❛t❤s✳ ❲❤❛t ✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡❄ ❆s ✇❡ ❦♥♦✇ ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✱ t✇♦ ❧✐♥❡s ❛♥❞✱ t❤❡r❡❢♦r❡✱ t✇♦ ✈❡❝t♦rs✱ ❞❡t❡r♠✐♥❡ ❛ ✳ ❚❤✐s ✐s ✇❤② ✇❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♦♣❡r❛t✐♦♥s✱ ❛s ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ t❤❡♠✱ ❛r❡ ❧✐♠✐t❡❞ t♦ ❛ ❝❡rt❛✐♥ ♣❧❛♥❡ ✇✐t❤✐♥ ❛ ♣♦ss✐❜❧② ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✿

♣❧❛♥❡

❊①❛♠♣❧❡ ✶✳✹✳✶✶✿ ✉♥✐ts

❖✉t ♦❢ ❝❛✉t✐♦♥✱ ✇❡ s❤♦✉❧❞ ❧♦♦❦ ❛t t❤❡ ✉♥✐ts ♦❢ t❤❡ s❝❛❧❛r✳ ❨❡s✱ t❤❡ ❢♦r❝❡ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✿ F = ma .

❍♦✇❡✈❡r✱ t❤❡s❡ t✇♦ ❤❛✈❡ ❞✐✛❡r❡♥t ✉♥✐ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝❛♥♥♦t ❜❡ ❛❞❞❡❞ t♦❣❡t❤❡r✦ ❆❧s♦✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✐s t❤❡ t✐♠❡ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ✈❡❧♦❝✐t②✿ ∆X = ∆t · V .

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✺

❇✉t t❤❡s❡ t✇♦ ❤❛✈❡ ❞✐✛❡r❡♥t ✉♥✐ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝❛♥♥♦t ❜❡ ❛❞❞❡❞✦ ❚❤❡② ❧✐✈❡ ✐♥ t✇♦ ❞✐✛❡r❡♥t s♣❛❝❡s✳ ❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡✱ t❤r♦✉❣❤♦✉t t❤❡ ❝❤❛♣t❡r✱ t♦ ✉s❡

❝❛♣✐t❛❧✐③❛t✐♦♥ t♦ ❤❡❧♣ t♦ t❡❧❧ ✈❡❝t♦rs ❢r♦♠ ♥✉♠❜❡rs✳ ❲❛r♥✐♥❣✦ ❚♦ ✐♥❞✐❝❛t❡ ✈❡❝t♦rs✱ ♠❛♥② s♦✉r❝❡s ✉s❡✿

•

❛♥ ❛rr♦✇ ❛❜♦✈❡ t❤❡ ❧❡tt❡r✱

•

t❤❡ ❜♦❧❞ ❢❛❝❡✱

~v ✱

♦r

v✳

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥tr♦❞✉❝❡❞ t♦ Rn ✱ ❛ s♣❛❝❡ ♦❢ ♣♦✐♥ts✱ ❛ ♥❡✇ ❡♥t✐t② ✕ ❛ ✈❡❝t♦r✳ ■t ✐s ❛♥ ♦r❞❡r❡❞ ♣❛✐r P Q ♦❢ ♣♦✐♥ts✱ P ❛♥❞ Q✱ ❧✐♥❦❡❞ ❜❛❝❦ t♦ ♣♦✐♥ts ❜② t❤✐s ❛❧❣❡❜r❛✿

P Q = Q − P ♦r Q = P + P Q ❚❤❡ ✈❡❝t♦rs ❝❛♥ ❜❡ ❛❞❞❡❞ t♦❣❡t❤❡r ❛♥❞ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ♥✉♠❜❡r ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r♦❝❡❞✉r❡s ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❚❤❡ ♦♣❡r❛t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

❚❤❡♦r❡♠ ✶✳✹✳✶✷✿ ❆①✐♦♠s ♦❢ ❆✣♥❡ ❙♣❛❝❡

❚❤❡ ♣♦✐♥ts ❛♥❞ t❤❡ ✈❡❝t♦rs ✐♥ t❤❡ ❛✣♥❡ s♣❛❝❡ ♦❢ Rn s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣✲ ❡rt✐❡s✿ ✶✳ ■❞❡♥t✐t②✿ ❋♦r ❡✈❡r② ♣♦✐♥t P ✱ ✇❡ ❤❛✈❡ ❢♦r s♦♠❡ ✈❡❝t♦r ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s 0 t❤❡ ❢♦❧❧♦✇✐♥❣✿ P +0=P .

✷✳ ❆ss♦❝✐❛t✐✈✐t②✿ ❋♦r ❡✈❡r② ♣♦✐♥t P ❛♥❞ ❛♥② ✈❡❝t♦rs V ❛♥❞ W st❛rt✐♥❣ ❛t P ✱ ✇❡ ❤❛✈❡✿ (P + V ) + W = P + (V + W ) .

✸✳ ❋r❡❡ ❛♥❞ tr❛♥s✐t✐✈❡ ❛❝t✐♦♥✿ ❋♦r ❡✈❡r② ♣♦✐♥t P ❛♥❞ ❡✈❡r② ♣♦✐♥t Q✱ t❤❡r❡ ✐s ❛ ✈❡❝t♦r V s✉❝❤ t❤❛t P + V = Q.

Pr♦♦❢✳ ✶✳ ❈❤♦♦s❡✿

0 = PP . ✷✳ ❈♦♠♣❛r❡✿

(P + P Q) + QR = R ❛♥❞ P + (P Q + QR) = P + P R = R .

✸✳ ❈❤♦♦s❡✿

V = PQ.

❊①❛♠♣❧❡ ✶✳✹✳✶✸✿ ❢❛♠✐❧② r❡❧❛t✐♦♥s ▲❡t✬s ✐♠❛❣✐♥❡ t❤❛t ❡✈❡r② ♣♦✐♥t ✐♥ t❤❡ s♣❛❝❡ st❛♥❞s ❢♦r ❛ ♣❡rs♦♥✳ ◆♦✇✱ ❡❛❝❤ ♣❡rs♦♥ ✐s ❧✐♥❦❡❞ ❜② ❛ ✈❡❝t♦r t♦ ♦♥❡✬s ❢❛♠✐❧②✿

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✻

❊✈❡r② ♣❡rs♦♥ ✐s t❤❡ ❝❡♥t❡r ♦❢ s✉❝❤ ❛ s②st❡♠ ♦❢ ❧✐♥❦s✳ ❆❧s♦✱ ♣♦t❡♥t✐❛❧❧②✱ ❡❛❝❤ ♣❡rs♦♥ ✐s ❧✐♥❦❡❞ t♦ ❛♥② ♦t❤❡r ♣❡rs♦♥✳ ◆♦✇✱ ❛

♠❛rr✐❛❣❡

✇✐❧❧ ❧✐♥❦ ♦♥❡ s✉❝❤ ❝❡♥t❡r t♦ ❛♥♦t❤❡r✱ ❛♥❞ r❡♥❛♠✐♥❣ ❝♦♠♠❡♥❝❡s✿

❚❤❡ ✇♦r❞ ✏❛✣♥❡✑ ♠❡❛♥s ✏✐♥✲❧❛✇✑✳

▲❡t✬s s✐♠♣❧✐❢②✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ✐♥ ♣❛rt✐❝✉❧❛r✿ ✶✳ ❚✇♦ ✈❡❝t♦rs ✇✐t❤ t❤❡ s❛♠❡ ♦r✐❣✐♥ ❝❛♥ ❜❡ ❛❞❞❡❞ t♦❣❡t❤❡r✱ ♣r♦❞✉❝✐♥❣ ❛♥♦t❤❡r ♦♥❡ ✇✐t❤ t❤❡ s❛♠❡ ♦r✐❣✐♥✳ ✷✳ ❆ ✈❡❝t♦r ❝❛♥ ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ♥✉♠❜❡r✱ ♣r♦❞✉❝✐♥❣ ❛♥♦t❤❡r ♦♥❡ ✇✐t❤ t❤❡ s❛♠❡ ♦r✐❣✐♥✳ ❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r♦❝❡❞✉r❡s ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ●✐✈❡♥ ♣♦✐♥ts ❛ r❡❛❧ ♥✉♠❜❡r

k✱

✇❡ ❤❛✈❡ ❢♦r s♦♠❡ ♣♦✐♥ts

R

❛♥❞

P

❛♥❞

Q

❛♥❞

S✿

PQ + PR = PR

❛♥❞

k · PQ = PS

■♥❞❡❡❞✱ t❤❡ r❡s✉❧t ✐s ❛♥♦t❤❡r ✈❡❝t♦r ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣♦✐♥t ❛s t❤❡ ♦r✐❣✐♥❛❧✭s✮✦ ❚❤✐s ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✭✐♥❝❧✉❞✐♥❣ t❤❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✮ ❝❛♥✱ t❤❡r❡❢♦r❡✱ ❜❡ ❝❛rr✐❡❞ ♦✉t s❡♣❛r❛t❡❧② ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✿

❆♥❞ t❤✐s ❛❧❣❡❜r❛ ✐s ✐❞❡♥t✐❝❛❧ ❢♦r ❡✈❡r② ❧♦❝❛t✐♦♥✦ ❚❤❡r❡❢♦r❡✱ ❛ s✐♥❣❧❡ ✐♥✐t✐❛❧ ♣♦✐♥t ✇✐❧❧ ❜❡ s✉✣❝✐❡♥t ❢♦r ♦✉r st✉❞② ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛✳

✶✳✹✳

❲❤❡r❡ ✈❡❝t♦rs ❝♦♠❡ ❢r♦♠

✹✼

❇✉t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t ✐s ❝r✉❝✐❛❧✳ ❚❤✐s ♣♦✐♥t ✇✐❧❧ ❜❡ ❛ss✉♠❡❞ t♦ ❜❡

O✱ t❤❡ ♦r✐❣✐♥✱ ✉♥❧❡ss ♦t❤❡r✇✐s❡

✐♥❞✐❝❛t❡❞✳ ❚❤✐s ✐s ✇❤❛t ♦✉r ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡❝t♦rs ❧♦♦❦s ❧✐❦❡✱ ❢♦r ❞✐♠❡♥s✐♦♥

❙✉❝❤ ❛ ✏s♣❛❝❡ ♦❢ ✈❡❝t♦rs✑ ✐s ❝❛❧❧❡❞ ❛

2✿

✈❡❝t♦r s♣❛❝❡

✳ ■t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t✇♦ ♦♣❡r❛t✐♦♥s✱ t❤❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥✿

✈❡❝t♦r

+

✈❡❝t♦r

=

✈❡❝t♦r

·

✈❡❝t♦r

=

✈❡❝t♦r

❛♥❞ t❤❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ♥✉♠❜❡r

❲❛r♥✐♥❣✦ ❚❤❡r❡ ❛r❡ ♥♦ ♣♦✐♥ts ✐♥ ❛ ✈❡❝t♦r s♣❛❝❡✳

▲❡t✬s ❛❞❞ ❛ ❈❛rt❡s✐❛♥ s②st❡♠ t♦ ♦✉r ♣❧❛♥❡ ✭✇✐t❤ t❤❡ ♦r✐❣✐♥ ❛❧r❡❛❞② ❝❤♦s❡♥✮✳ ❲❡ ❝❛♥ ✇❛t❝❤ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✈❡❝t♦rs ❛s ✇❡ ❝❛rr② ♦✉t ♦✉r ❛❧❣❡❜r❛✳ ❙✉♠ ✐♠♣❧✐❡s ❝♦♦r❞✐♥❛t❡✇✐s❡ ❛❞❞✐t✐♦♥✿

❙❝❛❧❛r ♣r♦❞✉❝t ✐♠♣❧✐❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ❜② t❤❡ s❛♠❡ ♥✉♠❜❡r✿

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✹✽

❲❡ ❛r❡ ✐♥ R2 ♥♦✇✳

✶✳✺✳ ❱❡❝t♦rs ✐♥

Rn

❲❡ ♥♦✇ ✉♥❞❡rst❛♥❞ ✈❡❝t♦rs✳ ❖r ❛t ❧❡❛st ✇❡ ✉♥❞❡rst❛♥❞ t❤❡♠ ✐♥ t❤❡ ❧♦✇❡r✲❞✐♠❡♥s✐♦♥❛❧ s❡tt✐♥❣✳ ◆♦✇✱ ✇❡ ♠♦✈❡ t♦ t❤❡ ♥❡①t st❛❣❡✿ ✶✳ ❲❡ ❛❞❞ ❛ ❈❛rt❡s✐❛♥ s②st❡♠ t♦ t❤❡s❡ s♣❛❝❡s✿ ❧✐♥❡✱ ♣❧❛♥❡✱ ❛♥❞ s♣❛❝❡✳ ✷✳ ❲❡ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ❛❜str❛❝t s♣❛❝❡s ♦❢ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥s✱ Rn ✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦r♠❡r t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ❛♣♣r♦❛❝❤ t♦ t❤❡ ❧❛tt❡r ♠❛❦❡s s❡♥s❡✳ ❆ ✈❡❝t♦r ✐s st✐❧❧ ❛ ♣❛✐r P Q ♦❢ ♣♦✐♥ts P ❛♥❞ Q ✐♥ Rn ✳ ◆♦✇✱ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ ♣♦✐♥ts ❝♦rr❡s♣♦♥❞s t♦ ❛ str✐♥❣ ♦❢ n ♥✉♠❜❡rs ❝❛❧❧❡❞ ✐ts ❝♦♦r❞✐♥❛t❡s✳ ❖♥ t❤❡ ❧✐♥❡ R1 ✱ ♣♦✐♥ts ❛r❡ ♥✉♠❜❡rs ❛♥❞ t❤❡ ✈❡❝t♦rs ❛r❡ s✐♠♣❧② ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ ♥✉♠❜❡rs✿ PQ = Q − P .

❖♥ t❤❡ ♣❧❛♥❡ R2 ✱ ✇❡ ♠✐❣❤t ❤❛✈❡✿

P = (1, 2) ❛♥❞ Q = (2, 5) .

❍♦✇ ❝❛♥ ✇❡ ❡①♣r❡ss ✈❡❝t♦r P Q ✐♥ t❡r♠s ♦❢ t❤❡s❡ ❢♦✉r ♥✉♠❜❡rs❄

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✹✾

❲❡ ❧♦♦❦ ❛t t❤❡ ❝❤❛♥❣❡ ❢r♦♠ P t♦ Q✿ • ❚❤❡ ❝❤❛♥❣❡ ✇✐t❤ r❡s♣❡❝t t♦ x✱ ✇❤✐❝❤ ✐s 2 − 1 = 1✳ • ❚❤❡ ❝❤❛♥❣❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✱ ✇❤✐❝❤ ✐s 5 − 2 = 3✳

❲❡ ❝♦♠❜✐♥❡ t❤❡s❡ ✐♥t♦ ❛ ♥❡✇ ♣❛✐r

♦❢ ♥✉♠❜❡rs ✭✇✐t❤ tr✐❛♥❣✉❧❛r ❜r❛❝❦❡ts t♦ ❞✐st✐♥❣✉✐s❤ t❤❡s❡ ❢r♦♠ ♣♦✐♥ts✮✿ P Q =< 1, 3 > .

❚❡❝❤♥✐❝❛❧❧②✱ ❤♦✇❡✈❡r✱ ✇❡ ❤❛✈❡ t♦ ♠❡♥t✐♦♥ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t P = (1, 2) ♦❢ t❤❡ ✈❡❝t♦r✳ ◆♦✇ ✐♥ R3 ✱ ✇❡ ♠✐❣❤t ❤❛✈❡✿

P = (x, y, z) ❛♥❞ Q = (x′ , y ′ , z ′ ) .

❍♦✇ ❝❛♥ ✇❡ ❡①♣r❡ss ✈❡❝t♦r P Q ✐♥ t❡r♠s ♦❢ t❤❡s❡ s✐① ♥✉♠❜❡rs❄

❚❤❡r❡ ❛r❡ t❤r❡❡ ❝❤❛♥❣❡s ✭❞✐✛❡r❡♥❝❡s✮ ❛❧♦♥❣ t❤❡ t❤r❡❡ ❛①❡s✱ ✐✳❡✳✱ ❛ tr✐♣❧❡ ✿ P Q =< x′ − x, y ′ − y, z ′ − z > .

❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✿ ✈❡❝t♦r ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts ❆ ✈❡❝t♦r P Q ✐♥ Rn ✇✐t❤ ✐ts ✐♥✐t✐❛❧ ♣♦✐♥t P = (x1 , x3 , ..., xn )

❛♥❞ ✐ts t❡r♠✐♥❛❧ ♣♦✐♥t

Q = (x′1 , x′2 , ..., x′n )

✐s ❣✐✈❡♥ ❜② t❤❡ str✐♥❣ ♦❢ n ♥✉♠❜❡rs ❝❛❧❧❡❞ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r✿ x′1 − x1 , x′2 − x3 , ..., x′n − xn .

❚❤❡ ❞❡✜♥✐t✐♦♥ ♠❛t❝❤❡s t❤❡ ♦♥❡ t❤❛t r❡❧✐❡s ♦♥ ❞✐r❡❝t❡❞ s❡❣♠❡♥ts✳ ❆ ✈❡❝t♦r ♠❛② ❡♠❡r❣❡ ❢r♦♠ ✐ts ✐♥✐t✐❛❧ ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥ts ♦r ✐♥❞❡♣❡♥❞❡♥t❧②✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ❛ss❡♠❜❧❡ t❤❡ ❝♦♠♣♦♥❡♥ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✳

❘♦✇ ❛♥❞ ❝♦❧✉♠♥ ✈❡❝t♦rs  a1  a2     ...  = < a1 , a2 , ..., an > an 

❚❤❡ ❢♦r♠❡r ✐s ♣r❡❢❡rr❡❞❀ t❤❡ ❧❛tt❡r ✐s ✐ts ❛❜❜r❡✈✐❛t✐♦♥✳

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✺✵

❖♥❝❡ ❛❣❛✐♥ ✇❡ ❝❛♥ ♦♥❧② ❝❛rr② ♦✉t ✈❡❝t♦r ❛❞❞✐t✐♦♥ ♦♥ ✈❡❝t♦rs ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣♦✐♥t✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t ✇❤✐❧❡ ❧❡❛✈✐♥❣ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦rs ✐♥t❛❝t❄ ◆♦t ♦♥❧② ✐s ❡❛❝❤ ✈❡❝t♦r ✏❝♦♣✐❡❞✑ ❜✉t s♦ ❛r❡ t❤❡ r❡s✉❧ts ♦❢ ❛❧❧ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❡② ❛r❡ t❤❡ s❛♠❡✱ ❥✉st s❤✐❢t❡❞ t♦ ❛ ♥❡✇ ❧♦❝❛t✐♦♥✿

❲❡ ❝❛♥ ❡✈❡♥ r♦t❛t❡ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✭r✐❣❤t✮✳ ■t ✐s t❤❡♥ s✉✣❝✐❡♥t t♦ ♣r♦✈✐❞❡ r❡s✉❧ts ❢♦r t❤❡ ✈❡❝t♦rs t❤❛t st❛rt ❛t t❤❡

■♥ t❤❛t ❝❛s❡✱

♦r✐❣✐♥ O ♦♥❧②✦ ❖♥❧② t❤❡s❡ ❛r❡ ❛❧❧♦✇❡❞✿

t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦rs ❛r❡ s✐♠♣❧② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✐ts ❡♥❞ ✿ P = (x1 , x3 , ..., xn ) =⇒ OP =< x1 , x2 , ..., xn >

❲❛r♥✐♥❣✦

❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ❛♥❞ ✈❡❝t♦rs ❧✐❡s ✐♥ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s t♦ ✇❤✐❝❤ t❤❡② ❛r❡ s✉❜❥❡❝t✳ ◆❡①t ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❜✉t t❤✐s t✐♠❡ t❤❡ ✈❡❝t♦rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r ❝♦♠✲ ♣♦♥❡♥ts✳ ❲❡ ❝❛rr② ♦✉t ♦♣❡r❛t✐♦♥s

❝♦♠♣♦♥❡♥t✇✐s❡✳

❲❡ ❞❡♠♦♥str❛t❡ t❤❡s❡ ♦♣❡r❛t✐♦♥s ❢♦r ❞✐♠❡♥s✐♦♥

n = 3

❛♥❞ ❢♦r ❜♦t❤ t❤❡ r♦✇ ❛♥❞ t❤❡ ❝♦❧✉♠♥ st②❧❡s ♦❢

♥♦t❛t✐♦♥✳ ❚❤❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②✿

A =< x, y, z > + B =< u, v, w > A + B =< x + u, y + v, z + w >

     x+u u x  y + v = y+v  z+w w z 

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✺✶

❚❤✐s ✐s ❤♦✇ ✇❡ ♣r♦❣r❡ss t♦ t❤❡ ❞❡✜♥✐t✐♦♥ t❤❛t ❞♦❡s♥✬t ✐♥✈♦❧✈❡ ♣♦✐♥ts✿ P = (x1 , ..., xn ) OP Q = (y1 , ..., yn ) OQ R = (x1 + y1 , ..., xn + yn ) OP + OQ = OR

→

U V U +V

=< x1 , ..., xn > =< y1 , ..., yn > =< x1 + y1 , ..., xn + yn >

❆ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤✐s ♦♣❡r❛t✐♦♥ ✐♥ R20 ✐s ❜❡❧♦✇✿

❚❤❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②✿ A =< x, y, z > × k kA =< kx, ky, kz >

   kx x k ·  y  =  ky  kz z 

■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ ❝♦♠♣♦♥❡♥ts ❛r❡ ❛❧✐❣♥❡❞✳ ❊✈❡♥ t❤♦✉❣❤ ❜♦t❤ s❡❡♠ ❡q✉❛❧❧② ❝♦♥✈❡♥✐❡♥t✱ t❤❡ ❢♦r♠❡r ✇✐❧❧ ❜❡ s❡❡♥ ❛s ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❧❛tt❡r✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♣r♦❣r❡ss t♦ t❤❡ ❞❡✜♥✐t✐♦♥ t❤❛t ❞♦❡s♥✬t ✐♥✈♦❧✈❡ ♣♦✐♥ts✿ k r❡❛❧ P = (x1 , ..., xn ) OP R = (kx1 , ..., kxn ) k · OP = OR

→

k r❡❛❧ U =< x1 , ..., xn > k · U =< kx1 , ..., kxn >

❆ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤✐s ♦♣❡r❛t✐♦♥ ✐♥ R20 ✐s ❜❡❧♦✇ ✭k = 1.3✮✿

❚❤❡ s❝❛❧❛r k ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❝♦♥st❛♥t

♠✉❧t✐♣❧❡✳

❊①❛♠♣❧❡ ✶✳✺✳✷✿ ✐♥✈❡st♠❡♥t ♣♦rt❢♦❧✐♦s

■❢ t❤❡r❡ ❛r❡ 10, 000 st♦❝❦s ♦♥ t❤❡ st♦❝❦ ♠❛r❦❡t✱ ❡✈❡r② ✐♥✈❡st♠❡♥t ♣♦rt❢♦❧✐♦ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ 10, 000✲ ❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r✳ ❚❤❡♥✱ ♠❡r❣✐♥❣ t✇♦ ♦r ♠♦r❡ ♣♦rt❢♦❧✐♦s ✇✐❧❧ ❛❞❞ t❤❡✐r ✈❡❝t♦rs✿

✶✳✺✳

❱❡❝t♦rs ✐♥

❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛✿

❙❡❝♦♥❞✱

✺✷

Rn

❂❘❈❬✲✷❪✰❘❈❬✲✶❪

♦r tr✐♣❧✐♥❣ ❛ ♣♦rt❢♦❧✐♦ ✇❤✐❧❡ ♣r❡s❡r✈✐♥❣ t❤❡ ♣r♦♣♦rt✐♦♥ ✭♦r ✇❡✐❣❤t✮ ♦❢ ❡❛❝❤ st♦❝❦ ✇✐❧❧ ✐ts ✈❡❝t♦r✿

❞♦✉❜❧✐♥❣

s❝❛❧❛r ♠✉❧t✐♣❧②

❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛✿

❂✷✯❘❈❬✲✶❪

❊✈❡♥ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❤♦❧❞✐♥❣s ❛r❡ s✉❜❥❡❝t t♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s✿ < 10000 t♦♥s ♦❢ ✇❤❡❛t , 20000 ❜❛rr❡❧s ♦❢ ♦✐❧ , ... > ,

♦r < $100000, U1000000, ... > .

❉❡✜♥✐t✐♦♥ ✶✳✺✳✸✿ s✉♠ ♦❢ ✈❡❝t♦rs ❋♦r t✇♦ ✈❡❝t♦rs ✐♥ Rn ✱ t❤❡✐r ❝♦♠♣♦♥❡♥t✇✐s❡ ❛❞❞✐t✐♦♥✿

s✉♠

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✈❡❝t♦r ❛❝q✉✐r❡❞ ❜② t❤❡✐r

A =< a1 , ..., an > B =< b1 , ..., bn > =⇒ A + B =< a1 + b1 , ..., an + bn >

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✺✸

❉❡✜♥✐t✐♦♥ ✶✳✺✳✹✿ s❝❛❧❛r ♣r♦❞✉❝t ❋♦r ❛ ♥✉♠❜❡r

k

Rn ✱

❛♥❞ ❛ ✈❡❝t♦r ✐♥

t❤❡✐r

s❝❛❧❛r ♣r♦❞✉❝t

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡

✈❡❝t♦r ❛❝q✉✐r❡❞ ❜② t❤❡✐r ❝♦♠♣♦♥❡♥t✇✐s❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

k r❡❛❧ A =< a1 , ..., an > =⇒ kA =< ka1 , ..., kan >

❲❡ t❤✉s ❤❛✈❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ❢♦r ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✦ ❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ ♣r♦✈❡♥ t♦ s❛t✐s❢② t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s ❛s t❤❡ ✈❡❝t♦rs ✐♥

R3

✭♥❡①t s❡❝t✐♦♥✮✳

❚❤❡

♣r♦♦❢ ✐s str❛✐❣❤t✲❢♦r✇❛r❞ ❛♥❞ r❡❧✐❡s ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt② ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ t♦ ♣r♦✈❡ t❤❡ ❝♦♠♠✉t❛t✐✈✐t② ♦❢ ✈❡❝t♦r ❛❞❞✐t✐♦♥✱ ✇❡ ✉s❡ t❤❡ ❝♦♠♠✉t❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥ ♦❢ ♥✉♠❜❡rs ❛s ❢♦❧❧♦✇s✿

            u x x x+u u+x u            A + B = y + v = y + v = v + y = v + y = B + A . w z z w z+w w+z

❆s ❛ r❡s✉❧t✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ tr❡❛t ✈❡❝t♦rs ❛s ✐❢ t❤❡② ✇❡r❡ ♥✉♠❜❡rs✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡

s♣❡❝✐❛❧ ✈❡❝t♦rs ❞❡s❡r✈❡ s♣❡❝✐❛❧ ❛tt❡♥t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✶✳✺✳✺✿ ③❡r♦ ✈❡❝t♦r ❚❤❡

③❡r♦ ✈❡❝t♦r ✐♥ Rn ❤❛s ♦♥❧② ③❡r♦ ❝♦♠♣♦♥❡♥ts✿ 0 =< 0, ..., 0 >

❉❡✜♥✐t✐♦♥ ✶✳✺✳✻✿ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ ✈❡❝t♦r ❚❤❡

♥❡❣❛t✐✈❡ ✈❡❝t♦r

t❤♦s❡ ♦❢

♦❢ ❛ ✈❡❝t♦r

A

✐♥

Rn

❤❛s ✐ts ❝♦♠♣♦♥❡♥ts t❤❡ ♥❡❣❛t✐✈❡s ♦❢

A✿

−A =< −a1 , ..., −an >

❊①❡r❝✐s❡ ✶✳✺✳✼ Pr♦✈❡ t❤❡ ❡✐❣❤t ❛①✐♦♠s ♦❢ ✈❡❝t♦r s♣❛❝❡s ❢♦r t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❞❡✜♥❡❞ t❤✐s ✇❛② ❢♦r ✭❛✮ Rn ✳

R3 ✱

✭❜✮

▲❡t✬s ❡①♣❧❛✐♥ t❤❡ r❡❛s♦♥ ❢♦r t❤❡ ✇♦r❞ ✏❝♦♠♣♦♥❡♥t✑✳ ❆ ✈❡❝t♦r

A

✐s

❞❡❝♦♠♣♦s❡❞

✐♥t♦ t❤❡ s✉♠ ♦❢ ♦t❤❡r ✈❡❝t♦rs✳

❲❡ ❝❤♦s❡ t❤♦s❡ ✈❡❝t♦rs t♦ ❜❡ s♣❡❝✐❛❧✿ ❊❛❝❤ ✐s

❛❧✐❣♥❡❞ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ❛①❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❞❡❝♦♠♣♦s❡✿

A =< 3, 2 > = < 3, 0 > + < 0, 2 > .

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✺✹

❚❤❡♥ t❤❡ t✇♦ ✈❡❝t♦rs ❛r❡ ❝❛❧❧❡❞ t❤❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

❝♦♠♣♦♥❡♥t ✈❡❝t♦rs

♦❢ A✳ ❲❡ t❛❦❡ t❤✐s ❛♥❛❧②s✐s ♦♥❡ st❡♣ ❢✉rt❤❡r ✇✐t❤

A =< 3, 2 > =< 3, 0 > + < 0, 2 >= 3< 1, 0 > + 2< 0, 1 > .

❚❤✐s ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ A✿

❙✐♠✐❧❛r❧②✱ ❛♥② ✈❡❝t♦r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s✉❝❤ ❛ ✇❛②✿ < a, b >= a < 1, 0 > +b < 0, 1 > .

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡s❡ s♣❡❝✐❛❧ ✈❡❝t♦rs ✐♥ R2 ✿ ❇❛s✐s ✈❡❝t♦rs ✐♥

R2

i =< 1, 0 >, j =< 0, 1 >

❚❤❡♥✱ ❛♥② ✈❡❝t♦r ✐s ❛s ❛ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦✿ < a, b >= ai + bj

❊①❛♠♣❧❡ ✶✳✺✳✽✿ ❞❡❝♦♠♣♦s✐t✐♦♥

❇❡❧♦✇ ✇❡ ♣r❡s❡♥t ❛ ♥❡✇ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❇❡❢♦r❡✱ ✇❡✬❞ ❝♦♥s✐❞❡r ❛ ♣♦✐♥t ❛♥❞ ✐ts ❝♦♦r❞✐♥❛t❡s✿ P = (4, 1) .

◆♦✇ ✇❡ ✇r✐t❡ ❛ ✈❡❝t♦r ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts✿ 4i + j =< 4, 1 > .

❚❤✉s✱ r❡♣r❡s❡♥t✐♥❣ ❛ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ ✐ts ❝♦♠♣♦♥❡♥ts ✐s ❥✉st ❛ ✇❛② ✭❛ s✐♥❣❧❡ ✇❛②✱ ✐♥ ❢❛❝t✮ t♦ r❡♣r❡s❡♥t ✐t ✐♥ t❡r♠s ♦❢ ❛ ♣❛✐r ♦❢ s♣❡❝✐✜❡❞ ✉♥✐t ✈❡❝t♦rs ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❛①✐s✳ ❲❡✱ ❢✉rt❤❡r♠♦r❡✱ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r s✉❝❤ ✈❡❝t♦rs ✐♥ R3 ✿

✶✳✺✳

❱❡❝t♦rs ✐♥ Rn

✺✺

❇❛s✐s ✈❡❝t♦rs ✐♥ R3 i =< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 >

❋♦r ❡✈❡r② ✈❡❝t♦r✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥✿ < a, b, c >= ai + bj + ck

❊①❛♠♣❧❡ ✶✳✺✳✾✿ ❞❡❝♦♠♣♦s✐t✐♦♥ ◆♦✇ ✇❡ ✇r✐t❡ ❛ ✈❡❝t♦r ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts✿ 2i + 4j + k =< 2, 4, 1 > .

❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✵✿ ❜❛s✐s ✈❡❝t♦rs ❚❤❡ ❜❛s✐s

✈❡❝t♦rs ✐♥ Rn ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s e1 =< 1, 0, 0, ..., 0 > e2 =< 0, 1, 0, ..., 0 > ... en =< 0, 0, 0, ..., 1 >

❚♦❣❡t❤❡r t❤❡② ❢♦r♠ ❛ ❜❛s✐s ♦❢ Rn ✳ ❚❤❡♥✱ ❛♥② ✈❡❝t♦r ✐s ❛s ❛ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ n ✈❡❝t♦rs✿ < x1 , x2 , ..., xn >= x1 e1 + x2 e2 + ... + xn en

❲❡ ❤❛✈❡ ❝♦♠❡ t♦ ❛ ♥❡✇ ✉♥❞❡rst❛♥❞✐♥❣✿ • ♦❧❞✿ ❈❛rt❡s✐❛♥ s②st❡♠ ❂ t❤❡ ❛①❡s

• ♥❡✇✿ ❈❛rt❡s✐❛♥ s②st❡♠ ❂ t❤❡ ♦r✐❣✐♥ ❛♥❞ t❤❡ ❜❛s✐s ✈❡❝t♦rs

❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❛ ❞✐✛❡r❡♥t ❈❛rt❡s✐❛♥ s②st❡♠ ❜② ❝❤♦♦s✐♥❣ ❛ ♥❡✇ s❡t ♦❢ ❜❛s✐s ✈❡❝t♦rs✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs ✐s ❞✐❝t❛t❡❞ ❜② t❤❡ ♣r♦❜❧❡♠ t♦ ❜❡ s♦❧✈❡❞✳

❊①❛♠♣❧❡ ✶✳✺✳✶✶✿ ❝♦♠♣♦✉♥❞ ♠♦t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ st✉❞② t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t s❧✐❞✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡✳ ❊✈❡♥ t❤♦✉❣❤ ❣r❛✈✐t② ✐s ♣✉❧❧✐♥❣ ✐t ✈❡rt✐❝❛❧❧② ❞♦✇♥✱ t❤❡ ♠♦t✐♦♥ ✐s r❡str✐❝t❡❞ t♦ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ s❧♦♣❡✳ ■t ✐s t❤❡♥ ❜❡♥❡✜❝✐❛❧ t♦ ❝❤♦♦s❡ t❤❡ ✜rst ❜❛s✐s ✈❡❝t♦r i t♦ ❜❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ s✉r❢❛❝❡ ❛♥❞ t❤❡ s❡❝♦♥❞ j ♣❡r♣❡♥❞✐❝✉❧❛r✿

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✺✻

❚❤❡ ❣r❛✈✐t② ❢♦r❝❡ ✐s t❤❡♥ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ s✉♠ ♦❢ t✇♦ ✈❡❝t♦rs✳ ■t ✐s t❤❡ ✜rst ♦♥❡ t❤❛t ❛✛❡❝ts t❤❡ ♦❜❥❡❝t ❛♥❞ ✐s t♦ ❜❡ ❛♥❛❧②③❡❞ ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✭◆❡✇t♦♥✬s ❙❡❝♦♥❞ ▲❛✇✮✳ ❚❤❡ s❡❝♦♥❞ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r ❛♥❞ ✐s ❝❛♥❝❡❧❡❞ ❜② t❤❡ r❡s✐st❛♥❝❡ ♦❢ t❤❡ s✉r❢❛❝❡ ✭◆❡✇t♦♥✬s ❚❤✐r❞ ▲❛✇✮✳ ❚❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s

1✲❞✐♠❡♥s✐♦♥❛❧✿

❊①❛♠♣❧❡ ✶✳✺✳✶✷✿ ✐♥✈❡st✐♥❣

❊✈❡♥ ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ t❤❡ ❛❜str❛❝t s♣❛❝❡s

Rn ✱

s✉❝❤ ❞❡❝♦♠♣♦s✐t✐♦♥s ♠❛② ❜❡ ✉s❡❢✉❧✳

❋♦r ❡①❛♠♣❧❡✱ ❛♥ ✐♥✈❡st♠❡♥t ❛❞✈✐❝❡ ♠✐❣❤t ❜❡ t♦ ❤♦❧❞ t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ st♦❝❦s ❛♥❞ ❜♦♥❞s ♣❧♦t ❡❛❝❤ ♣♦ss✐❜❧❡ ♣♦rt❢♦❧✐♦ ❛s ❛ ♣♦✐♥t ♦♥ t❤❡

xy ✲♣❧❛♥❡✱

✇❤❡r❡

x

❛♠♦✉♥t ♦❢ ❜♦♥❞s ✐♥ ✐t✳ ❚❤❡♥ t❤❡ ✏✐❞❡❛❧✑ ♣♦rt❢♦❧✐♦s ❧✐❡ ♦♥ t❤❡ ❧✐♥❡

y = 2x✳

❲❡ t❤❡

❋✉rt❤❡r♠♦r❡✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡

t♦ ❡✈❛❧✉❛t❡ ❤♦✇ ✇❡❧❧ ♣♦rt❢♦❧✐♦s ❢♦❧❧♦✇ t❤✐s ❛❞✈✐❝❡✳ ❲❡ ❝❤♦♦s❡ t❤❡ ✜rst ❜❛s✐s ✈❡❝t♦r t♦ ❜❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐t✱

1✲t♦✲2✳ y ✐s

✐s t❤❡ ❛♠♦✉♥t ♦❢ st♦❝❦s ❛♥❞

i =< 2, 1 >

j =< −1, 2 >✿

❚❤❡♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ✕ ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ♥❡✇ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✕ ♦❢ ②♦✉r ♣♦rt❢♦❧✐♦ r❡✢❡❝ts ❤♦✇ ❢❛r ②♦✉ ❤❛✈❡ ❢♦❧❧♦✇❡❞ t❤❡ ❛❞✈✐❝❡✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ❤♦✇ ♠✉❝❤ ②♦✉✬✈❡ ❞❡✈✐❛t❡❞ ❢r♦♠ ✐t✳ ◆♦✇ ✇❡ ❥✉st ♥❡❡❞ t♦ ❧❡❛r♥ ❤♦✇ t♦ ❝♦♠♣✉t❡ ❞✐st❛♥❝❡s ❛♥❞ ❛♥❣❧❡s ✐♥ s✉❝❤ ❛ s♣❛❝❡✳

✶✳✻✳ ❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

❲❡ ✇✐❧❧ ❧♦♦❦ ❢♦r s✐♠✐❧❛r✐t✐❡s ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ♥✉♠❜❡rs✿ t❤❡

❧❛✇s ♦❢ ❛❧❣❡❜r❛✳

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✺✼

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❢r❡❡❧② ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt❝✉t ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ ♥✉♠❜❡rs✿

x · (y + z) = (x · y) + (x · z) . ■s t❤❡r❡ ❛ s✐♠✐❧❛r r✉❧❡ ❢♦r ✈❡❝t♦rs❄ ❨❡s✱ ✐♥ ❛ s❡♥s❡✳ ❚❤❡ ❝♦♠♣❧❡①✐t② ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡s❡ ❧❛✇s ✇✐❧❧ ❜❡ ❤✐❣❤❡r ❜❡❝❛✉s❡ t❤❡ ♣❛rt✐❝✐♣❛♥ts ❛r❡ ♦❢ t✇♦ ❞✐✛❡r❡♥t t②♣❡s✿ ♥✉♠❜❡rs ❛♥❞ ✈❡❝t♦rs✳ ❚❤❡② ❛r❡ ❛❧s♦ ✐♥t❡r♠✐①❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠❛② ✇r✐t❡ ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s ❛s ❢♦❧❧♦✇s✿

x · (Y + Z) = (x · Y ) + (x · Z) . ❍❡r❡

x

✐s st✐❧❧ ❛ ♥✉♠❜❡r✱ ❜✉t

Y

❛♥❞

Z

❛r❡ ✈❡❝t♦rs✳ ❚❤❡ ❢♦r♠✉❧❛ ❝❛♥✱ ❤♦✇❡✈❡r✱ ❜❡ ❡❛s✐❧② ✈❡r✐✜❡❞ ✐♥ s♣❡❝✐✜❝

s✐t✉❛t✐♦♥s✳ ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿

2 · (< 3, 4 > + < 5, 6 >) = 2 · (< 3 + 5, 4 + 6 >= 2· < 8, 10 >=< 16, 20 > . ❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

2· < 3, 4 > +2· < 5, 6 >=< 2·3, 2·4 > + < 2·5, 2·6 >=< 6, 8 > + < 10, 12 >=< 6+10, 8+12 >=< 16, 20 > . ❲❡ ✇✐❧❧ ✜rst ❡①♣❧♦r❡ t❤❡s❡ r✉❧❡s ❛♥❞ s❤♦rt✲❝✉ts ❛s t❤❡② ❛♣♣❡❛r ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❝♦♠♣♦♥❡♥t✇✐s❡ r❡♣r❡s❡♥✲ t❛t✐♦♥s ♦❢ ✈❡❝t♦rs✳ ■♥ ❞✐♠❡♥s✐♦♥

1✱

✇❡ ❞❡❛❧ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❞✐r❡❝t❡❞ s❡❣♠❡♥ts✳ ❆s t❤❡s❡ s❡❣♠❡♥ts ♥♦✇ ❛❧❧ st❛rt ❛t

0✱

t❤✐s

✐s

t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❡ ❦❡❡♣ t❤❡ t✇♦ t②♣❡s ❛♣❛rt✱ ✇✐t❤ ❛♥ ❡②❡ ♦♥ t❤❡ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ✐❞❡❛ ❝♦♥♥❡❝ts ✈❡❝t♦r ❛❞❞✐t✐♦♥ t♦ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

A + A = 2A . ■ts ❣❡♥❡r❛❧✐③❛t✐♦♥ ✐s t❤❡

❋✐rst ❉✐str✐❜✉t✐✈✐t② Pr♦♣❡rt② ♦❢ ❱❡❝t♦r ❆❧❣❡❜r❛ ✿ aA + bA = (a + b)A

■t✬s ❥✉st ❢❛❝t♦r✐♥❣✿

◮

❲❡ ❢❛❝t♦r ❛

✈❡❝t♦r ♦✉t✳

❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s ❛❞❞ ❛ ❞♦✉❜❧❡ t♦ ❛ tr✐♣❧❡✿

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡ ✐❢ ✇❡ q✉✐♥t✉♣❧❡ t❤❡ ♦r✐❣✐♥❛❧ ✈❡❝t♦r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❞✐str✐❜✉t❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ■❢ ✇❡ ❛r❡ t♦ ❜❡ ♣r❡❝✐s❡✱ t❤❡ s②♠❜♦❧ ✏ +✑ st❛♥❞s ❢♦r t✇♦ ❞✐✛❡r❡♥t t❤✐♥❣s ❛❜♦✈❡✿

• aA + bA✱ • a + b✱

❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs

❛❞❞✐t✐♦♥ ♦❢ ♥✉♠❜❡rs

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✺✽

◆❡①t✱ ✇❡ ❝❛♥ ❛❧s♦ ❞✐str✐❜✉t❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ r❡❛❧ ♥✉♠❜❡rs ♦✈❡r ❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs✳ ❚❤❡

Pr♦♣❡rt② ♦❢ ❱❡❝t♦r ❆❧❣❡❜r❛ ✐s✿

❙❡❝♦♥❞ ❉✐str✐❜✉t✐✈✐t②

aA + aB = a(A + B) ■t✬s ❥✉st ❢❛❝t♦r✐♥❣ ❛❣❛✐♥✿

◮

❲❡ ❢❛❝t♦r ❛

♥✉♠❜❡r ♦✉t✳

❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s ❞♦✉❜❧❡ t❤❡ s✉♠✿

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡ ✐❢ ✇❡ ❛❞❞ t❤❡ ❞♦✉❜❧❡s✳ ❉✐♠❡♥s✐♦♥

2✳

❇❡❧♦✇✱ ✇❡ ❛❞❞ t✇♦ ✈❡❝t♦rs ❛♥❞ t❤❡♥ str❡t❝❤ t❤❡ r❡s✉❧t ✭❧❡❢t✮ ❛♥❞ ✇❡ str❡t❝❤ t✇♦ ✈❡❝t♦rs ❛♥❞ t❤❡♥ ❛❞❞ t❤❡♠ ✭r✐❣❤t✮✿

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✳ ❊①❡r❝✐s❡ ✶✳✻✳✶

❊①♣❧❛✐♥ ✇❤② t❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡✳ ❍✐♥t✿ ❙✐♠✐❧❛r tr✐❛♥❣❧❡s✳

❚❤✐s ❛❧❣❡❜r❛✐❝ r✉❧❡✱ ❛♥❞ ♦t❤❡rs st✐❧❧ t♦ ❝♦♠❡✱ ❤❛s ❜❡❡♥ ❥✉st✐✜❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❢❛♠✐❧✐❛r ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② 2 3 ♦❢ t❤❡ ✏♣❤②s✐❝❛❧ s♣❛❝❡✑ R ❛♥❞ R ✳ ❍♦✇❡✈❡r✱ t❤❡② ❛❧s♦ s❡r✈❡ ❛s ❛ ❢♦r ❢✉rt❤❡r ❞❡✈❡❧♦♣♠❡♥t ♦❢ n ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❞❡✜♥❡❞ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❛❜str❛❝t s♣❛❝❡ R ❛♥❞ ♥♦✇ ❞❡♠♦♥str❛t❡

st❛rt✐♥❣ ♣♦✐♥t

t❤❛t t❤❡s❡ ✏❛①✐♦♠s✑ ❛r❡ st✐❧❧ s❛t✐s✜❡❞✳ ❇❡❧♦✇ ✐s t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❢♦r ❞✐♠❡♥s✐♦♥

2✱

♣r❡s❡♥t❡❞ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦rs✿

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✺✾

❚❤❡ ❢♦r♠✉❧❛ ✐s ✐❧❧✉str❛t❡❞ ❢♦r ❞✐♠❡♥s✐♦♥ 40✿

❘❡❝❛❧❧ t❤❛t t♦ ✜♥❞ A + B ✱ ✇❡ ♠❛❦❡ ❛ ❝♦♣② B ′ ♦❢ B ✱ ❛tt❛❝❤ ✐t t♦ t❤❡ ❡♥❞ ♦❢ A✱ ❛♥❞ t❤❡♥ ❝r❡❛t❡ ❛ ♥❡✇ ✈❡❝t♦r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t t❤❛t ♦❢ A ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥t t❤❛t ♦❢ B ′ ✳ ◆♦✇✱ t♦ ✜♥❞ B + A✱ ✇❡ ♠❛❦❡ ❛ ❝♦♣② A′ ♦❢ A✱ ❛tt❛❝❤ ✐t t♦ t❤❡ ❡♥❞ ♦❢ B ✱ ❛♥❞ t❤❡♥ ❝r❡❛t❡ ❛ ♥❡✇ ✈❡❝t♦r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t t❤❛t ♦❢ B ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥t t❤❛t ♦❢ A′ ✿

❚❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡✳ ❊①❡r❝✐s❡ ✶✳✻✳✷

❊①♣❧❛✐♥ ✇❤② t❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡✳ ❍✐♥t✿ ❙✐♠✐❧❛r tr✐❛♥❣❧❡s✳

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡

✻✵

❈♦♠♠✉t❛t✐✈✐t② Pr♦♣❡rt② ♦❢ ❱❡❝t♦r ❆❞❞✐t✐♦♥

✿

A+B =B+A

◆♦✇ ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♣r❡s❡♥t✱ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦rs ❛r❡ ❝♦♠❜✐♥❡❞ ❛s ❢♦❧❧♦✇s✿

◆❡①t✱ ✇❡ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ✐❣♥♦r❡ t❤❡ ♣❛r❡♥t❤❡s❡s ✇❤❡♥ ✇❡ ❛r❡ ❛❞❞✐♥❣ ♥✉♠❜❡rs✿

(1 + 2) + 3 = 1 + (2 + 3) = 1 + 2 + 3 . ■❞❡♥t✐❝❛❧ ✐s t❤❡

❆ss♦❝✐❛t✐✈✐t② Pr♦♣❡rt② ♦❢ ❱❡❝t♦r ❆❞❞✐t✐♦♥

✿

A + (B + C) = (A + B) + C ❚❤❡ ♦r❞❡r ♦❢ ❛❞❞✐t✐♦♥ ❞♦❡s♥✬t ♠❛tt❡r✦ ❚❤✐s ✐s t❤❡ ♣r♦♣❡rt② ♦❢ ❞✐♠❡♥s✐♦♥

❚❤✐s ✐s ❞✐♠❡♥s✐♦♥

2✿

1✿

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✻✶

▲❡t✬s ❝♦♥s✐❞❡r ♥❡①t ❤♦✇ ✇❡ ❝❛♥ ❛♣♣❧② t✇♦ ✭♦r ♠♦r❡✮ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ✐♥ ❛ r♦✇✳ ●✐✈❡♥ ❛ ✈❡❝t♦r ❛♥❞

b✱

✇❡ ❝❛♥ ❝r❡❛t❡ s❡✈❡r❛❧ ♥❡✇ ✈❡❝t♦rs✿

• B = aA

❢r♦♠

A

❛♥❞ t❤❡♥

C = bB

❢r♦♠

B

• D = bA

❢r♦♠

A

❛♥❞ t❤❡♥

C = aD

❢r♦♠

D

r❡❛❧ ♥✉♠❜❡rs

a

• C = (ab)A

❞✐r❡❝t❧② ❢r♦♠

A

❛♥❞

A

❚❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❜❡❧♦✇ ✇❡ ❞♦✉❜❧❡✱ t❤❡♥ tr✐♣❧❡✿

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡ ✐❢ ✇❡ s❡①t✉♣❧❡✳ ■❢ ✇❡ ❛r❡ t♦ ❜❡ ♣r❡❝✐s❡✱ t❤❡ ♠✐ss✐♥❣ s②♠❜♦❧ ✏ ·✑ st❛♥❞s ❢♦r t✇♦ ❞✐✛❡r❡♥t t❤✐♥❣s ❛❜♦✈❡✿

• B = a · A✱

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r

• D = b · A✱

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r

• C = (a · b)A✱

♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♥✉♠❜❡rs

❖✉r ❝♦♥❝❧✉s✐♦♥ ✐s t❤❡

❆ss♦❝✐❛t✐✈✐t② Pr♦♣❡rt② ♦❢ ❙❝❛❧❛r ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✿ b(aA) = (ba)A

❊①❡r❝✐s❡ ✶✳✻✳✸

Pr♦✈✐❞❡ ❛♥ ✐❧❧✉str❛t✐♦♥ ❢♦r ❞✐♠❡♥s✐♦♥

❚❤❡r❡ ❛r❡ s♦♠❡

❆r❡ t❤❡r❡

2✳

s♣❡❝✐❛❧ ♥✉♠❜❡rs ✿ 0 ❛♥❞ 1✳ ❚❤❡s❡ ❢♦r♠✉❧❛s ✐s ✇❤❛t ♠❛❦❡s t❤❡♠ s♣❡❝✐❛❧✿ 0 + x = x, 0 · x = 0, 1 · x = x .

s♣❡❝✐❛❧ ✈❡❝t♦rs ❄

❈♦♥s✐❞❡r ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ ❚❤❡ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s t❤❡ ❞♦t

O

③❡r♦ ✈❡❝t♦r

✐s s♣❡❝✐❛❧✳ ■t ❤❛s ♥♦ ♠❛❣♥✐t✉❞❡ ♥♦r ❞✐r❡❝t✐♦♥ ❛♥❞ ✇♦✉❧❞ ❤❛✈❡ t♦ 40 ✐ts❡❧❢✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ③❡r♦ ✈❡❝t♦r ✐s ✐♥ R ✿

✶✳✻✳

❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✻✷

❚❤❡r❡ ✐s 0✱ t❤❡ r❡❛❧ ♥✉♠❜❡r✱ ❛♥❞ t❤❡♥ t❤❡r❡ ✐s 0✱ t❤❡ ✈❡❝t♦r✳ ❚❤❡ ❧❛tt❡r ❝❛♥ ♠❡❛♥ ♥♦ ❞✐s♣❧❛❝❡♠❡♥t✱ ♥♦ ♠♦t✐♦♥ ✭③❡r♦ ✈❡❧♦❝✐t②✮✱ ❢♦r❝❡s t❤❛t ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✱ ❡t❝✳ ❚❤❡ t✇♦ ❛r❡ r❡❧❛t❡❞✿

0·A=0 ❚❤✐s ✐s ❛ s✐♠♣❧❡ ❡①♣r❡ss✐♦♥ ✇✐t❤ ❛ tr✐❝❦② ❛❧❣❡❜r❛✐❝ ♠❡❛♥✐♥❣✿ ♥✉♠❜❡r

·

✈❡❝t♦r

=

✈❡❝t♦r

❖❢ ❝♦✉rs❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛❧❧ ✈❡❝t♦rs✿

A+0=A ◆♦✇✱ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❈♦♥s✐❞❡r✿

1·A=A ❲❡ ❤❛✈❡ t❤❡ s❛♠❡ ♣❛rt✐❝✐♣❛♥ts ❤❡r❡ ❛s ❛❜♦✈❡✿ ♥✉♠❜❡r ❙♦✱

1

·

✈❡❝t♦r

=

✈❡❝t♦r

r❡♠❛✐♥s s♣❡❝✐❛❧ ✐♥ ✈❡❝t♦r ❛❧❣❡❜r❛✳

◆❡①t✱ s✐♥❝❡

P Q = −QP ✱

✇❡ ❤❛✈❡ t❤❡

♥❡❣❛t✐✈❡

−A

♦❢ ❛ ✈❡❝t♦r

❚❤❡② ❛r❡ ❛❝q✉✐r❡❞ ❜② t❤❡ ❝❡♥tr❛❧ s②♠♠❡tr② ♦❢ t❤❡ ♣❧❛♥❡✿

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✐♥

R40 ✿

A✱

❛s t❤❡ ✈❡❝t♦r t❤❛t ❣♦❡s ✐♥ r❡✈❡rs❡ ♦❢

A✳

✶✳✻✳ ❆❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs

✻✸

❋r♦♠ t❤❡ ❛❧❣❡❜r❛✱ ✇❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t −A = (−1) · A

❆s ❛ s✉♠♠❛r②✱ t❤✐s ✐s t❤❡ ❝♦♠♣❧❡t❡ ❧✐st ♦❢ r✉❧❡s ♦♥❡ ♥❡❡❞s t♦ ❝❛rr② ♦✉t ❛❧❣❡❜r❛ ✇✐t❤ ✈❡❝t♦rs✿ ❚❤❡♦r❡♠ ✶✳✻✳✹✿ ❆①✐♦♠s ♦❢ ❱❡❝t♦r ❙♣❛❝❡

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✕ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦rs ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❜② ❛ s❝❛❧❛r ✕ ✐♥ Rn s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ✶✳ X + Y = Y + X ❢♦r ❛❧❧ X ❛♥❞ Y ✳ ✷✳ X + (Y + Z) = (X + Y ) + Z ❢♦r ❛❧❧ X ✱ Y ✱ ❛♥❞ Z ✳ ✸✳ X + 0 = X = 0 + X ❢♦r s♦♠❡ ✈❡❝t♦r 0 ❛♥❞ ❛❧❧ X ✳ ✹✳ X + (−X) = 0 ❢♦r ❛♥② ❳ ❛♥❞ s♦♠❡ ✈❡❝t♦r −X ✳ ✺✳ a(bX) = (ab)X ❢♦r ❛❧❧ X ❛♥❞ ❛❧❧ s❝❛❧❛rs a, b✳ ✻✳ 1X = X ❢♦r ❛❧❧ X ✳ ✼✳ a(X + Y ) = aX + aY ❢♦r ❛❧❧ X ❛♥❞ Y ✳ ✽✳ (a + b)X = aX + bX ❢♦r ❛❧❧ X ❛♥❞ ❛❧❧ s❝❛❧❛rs a, b✳

❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ ♣r♦♣❡rt✐❡s ♦❢ ✈❡❝t♦rs ♠❛t❝❤ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ♥✉♠❜❡rs ♣❡r❢❡❝t❧②✦ ❲❡ ♣✉t ❢♦r✇❛r❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡❛✿ ◮ ❆❧❧ ♠❛♥✐♣✉❧❛t✐♦♥s ♦❢ ❛❧❣❡❜r❛✐❝ ❡①♣r❡ss✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ ✇✐t❤ ♥✉♠❜❡rs ❛r❡ ♥♦✇ ❛❧❧♦✇❡❞ ✇✐t❤ ✈❡❝t♦rs ✕ ❛s ❧♦♥❣ ❛s t❤❡ ❡①♣r❡ss✐♦♥ ✐ts❡❧❢ ♠❛❦❡s s❡♥s❡✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❥✉st ♥❡❡❞ t♦ ❛✈♦✐❞ ♦♣❡r❛t✐♦♥s t❤❛t ❤❛✈❡♥✬t ❜❡❡♥ ❞❡✜♥❡❞✿ ♥♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs✱ ♥♦ ❞✐✈✐s✐♦♥ ♦❢ ✈❡❝t♦rs✱ ♥♦ ❛❞❞✐♥❣ ♥✉♠❜❡rs t♦ ✈❡❝t♦rs ✭♦❢ ❝♦✉rs❡✦✮✱ ❡t❝✳ ❆❧❧ s♣❛❝❡s ♦❢ ✈❡❝t♦rs✱ ✈❡❝t♦r s♣❛❝❡s ✐❢ ②♦✉ ❧✐❦❡✱ ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ❤❛✈❡ ❜❡❡♥ ♦♥❧② Rn ✳ ❆r❡ t❤❡r❡ ♦t❤❡rs❄ ❊①❛♠♣❧❡ ✶✳✻✳✺✿ s✉❜s❡ts ❛♥❞ s✉❜s♣❛❝❡s

▲❡t✬s ✜① t❤❡ ❧❛st ❝♦♦r❞✐♥❛t❡ ✐♥ Rn ❛♥❞ ❧♦♦❦ ❛t t❤❡ ❛❧❣❡❜r❛✿ < a1 a2 ... an−1 an > + < b1 b2 ... bn−1 bn > < a1 + b1 a2 + b2 ... an−1 + bn−1 an + bn >

■t ✇♦r❦s ❡①❛❝t❧② t❤❡ s❛♠❡✦

→

< a1 a2 ... an−1 0> + < b1 b2 ... bn−1 0> < a1 + b1 a2 + b2 ... an−1 + bn−1 0 >

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✻✹

❍♦✇ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇♦r❦s ✐s ❛❧s♦ ♠❛t❝❤❡❞✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ✇♦✉❧❞ ❛♥t✐❝✐♣❛t❡ t❤❛t t❤❡ ❡✐❣❤t ♣r♦♣❡rt✐❡s ✐♥ t❤❡ t❤❡♦r❡♠ ❛r❡ s❛t✐s✜❡❞✳ ▲❡t✬s ❞❡♥♦t❡ t❤✐s s❡t ❛s ❢♦❧❧♦✇s✿ Rn0 = {< x1 , x2 , ..., xn−1 , 0 >} ⊂ Rn .

■❢ ✇❡ ❞r♦♣ t❤❡ r❡❞✉♥❞❛♥t 0✬s✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤✐s ✐s ❥✉st ❛ ❝♦♣② ♦❢ Rn−1 ✿

→

< a1 a2 ... an−1 > + < b1 b2 ... bn−1 > < a1 + b1 a2 + b2 ... an−1 + bn−1 >

❉❡✜♥✐t✐♦♥ ✶✳✻✳✻✿ ✈❡❝t♦r s♣❛❝❡ ❆♥② s❡t ✇✐t❤ t✇♦ ♦♣❡r❛t✐♦♥s t❤❛t s❛t✐s❢② t❤❡ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ❝❛❧❧❡❞ ❛ ❛ ✈❡❝t♦r s♣❛❝❡✳ ❙♦✱ Rn−1 ✐s ❛ ✈❡❝t♦r s♣❛❝❡✱ ❛ s✉❜s♣❛❝❡ ♦❢ Rn ✳ 0 ❊✈❡r② s✉❜s❡t ♦❢ Rn ✐s s✉❜❥❡❝t t♦ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡✳ ❍♦✇ ❞♦ ✇❡ ❞❡t❡r♠✐♥❡ ✇❤❡♥ t❤✐s ✐s ❛ ✈❡❝t♦r s♣❛❝❡❄ ❲❡ ❥✉st ♥❡❡❞ t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ❛❧❣❡❜r❛ ♠❛❦❡s s❡♥s❡✿

❚❤❡♦r❡♠ ✶✳✻✳✼✿ ❙✉❜s♣❛❝❡s ❙✉♣♣♦s❡ U ✐s ❛ s✉❜s❡t ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ t❤❛t s❛t✐s✜❡s✿ ✶✳ ■❢ X ❛♥❞ Y ❜❡❧♦♥❣ t♦ U ✱ t❤❡♥ s♦ ❞♦❡s X + Y ✳ ✷✳ ■❢ X ❜❡❧♦♥❣s t♦ U ✱ t❤❡♥ s♦ ❞♦❡s kX ❢♦r ❛♥② ♥✉♠❜❡r k ✳ ❚❤❡♥ U ✐s ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✻✳✽ ❋♦r t❤❡ ❧❛st ❡①❛♠♣❧❡✱ s❤♦✇ t❤❛t s❡tt✐♥❣ t❤❡ ❧❛st ❝♦♦r❞✐♥❛t❡ t♦ ❛ ♥♦♥✲③❡r♦ ♥✉♠❜❡r ✇♦♥✬t ❝r❡❛t❡ ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✻✳✾ ❋♦r t❤❡ ❧❛st ❡①❛♠♣❧❡✱ s❤♦✇ t❤❛t s❡tt✐♥❣ t❤❡ s❡✈❡r❛❧ ❝♦♦r❞✐♥❛t❡s t♦ ③❡r♦ ✇✐❧❧ ❝r❡❛t❡ ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✻✳✶✵ Pr♦✈❡ t❤❛t ❛ ❧✐♥❡ t❤r♦✉❣❤ 0 ♦♥ t❤❡ ♣❧❛♥❡ ✐s ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✻✳✶✶ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✶✳✼✳ ❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs ❚❤❡ ❛✈❡r❛❣❡ ♦❢ t✇♦ ♥✉♠❜❡rs ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡✐r ❤❛❧❢ s✉♠✿ x+y 1 1 = x+ y. 2 2 2

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

❙✐♥❝❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ♠✐♠✐❝s t❤❛t ♦❢ ♥✉♠❜❡rs✱ ♥♦t❤✐♥❣ st♦♣s ✉s ❢r♦♠ ❞❡✜♥✐♥❣ t❤❡ U ❛♥❞ V ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✿

✻✺

❛✈❡r❛❣❡ ♦❢ ✈❡❝t♦rs

1 1 1 U + V = (U + V ) . 2 2 2 ■t ✐s ❛ ❝♦♥✈❡♥✐❡♥t ❝♦♥❝❡♣t ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ❢♦r ❞✐♠❡♥s✐♦♥ 2✿

❚❤❡ ❛✈❡r❛❣❡ ♦❢ t✇♦ ✈❡❝t♦rs ✐s t❤❡ ✈❡❝t♦r t❤❛t ❣♦❡s t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❢♦r♠❡❞ ❜② t❤❡ t✇♦✳ ❊①❡r❝✐s❡ ✶✳✼✳✶

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ■♥ ❞✐♠❡♥s✐♦♥ 40✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❧✐❡s ❤❛❧❢✲✇❛② ✭✈❡rt✐❝❛❧❧②✮ ❜❡t✇❡❡♥ t❤❡ t✇♦✿

▲❡t✬s t❛❦❡ t❤✐s ✐❞❡❛ ♦♥❡ st❡♣ ❢✉rt❤❡r✳ ❚❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t✇♦ ♥✉♠❜❡rs ✐s ❞❡✜♥❡❞ t♦ ❜❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ❧✐❦❡ t❤✐s✿ αx + βy ,

✇❤❡r❡ α ≥ 0 ❛♥❞ β ≥ 0 ❛❞❞ ✉♣ t♦ 1✿ ❙✐♠✐❧❛r❧②✱ t❤❡ ✇❡✐❣❤t❡❞

α + β = 1.

❛✈❡r❛❣❡ ♦❢ ✈❡❝t♦rs U ❛♥❞ V

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

αU + βV .

❲❡ ❝❛♥ s❡❡ ✐♥ ❞✐♠❡♥s✐♦♥ 2 ❤♦✇ ♦♥❡ ✈❡❝t♦r ✐s ❣r❛❞✉❛❧❧② tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ ♦t❤❡r ❛s alpha r✉♥s ❢r♦♠ 1 t♦ 0 ✭❛♥❞ β ❢r♦♠ 0 t♦ 1✮✿

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✻✻

❚❤❡s❡ ✐♥t❡r♠❡❞✐❛t❡ st❛❣❡s ❛r❡ ❛❧s♦ ❝❛❧❧❡❞ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ t✇♦ ✈❡❝t♦rs✳ ❚♦❣❡t❤❡r t❤❡✐r ❡♥❞s ❢♦r♠ ❛ ❧✐♥❡ s❡❣♠❡♥t❀ ✐t r✉♥s ❢r♦♠ t❤❡ ❡♥❞ ♦❢ U t♦ t❤❡ ❡♥❞ ♦❢ V ✳ ■♥ ❞✐♠❡♥s✐♦♥ 40✱ ✇❡ ❝❛♥ ❛❧s♦ s❡❡ ❛ ❣r❛❞✉❛❧ tr❛♥s✐t✐♦♥ ❢r♦♠ ♦♥❡ ✈❡❝t♦r t♦ t❤❡ ♦t❤❡r✿

◆♦✇✱ ✇❡ s❛✇ ✐♥ ❞✐♠❡♥s✐♦♥ 2 t❤❛t t❤❡ ❛✈❡r❛❣❡ ✐s ❛❧s♦ ❛ str❛✐❣❤t s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❡❝t♦rs✳ ❖❢ ❝♦✉rs❡✱ ✐t ✐s ❛ str❛✐❣❤t s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢♦r ❞✐♠❡♥s✐♦♥ 3 ♦r ❛♥② ❞✐♠❡♥s✐♦♥ t❤❛t ✇❡ ❝❛♥ ✈✐s✉❛❧✐③❡✳ ■❢ ✇❡ r❡♠♦✈❡ t❤❡ r❡str✐❝t✐♦♥ α ≥ 0 ❛♥❞ β ≥ 0✱ ♦✉r ❝♦♠❜✐♥❛t✐♦♥s αU + βV

❛r❡ ❝❛❧❧❡❞ t❤❡ ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ U ❛♥❞ V ✳ ❚♦❣❡t❤❡r✱ t❤❡✐r ❡♥❞s ❢♦r♠ ❛ ✇❤♦❧❡ ❧✐♥❡❀ ✐t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❡♥❞ ♦❢ U ❛♥❞ t❤❡ ❡♥❞ ♦❢ V ✿

▲❡t✬s r❡❝❛❧❧ t❤❡ ♣r♦❜❧❡♠ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ t♦ ✐❧❧✉str❛t❡ ♠❛♥② ♥❡✇ ✐❞❡❛s✳

Pr♦❜❧❡♠✿ ❲❡ ❛r❡ ❣✐✈❡♥ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❛t $2 ♣❡r ♣♦✉♥❞ ❛♥❞ t❤❡ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡ ❛t $3 ♣❡r ♣♦✉♥❞✳ ❍♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡ 6 ♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ t❤❡ t♦t❛❧ ♣r✐❝❡ ♦❢ $14❄

❲❡ ❧❡t x ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❛♥❞ ❧❡t y ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡✳ ❚❤❡♥ t❤❡ t♦t❛❧ ♣r✐❝❡ ♦❢ t❤❡ ❜❧❡♥❞ ✐s $14✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ s②st❡♠✿ x + y =6 2x + 3y = 14

❚❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ s②st❡♠ ❛s ♣r❡s❡♥t❡❞ ✐♥✐t✐❛❧❧② ❤❛❞ ❛ ❝❧❡❛r ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣✳ ❲❡ t❤♦✉❣❤t ♦❢ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛s ❡q✉❛t✐♦♥s ❛❜♦✉t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts✱ (x, y)✱ ✐♥ t❤❡ ♣❧❛♥❡✳ ■♥ ❢❛❝t✱ ❡✐t❤❡r ❡q✉❛t✐♦♥ ✐s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ (x, y) = (4, 2) ✐s t❤❡ ♣♦✐♥t ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✻✼

❢✉♥❝t✐♦♥

❚❤❡ s❡❝♦♥❞ ✐♥t❡r♣r❡t❛t✐♦♥ ✇❛s ✐♥ t❡r♠s ♦❢ ❛ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣❧❛♥❡✳ ❆ ❢✉♥❝t✐♦♥ F : R2 → R2 ✐s ❣✐✈❡♥ ❜②✿ F (x, y) = (x + y, 2x + 3y) . ❚❤❡♥ ♦✉r s♦❧✉t✐♦♥ ✐s✿

(x, y) = F −1 (6, 14) .

❲❡ ♥♦✇ ❤❛✈❡ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥ ✕ ✐♥ t❡r♠s ♦❢

✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡✳

❲❡ r❡✇r✐t❡ t❤❡ s②st❡♠ ❛s ❛ ✈❡❝t♦r ❡q✉❛t✐♦♥✿

x + y =6 2x + 3y = 14

=⇒



   6 x + y . = 14 2x + 3y

❚❤❡ ✜rst ✈❡❝t♦r✬s ❝♦♠♣♦♥❡♥ts ❛r❡ ❝♦♠♣✉t❡❞ ✈✐❛ s♦♠❡ ❛❧❣❡❜r❛✳ ❲❡ ✇✐❧❧ tr② t♦ ✐♥t❡r♣r❡t t❤✐s ❛❧❣❡❜r❛ ♦❢ ♥✉♠❜❡rs ✐♥ t❡r♠s ♦❢ ♦✉r ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs✳ ❲❡ s♣❧✐t t❤❡ ✈❡❝t♦r ✉♣✿



❲❡ ❢❛❝t♦r t❤❡ r❡♣❡❛t❡❞ ❝♦❡✣❝✐❡♥ts ♦✉t✿ 

     y x x + y + = 3x 2x 2x + 3y

     1 1 x + y . +y =x 3 2 2x + 3y

❖✉r s②st❡♠ ❤❛s ❜❡❡♥ r❡❞✉❝❡❞ t♦ ❛ s✐♥❣❧❡

✈❡❝t♦r ❡q✉❛t✐♦♥ ✿

      6 1 1 . = +y x 14 3 2 ▲❡t✬s ❛♥❛❧②③❡ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ✐t ♣r❡s❡♥ts✳ ●✐✈❡♥ t✇♦ ✈❡❝t♦rs✿

  1 U= 2

✜♥❞ t✇♦ ♥✉♠❜❡rs x ❛♥❞ y s♦ t❤❛t ✇❡ ❤❛✈❡✿

  1 , ❛♥❞ V = 3

  6 . xU + yV = 14

❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦rs ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✳

■t ♠❛② ❛♣♣❡❛r t❤❛t ✇❡ ❥✉st ♥❡❡❞ t♦ r❡♣r❡s❡♥t t❤❡ ✈❡❝t♦r < 6, 14 > ❛s ❛♥ U ❛♥❞ V ✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ♥♦ r❡str✐❝t✐♦♥ t❤❛t x ❛♥❞ y ♠✉st ❛❞❞ ✉♣ t♦ 1✳ ❲❡ s♣❡❛❦ ♦❢

str❡t❝❤

❙♦✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ✇❛② t♦ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs s♦ t❤❛t t❤❡✐r s✉♠ ✐s t❤❡ t❤✐r❞ ✈❡❝t♦r✳ ❚❤❡ s❡t✉♣ ✐s ♦♥ t❤❡ ❧❡❢t ❢♦❧❧♦✇❡❞ ❜② ❛ tr✐❛❧✲❛♥❞✲❡rr♦r✿

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✻✽

❏✉st ❛❞❞✐♥❣ t❤❡ t✇♦ ✈❡❝t♦rs ♦r ❛❞❞✐♥❣ t❤❡✐r ♣r♦♣♦rt✐♦♥❛❧ ♠✉❧t✐♣❧❡s ❢❛✐❧s❀ ✐t ✐s ❝❧❡❛r t❤❛t t❤❡ ❛♥❣❧❡ ❝❛♥✬t ♠❛t❝❤✳ ❍②♣♦t❤❡t✐❝❛❧❧②✱ ✇❡ ❣♦ t❤r♦✉❣❤ ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs t♦ ✜♥❞ ♦♥❡ t❤❛t ✐s ❥✉st r✐❣❤t✳ ❙♦✱ t❤❡ ♥❡✇ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♠✐①t✉r❡s ✐s ❞✐✛❡r❡♥t✿ ◮ ■♥st❡❛❞ ♦❢ t❤❡

❧♦❝❛t✐♦♥s✱ ✇❡ ❛r❡ ❛❢t❡r t❤❡ ❞✐r❡❝t✐♦♥s✳

■♥ ❣❡♥❡r❛❧✱ t❤❡s❡ ❛r❡ ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s αU + βV ✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥s✿ α\β ... −2 −1 0 1 2 ...

... −2 ... ... ... −2U − 2V ... −U − 2V ... −2V ... U − 2V ... 2U − 2V ... ...

−1 ... −2U − V −U − V −V U −V 2U − V ...

0 1 ... ... −2U −2U + V −U −U + V 0 V U U +V 2U 2U + V ... ...

2 ... −2U + 2V −U + 2V 2V U + 2V 2U + 2V ...

... ... ... ... ... ... ... ...

❚❤❡s❡ ❛r❡ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ✇✐t❤ ✐♥t❡❣❡r ❝♦❡✣❝✐❡♥ts ♦❢ U =< 2, 1 > ❛♥❞ V =< −1, 1 >✿

❚❤❡② s❡❡♠ t♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✳ ❊①❡r❝✐s❡ ✶✳✼✳✷

■s ✐t tr✉❡❄ ■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡s❡ t❤r❡❡ ❝♦♠❜✐♥❛t✐♦♥s✿

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✻✾

❉❡✜♥✐t✐♦♥ ✶✳✼✳✸✿ ❧✐♥❡❛r✱ ❛✣♥❡✱ ❛♥❞ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s

✶✳ ❆ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦rs U ❛♥❞ V ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ✇✐t❤ ❛♥② r❡❛❧ ♥✉♠❜❡rs α ❛♥❞ β ❝❛❧❧❡❞ ✐ts ❝♦❡✣❝✐❡♥ts✿ αU + βV .

✷✳ ❆♥ ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦rs U ❛♥❞ V ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡✐r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ❝♦❡✣❝✐❡♥ts α ❛♥❞ β t❤❛t ❛❞❞ ✉♣ t♦ 1✿ α + β = 1.

✸✳ ❆ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦rs U ❛♥❞ V ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡✐r ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ❝♦❡✣❝✐❡♥ts α ❛♥❞ β t❤❛t ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✿ α ≥ 0, β ≥ 0 .

❊①❛♠♣❧❡ ✶✳✼✳✹✿ ❤✉❧❧s

❚❤❡s❡ ❝♦♥❝❡♣ts ❤❛✈❡ ❛♥❛❧♦❣s ❢♦r ♣♦✐♥ts✳ ❲❡ ❞❡✜♥❡ t❤r❡❡ ❤✉❧❧s ✿

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ❝♦♥✈❡① ❛♥❞ t❤❡ ❛✣♥❡ ❤✉❧❧s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥✳

❊①❡r❝✐s❡ ✶✳✼✳✺ ❲❤❛t ❛❜♦✉t t❤❡ ❧✐♥❡❛r ❤✉❧❧❄ ■♥ t❤❡ ❣❡♥❡r❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ r❡s✉❧ts ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ t❤❡ t✇♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s t❤❛t ✇❡ ❤❛✈❡✿ ✈❡❝t♦r ❛❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❚❤✐s ✐s ✇❤❛t ❛ ❢❡✇ ♦❢ t❤❡♠ ✭αf + βg ✮ ❧♦♦❦ ❧✐❦❡ ✐♥ R40 ✿

✶✳✼✳

❈♦♥✈❡①✱ ❛✣♥❡✱ ❛♥❞ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✈❡❝t♦rs

✼✵

❊①❡r❝✐s❡ ✶✳✼✳✻ ▲❛❜❡❧ t❤❡s❡✳

❚❤❡ ♥❡①t ❞❡✜♥✐t✐♦♥ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ✐❞❡❛ t♦ ❛♥ ✉♥❧✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ✈❡❝t♦rs✿

❉❡✜♥✐t✐♦♥ ✶✳✼✳✼✿ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❙✉♣♣♦s❡

V1 , ..., Vm

V1 , ..., Vm

✇✐t❤ ❝♦❡✣❝✐❡♥ts

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦rs

Rn ✳ ❚❤❡♥✱ t❤❡ r1 , ..., rm ✐s t❤❡ ❢♦❧❧♦✇✐♥❣

❛r❡ ✈❡❝t♦rs ✐♥

✈❡❝t♦r✿

r1 V1 + ... + rm Vm . ❚❤❡♥✱ t❤❡ s❡t ♦❢ ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❛ s✐♥❣❧❡ ✈❡❝t♦r ✐s s✐♠♣❧② t❤❡ s❡t ♦❢ ✐ts ♠✉❧t✐♣❧❡s ✭❛ ❧✐♥❡✮✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t✇♦ ✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡

R2

✐s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✱ ✉♥❧❡ss t❤❡ t✇♦ ❛r❡

♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳

❊①❡r❝✐s❡ ✶✳✼✳✽ Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳

❊①❡r❝✐s❡ ✶✳✼✳✾ ❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡✿ ✏❚❤❡ s❡t ♦❢ ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤r❡❡ ✈❡❝t♦rs ✐♥ t❤❡

3✲s♣❛❝❡✱

3✲s♣❛❝❡

✉♥❧❡ss ❴❴❴❴❴❴❴❴✑✳

❆♥ ✐♠♣♦rt❛♥t ❢❛❝t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✶✳✼✳✶✵✿ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇❛s✐s ❱❡❝t♦rs ❊✈❡r② ✈❡❝t♦r ✐♥

Rn

✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs✿

< a1 , ..., an >= a1 e1 + ... + an en .

✐s t❤❡ ✇❤♦❧❡

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✶

❊①❡r❝✐s❡ ✶✳✼✳✶✶ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✶✳✼✳✶✷✿ ♣♦❧②♥♦♠✐❛❧s ❆ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✿ a0 + a1 x1 + ... + an xn .

■♥ t❤✐s s❡♥s❡✱ t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✉♣ t♦ n ✐s ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ Rn+1 ✳

❊①❡r❝✐s❡ ✶✳✼✳✶✸ ❙❤♦✇ t❤❛t t❤❡ ♠✉❧t✐♣❧❡s ♦❢ ❛ ❣✐✈❡♥ ✈❡❝t♦r ✐♥ ❛ ✈❡❝t♦r s♣❛❝❡ ❢♦r♠ ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✼✳✶✹ ❙❤♦✇ t❤❛t t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❛ ❣✐✈❡♥ ♣❛✐r ♦❢ ✈❡❝t♦r ✐♥ ❛ ✈❡❝t♦r s♣❛❝❡ ❢♦r♠ ❛ ✈❡❝t♦r s♣❛❝❡✳

❊①❡r❝✐s❡ ✶✳✼✳✶✺ ❲❤❛t ✐s t❤❡ ♥❡①t st❛t❡♠❡♥t ✐♥ t❤✐s s❡q✉❡♥❝❡❄

✶✳✽✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

❆ ✈❡❝t♦r ✐s ❛ ❞✐r❡❝t❡❞ s❡❣♠❡♥t✳ ■ts ❛ttr✐❜✉t❡s ❛r❡✱ t❤❡r❡❢♦r❡✱ t❤❡ ❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡✳ ■t ♠❛② ❜❡ ❤❛r❞ t♦ ❡①♣❧❛✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ♠❡❛♥s ✇✐t❤♦✉t r❡❢❡rr✐♥❣✱ ❝✐r❝✉❧❛r❧②✱ t♦ ✈❡❝t♦rs✳ ❚❤❛t ✐s ✇❤② ✇❡ ❧♦♦❦ ❛t t❤❡ ♠❛❣♥✐t✉❞❡ ✜rst✿ ◮ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r ✐s ✇❤❛t✬s ❧❡❢t ♦❢ ✐t ✇❤❡♥ ✐t✬s str✐♣♣❡❞ ♦✛ ✐ts ❞✐r❡❝t✐♦♥✳

❲❤❡♥ ✇❡ ✐♥t❡r♣r❡t t❤❡ ✈❡❝t♦r ❛s ❛ ❞✐s♣❧❛❝❡♠❡♥t✱ ✇❡✬❞ r❛t❤❡r t❛❧❦ ❛❜♦✉t ✐ts ❧❡♥❣t❤✳ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤✐s ♥✉♠❜❡r ✐s ❝❧❡❛r ✇❤❡♥ t❤❡ ✈❡❝t♦r ✐s ❣✐✈❡♥ ❜② t✇♦ ♣♦✐♥ts✱ P Q✳ ❚❤❡ ❧❡♥❣t❤ ✐s t❤❡ ❞✐st❛♥❝❡ d(P, Q) ❜❡t✇❡❡♥ t❤❡♠✿

❲❡ ✐♥t❡♥t✐♦♥❛❧❧② ♠❛❦❡ ♥♦ r❡❢❡r❡♥❝❡ t♦ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✿

❉❡✜♥✐t✐♦♥ ✶✳✽✳✶✿ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦r t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ✈❡❝t♦r ✐♥ Rn ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✐ts ✐♥✐t✐❛❧ ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥ts✿ ||P Q|| = d(P, Q)

❚❤✐s ♥✉♠❜❡r ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♥♦r♠ ♦❢ t❤❡ ✈❡❝t♦r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡✿

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✷

✶✳ ■❢ d(P, Q) st❛♥❞s ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝✱ ||P Q|| ✐s ❝❛❧❧❡❞ t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠✳ ✷✳ ■❢ d(P, Q) st❛♥❞s ❢♦r t❤❡ t❛①✐❝❛❜ ♠❡tr✐❝✱ ||P Q|| ✐s ❝❛❧❧❡❞ t❤❡ t❛①✐❝❛❜ ♥♦r♠✳ ❚❤❡ ♥♦t❛t✐♦♥ r❡s❡♠❜❧❡s t❤❡ ❛❜s♦❧✉t❡ R✳

✈❛❧✉❡ ❛♥❞ ♥♦t ❜② ❛❝❝✐❞❡♥t❀ t❤❡② ❛r❡ t❤❡ s❛♠❡ ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱

❲❡ ❛❧s♦ ✐♥t❡♥t✐♦♥❛❧❧② ♠❛❦❡ ♥♦ r❡❢❡r❡♥❝❡ t♦ ❛ s♣❡❝✐✜❝ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ❛s ❢♦❧❧♦✇s✿

◮ ❲❡ ♥♦✇ ❧♦♦❦ ❛t ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❆①✐♦♠s tr❛♥s❧❛t❡ ✐t ✐♥t♦ ❛ ♣r♦♣❡rt② ♦❢ ✈❡❝t♦rs✳ ❋✐rst✱ t❤❡

♦❢ ▼❡tr✐❝ ❙♣❛❝❡

❛♥❞ ✕ ✉s✐♥❣ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ ✕

P♦s✐t✐✈✐t② ✿ d(P, Q) ≥ 0;

❛♥❞ d(P, Q) = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ P = Q

❲❡ r❡✇r✐t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❜♦✈❡✿

||P Q|| ≥ 0;

❛♥❞ ||P Q|| = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ P = Q

❇✉t t❤❡ ✈❡❝t♦r P P ✐s ❥✉st t❤❡ ③❡r♦ ✈❡❝t♦r✦ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤✐s ♥❡✇ ❢♦r♠ ♦❢ t❤❡ ♣r♦♣❡rt②✿

||A|| ≥ 0; ❙❡❝♦♥❞✱ t❤❡

❛♥❞ ||A|| = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ A = 0

❙②♠♠❡tr② ✿ d(P, Q) = d(Q, P )

❲❡ r❡✇r✐t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❜♦✈❡✿

||P Q|| = ||QP || ❇✉t t❤❡ ✈❡❝t♦r P Q ✐s t❤❡ ♥❡❣❛t✐✈❡ ♦❢ QP ✿

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤✐s ♥❡✇ ❢♦r♠ ♦❢ t❤❡ ♣r♦♣❡rt②✿

||A|| = || − A|| ■ts ♠❡❛♥✐♥❣ ✐s ✈✐s✉❛❧✐③❡❞ ❢♦r R40 ❜❡❧♦✇✿

✶✳✽✳

✼✸

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

❚❤❡ ♥♦r♠ ✐s t❤❡ ♣✉r♣❧❡ ❛r❡❛ ❛t t❤❡ ❜♦tt♦♠✳ ❚❤✐r❞✱ t❤❡ ❚r✐❛♥❣❧❡

■♥❡q✉❛❧✐t② ✿

d(P, Q) + d(Q, R) ≥ d(P, R)

❲❡ r❡✇r✐t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❜♦✈❡✿

||P Q|| + ||QR|| ≥ ||P R||

❇✉t P Q + QR = P R✿

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤✐s ♥❡✇ ❢♦r♠ ♦❢ t❤❡ ♣r♦♣❡rt②✿ ||A|| + ||B|| ≥ ||A + B||

❙♦✱ ✇❡ ❤❛✈❡ ♠♦✈❡❞ ❢r♦♠ t❤❡ ❣❡♦♠❡tr②

♦❢ ♣♦✐♥ts

t♦ t❤❡

❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs✳

■♥ s✉♠♠❛r②✿ ✶✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ❝❛♥♥♦t ❜❡ ♥❡❣❛t✐✈❡✱ ❛♥❞ ♦♥❧② t❤❡ ③❡r♦ ✈❡❝t♦r ❤❛s ❛ ③❡r♦ ♠❛❣♥✐t✉❞❡✳ ✷✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ ❛ ✈❡❝t♦r ✐s ❡q✉❛❧ t♦ t❤❛t ♦❢ t❤❡ ✈❡❝t♦r✳ ✸✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ✈❡❝t♦rs ✐s ❧❛r❣❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ t❤❛t ♦❢ t❤❡✐r s✉♠✳ ❚❤❛t ✐s ❤♦✇ t❤❡ ♠❛❣♥✐t✉❞❡ ✐♥t❡r❛❝ts ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt② ♦❢ ❞✐st❛♥❝❡s✳

✈❡❝t♦r ❛❞❞✐t✐♦♥✳

❲❤❛t ❛❜♦✉t

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❄

❚❤❡r❡ ✐s ♥♦

❆❧❧ ✈❡❝t♦rs ❞♦✉❜❧❡ ✐♥ ❧❡♥❣t❤ ✇❤❡t❤❡r ✇❡ ♠✉❧t✐♣❧② ❜② 2 ♦r −2✱ ✇❤✐❧❡ t❤❡✐r ❞✐r❡❝t✐♦♥s ❛r❡ ♣r❡s❡r✈❡❞ ♦r ✢✐♣♣❡❞✳ ❲❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ❞♦❡s♥✬t ♠❛tt❡r✱ ✇❡ ❥✉st ♠✉❧t✐♣❧② t❤❡ ❝♦♠♣♦♥❡♥ts ❜② t❤✐s str❡t❝❤✐♥❣ ❢❛❝t♦r✱ 2 = |2| = |−2|✿

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✹

❚❤❡ r❡s✉❧t ✐s ❛♥♦t❤❡r ❝♦♥✈❡♥✐❡♥t ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✶✳✽✳✷✿ ❍♦♠♦❣❡♥❡✐t② ♦❢ ◆♦r♠ ❇♦t❤ t❤❡ ❊✉❝❧✐❞❡❛♥ ❛♥❞ t❤❡ t❛①✐❝❛❜ ♥♦r♠s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r ❛♥② ✈❡❝t♦r

A

❛♥❞ ❛♥② s❝❛❧❛r

k✿ ||k · A|| = |k| · ||A||

Pr♦♦❢✳ ❋♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ✐♥

R2 ✿

||k < a, b > || = || < ka, kb > || = ❋♦r t❤❡ t❛①✐❝❛❜ ♥♦r♠ ✐♥

R2 ✿

p √ (ka)2 + (kb)2 = |c| · a2 + b2 = |k| · || < a, b > || .

||k < a, b > || = || < ka, kb > || = |ka| + |kb| = |k| · |a| + |k| · |b| = |k| · (|a| + |b|) = k · || < a, b > || .

❲❛r♥✐♥❣✦ ❚❤❡

❙②♠♠❡tr②

❛❜♦✈❡ ✐s ♥♦✇ r❡❞✉♥❞❛♥t ❛s ✐t ✐s ✐♥✲

❝♦r♣♦r❛t❡❞ ✐♥t♦ t❤✐s ♥❡✇ ♣r♦♣❡rt②✿

−A = (−1)A . ❚❤❡s❡ ♣r♦♣❡rt✐❡s ❛r❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛❧❧ ❞✐♠❡♥s✐♦♥s ❛♥❞ ❛r❡ ✉s❡❞ t♦ ♠❛♥✐♣✉❧❛t❡ ✈❡❝t♦r ❡①♣r❡ss✐♦♥s✳ ❲❡ ♥♦✇ t✉r♥ ❛r♦✉♥❞ ❛♥❞ ❛s❦✿

◮

❲❤❛t✬s ❧❡❢t ♦❢ ❛ ✈❡❝t♦r ✇❤❡♥ ✐ts ♠❛❣♥✐t✉❞❡ ✐s str✐♣♣❡❞ ♦✛ ❄

■❢ ✇❡ ✏r❡♠♦✈❡✑ t❤❡ ♠❛❣♥✐t✉❞❡ ❢r♦♠ ❝♦♥s✐❞❡r❛t✐♦♥✱ ✇❡ ❛r❡ ❧❡❢t ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t t❤❡ ❞✐r❡❝t✐♦♥✳ ❲❡ ❝❛♥ ♦♥❧② s❛② t❤✐s✿

◮

❱❡❝t♦rs ✇✐t❤ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ❛r❡ ✭♣♦s✐t✐✈❡✮ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳

❇✉t ✇❤❛t ✐s t❤❡ s✐♠♣❧❡st ✈❡❝t♦r ❛♠♦♥❣ t❤♦s❡❄ ❚♦ st✉❞② ❞✐r❡❝t✐♦♥s✱ ✇❡ ❧✐♠✐t ♦✉r ❛tt❡♥t✐♦♥ t♦ s♦♠❡ s♣❡❝✐❛❧ ✈❡❝t♦rs✿

❉❡✜♥✐t✐♦♥ ✶✳✽✳✸✿ ✉♥✐t ✈❡❝t♦r ❊✈❡r② ✈❡❝t♦r ✇✐t❤ ♠❛❣♥✐t✉❞❡ ❡q✉❛❧ t♦

1

✐s ❝❛❧❧❡❞ ❛

✉♥✐t ✈❡❝t♦r ✿

||X|| = 1 . ❲❡ ❝❛♥ ♠❛❦❡ s✉❝❤ ❛ ✈❡❝t♦r ❢r♦♠ ❛♥② ✈❡❝t♦r ✕ ✏♥♦r♠❛❧✐③❡✑ ✐t ✕ ❡①❝❡♣t

0✱

❜② ❞✐✈✐❞✐♥❣ ❜② ✐ts ♠❛❣♥✐t✉❞❡✿

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✺

❚❤❡♦r❡♠ ✶✳✽✳✹✿ ◆♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❱❡❝t♦rs ❋♦r ❛♥② ✈❡❝t♦r

X 6= 0✱

t❤❡ ✈❡❝t♦r

Y =

X ||X||

✐s ❛ ✉♥✐t ✈❡❝t♦r✳

■t✬s s✐♠♣❧② ❛ r❡✲s❝❛❧❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧✳ ❚❤✐s ✐s ✇❤❛t ✐t ❧♦♦❦s ❧✐❦❡ ✐♥ R40 ✿

❊①❡r❝✐s❡ ✶✳✽✳✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❡✛❡❝t ♦❢ ♥♦r♠❛❧✐③❛t✐♦♥ ✐s t❤❛t t❤❡ ✈❡❝t♦rs t❤❛t ❛r❡ t♦♦ ❧♦♥❣ ❛r❡ s❤r✉♥❦ ❛♥❞ t❤❡ ♦♥❡s t❤❛t ❛r❡ t♦♦ s❤♦rt ❛r❡ str❡t❝❤❡❞ ✕ r❛❞✐❛❧❧② ✕ t♦✇❛r❞ t❤❡ ✉♥✐t ❝✐r❝❧❡✿

❯♥✐t ✈❡❝t♦rs ❝❛♣t✉r❡ ♥♦t❤✐♥❣ ❜✉t t❤❡ ❞✐r❡❝t✐♦♥✿ ❚❤❡♦r❡♠ ✶✳✽✳✻✿ ▼✉❧t✐♣❧❡s ♦❢ ❱❡❝t♦rs ❙✉♣♣♦s❡ t✇♦ ✈❡❝t♦rs ❤❛✈❡ ❡q✉❛❧ ♦r ♦♣♣♦s✐t❡ ✉♥✐t ✈❡❝t♦rs✿

V W =± . ||V || ||W || ❚❤❡♥ t❤❡② ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿

V = kW . ❊①❡r❝✐s❡ ✶✳✽✳✼

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✻

❊①❛♠♣❧❡ ✶✳✽✳✽✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②

❘❡❝❛❧❧ t❤❡ ❧❛✇ ♦❢ ❣r❛✈✐t②✳ ■ts ❢♦r❝❡ ♣✉❧❧s t✇♦ ♦❜❥❡❝ts t❤❡ ❤❛r❞❡r t❤❡ ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r t❤❡② ❛r❡✿

❚❤❡ ❣r❛✈✐t② ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ x, y ✱ t❤❛t ❣✐✈❡ t❤❡ ❧♦❝❛t✐♦♥✱ ♦r ❜❡tt❡r✱ ✐t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♣♦✐♥ts✱ P ✱ ✐♥ R2 ✇✐t❤ r❡❛❧ ✈❛❧✉❡s✿ f : R2 → R .

❆❧❣❡❜r❛✐❝❛❧❧②✱ t❤❡ ❧❛✇ s❛②s t❤❛t t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ♦❢ ♠❛ss❡s M ❛♥❞ m ❧♦❝❛t❡❞ ❛t ♣♦✐♥ts O ❛♥❞ P ✐s ❣✐✈❡♥ ❜②✿ f (P ) = G

mM d(O, P )2

◆❡①t✱ t❤❡ ❧❛✇✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❢♦r♠✉❧❛✱ ✐♥❝❧✉❞❡s t❤❡ st❛t❡♠❡♥t t❤❛t t❤❡ ❢♦r❝❡ ✐s ❞✐r❡❝t❡❞ ❢r♦♠ P t♦ O✳ ❚❤✐s ✐s ✐♠♣❧✐❝✐t❧② t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ✈❡❝t♦rs ✿ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r❝❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✈❡❝t♦r✳ ▲❡t✬s s♦rt t❤✐s ♦✉t✳ ▲❡t✬s t❛❦❡ ❝❛r❡ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡s ✜rst✳ ❲❡ ❤❛✈❡ t✇♦ ✈❡❝t♦rs✿ • ●r❛✈✐t② ✐s ❛ ❢♦r❝❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛ ✈❡❝t♦r✳ • ❚❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t ✐s ✐ts ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ✜rst ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛ ✈❡❝t♦r✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❤❛✈❡ ❜♦t❤ ✈❡❝t♦r ✐♥♣✉ts ❛♥❞ ✈❡❝t♦r ♦✉t♣✉ts✿ F : R2 → R2 .

❲❡ ♣❧❛❝❡ t❤❡ ♦r✐❣✐♥ O ❛t t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ✜rst ♦❜❥❡❝t ✭♠❛②❜❡ t❤❡ ❙✉♥✮✳ ❚❤❡♥ OP = X ✱ ❛♥❞ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ❛s ❢♦❧❧♦✇s✿ ||F (X)|| = G

mM ||X||2

◆❡①t✱ ✇❤❛t ❛❜♦✉t t❤❡ ❞✐r❡❝t✐♦♥s❄ ▲❡t✬s ❞❡r✐✈❡ t❤❡ ✈❡❝t♦r ❢♦r♠ ♦❢ t❤❡ ❧❛✇✳ ❚❤❡ ❧❛✇ st❛t❡s✿ ◮ ❚❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❛✛❡❝t✐♥❣ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts ✐s ❞✐r❡❝t❡❞ t♦✇❛r❞s t❤❡ ♦t❤❡r ♦❜❥❡❝t✳ ❲❡ s❡❡ t❤✐s ❜❡❧♦✇✿

■♥ ♦t❤❡r ✇♦r❞s✱ F ♣♦✐♥ts ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥ t♦ X ✱ ✐✳❡✳✱ ✐t✬s ❞✐r❡❝t✐♦♥ ✐s t❤❛t ♦❢ −X ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✉♥✐t ✈❡❝t♦rs ♦❢ F (X) ❛♥❞ −X ❛r❡ ❡q✉❛❧✿ F X =− . ||F || ||X||

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✼

❚❤❡r❡❢♦r❡✱ t❤❡ ✈❡❝t♦rs t❤❡♠s❡❧✈❡s ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿

F (X) = c(−X) ❚❤❛t✬s ❛❧❧ ✇❡ ♥❡❡❞ ❡①❝❡♣t ❢♦r t❤❡ ❝♦❡✣❝✐❡♥t✳ ❲❡ ♥♦✇ ✉s❡

G

❍♦♠♦❣❡♥❡✐t②

t♦ ✜♥❞ ✐t✿

mM = ||F || = |c| · ||(−X)|| = |c| · ||X|| . ||X||2

❚❤❡ ✜♥❛❧ ❢♦r♠ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

F (X) = −G ◆♦✇ t❤❛t ❜♦t❤ t❤❡ ✐♥♣✉t ❛♥❞ t❤❡ ♦✉t♣✉t ❛r❡

mM X ||X||3

2✲❞✐♠❡♥s✐♦♥❛❧

✈❡❝t♦rs✱ ❤♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤✐s ❦✐♥❞ ♦❢ 2 3 ❢✉♥❝t✐♦♥❄ ❊✈❡♥ t❤♦✉❣❤ t❤✐s ✐s ❥✉st ❛ ✭♥♦♥✲❧✐♥❡❛r✮ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ R ✭♦r R ✮✱ t❤❡r❡ ✐s ❛ ❜❡tt❡r ✇❛②✳ ❋✐rst✱ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ✐♥♣✉t ❛s ❛ ❢♦r♠❡r✳ ❇❡❧♦✇✱ ✇❡ ♣❧♦t ✈❡❝t♦r

■t ✐s ❝❛❧❧❡❞ ❛

♣♦✐♥t

F (X)

❛♥❞ t❤❡ ♦✉t♣✉t ❛s ❛

st❛rt✐♥❣ ❛t ❧♦❝❛t✐♦♥

✈❡❝t♦r

X

❛♥❞ t❤❡♥ ✇❡ ❛tt❛❝❤ t❤❡ ❧❛tt❡r t♦ t❤❡

♦♥ t❤❡ ♣❧❛♥❡✿

✈❡❝t♦r ✜❡❧❞✳

❊①❡r❝✐s❡ ✶✳✽✳✾

❙✉❣❣❡st ♦t❤❡r ❡①❛♠♣❧❡s ♦❢ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t ❢♦r❝❡s✳

❚❤❡s❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ❤❛✈❡ ❜❡❡♥ ❥✉st✐✜❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❢❛♠✐❧✐❛r ❣❡♦♠❡tr② ♦❢ t❤❡ ✏♣❤②s✐❝❛❧ s♣❛❝❡✑ ❍♦✇❡✈❡r✱ t❤❡② ❛❧s♦ s❡r✈❡ ❛s ❛

st❛rt✐♥❣ ♣♦✐♥t

❢♦r ❛ ❢✉rt❤❡r ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳

❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝s✱

❲❤❡♥ ❛ ❈❛rt❡s✐❛♥ s②st❡♠ ✐s ♣r♦✈✐❞❡❞✱ ✇❡ ❤❛✈❡ t❤❡ Q ✐♥ R3 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s (x, y, z) ❛♥❞ (x′ , y ′ , z ′ ) r❡s♣❡❝t✐✈❡❧② ✐s

✐✳❡✳✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts

❛♥❞

d(P, Q) =

p

R3 ✳

(x − x′ )2 + (y − y ′ )2 + (z − z ′ )2 .

❚❤❡ t❤r❡❡ t❡r♠s ❛r❡ r❡❝♦❣♥✐③❡❞ ❛s t❤❡ t❤r❡❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r

P Q✿

P

✶✳✽✳

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r

✼✽

P = (a, b, c)✱ ✇❡ ❝♦♥❝❧✉❞❡✿ √ || < a, b, c > || = a2 + b2 + c2 .

■❢ ✇❡ ❛♣♣❧② t❤✐s ❝♦♠♣✉t❛t✐♦♥ t♦ ❛ ✈❡❝t♦r

OP

✇✐t❤

▼❡❛♥✇❤✐❧❡✱ t❤❡ t❛①✐❝❛❜ ♥♦r♠ ♦❢ t❤✐s ✈❡❝t♦r ✐s

|| < a, b, c > || = |a| + |b| + |c| . ■♥ ❣❡♥❡r❛❧ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✽✳✶✵✿ ▼❛❣♥✐t✉❞❡ ♦❢ ❱❡❝t♦r

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r A =< a1 , ..., an >

✐♥ Rn ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ A ✐s ❡q✉❛❧ t♦ q ||A|| = a21 + ... + a2n

✷✳ ❚❤❡ t❛①✐❝❛❜ ♥♦r♠ ♦❢ ❛ ✈❡❝t♦r A ✐s ❡q✉❛❧ t♦ ||A|| = |a1 | + ... + |an | ■♥ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✱ ✇❡ ❤❛✈❡✱ r❡s♣❡❝t✐✈❡❧②✿

v u n uX a2 , ||A|| = t k

k=1

❛♥❞

||A|| =

n X k=1

|ak | .

❆s ❛ s✉♠♠❛r②✱ t❤❡s❡ ❛r❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ ✈❡❝t♦rs✳ ❚❤❡♦r❡♠ ✶✳✽✳✶✶✿ ❆①✐♦♠s ♦❢ ◆♦r♠❡❞ ❙♣❛❝❡

❋♦r ❛♥② ✈❡❝t♦rs A, B ✐♥ Rn ❛♥❞ ❛♥② r❡❛❧ k ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❛r❡ s❛t✐s✜❡❞ ❜② ❜♦t❤ t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ❛♥❞ t❤❡ t❛①✐❝❛❜ ♥♦r♠✿ ✶✳ P♦s✐t✐✈✐t②✿ ||A|| ≥ 0❀ ❛♥❞ ||A|| = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ A = 0 . ✷✳ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✿ ||A|| + ||B|| ≥ ||A + B|| . ✸✳ ❍♦♠♦❣❡♥❡✐t②✿ ||k · A|| = |k| · ||A|| . ❊①❡r❝✐s❡ ✶✳✽✳✶✷

❉❡♠♦♥str❛t❡ t❤❛t t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ♥♦r♠s s❛t✐s❢② t❤♦s❡ t❤r❡❡ ♣r♦♣❡rt✐❡s✳ ❊①❛♠♣❧❡ ✶✳✽✳✶✸✿ ✐♥✈❡st♠❡♥t ♣♦rt❢♦❧✐♦s

❆ ♣♦rt❢♦❧✐♦ ♦❢ st♦❝❦s ❝❛♥ ❜❡ s✉❜❥❡❝t t♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s✳ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② t❤❡s❡ ❛✈❛✐❧❛❜❧❡✱ ❛❧❧ ♣♦rt❢♦❧✐♦s ❛r❡ ✈❡❝t♦rs ✭♦r ♣♦✐♥ts✮ ✐♥

R

10

✿

10

st♦❝❦s

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✼✾

❚❤❡ t❛①✐❝❛❜ ♥♦r♠ ✭②❡❧❧♦✇✮ ✐s ❥✉st t❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ♣♦rt❢♦❧✐♦✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ✐s ✐♥ ♣✐♥❦✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ✏❞✐r❡❝t✐♦♥✑ ♦❢ t❤✐s ♣♦rt❢♦❧✐♦✳ ❲❡ ♥♦r♠❛❧✐③❡ t❤✐s ✈❡❝t♦r ❜② ❞✐✈✐❞✐♥❣ ❜② t❛①✐❝❛❜ ♥♦r♠ ❛♥❞ ❜②

38.0

88.1

❢♦r t❤❡

❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠✿

❚❤❡ ❢♦r♠❡r s✐♠♣❧② ❝♦♥s✐sts ♦❢ t❤❡ ♣❡r❝❡♥t❛❣❡s ♦❢ t❤❡ st♦❝❦s ✇✐t❤✐♥ t❤❡ ♣♦rt❢♦❧✐♦✳

❲❛r♥✐♥❣✦ ■t ✇♦✉❧❞♥✬t ♠❛❦❡ s❡♥s❡ t♦ ❤❛✈❡ t❤❡ ♥♦r♠ ♦❢ ❛ ♣♦rt✲ ❢♦❧✐♦ ♦❢

♥♦♥✲❤♦♠♦❣❡♥❡♦✉s

✐t❡♠s✱ s✉❝❤ ❛s ❝♦♠♠♦❞✐✲

t✐❡s✿

< 10000

, 20000

t♦♥s ♦❢ ✇❤❡❛t

♦r ❝✉rr❡♥❝✐❡s✿

< $100000, U1000000, ... > . ❊①❡r❝✐s❡ ✶✳✽✳✶✹ ❲❤❡♥ ✐s t❤❡ ♥♦r♠ ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❝♦♠♣♦♥❡♥ts❄

✶✳✾✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❋✉♥❝t✐♦♥s ♠❛② ♣r♦❝❡ss ❛♥ ✐♥♣✉t ♦❢ ❛♥② ♥❛t✉r❡ ❛♥❞ ♣r♦❞✉❝❡ ❛♥ ♦✉t♣✉t ♦❢ ❛♥② ♥❛t✉r❡✳

, ... > ,

❜❛rr❡❧s ♦❢ ♦✐❧

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✵

❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ✐♥♣✉t x

❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿

❢✉♥❝t✐♦♥ 7→

f

♦✉t♣✉t 7→

y

❈❖◆❱❊◆❚■❖◆

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✉♣♣❡r ❝❛s❡ ❧❡tt❡rs ❢♦r t❤❡ ❢✉♥❝t✐♦♥s t❤❡ ♦✉t♣✉ts ♦❢ ✇❤✐❝❤ ❛r❡ ✭♦r ♠❛② ❜❡✮ ♠✉❧t✐❞✐♠❡♥✲ s✐♦♥❛❧✱ s✉❝❤ ❛s ♣♦✐♥ts ❛♥❞ ✈❡❝t♦rs✿ F, G, P, Q, ...

❲❡ ✇✐❧❧ ✉s❡ t❤❡

❧♦✇❡r ❝❛s❡ ❧❡tt❡rs ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ♥✉♠❡r✐❝❛❧ ♦✉t♣✉ts✿ f, g, h, ...

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s t❛❦❡ ♣♦✐♥ts ♦r ✈❡❝t♦rs ❛s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡ ♣♦✐♥ts ♦r ✈❡❝t♦rs ♦❢ ✈❛r✐♦✉s ❞✐♠❡♥s✐♦♥s ❛s t❤❡ ♦✉t♣✉t✳ ❲❡ ❝❛♥ s❛② t❤❛t t❤❡ ✐♥♣✉t X ✐s ✐♥ Rn ❛♥❞ t❤❡ ♦✉t♣✉t U = F (X) ♦❢ X ✐s ✐♥ Rm ✿ F : P

✐♥ R

n

7→ U

✐♥ Rm

❚❤❡♥✱ t❤❡ ❞♦♠❛✐♥ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ✐♥ Rn ❛♥❞ t❤❡ r❛♥❣❡ ✭✐♠❛❣❡✮ ✐s ✐♥ Rm ✳ ❚❤❡ ❞♦♠❛✐♥ ❝❛♥ ❜❡ ❧❡ss t❤❛♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❢♦✉r ✭❧✐♥❡❛r✮ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r n = 1, 2 ❛♥❞ m = 1, 2✿

❲❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ✐♥ t❤✐s s❡❝t✐♦♥ ♦♥ t❤❡ ✜rst ✭✐♥✜♥✐t❡✮ ❝♦❧✉♠♥✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ ❛s ❛

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t♦

• ❛♥② ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r❡❛❧ ✈❛r✐❛❜❧❡✱ ✐✳❡✳✱ t❤❡ ❞♦♠❛✐♥ ❧✐❡s ✐♥s✐❞❡ R✱ ❛♥❞

• ✇✐t❤ ✐ts ✈❛❧✉❡s ✐♥ Rm ❢♦r s♦♠❡ m = 1, 2, 3, ...✳

✶✳✾✳

✽✶

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❘❡❝❛❧❧ ❢r♦♠ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r ❤♦✇ str❛✐❣❤t ❧✐♥❡s ❛♣♣❡❛r ❛s ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ t✇♦ ✈❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡✿

❆♥❞ t❤✐s ✐s t❤❡ ❧✐♥❡ ✐♥ R40 t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ t✇♦ ♣♦✐♥ts s❤♦✇♥ ✐♥ r❡❞ ❛♥❞ ❣r❡❡♥✿

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✈✐❛ ♠♦t✐♦♥✳ ❚❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s t❤❡ t✐♠❡✱ ❛♥❞ t❤❡ ✈❛❧✉❡ ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ❆ ♣♦✐♥t ✐s t❤❡ s✐♠♣❧❡st ❝✉r✈❡✳ ❙✉❝❤ ❛ ❝✉r✈❡ ✇✐t❤ ♥♦ ♠♦t✐♦♥ ✐s ♣r♦✈✐❞❡❞ ❜② ❛ ❝♦♥st❛♥t ❆ str❛✐❣❤t

❧✐♥❡

❢✉♥❝t✐♦♥✳

✐s t❤❡ s❡❝♦♥❞ s✐♠♣❧❡st ❝✉r✈❡✳

❲❡ st❛rt ✇✐t❤ ❧✐♥❡s ✐♥ R2 ✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ ❤♦✇ t♦ r❡♣r❡s❡♥t str❛✐❣❤t ❧✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡✿ ✶✳ ❚❤❡ ✜rst ♠❡t❤♦❞ ✐s t❤❡ s❧♦♣❡✲✐♥t❡r❝❡♣t

❢♦r♠ ✿

y = mx + b .

❚❤✐s ♠❡t❤♦❞ ❡①❝❧✉❞❡s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡s✦ ❚❤✐s ✐s t♦♦ ❧✐♠✐t✐♥❣ ❜❡❝❛✉s❡ ✐♥ ♦✉r st✉❞② ♦❢ ❝✉r✈❡s✱ t❤❡r❡ ❛r❡ ♥♦ ♣r❡❢❡rr❡❞ ❞✐r❡❝t✐♦♥s✳ ✷✳ ❚❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ ✐s ✐♠♣❧✐❝✐t ✿ px + qy = r .

❚❤❡ ❝❛s❡ ♦❢ p 6= 0, q = 0 ❣✐✈❡s ✉s ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✳

✸✳ ❚❤❡ t❤✐r❞ ♠❡t❤♦❞ ✐s ♣❛r❛♠❡tr✐❝✳ ■t ❤❛s ❛ ❞②♥❛♠✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ✭❜❡❧♦✇✮✳ ❊①❛♠♣❧❡ ✶✳✾✳✶✿ str❛✐❣❤t ♠♦t✐♦♥

❙✉♣♣♦s❡ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ tr❛❝❡ t❤❡ ❧✐♥❡ t❤❛t st❛rts ❛t t❤❡ ♣♦✐♥t (1, 3) ❛♥❞ ♣r♦❝❡❡❞s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < 2, 3 >✳

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✷

❲❡ ✉s❡ ♠♦t✐♦♥ ❛s ❛ st❛rt✐♥❣ ♣♦✐♥t ❛♥❞ ❛s ✇❡❧❧ ❛s ❛ ♠❡t❛♣❤♦r ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ st❛rt ♠♦✈✐♥❣✿ • ❢r♦♠ t❤❡ ♣♦✐♥t P0 = (1, 3)✱ • ✉♥❞❡r ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ♦❢ V =< 2, 3 >✳ ❚♦ ❣❡t t❤❡ r❡st ♦❢ t❤❡ ♣❛t❤✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✱ t✐♠❡ t✳ ❲❤❡♥ t = 1, 2, 3, ... ✐s ✐♥❝r❡❛s✐♥❣ ✐♥❝r❡♠❡♥t❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡✿

❋♦r t❤❡ ♥❡❣❛t✐✈❡ t✬s✱ ✇❡ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳ ▲❡t✬s ✐♥✐t✐❛❧❧② tr❡❛t x ❛♥❞ y s❡♣❛r❛t❡❧②✿ • ❍♦r✐③♦♥t❛❧✿ ❲❡ ♠♦✈❡ ❢r♦♠ 1 ❛t 2 ❢❡❡t ♣❡r s❡❝♦♥❞✳ • ❱❡rt✐❝❛❧✿ ❲❡ ♠♦✈❡ ❢r♦♠ 3 ❛t 3 ❢❡❡t ♣❡r s❡❝♦♥❞✳

▲❡t✬s ✜♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ t✇♦✳ ❚❤❡s❡ ❛r❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✿ x(0) = 1, x(1) = 3 ❛♥❞ y(0) = 3, y(1) = 6 .

❚❤❡ ❢✉♥❝t✐♦♥s x ❛♥❞ y ♠✉st ❜❡ ❧✐♥❡❛r✿ x(t) = 1 + 2t ❛♥❞ y(t) = 3 + 3t .

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✸

❈♦♠❜✐♥❡❞✱ t❤✐s ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ◆♦✇✱ ❧❡t✬s tr❛♥s❧❛t❡ t❤❡s❡ ❢♦r♠✉❧❛s ✐♥t♦ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ✈❡❝t♦rs✳ ■♥ t❡r♠s ♦❢ ✈❡❝t♦rs✱ ✐❢ ✇❡ ❛r❡ ❛t ♣♦✐♥t P ♥♦✇✱ ✇❡ ✇✐❧❧ ❜❡ ❛t ♣♦✐♥t P + V ❛❢t❡r ♦♥❡ s❡❝♦♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❛r❡ ❛t P1 = P0 + V = (1, 3)+ < 2, 3 >= (3, 6) ❛t t✐♠❡ t = 1✳ ❲❡ ❞❡✜♥❡ t❤✐s ❢✉♥❝t✐♦♥✿ P : R → R2 .

■t ✐s ♠❛❞❡ ♦❢ t✇♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿ P (t) = (x(t), y(t)) .

❲❡ ❛❧r❡❛❞② ❤❛✈❡ t✇♦ ♣♦✐♥ts ♦♥ ♦✉r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P ✿ P (0) = P0 = (1, 3) ❛♥❞ P (1) = P1 = (3, 6) .

❲❤❛t ✐s ✐ts ❢♦r♠✉❧❛❄ ❲❡ ♥❡❡❞ t♦ ❝♦♥✈❡rt t❤✐s t♦ ✈❡❝t♦rs✿ x(t) = 1 + 2t, y(t) = 3 + 3t .

▲❡t✬s ❛ss❡♠❜❧❡ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ ❢✉♥❝t✐♦♥s ✐♥t♦ ♦♥❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ P (t) = (x(t), y(t)) = (1 + 2t, 3 + 3t) .

❚❤✐s ✐s st✐❧❧ ♥♦t ❣♦♦❞ ❡♥♦✉❣❤❀ ✇❡✬❞ r❛t❤❡r s❡❡ t❤❡ P0 ❛♥❞ V ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❲❡ ❝♦♥t✐♥✉❡ ❜② ✉s✐♥❣ ✈❡❝t♦r ❛❧❣❡❜r❛✿ ❲❡ ✉♥❞♦ ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ P (t) = (1 + 2t, 3 + 3t) = (1, 3)+ < 2t, 3t > ❚❤❡♥ ✇❡ ✉♥❞♦ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ = (1, 3) + t < 2, 3 > ❆♥❞ ✜♥❛❧❧② ✇❡ ❤❛✈❡ t❤❡ ❛♥s✇❡r✳ = P0 + tV . ❙♦✱ t❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts✱ ♦❢ ❝♦✉rs❡✱ ❝♦♠❡ ❢r♦♠ t❤❡ s♣❡❝✐✜❝ ♥✉♠❜❡rs t❤❛t ❣✐✈❡ ✉s P0 ❛♥❞ V ✳ ❊①❡r❝✐s❡ ✶✳✾✳✷

❚❤❡ ❧✐♥❡ ✐s ♥♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ P ❜✉t ✐ts ❴❴❴❴❴ ✳ ❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ❛ ✈❡❝t♦r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ✉♥✐❢♦r♠ ♠♦t✐♦♥✳ ❚❤❡ ❧♦❝❛t✐♦♥ P ✐s ❣✐✈❡♥ ❜②✿ P (t) = P0 + tV ,

✇❤❡r❡ P0 ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ V ✐s t❤❡ ✭❝♦♥st❛♥t✮ ✈❡❧♦❝✐t②✳ ❚❤❡♥ tV ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❲❛r♥✐♥❣✦ ❖♥❡ ❝❛♥✱ ♦❢ ❝♦✉rs❡✱ ♠♦✈❡ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛

✈❛r✐❛❜❧❡

✈❡❧♦❝✐t②✳

❙♦✱ ✇❡ ❤❛✈❡✿ ♣♦s✐t✐♦♥ ❛t t✐♠❡ t = ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ + t · ✈❡❧♦❝✐t② ❲❡ ✉s❡❞ t❤✐s ❛♣♣r♦❛❝❤ ❢♦r ❞✐♠❡♥s✐♦♥ 1❀ ♦♥❧② t❤❡ ❝♦♥t❡①t ❤❛s ❝❤❛♥❣❡❞✳

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✹

❚❤❡ ♣❛tt❡r♥ ❜❡❝♦♠❡s ❝❧❡❛r✳ ❚❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t (a, b) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < u, v > ✐s r❡♣r❡s❡♥t❡❞ ♣❛r❛♠❡tr✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✿ P (t) = (a, b) + t < u, v > .

❙✐♠✐❧❛r❧② ❢♦r ❞✐♠❡♥s✐♦♥ 3✱ t❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t (a, b, c) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < u, v, w > ✐s r❡♣r❡s❡♥t❡❞ ❛s ❢♦❧❧♦✇s✿ P (t) = (a, b, c) + t < u, v, w > .

❆♥❞ s♦ ♦♥✳ ❆t t❤❡ ♥❡①t ❧❡✈❡❧✱ ✇❡✬❞ r❛t❤❡r ❤❛✈❡ ♥♦ r❡❢❡r❡♥❝❡s t♦ ♥❡✐t❤❡r t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ♥♦r t❤❡ s♣❡❝✐✜❝ ❝♦♦r❞✐♥❛t❡s✿

❉❡✜♥✐t✐♦♥ ✶✳✾✳✸✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥ t❤r♦✉❣❤ P0 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ V ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙✉♣♣♦s❡ P0 ✐s ❛ ♣♦✐♥t ✐♥ Rm ❛♥❞ V ✐s ❛ ✈❡❝t♦r✳ ❚❤❡♥ t❤❡

P (t) = P0 + tV

❚❤❡♥✱ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ P0 ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳

✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢

V ✐s t❤❡ ♣❛t❤ ✭✐♠❛❣❡✮ ♦❢ t❤✐s

❙t❛t❡❞ ❢♦r ❞✐♠❡♥s✐♦♥ m = 1✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❢❛♠✐❧✐❛r ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✿ P (t) = P0 + tV .

■♥❞❡❡❞✱ P0 ✐s t❤❡ y ✲✐♥t❡r❝❡♣t ❛♥❞ V ✐s t❤❡ s❧♦♣❡✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ s✐♥❣❧❡ ♥✉♠❜❡r ❜❡❝❛✉s❡ t❤❡ ❝❤❛♥❣❡ ✐s ❡♥t✐r❡❧② ✇✐t❤✐♥ t❤❡ y ✲❛①✐s✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ ✐s t❤❡ ❝♦♥t❡①t ❛s t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ✐♥ R2 ❢♦r ❝❤❛♥❣❡✿

❚❤❛t ✐s ✇❤② t❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ ✈❡❝t♦r✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ str❛✐❣❤t ❧✐♥❡s st❡♠s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t✱ ✉♥❞❡r ❝♦♠♠♦♥ r❡str✐❝t✐♦♥s✱ ❡✈❡r② ❝✉r✈❡ ✐s ❧✐❦❡❧② t♦ ❧♦♦❦ ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡ ✐♥ t❤❡ s❤♦rt t❡r♠✿

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✺

❊①❛♠♣❧❡ ✶✳✾✳✹✿ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s

❚❤❡s❡ ❛r❡ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s t❤❛t ❣✐✈❡ t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t ✭k = 0, 1, ...✮✿ x : pk+1 = pk +v∆t y : qk+1 = qk +u∆t

❚❤❡s❡ ♣♦✐♥ts ❛r❡ ♣❧♦tt❡❞ ♦♥ t❤❡ r✐❣❤t ❢♦r p0 = 3✱ q0 = 3✱ v = 1✱ u = 3✱ ∆t = 1/5✿

❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ♥♦✇ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♣♦✐♥ts ❛♥❞ ✈❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡✿ Pk = (pk , qk ), V =< v, u > .

❚❤❡ ❡q✉❛t✐♦♥s t❛❦❡ ❛ ✈❡❝t♦r ❢♦r♠ t♦♦✿ Pk+1 = Pk + V ∆t . ❊①❡r❝✐s❡ ✶✳✾✳✺

❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s♥✬t ❝♦♥st❛♥t✳ ❊①❛♠♣❧❡ ✶✳✾✳✻✿ ♣r✐❝❡ ❞②♥❛♠✐❝s

❚❤❡ ❞❡✜♥✐t✐♦♥ ❛♣♣❧✐❡s t♦ s♣❛❝❡s ♦❢ ❞❛t❛✳ ❙✉♣♣♦s❡ Rm ✐s t❤❡ s♣❛❝❡ ♦❢ ♣r✐❝❡s ✭♦❢ st♦❝❦s ♦r ❝♦♠♠♦❞✐t✐❡s✮❀ ✇❡ ♠✐❣❤t ❤❛✈❡ m = 10, 000✳

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✻

❚❤❡ ♣r✐❝❡s r❡❝♦r❞❡❞ ❝♦♥t✐♥✉♦✉s❧② ♦r ✐♥❝r❡♠❡♥t❛❧❧② ✇✐❧❧ ♣r♦❞✉❝❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ t❤✐s ❝✉r✈❡ ♠✐❣❤t ❜❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❚❤✐s ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ♣r✐❝❡s ❛r❡ ❣r♦✇✐♥❣ ✭♦r ❞❡❝❧✐♥✐♥❣✮

♣r♦♣♦rt✐♦♥❛❧❧②

❜✉t✱

♣♦ss✐❜❧②✱ ❛t ❞✐✛❡r❡♥t r❛t❡s✳ ❘❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❛r❡ ❡s♣❡❝✐❛❧❧② ❡❛s② t♦ ✐♠♣❧❡♠❡♥t ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ■♥ ❡❛❝❤ ❝♦❧✉♠♥✱ ✇❡ ✉s❡ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❢♦r t❤❡

k t❤

♣r✐❝❡✿

xk (t + ∆t) = xk (t) + vk ∆t , ✇❤❡r❡

vk

✐s t❤❡

k t❤

r❛t❡ ♦❢ ❝❤❛♥❣❡ s❤♦✇♥ ❛t t❤❡ t♦♣✿

❚❤❡ t❛❜❧❡ ❣✐✈❡s ✉s ♦✉r ❝✉r✈❡✳ ■t ❧✐❡s ✐♥ t❤❡

10, 000✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✳ ❈❛♥ ✇❡ ✈✐s✉❛❧✐③❡ s✉❝❤ ❛ ❝✉r✈❡

✐♥ ❛♥② ✇❛②❄ ❱❡r② ✐♠♣❡r❢❡❝t❧②✳ ❲❡ ♣✐❝❦ t✇♦ ❝♦❧✉♠♥s ❛t ❛ t✐♠❡ ❛♥❞ ♣❧♦t t❤❛t ❝✉r✈❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❙✐♥❝❡ t❤❡s❡ ❝♦❧✉♠♥s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛①❡s✱ ✇❡ ❛r❡ ♣❧♦tt✐♥❣ ❛ ✏s❤❛❞♦✇✑ ✭❛ ♣r♦❥❡❝t✐♦♥✮ ♦❢ ♦✉r ❝✉r✈❡ ❝❛st ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✳ ❚❤❡② ❛r❡ ❛❧❧ str❛✐❣❤t ❧✐♥❡s✳ ❆ s✐♠✐❧❛r ✭s❤♦rt✲t❡r♠✮ ❞②♥❛♠✐❝s ♠❛② ❜❡ ❡①❤✐❜✐t❡❞ ❜② ♦t❤❡r ❞❛t❛ s✉❝❤ ❛s✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ✈✐t❛❧s ♦❢ ❛ ♣❡rs♦♥✿ ✶✳ ❜♦❞② t❡♠♣❡r❛t✉r❡ ✷✳ ❜❧♦♦❞ ♣r❡ss✉r❡ ✸✳ ♣✉❧s❡ ✭❤❡❛rt r❛t❡✮ ✹✳ ❜r❡❛t❤✐♥❣ r❛t❡

❊①❡r❝✐s❡ ✶✳✾✳✼

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts

P

❛♥❞

Q✳

■♥ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡✱ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛♥ ♦❜❥❡❝t ✇❤❡♥ t❤❡r❡ ❛r❡ ♥♦ ❢♦r❝❡s ❛t ♣❧❛②✳

❊✈❡♥ ❛

❝♦♥st❛♥t ❢♦r❝❡ ❧❡❛❞s t♦ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤✐❝❤ ♠❛② ❝❤❛♥❣❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ♥♦ ❧♦♥❣❡r ❛ ❝♦♥❝❡r♥✦ ❊①❛♠♣❧❡ ✶✳✾✳✽✿ ❢r♦♠ r❡❧❛t✐♦♥ t♦ ♣❛r❛♠❡tr✐❝

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡ ❣✐✈❡♥ ❜② ✐ts r❡❧❛t✐♦♥✿

y − 3 = 2(x − 1) . ❲❤❛t ✐s ✐ts ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥❄ ▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥✳ ❋r♦♠ ✐ts t❤❡ s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠ ✇❡ ❞❡r✐✈❡✿ ✶✳ ✷✳

2 ✐s t❤❡ (1, 3) ✐s

s❧♦♣❡✳ t❤❡ ♣♦✐♥t✳

❙♦✱ ❧❡t✬s ❥✉st ♠♦✈❡ ✶✳ ❢r♦♠ t❤❡ ♣♦✐♥t

(1, 3)✱ < 1, 2 >

✷✳ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r

❡✈❡r② s❡❝♦♥❞✳

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✼

❲❡ ❤❛✈❡✿

(x, y) = (1, 3) + t < 1, 2 > . ❊①❡r❝✐s❡ ✶✳✾✳✾

❲❤❛t ✐❢ ✇❡ ♠♦✈❡ ❢❛st❡r❄ ❚❤❡ ❡①❛♠♣❧❡ s✉❣❣❡sts ❛ s❤♦rt❝✉t ❢♦r R2 ✿ s❧♦♣❡ =

r✐s❡ =⇒ ❞✐r❡❝t✐♦♥ = < r✉♥, r✐s❡ > r✉♥

❊①❛♠♣❧❡ ✶✳✾✳✶✵✿ t❤r♦✇♥ ❜❛❧❧

▲❡t✬s r❡✈✐❡✇ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ t❤r♦✇♥ ❜❛❧❧✳ ❆ ❝♦♥st❛♥t ❢♦r❝❡ ❝❛✉s❡s t❤❡ ✈❡❧♦❝✐t② t♦ ❝❤❛♥❣❡ ❧✐♥❡❛r❧②✱ ❥✉st ❛s t❤❡ ❧♦❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❍♦✇ ❞♦❡s t❤❡ ❧♦❝❛t✐♦♥ ❝❤❛♥❣❡ t❤✐s t✐♠❡❄ ■♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❤❡✐❣❤t ♦♥ t❤❡ t✐♠❡ ✐s q✉❛❞r❛t✐❝✳ ❚❤❡ ♣❛t❤ ♦❢ t❤❡ ❜❛❧❧ ✇✐❧❧ ❛♣♣❡❛r t♦ ❛♥ ♦❜s❡r✈❡r ✕ ❢r♦♠ t❤❡ r✐❣❤t ❛♥❣❧❡ ✕ ❛s ❛ ❝✉r✈❡✿

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s s✉❜❥❡❝t t♦ t❤❡s❡ ❛❝❝❡❧❡r❛t✐♦♥s✱ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✿

x : ak+1 = 0;

y : ak+1 = −g .

◆♦✇ r❡❝❛❧❧ t❤❡ s❡t✉♣ ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧②✿ ❢r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✱ x : p0 = 0 ❛♥❞ y : p0 = 200✳ • ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✱ x : v0 = 200 ❛♥❞ y : v0 = 0✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♣❛✐rs ♦❢ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥s ✕ ❢♦r t❤❡ ❧♦❝❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ✐♥ t❡r♠s ♦❢ ❛❝❝❡❧❡r❛t✐♦♥ ✕ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✿

x : vk+1 = v0 , pk+1 = pk +vk ∆t y : uk+1 = vk −g∆t, qk+1 = qk +uk ∆t

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✽

❚❤❡s❡ ❛r❡ t❤❡ ❢♦r♠✉❧❛s ✐♥ t❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥✿

Vk+1 = Vk +A ·∆t Pk+1 = Pk +Vk+1 ·∆t ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ♥♦ ❧♦♥❣❡r ❛ ❝♦♥❝❡r♥✦ ❊①❛♠♣❧❡ ✶✳✾✳✶✶✿ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s

■♥ ❞✐♠❡♥s✐♦♥ 2✱ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ❛❧✐❣♥ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t❤r♦✇ ❛♥❞ ✐♥ ❞✐♠❡♥s✐♦♥ 3 ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ❛❧✐❣♥ t❤❡ z ✲❛①✐s ✇✐t❤ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥✳ ◆♦♥❡t❤❡❧❡ss✱ ❧❡t✬s st❛rt ✇✐t❤ ❢♦r♠❡r ❝❛s❡✳ ❆ 6✲❢♦♦t ♠❛♥ t❤r♦✇s ✕ str❛✐❣❤t ❢♦r✇❛r❞ ✕ ❛ ❜❛❧❧ ✇✐t❤ t❤❡ s♣❡❡❞ ♦❢ 100 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ■❢ t❤❡ t❤r♦✇ ✐s ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s ✐s ✈❡rt✐❝❛❧✱ ✇❡ ❤❛✈❡✿

A =< 0, −32 >, V0 =< 0, 100 >, P0 = (6, 0) . ❚❤✐s ❞❛t❛ ❣♦❡s ✐♥t♦ t❤❡ ✜rst r♦✇ ♦❢ ♦✉r t❛❜❧❡ ❢♦r t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ x′′ , y ′′ ✱ x′ , y ′ ✱ ❛♥❞ x, y r❡s♣❡❝t✐✈❡❧②✳

❲❡ ❛♣♣❧② t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❣✐✈❡♥ ❛❜♦✈❡✳ ■♥ t❤❡ s♣r❡❛❞s❤❡❡t✱ • ❚❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ • ❚❤❡ ❧♦❝❛t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ♦✉r ✈❡❝t♦rs ❛♣♣r♦❛❝❤ ❢r♦♠ ♦✉r ♣r❡✈✐♦✉s tr❡❛t♠❡♥t ♦❢ t❤❡ ✢✐❣❤t ♦❢ ❛ ❜❛❧❧❄ ■♥st❡❛❞ ♦❢ t❤r❡❡ ❝♦❧✉♠♥s ❢♦r x′′ , x′ , x ❛♥❞ t❤❡♥ t❤r❡❡ ❝♦❧✉♠♥s ❢♦r y ′′ , y ′ , y ✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ♦❢ ❛❝❝❡❧❡r❛t✐♦♥✱ ✈❡❧♦❝✐t②✱ ❛♥❞ ❧♦❝❛t✐♦♥ ❛r❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ✈❡❝t♦rs ❝♦♥t❛✐♥❡❞ ✐♥ t✇♦ ❝♦❧✉♠♥s ❡❛❝❤✿ x′′ , y ′′ ✱ t❤❡♥ x′ , y ′ ✱ t❤❡♥ x, y ✳ ❚❤❡ ❢♦r♠✉❧❛ ✐s ❛❧♠♦st t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✿

❂❘❬✲✶❪❈✰✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✯❘✷❈✶ ◆❡①t ❛♥ ❛♥❣❧❡❞ t❤r♦✇✳✳✳ ❚❤❡ ♦♥❧② ❝❤❛♥❣❡ ✐s t❤❡ ✈❡❝t♦r ♦❢ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿

A =< 0, −32 >, V0 =< 100 cos α, 100 sin α >, P0 = (6, 0) , ✇❤❡r❡ α ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ t❤r♦✇✳

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✽✾

❊①❛♠♣❧❡ ✶✳✾✳✶✷✿ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥ ◆♦✇ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳ ❙t❛rt✐♥❣ ✇✐t❤ t❤❡ ♣❤②s✐❝s✱



✇❡ ✐♥t❡❣r❛t❡ ✕ ❝♦♦r❞✐♥❛t❡✇✐s❡ ✕ ♦♥❝❡✿  ′ x = vx , y ′ = −gt + vy , ❛♥❞ t✇✐❝❡✿ 

x′ (0) = vx ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t②, y ′ (0) = vy ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②;

x = vx t + px , 1 2 y = − 2 gt + vy t + py ,

❚❤✉s✱ ✇❡ ❤❛✈❡✿  ❞❡♣t❤ = ❤❡✐❣❤t =

✐♥✐t✐❛❧ ❞❡♣t❤ ✐♥✐t✐❛❧ ❤❡✐❣❤t

x′′ = 0, y ′′ = −g,

+ +

x(0) = px ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥, y(0) = py ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥.

✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②

· t✐♠❡ , · t✐♠❡ − 21 g · t✐♠❡ 2 .

❲❡ t❛❦❡ t❤✐s s♦❧✉t✐♦♥ t♦ t❤❡ ♥❡①t ❧❡✈❡❧ ❜② ❛ss❡♠❜❧✐♥❣ t❤❡s❡ ❝♦♠♣♦♥❡♥ts ✐♥t♦ ✈❡❝t♦rs ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❧♦❝❛t✐♦♥ =

✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ +

✐♥✐t✐❛❧ ✈❡❧♦❝✐t②

· t✐♠❡

+ < 0, − 21 g · t✐♠❡ 2 > .

❚❤❡ ❧❛st t❡r♠ ♥❡❡❞s ✇♦r❦✳ ❚❤❡ ③❡r♦ r❡♣r❡s❡♥ts t❤❡ ③❡r♦ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤✐❧❡ −g ✐s t❤❡ ✈❡rt✐❝❛❧ t2 ❛❝❝❡❧❡r❛t✐♦♥✳ ❚❤❡♥ t❤❡ ❧❛st t❡r♠ ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ t✐♠❡s ✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿ 2         t2 px x vx 0 = + · . ·t + py vy y −g 2 ❚❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ✐rr❡❧❡✈❛♥t❀ ✇❡ ♦♥❧② ♥❡❡❞ ✐t t♦ ❜❡ ❝♦♥st❛♥t✳

❉❡✜♥✐t✐♦♥ ✶✳✾✳✶✸✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥ ❙✉♣♣♦s❡ P0 ✐s ❛ ♣♦✐♥t ✐♥ Rm ❛♥❞ V0 , A ❛r❡ ✈❡❝t♦rs✳ ❚❤❡♥ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥ t❤r♦✉❣❤ P0 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ V ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥ A ✐s✿ t2 P (t) = P0 + V0 · t + A · . 2 ❲❡ ❤❛✈❡ ❛♥ ❡①tr❛ t❡r♠✱ t❤❛t ❞✐s❛♣♣❡❛rs ✇❤❡♥ A = 0✱ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥✳ ❏✉st ❛s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ❛ ❝♦♥st❛♥t ❛❝❝❡❧❡r❛t✐♦♥ ♣r♦❞✉❝❡s ❛ q✉❛❞r❛t✐❝ ♠♦t✐♦♥✦

❊①❡r❝✐s❡ ✶✳✾✳✶✹ ❙❤♦✇ t❤❛t t❤❡ ♣❛t❤ ♦❢ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛ ♣❛r❛❜♦❧❛✳ ❚❤❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❧✐❡ ✐♥ Rm ❛s ♣♦✐♥ts ❜✉t ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s ✈❡❝t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❢❛♠✐❧✐❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ♣♦✐♥ts✿

P (t) = P0 + V0 · t + A ·

t2 , 2

R(t) = R0 + V0 · t + A ·

t2 . 2

❛s ♦♥❡ ♦❢ ✈❡❝t♦rs✿

✶✳✾✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✾✵

■♥st❡❛❞ ♦❢ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t

P0

✐t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❡♥❞ ♣♦✐♥t ♦❢ ✈❡❝t♦r

t❤✐♥❣✳ ❆♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❡♥❞ ♦❢ ✈❡❝t♦r

R(t) ✐s t❤❡ ♣♦✐♥t P (t)✳

R0 = OP0 ✱

✇❤✐❝❤ ✐s t❤❡ s❛♠❡

❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❧❛tt❡r ❛♣♣r♦❛❝❤ ✐s t❤❛t

✐t ❛❧❧♦✇s ✉s t♦ ❛♣♣❧② ✈❡❝t♦r ♦♣❡r❛t✐♦♥s t♦ t❤❡ ❝✉r✈❡s✳ ❆ ♠♦r❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤ t♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❛s ✇❡❧❧ ❛s t❤❡✐r ❝❛❧❝✉❧✉s✱ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷✳ ❊①❛♠♣❧❡ ✶✳✾✳✶✺✿ ❝✐r❝❧❡ tr❛♥s❢♦r♠❡❞

❘❡❝❛❧❧ ❤♦✇ ✇❡ ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✉s✐♥❣ t❤❡ ❛♥❣❧❡ ❛s t❤❡ ♣❛r❛♠❡t❡r✳ ❍❡r❡✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ❛t ❛♥❣❧❡

t

✐s

cos t

❛♥❞

sin t

x✲

❛♥❞

y✲

r❡s♣❡❝t✐✈❡❧②✿

x = cos t, y = sin t . ❚❤❡ ✈❛❧✉❡s ♦❢

t

♠❛② ❜❡ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ s✉❝❤ ❛s

[0, 2π]

♦r r✉♥ t❤r♦✉❣❤ t❤❡ ✇❤♦❧❡

✐♥t❡r✈❛❧✳

❲❡ ❝❛♥ ❛❧s♦ ❧♦♦❦ ❛t t❤✐s ❢♦r♠✉❧❛ ❛s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✳ ❚❤❡♥ t❤✐s ✐s ❛ r❡❝♦r❞ ♦❢ ♠♦t✐♦♥ ✇✐t❤ ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ ❛ ❝♦♥st❛♥t

❛♥❣✉❧❛r

✈❡❧♦❝✐t②✳ ◆♦✇✱ t❤✐s ✐s t❤❡ ✈❡❝t♦r

r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ❝✉r✈❡✿

R(t) =< cos t, sin t > . ❙♦✱ ❛♣♣❧②✐♥❣ ✈❡❝t♦r ♦♣❡r❛t✐♦♥s t♦ t❤✐s ❝✉r✈❡ ✇✐❧❧ ❣✐✈❡ ❛s ♥❡✇ ❝✉r✈❡s✱ ❥✉st ❛s ✐♥ t❤❡ ✭❈❤❛♣t❡r ✶P❈✲✹✮✳ ❋♦r ❡①❛♠♣❧❡✱ ✉s✐♥❣ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

1✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡

2 ♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s str❡t❝❤✐♥❣ r❛❞✐❛❧❧②

t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❲❡ t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿

Q(t) = 2R(t) = 2 < cos t, sin t > , ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

❙✐♠✐❧❛r❧②✱ ✉s✐♥❣ ✈❡❝t♦r ❛❞❞✐t✐♦♥ ✇✐t❤

2✳

W =< 3, 1 >

♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s

t❤✐s ✈❡❝t♦r✳ ❲❡ t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿

S(t) = W + 2R(t) = (1, 2) + 2 < cos t, sin t > , ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

2

❝❡♥t❡r❡❞ ❛t

(1, 2)✳

s❤✐❢t✐♥❣

t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❜②

✶✳✶✵✳

✾✶

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❆♥❞ s♦ ♦♥ ✇✐t❤ ♦t❤❡r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡✳

✶✳✶✵✳ ❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❘❡❝❛❧❧ t❤❛t ❛ ❈❛rt❡s✐❛♥ s②st❡♠ ♣r❡✲♠❡❛s✉r❡s t❤❡ s♣❛❝❡ Rn s♦ t❤❛t ✇❡ ❝❛♥ ❞♦ ❛♥❛❧②t✐❝

❣❡♦♠❡tr②

✿

◮ ❯s✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts ❛♥❞ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦rs✱ ✇❡ ❝♦♠♣✉t❡ ❞✐st❛♥❝❡s ❛♥❞

❛♥❣❧❡s✳

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❛♣♣❧✐❡❞ t❤✐s ✐❞❡❛ t♦ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ♣♦✐♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ t♦ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ ✈❡❝t♦rs✳ ❲❤❛t ❛❜♦✉t t❤❡ ❛♥❣❧❡s ❄ ▲❡t✬s ✜rst r❡✈✐❡✇ ✇❤❛t ✇❡ ❞✐❞ ❢♦r ❞✐♠❡♥s✐♦♥s 1 ❛♥❞ 2✳ ❉✐♠❡♥s✐♦♥ 1 ✜rst✳ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs OP ❛♥❞ OQ ✭P, Q ❛r❡ ♥♦t ❡q✉❛❧ t♦ O✮ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♠♣♦♥❡♥ts x ❛♥❞ x′ ❄ ❚❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ • ■❢ P ❛♥❞ Q ❛r❡ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ O t❤❡♥ t❤❡

✱

❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡

• ■❢ P ❛♥❞ Q ❛r❡ ♦♥ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ O t❤❡♥ t❤❡

❚❤❡♥ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t t❤❡

❞✐r❡❝t✐♦♥s ❢♦r ❞✐♠❡♥s✐♦♥

OP ❛♥❞ OQ ✇✐t❤ ❝♦♠♣♦♥❡♥ts x 6= 0 ❛♥❞ x′ 6= 0 ✐s

✳

❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ ♦♣♣♦s✐t❡

1 ✐s st❛t❡❞ ❛s ❢♦❧❧♦✇s✿❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs

• 0 ✇❤❡♥ x · x′ > 0❀ ❛♥❞ • π ✇❤❡♥ x · x′ < 0✳

❍♦✇❡✈❡r✱ ✇❡ ❤❛✈❡ ♠❛❞❡ s♦♠❡ ♣r♦❣r❡ss s✐♥❝❡ ✇❡ ❢❛❝❡❞ t❤✐s t❛s❦✳ ▼❛✐♥❧②✱ ✐t ✐s t❤✐s r❡❛❧✐③❛t✐♦♥✿ ◮ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r

✐s

✐ts ♥♦r♠❛❧✐③❛t✐♦♥✱ ❛ ✉♥✐t ✈❡❝t♦r✳

■♥❞❡❡❞✱ ♦♥❧② t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rs ♠❛tt❡r ❛♥❞ ♥♦t t❤❡ s✐③❡s✦ ❲❡ ❝❛♥ t❤❡♥ ♠❛❦❡ t❤❡ s❛♠❡ st❛t❡♠❡♥t ❜✉t ❛❜♦✉t t❤❡ ✉♥✐t ✈❡❝t♦rs✿ x x′ ❛♥❞ . ′ |x|

|x |

❚❤❡ ❛❞✈❛♥t❛❣❡ ✐s t❤❛t t❤❡② ❝❛♥ ♦♥❧② t❛❦❡ t✇♦ ♣♦ss✐❜❧❡ ✈❛❧✉❡s✱ 1 ❛♥❞ −1✱ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥✳ ❆♥❞ s♦ ❞♦❡s t❤❡✐r ♣r♦❞✉❝t✿

✶✳✶✵✳

✾✷

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❲❡ ❝❛♥ t❤❡♥ r❡st❛t❡ t❤❡ r❡s✉❧t✿ ❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ✈❡❝t♦rs OP ❛♥❞ OQ ✇✐t❤ ❝♦♠♣♦♥❡♥ts x 6= 0 ❛♥❞ x′ 6= 0 ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ♣r♦❞✉❝t ❛♥❣❧❡

♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ x · ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ y

▼❛t❝❤✐♥❣ t❤❡s❡ ❢♦✉r ♥✉♠❜❡rs✱

x |x| x |x|

✇❡ r❡❛❧✐③❡ t❤❛t t❤✐s ✐s t❤❡ ❝♦s✐♥❡✿

· ·

x′ |x′ | x′ |x′ |

=

1

0

=

−1

π

0 7→ 1 ❛♥❞ π 7→ −1 , cos 0 = 1 ❛♥❞

❲❡ t❤❡♥ ❤❛✈❡ ❛ ♥❡✇ ✈❡rs✐♦♥ ♦❢ ♦✉r t❤❡♦r❡♠✿

cos π = −1 .

❚❤❡♦r❡♠ ✶✳✶✵✳✶✿ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✶

■❢ θ ✐s t❤❡ ❛♥❣❧❡ x′ = 6 0 t❤❡♥

❜❡t✇❡❡♥ ✈❡❝t♦rs

OP

❛♥❞

cos θ =

OQ

✐♥

R

✇✐t❤ ❝♦♠♣♦♥❡♥ts

x 6= 0

❛♥❞

x x′ · |x| |x′ |

◆♦✇ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 2✳ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ❢r♦♠ t❤❡ ♦r✐❣✐♥ O t♦✇❛r❞ ♣♦✐♥ts P ❛♥❞ Q ✭♦t❤❡r t❤❛♥ O✮ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s (x, y) ❛♥❞ (x′ , y ′ )❄ ❲❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥♦♥✲③❡r♦ ✈❡❝t♦rs✿ OP =< x, y > ❛♥❞ OQ =< x′ , y ′ > . ❲❛r♥✐♥❣✦ ■❢ ♦♥❡ ♦❢ t❤❡ ✈❡❝t♦rs ✐s ③❡r♦✱ t❤❡r❡ ✐s ♥♦ ❛♥❣❧❡ ❜❡✲ ❝❛✉s❡ t❤❡r❡ ✐s ♥♦ ❞✐r❡❝t✐♦♥✳

❲❡ ❦♥♦✇ ❤♦✇ t♦ ✜♥❞ t❤❡ ❛♥❣❧❡s ✇✐t❤ t❤❡ x✲❛①✐s✱ α ❛♥❞ β ✿

✶✳✶✵✳ ❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

✾✸

❚❤❡ ❛♥❣❧❡ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ✐s✿

[ = β −α. θ = QOP ❚❤❡ ❝♦s✐♥❡ ♦❢ t❤✐s ❛♥❣❧❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ❢r♦♠ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤❡s❡ t✇♦ ❛♥❣❧❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

cos θ = cos(β − α) = cos β cos α + sin β sin α . ▲❡t✬s ❡①❝❧✉❞❡ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ t❤❡ ✈❡❝t♦rs ❢r♦♠ ❝♦♥s✐❞❡r❛t✐♦♥✿

■♥st❡❛❞ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✈❡❝t♦rs

OP

❛♥❞

OQ✱

✇❡ ❧♦♦❦ t❤❡✐r ♥♦r♠❛❧✐③❛t✐♦♥s✱

U

❛♥❞

V✱

r❡s♣❡❝t✐✈❡❧②✿

  < x, y > y x OP =< x, y > =⇒ U = = , || < x, y > || || < x, y > || || < x, y > ||   y x < x′ , y ′ > = , OQ =< x′ , y ′ > =⇒ V = ′ ′ ′ ′ ′ || < x , y > || || < x , y > || || < x , y ′ > || ❚❤❡ s✐♥❡s ❛♥❞ ❝♦s✐♥❡s ♦❢ t❤❡s❡ ❛♥❣❧❡s ❛r❡ ❢♦✉♥❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❢♦✉r ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs ❛♥❞

V✳

U

❚❤❡s❡ s✐♥❡s ❛♥❞ ❝♦s✐♥❡s ❛r❡ ❡①❛❝t❧② t❤❡s❡ ❝♦♠♣♦♥❡♥ts ❜❡❝❛✉s❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❝t♦r ❛♥❞✱

t❤❡r❡❢♦r❡✱ t❤❡ ❤②♣♦t❡♥✉s❡ ✐s

1

✐♥ ❡✐t❤❡r ❝❛s❡✿

x || < x, y > || x′ cos β = || < x′ , y ′ > || cos α =

y || < x, y > || y′ sin β = || < x′ , y ′ > || sin α =

❚❤❡r❡❢♦r❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ❤❛✈❡✿

x′ y y′ x · + · || < x, y > || || < x′ , y ′ > || || < x, y > || || < x′ , y ′ > || xx′ + yy ′ . = || < x, y > || · || < x′ , y ′ > ||

cos θ =

❲❡ ✇✐❧❧ ❤❛✈❡ ❛ s♣❡❝✐❛❧ ♥❛♠❡ ❢♦r t❤❡ ♥✉♠❡r❛t♦r ♦❢ t❤✐s ❢r❛❝t✐♦♥✿

✶✳✶✵✳

✾✹

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✷✿ ❞♦t ♣r♦❞✉❝t ❚❤❡

❞♦t ♣r♦❞✉❝t

♦❢ ✈❡❝t♦rs < x, y > ❛♥❞ < x′ , y ′ > ✐♥ R2 ✐s ❞❡✜♥❡❞ ❜②✿ < x, y > · < x′ , y ′ >= xx′ + yy ′

❚❤✉s✱ t❤❡ ❞♦t ♣r♦❞✉❝t ✐s ❝♦♠♣✉t❡❞✱ ❛s ♦t❤❡r ✈❡❝t♦r ♦♣❡r❛t✐♦♥s✱ ❝♦♠♣♦♥❡♥t✇✐s❡✳ ❲❡ ♥♦✇ r❡✲st❛t❡ ♦✉r t❤❡♦r❡♠ ❛❜♦✉t t❤❡ ❞✐r❡❝t✐♦♥s✿

❚❤❡♦r❡♠ ✶✳✶✵✳✸✿ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✷ ■❢

θ

✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ✈❡❝t♦rs

A 6= 0

cos θ =

❛♥❞

B 6= 0

✐♥

R2 ✱

t❤❡♥✿

A·B ||A|| ||B||

❲❛r♥✐♥❣✦ ■t ♠❛❦❡s s❡♥s❡ ♥♦t t♦ ♣✉t ✏ ·✑ ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r t♦ ❛✈♦✐❞ ❝♦♥❢✉s✐♦♥✳

❚❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡s ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r s✉❣❣❡sts ✭t♦ ❜❡ ♣r♦✈❡♥ ❧❛t❡r✮ t❤❛t t❤❡ r❡s✉❧t ✐s✱ ❛s ❡①♣❡❝t❡❞✱ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ❞✐r❡❝t✐♦♥s✳

❊①❛♠♣❧❡ ✶✳✶✵✳✹✿ s✐♠♣❧❡ ✈❡❝t♦rs ▲❡t✬s t❡st t❤❡ t❤❡♦r❡♠ ♦♥ s✐♠♣❧❡ ✈❡❝t♦rs✳ ❋✐rst t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs✿ i =< 1, 0 >, j =< 0, 1 > =⇒ i · j = 1 · 0 + 0 · 1 = 0 .

■♥❞❡❡❞✱ t❤❡② ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♥❞ cos π/2 = 0✳ ❙✐♠✐❧❛r❧②✱ < 1, 1 > · < −1, 1 >= 1 · (−1) + 1 · 1 = 0 .

❍♦✇❡✈❡r✱ < 1, 0 > · < 1, 1 >= 1 · 1 + 0 · 1 = 1 .

❚♦ s❡❡ t❤❡ ❝♦rr❡❝t ❛♥❣❧❡ ♦❢ 45 ❞❡❣r❡❡s✱ ✇❡ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ t❤❡♦r❡♠✿ √ < 1, 0 > · < 1, 1 > 2 1 = √ = . cos θ = || < 1, 0 > || || < 1, 1 > || 2 1 2

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ✈❡r② ❝♦♥✈❡♥✐❡♥t r❡s✉❧t✿

❈♦r♦❧❧❛r② ✶✳✶✵✳✺✿ ❘✐❣❤t ❆♥❣❧❡✱ ❩❡r♦ ❉♦t Pr♦❞✉❝t ❚✇♦ ♥♦♥✲③❡r♦ ✈❡❝t♦rs ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡✐r ❞♦t ♣r♦❞✉❝t ✐s ③❡r♦❀ ✐✳❡✳✱

A ⊥ B ⇐⇒ A · B = 0

✶✳✶✵✳

✾✺

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❊①❛♠♣❧❡ ✶✳✶✵✳✻✿ ❧✐♥❡s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❧✐♥❡s ❣✐✈❡♥ ❜② t❤❡✐r r❡❧❛t✐♦♥s✿ y − 3 = 2(x − 1) ❛♥❞ y + 1 = −3(x − 3) .

❲❤❛t ✐s t❤❡ ❛♥❣❧❡ θ ❜❡t✇❡❡♥ t❤❡♠❄ ❉♦ ✇❡ ♥❡❡❞ t❤❡✐r ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥s❄ ◆♦✱ ❥✉st t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦rs✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ ✜rst ✐s 2✱ s♦ ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r t♦ ❜❡ V =< 1, 2 >✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝♦♥❞ ✐s −3✱ s♦ ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r t♦ ❜❡ U =< 1, −3 >✳ ❚❤❡r❡❢♦r❡✱ cos θ =

5 1−6 < 1, 2 > · < 1, −3 > = √ √ = −√ . || < 1, 2 > || || < 1, −3 > || 5 10 50

❲❡ st❛rt t♦ ❝❧✐♠❜ t❤❡ ❞✐♠❡♥s✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✼✿ ❞♦t ♣r♦❞✉❝t ❚❤❡

❞♦t ♣r♦❞✉❝t

♦❢ ✈❡❝t♦rs < x, y, z > ❛♥❞ < x′ , y ′ , z ′ > ✐♥ R3 ✐s ❞❡✜♥❡❞ ❜②✿ < x, y, z > · < x′ , y ′ , z ′ >= xx′ + yy ′ + zz ′

❚❤❡ ❞♦t ♣r♦❞✉❝t ✐s ❝♦♠♣♦♥❡♥t✇✐s❡ ♦♣❡r❛t✐♦♥✿ A =< x, y, z > · B =< u, v, w > A · B = x · u+ y · v+ z · w

   x · u+ u x     = y · v+ A·B = y · v z·w w z 

❖✉r t❤❡♦r❡♠ ❛❜♦✉t t❤❡ ❞✐r❡❝t✐♦♥s r❡♠❛✐♥s ✈❛❧✐❞✿

❚❤❡♦r❡♠ ✶✳✶✵✳✽✿ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✸ ■❢

θ

✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ✈❡❝t♦rs

A 6= 0

cos θ =

❛♥❞

B 6= 0

✐♥

R3 ✱

t❤❡♥✿

A·B ||A|| ||B||

Pr♦♦❢✳ ■♥st❡❛❞ ♦❢ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛s ✇❡ ✉s❡❞ ❢♦r ❝❛s❡ n = 2✱ ✇❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ▲❛✇ ♦❢ ❈♦s✐♥❡s ✭❝♦s✐♥❡ ✐s ✇❤❛t ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥②✇❛②✮ ✇❤✐❝❤ st❛t❡s✿ t❤❡ ❞♦t ♣r♦❞✉❝t

c2 = a2 + b2 − 2ab cos θ ,

❢♦r ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s a, b, c ❛♥❞ ❛♥❣❧❡ θ ❜❡t✇❡❡♥ a ❛♥❞ b✳

❲❡ ✐♥t❡r♣r❡t t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ ✈❡❝t♦rs✿ a = ||A||, b = ||B||, c = ||A − B|| .

✶✳✶✵✳

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❚❤❡♥ ✇❡ tr❛♥s❧❛t❡ t❤❡ ❧❛✇ ✐♥t♦ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ✈❡❝t♦rs✿ ||A − B||2 = ||A||2 + ||B||2 − 2||A|| ||B|| cos θ .

■♥st❡❛❞ ♦❢ s♦❧✈✐♥❣ ❢♦r cos γ ✱ ✇❡ ❡①♣❛♥❞ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ ◆♦r♠❛❧✐③❛t✐♦♥✳ ||A − B||2 = (A − B) · (A − B) = A · A + A · (−B) + (−B) · A + (−B) · (−B) ❉✐str✐❜✉t✐✈✐t②✳ = ||A||2 − 2A · B + ||B||2 ❆ss♦❝✐❛t✐✈✐t② ❛♥❞ ◆♦r♠❛❧✐③❛t✐♦♥✳ ❚❤❡ ▲❛✇ ♦❢ ❈♦s✐♥❡s t❤❡♥ t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ||A||2 − 2A · B + ||B||2 = ||A||2 + ||B||2 − 2||A|| ||B|| cos θ .

◆♦✇ ✇❡ ❝❛♥❝❡❧ t❤❡ r❡♣❡❛t❡❞ t❡r♠s ✐♥ t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ −2A · B = −2||A|| ||B|| cos θ . ❊①❛♠♣❧❡ ✶✳✶✵✳✾✿ ❜❛s✐s ✈❡❝t♦rs

■t ✐s ♦♥❝❡ ❛❣❛✐♥ ❡❛s② t♦ ❝♦♥✜r♠ t❤❛t t❤❡ ❜❛s✐s ✈❡❝t♦rs ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✿

❚❤❡ 1 ❛♥❞ 0 ❛r❡ ♠✐s♠❛t❝❤❡❞✿ i · j =< 1, 0, 0 > · < 0, 1, 0 > = 1 · 0 + 0 · 1 + 0 · 0 = 0 j · k =< 0, 1, 0 > · < 0, 0, 1 > = 0 · 0 + 1 · 0 + 0 · 1 = 0 k · i =< 0, 0, 1 > · < 1, 0, 0 > = 0 · 1 + 0 · 0 + 1 · 0 = 0 ❊①❛♠♣❧❡ ✶✳✶✵✳✶✵✿ ❞✐❛❣♦♥❛❧s

❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ s✐❞❡s ❛♥❞ t❤❡ ❞✐❛❣♦♥❛❧ ✐♥ ❛ sq✉❛r❡ ✐s 45 ❞❡❣r❡❡s✳ ◆♦✇✱ ✇❤❛t ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ ❝✉❜❡ ❛♥❞ ❛♥② ♦❢ ✐ts ❡❞❣❡s❄ ❚r② t♦ ❣✉❡ss ❢r♦♠ t❤❡ ♣✐❝t✉r❡ ✐❢ t❤❡ ❛♥❣❧❡ ✐s 45 ❞❡❣r❡❡s✿

❆ ❤❛r❞ tr✐❣♦♥♦♠❡tr✐❝ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❡❛s✐❧② ✇✐t❤ t❤❡ ❞♦t✲♣r♦❞✉❝t✳ ❋✐rst ✇❡ ❝❤♦♦s❡ ✈❡❝t♦rs t♦ r❡♣r❡s❡♥t t❤❡ ❡❞❣❡s✿ t❤❡ t❤r❡❡ ❜❛s✐s ✈❡❝t♦rs ❢♦r t❤❡ ♦✉ts✐❞❡ ❡❞❣❡s ❛♥❞ A =< 1, 1, 1 > ❢♦r t❤❡ ❞✐❛❣♦♥❛❧✳ ❲❡ ❤❛✈❡ ❢♦r t❤❡ ❛♥❣❧❡✿ cos θ =

< 1, 1, 1 > · < 1, 0, 0 > 1 =√ . || < 1, 1, 1 > || || < 1, 0, 0 > || 3

✾✻

✶✳✶✵✳ ❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❲❤❛t ✐s t❤✐s ❛♥❣❧❡❄ ❲❡ ♦♥❧② ❦♥♦✇ t❤❛t

✾✼

1 1 √

||A||2 = a2 + b2

3

A

< a, b, c >

||A||2 = a2 + b2 + c2

...

...

...

...

n

A

< a1 , a2 , ..., an > ||A||2 = a21 + a22 + ... + a2n

❲❡ ❞♦ t❤❡ s❛♠❡ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t✿

❞✐♠❡♥s✐♦♥

✈❡❝t♦rs

❝♦♠♣♦♥❡♥ts

1

A

a

B

u

A

< a, b >

B

< u, v >

A

< a, b, c >

B

< u, v, w >

A·B =a·u+b·v+c·w

...

...

...

...

n

A

< a1 , a2 , ..., an >

B

< b1 , b2 , ..., bn >

2 3

❞♦t ♣r♦❞✉❝t

A·B =a·u A·B =a·u+b·v

A · B = a1 · b1 + a2 · b2 + ... + an · bn

❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✶✷✿ ❞♦t ♣r♦❞✉❝t ✐♥ ♥✲s♣❛❝❡ ❚❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs

A

❛♥❞

B

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝ts ♦❢

✶✳✶✵✳

✾✽

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

t❤❡✐r ❝♦♠♣♦♥❡♥ts✿ A =< a1 ,

a2 ,

...,

an

>

B =< b1 ,

b2 ,

...,

bn

>

A·B =

a1 b1 +a2 b2 +... +an bn

■♥ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿ A·B =

n X

ak bk

k=1

❊①❡r❝✐s❡ ✶✳✶✵✳✶✸

❲❤❛t ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ 4✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡ ❛♥❞ ❛♥② ♦❢ ✐ts ❡❞❣❡s❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✹

❲❤❛t ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ n✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡ ❛♥❞ ❛♥② ♦❢ ✐ts ❡❞❣❡s❄ ❲❤❛t ✈❛❧✉❡ ❞♦❡s t❤❡ ❛♥❣❧❡ ❛♣♣r♦❛❝❤ ✇❤❡♥ n ❛♣♣r♦❛❝❤❡s ✐♥✜♥✐t②❄ ❇❡❧♦✇ ✇❡ s❡❡ ❤♦✇ t❤✐s ♥❡✇ ♦♣❡r❛t✐♦♥ ✭❧❛st r♦✇✮ ❝♦♠♣❛r❡s ✇✐t❤ t❤❡ ♦t❤❡r ✈❡❝t♦r ♦♣❡r❛t✐♦♥s✿ ✈❡❝t♦r ❛❞❞✐t✐♦♥

A

+

✈❡❝t♦r s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥

c

♥✉♠❜❡r ❞♦t ♣r♦❞✉❝t

A

✈❡❝t♦r

B

=

✈❡❝t♦r · ·

A

✈❡❝t♦r =

✈❡❝t♦r B

C C

✈❡❝t♦r =

✈❡❝t♦r

s

♥✉♠❜❡r

❲❛r♥✐♥❣✦ ❚❤❡ ❧❛st t✇♦ ♠✐❣❤t ❜❡ ❝♦♥❢✉s✐♥❣ ✇✐t❤♦✉t ❛

❝♦♥t❡①t ❀

❢♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡ ♠❡❛♥✐♥❣s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿

0 · A = 0.

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ♣r♦♣❡rt✐❡s

✳

♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t

■❢ ✇❡ ❥✉st s❡t Y = X ✱ ✇❡ ❤❛✈❡ t❤❡ s♦✲❝❛❧❧❡❞ ◆♦r♠❛❧✐③❛t✐♦♥ ✿ ||X||2 = X · X

❖♥❡ ❝❛♥✱ t❤❡r❡❢♦r❡✱ r❡❝♦✈❡r t❤❡ ♠❛❣♥✐t✉❞❡ ❢r♦♠ t❤❡ ❞♦t ♣r♦❞✉❝t ❥✉st ❛s ❜❡❢♦r❡✳ ❙♦✱ ♦♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ❞♦t ♣r♦❞✉❝t✱ ✇❡ ❞♦♥✬t ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ♠❛❣♥✐t✉❞❡ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❚❤❡ P♦s✐t✐✈✐t② ♦❢ t❤❡ ♥♦r♠ t❤❡♥ r❡q✉✐r❡s ❛ s✐♠✐❧❛r ♣r♦♣❡rt② ❢♦r t❤❡ ❞♦t ♣r♦❞✉❝t✿ V · V ≥ 0;

❛♥❞ V · V = 0 ⇐⇒ V = 0

✶✳✶✵✳ ◆❡①t✱

✾✾

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❈♦♠♠✉t❛t✐✈✐t②

♦r

❙②♠♠❡tr②

✿

A·B =B·A ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❛♥❣❧❡ ✐s ◆❡①t✱

❆ss♦❝✐❛t✐✈✐t②

❜❡t✇❡❡♥

A ❛♥❞ B ❀ ✐✳❡✳✱ t❤❡ s❛♠❡ ❢r♦♠ A t♦ B ❛s ❢r♦♠ B t♦ A✳

✿

(kA) · B = k(A · B) = A · (kB) ❙♦✱ t❤❡ ❡✛❡❝t ♦❢ str❡t❝❤✐♥❣ ♦♥ t❤❡ ❞♦t ♣r♦❞✉❝t ✐s ❛ ♠✉❧t✐♣❧❡ ❛♥❞ t❤❡ ❛♥❣❧❡ ❞♦❡s♥✬t ❝❤❛♥❣❡ ❢♦r k > 0 ♦r ✐s r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ ✇❤❡♥ k < 0✳ ❲❡ ❝❛♥ s❡❡ ♥♦✇ t❤❛t ♦♥❧② t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥s ♠❛tt❡r ❢♦r t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ ✈❡❝t♦rs✳ ❲❡ ❥✉st ❝❤♦♦s❡ t❤❡s❡ ✈❛❧✉❡s ❢♦r k ✐♥ t❤❡ ❧❛st ❢♦r♠✉❧❛✿

1 1 , , ||A|| ||B|| ||A||

❛♥❞

1 ||B||

t♦ r❡✇r✐t❡ ♦✉r ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✿

cos θ =

A·B 1 1 1 A B = A·B = A· B= · . ||A|| ||B|| ||A|| ||B|| ||A|| ||B|| ||A|| ||B||

❚❤❡ r❡s✉❧t s✉❣❣❡sts t❤❛t t❤❡ ❞♦t ♣r♦❞✉❝t ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❈❡rt❛✐♥❧②✱ t❤✐s s②st❡♠ ✐s ❥✉st ❛ t♦♦❧ t❤❛t ✇❡ ✐♥tr♦❞✉❝❡ ✐♥t♦ t❤❡ s♣❛❝❡ t❤❡ ❣❡♦♠❡tr② ♦❢ ✇❤✐❝❤ ✇❡ st✉❞②✱ ❛♥❞ ✇❡ ❞♦♥✬t ❡①♣❡❝t t❤❛t ❝❤❛♥❣✐♥❣ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦rs ✇✐❧❧ ❛❧s♦ ❝❤❛♥❣❡ t❤❡ ❞✐st❛♥❝❡s ❛♥❞ t❤❡ ❛♥❣❧❡s✳ ❇✉t ✐t✬s ❛❧s♦ tr✉❡ ✐♥ Rn ✦ ◆❡①t✱

❉✐str✐❜✉t✐✈✐t②

♦r

▲✐♥❡❛r✐t②

✿

A · (B + C) = A · B + A · C ❚r❡❛t❡❞ ❝♦♠♣♦♥❡♥t✇✐s❡✱ t❤❡ ❈♦♠♠✉t❛t✐✈✐t②✱ ❆ss♦❝✐❛t✐✈✐t②✱ ❉✐str✐❜✉t✐✈✐t② ♣r♦♣❡rt✐❡s ❢♦r t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❈♦♠♠✉t❛t✐✈✐t②✱ ❆ss♦❝✐❛t✐✈✐t②✱ ❉✐str✐❜✉t✐✈✐t② ❢♦r ♥✉♠❜❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s t❤❡ ✇❤♦❧❡ ♣r♦♦❢ ♦❢ t❤❡ ❈♦♠♠✉t❛t✐✈✐t② ❢♦r n = 2✿         a u u a   ·   = au + bv = ua + vb =   ·   b v v b ❖♥❝❡ ❛❣❛✐♥✱ t❤❡s❡ ♣r♦♣❡rt✐❡s ❛❧❧♦✇ ✉s t♦ ✉s❡ t❤❡ ✉s✉❛❧ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥ st❡♣s ❢♦r ♥✉♠❜❡rs ❛s ❧♦♥❣ ❛s t❤❡ ❡①♣r❡ss✐♦♥s ♠❛❦❡ s❡♥s❡ t♦ ❜❡❣✐♥ ✇✐t❤✳ ❍♦✇ ❞♦ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤✐s ♠❛❞❡✲✉♣ s♣❛❝❡❄ ❆❢t❡r ❛❧❧✱ ✇❤❡♥ n > 3✱ t❤❡r❡ ✐s ♥♦ r❡❛❧✐t② t❡st ❢♦r t❤✐s ❝♦♥❝❡♣t ❛♥❞ ✇❡ ❝❛♥✬t ✈❡r✐❢② t❤❡ ❢♦r♠✉❧❛s ✇❡ ❛r❡ t♦ ✉s❡✦ ❲❡ ❤❛✈❡ ❝♦♠❡ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❞✐st❛♥❝❡s ✐♥ Rn ❛♥❞ ♥♦✇ ❛s❦✿

◮ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ ✈❡❝t♦rs A ❛♥❞ B ✐♥ Rn ❄ ❚❤❡ ❛♥s✇❡r ✐s t♦ r❡❞✉❝❡ t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ t♦ t❤❡ ❝❛s❡ ♦❢ n = 2✳ ■♥❞❡❡❞✱ ❡✈❡r② t✇♦ ✈❡❝t♦rs ❞❡✜♥❡ ❛ ♣❧❛♥❡ ❛♥❞ t❤✐s ♣❧❛♥❡ ❤❛s t❤❡ s❛♠❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦♣❡r❛t✐♦♥s ✕ ✐♥❝❧✉❞✐♥❣ t❤❡ ❞♦t ♣r♦❞✉❝t ✕ ❛s t❤❡ ❛♠❜✐❡♥t s♣❛❝❡ Rn ✿

✶✳✶✵✳

✶✵✵

❚❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs❀ t❤❡ ❞♦t ♣r♦❞✉❝t

❚❤❡ ❉✐str✐❜✉t✐✈✐t② ✇✐❧❧ r❡q✉✐r❡ 3 ❞✐♠❡♥s✐♦♥s✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ♣❧❛♥❡ ❤❛s t❤❡ ✇❡❧❧✲✉♥❞❡rst♦♦❞ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✿ ❚❤❡ ❧❡♥❣t❤s ♦❢ ✈❡❝t♦rs ❛♥❞ t❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ ✈❡❝t♦rs ❝❛♥ ❜❡ ♠❡❛s✉r❡❞✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛❜str❛❝t ❜✉t ✐t ♠❛t❝❤❡s t❤❡ ❧♦✇❡r ❞✐♠❡♥s✐♦♥s n = 1, 2, 3✿

❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✶✺✿ ❛♥❣❧❡s ❢♦r ❞✐♠❡♥s✐♦♥ n ❚❤❡ ❛♥❣❧❡ θ ❜❡t✇❡❡♥ ✈❡❝t♦rs A 6= 0 ❛♥❞ B 6= 0 ✐♥ Rn ✐s ❞❡✜♥❡❞ t♦ s❛t✐s❢②✿ cos θ =

A·B ||A|| ||B||

❋♦r t❤❡ r❡❝♦r❞✱ ✇❡ s✉♠♠❛r✐③❡ t❤❡ r✉❧❡s ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t✿

❚❤❡♦r❡♠ ✶✳✶✵✳✶✻✿ ❆①✐♦♠s ♦❢ ■♥♥❡r Pr♦❞✉❝t ❙♣❛❝❡ ❚❤❡ ❞♦t ♣r♦❞✉❝t ✐♥ Rn s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s ❢♦r ❛❧❧ ✈❡❝t♦rs U, V, W ❛♥❞ ❛❧❧ s❝❛❧❛rs a, b✿ • • •

❙②♠♠❡tr②✿ U · V = V · U . ▲✐♥❡❛r✐t②✿ U · (aV + bW ) = a(U · V ) + b(U · W ) . P♦s✐t✐✈❡✲❞❡✜♥✐t❡♥❡ss✿ V · V ≥ 0❀ ❛♥❞ V · V = 0 ⇐⇒ V = 0 .

❲❡ ❤❛✈❡ ❛❞❞❡❞ ❛ t❤✐r❞ ✈❡❝t♦r ♦♣❡r❛t✐♦♥ t♦ t❤❡ t♦♦❧❦✐t ❜✉t ✈❡❝t♦r ❛❧❣❡❜r❛ st✐❧❧ ❧♦♦❦s ❧✐❦❡ t❤❛t ♦❢ ♥✉♠❜❡rs✦ ❋r♦♠ t❤❡ ✐♥❡q✉❛❧✐t② ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

| cos θ| ≤ 1 ,

❈♦r♦❧❧❛r② ✶✳✶✵✳✶✼✿ ❈❛✉❝❤② ■♥❡q✉❛❧✐t② ❋♦r ❛♥② ♣❛✐r ♦❢ ✈❡❝t♦rs A 6= 0 ❛♥❞ B 6= 0 ✐♥ Rn ✱ ✇❡ ❤❛✈❡✿ |A · B| ≤ ||A|| ||B||

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ ✇❡ r♦t❛t❡ ♦♥❡ ♦❢ t❤❡ ✈❡❝t♦rs✱ t❤❡ ❞♦t ♣r♦❞✉❝t r❡❛❝❤❡s ✐ts ♠❛①✐♠✉♠ ✇❤❡♥ t❤❡② ❛r❡ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❙♦✱ ✇❤❛t ✐s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ✈❡❝t♦rs❄ ❚✇♦ ♣❛rt✐❛❧ ❛♥s✇❡rs✿ ✶✳ ❚❤❡ ❞♦t ♣r♦❞✉❝t ✐s t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ✇❤❡♥ t❤❡ ✈❡❝t♦rs ❛r❡ ✉♥✐t ✈❡❝t♦rs✳ ✷✳ ❚❤❡ ❞♦t ♣r♦❞✉❝t ✐s t❤❡ sq✉❛r❡ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ ✇❤❡♥ t❤❡ ✈❡❝t♦rs ❛r❡ ❡q✉❛❧✳

✶✳✶✶✳

Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

✶✵✶

❆ ❝♦♠♣❧❡t❡✱ ❣❡♦♠❡tr✐❝ ❛♥s✇❡r ♠❛② s✐♠♣❧② ❜❡ ♦✉r ❢♦r♠✉❧❛✿ A · B = ||A|| ||B|| cos θ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✽✿ ✐♥♥❡r ♣r♦❞✉❝t ❢♦r t❛①✐❝❛❜❄

❖♥❝❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❞♦t ♣r♦❞✉❝t ❡♠❡r❣❡❞ ❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝ ❛♥❞ ♥♦r♠✿ ❞✐♠❡♥s✐♦♥ ✈❡❝t♦rs ❝♦♠♣♦♥❡♥ts

❊✉❝❧✐❞❡❛♥ ♥♦r♠ ❛♥❞ ❞♦t ♣r♦❞✉❝t

n

A

< a1 , a2 , ..., an > ||A||2 = a1 · a1 + a2 · a2 + ... + an · an

n

A

< a1 , a2 , ..., an >

B

< b1 , b2 , ..., bn >

A · B = a1 · b1 + a2 · b2 + ... + an · bn

❇✉t ✇❤❛t ❛❜♦✉t t❤❡ t❛①✐❝❛❜ ♠❡tr✐❝❄ ❲❡ ❝❛♥ s✉❣❣❡st t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ ✐♥♥❡r ♣r♦❞✉❝t✿ ❞✐♠❡♥s✐♦♥ ✈❡❝t♦rs ❝♦♠♣♦♥❡♥ts

t❛①✐❝❛❜ ♥♦r♠ ❛♥❞ ✐♥♥❡r ♣r♦❞✉❝t❄

n

A

< a1 , a2 , ..., an > ||A|| = |a1 | + |a2 | + ... + |an |

n

A

< a1 , a2 , ..., an >

B

< b1 , b2 , ..., bn >

A·B =

p p p |a1 | · |b1 | + |a2 | · |b2 | + ... + |an | · |bn |

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ▲✐♥❡❛r✐t② ❢❛✐❧s✦ ❲❡✱ t❤❡r❡❢♦r❡✱ ✇✐❧❧ ❜❡ ✉♥❛❜❧❡ t♦ ❞✐s❝✉ss ❛♥❣❧❡s ✐♥ t❤✐s s♣❛❝❡✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✾

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t ❢♦r n = 2✳

✶✳✶✶✳ Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

❚♦ ✜♥❞ t❤❡ x✲❝♦♦r❞✐♥❛t❡ ♦❢ ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡✱ ✇❡ ❣♦ ✈❡rt✐❝❛❧❧② ❢r♦♠ t❤❛t ♣♦✐♥ts ✉♥t✐❧ ✇❡ r❡❛❝❤ t❤❡ x✲❛①✐s✿

■t✬s ❛ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥✿

F : R2 → R, < x, y >7→ x .

❚❤❡ r❡s✉❧t r❡s❡♠❜❧❡s s❤❛❞♦✇s ❝❛st ❜② ♣♦✐♥ts✱ ✈❡❝t♦rs✱ ♦r s✉❜s❡ts ♦♥t♦ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ ❧✐❣❤t ❝❛st ❢r♦♠ ❛❜♦✈❡ ✭♦r ❢r♦♠ t❤❡ r✐❣❤t ❢♦r t❤❡ y ✲❛①✐s✮✿

✶✳✶✶✳

Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

✶✵✷

■t ✐s ❝❛❧❧❡❞ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t ♦♥ t❤❡ x✲❛①✐s✳ ❙❛♠❡ ❢♦r t❤❡ y ✲❛①✐s✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ♦♥ t❤❡ x✲❛①✐s ❣✐✈❡s ✐ts x✲❝♦♠♣♦♥❡♥t✳ ■❢ s❡✈❡r❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❝♦✲❡①✐st✱ ❛ tr❛♥s✐t✐♦♥ ❢r♦♠ ♦♥❡ t♦ ❛♥♦t❤❡r ✇✐❧❧ r❡q✉✐r❡ ❡①♣r❡ss✐♥❣ t❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ♦r t❤❡ ♥❡✇ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ t❤❡ ♦❧❞✳ ❲❡ ❞♦ t❤❛t ♦♥❡ ❛①✐s ♦r ❜❛s✐s ✈❡❝t♦r ❛t ❛ t✐♠❡✳ ❆s ❛ r❡❛s♦♥ ❢♦r t❤✐s st✉❞② ❤❡r❡ ✐s t❤❡ ❡①❛♠♣❧❡ ♦❢ ❝♦♠♣♦✉♥❞ ♠♦t✐♦♥ ❢r♦♠ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✱ ❛♥ ♦❜❥❡❝t s❧✐❞✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡✿

■♥ ♦r❞❡r t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ r❡❧❡✈❛♥t ♣❛rt ♦❢ t❤❡ ♠♦t✐♦♥✱ ♦♥❡ ✇♦✉❧❞ ❝❤♦♦s❡ t❤❡ ✜rst ❜❛s✐s ✈❡❝t♦r t♦ ❜❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ s✉r❢❛❝❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r✳ ❋♦r ✈❡❝t♦rs✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ✇✐❧❧ r❡q✉✐r❡ ❝❤♦♦s✐♥❣ ❛ r❡♣r❡s❡♥t❛t✐✈❡ ✈❡❝t♦r ♦♥ ✐t❀ ✐t✬s i✳ ❆♥② ✈❡❝t♦r A ✐s ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ i ❜② ✜♥❞✐♥❣ A✬s ❝♦♠♣♦♥❡♥t P ♦♥ t❤❡ x✲❛①✐s✳ ❋♦r ❡①❛♠♣❧❡✱ ❜❡❧♦✇ t❤❡ ❝♦♠♣♦♥❡♥t ✐s 3✿

❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ i ❛♥❞ j ✿ A =< 3, 4 >= 3i + 4j .

❲❤❛t ✐❢ ✐♥st❡❛❞ ♦❢ i ✇❡ ❤❛✈❡ ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r V ❄ ❇✉t ✜rst✱ ✇❡ ❝♦♥s✐❞❡r ❛ s❤♦rt❝✉t ❢♦r ✜♥❞✐♥❣ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r

✈❡❝t♦r ✿

✶✳✶✶✳

Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

❚❤❡ ♣r♦❜❧❡♠ ♦❢ r♦t❛t✐♥❣ ❛ ❣✐✈❡♥ ✈❡❝t♦r

V

✶✵✸

✐♥ t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤

❚❤❡♦r❡♠ ✶✳✶✶✳✶✿ ❖rt❤♦❣♦♥❛❧✐t② ✐♥ ❋♦r ❛♥② ✈❡❝t♦r

V =< u, v >

π/2

❤❛s ❛♥ ❡❛s② s♦❧✉t✐♦♥✿

2✲s♣❛❝❡

♦♥ t❤❡ ♣❧❛♥❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r✱ ❝❛❧❧❡❞ ❛

✈❡❝t♦r ♦❢ V ✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s V

⊥

✱ ❤❛s t❤❡ ❛♥❣❧❡ ♦❢

π/2

✇✐t❤

V✿

♥♦r♠❛❧

V ⊥ =< u, v >⊥ =< −v, u > Pr♦♦❢✳ ❚❤✐s ✐s ❡❛s② t♦ ❝♦♥✜r♠✿

V · V ⊥ =< u, v > · < −v, u >= u(−v) + vu = 0 . ❲❡ ❤❛✈❡ t❤❡♥ ❛ s♣❡❝✐❛❧ ♦♣❡r❛t✐♦♥ ♦♥ ✈❡❝t♦rs ✐♥ ❞✐♠❡♥s✐♦♥

2✳

❊✈❡r② ✈❡❝t♦r ❤❛s ❡①❛❝t❧② t✇♦ ♥♦r♠❛❧

✉♥✐t

✈❡❝t♦rs✳

❊①❛♠♣❧❡ ✶✳✶✶✳✷✿ ✐♥✈❡st✐♥❣ ❛❞✈✐❝❡ ❆♥ ✐♥✐t✐❛❧❧② ✐♥✈❡st♠❡♥t ❛❞✈✐❝❡ ♠✐❣❤t ❜❡ ✈❡r② s✐♠♣❧❡✱ ❢♦r ❡①❛♠♣❧❡✿ ❤♦❧❞ t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ st♦❝❦s ❛♥❞

1✲t♦✲2✳ ■❢ x ✐s t❤❡ ❛♠♦✉♥t ♦❢ st♦❝❦s ❛♥❞ y ✐s t❤❡ ❛♠♦✉♥t ♦❢ ❜♦♥❞s ✐♥ ✐t✱ t❤❡ ✏✐❞❡❛❧✑ t❤❡ ❧✐♥❡ y = 2x✱ ✐✳❡✳✱ t❤❡② ❛r❡ t❤❡ ❡♥❞s ♦❢ ✈❡❝t♦rs t❤❛t ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ V =< 2, 1 >✿

❜♦♥❞s ♦♥

❍♦✇ ❞♦ ✇❡ ❞❡t❡r♠✐♥❡ ❤♦✇ ❝❧♦s❡ ✐s ❡❛❝❤ ♣♦rt❢♦❧✐♦ t♦ t❤❡ ✐❞❡❛❧❄ ❲❡ ❝❛♥ ✉s❡ ❲❤❛t ❛❜♦✉t t❤❡ ❜❛❞❄ ❲❡ ❝❛♥ ✜♥❞ ❛ ✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦

V

V

♣♦rt❢♦❧✐♦s ❧✐❡

❛s ❛ ♠❡❛s✉r✐♥❣ st✐❝❦✳

✈✐❛ t❤❡ ❧❛st t❤❡♦r❡♠✳ ❚❤♦✉❣❤ ♥♦t ✉♥✐t

✈❡❝t♦rs✱ t❤❡s❡ t✇♦ ✈❡❝t♦rs ✇✐❧❧ ❣✐✈❡ t❤❡ ❣♦♦❞ ❛♥❞ t❤❡ ❜❛❞ ❝♦♠♣♦♥❡♥ts ♦❢ ❛♥② ♣♦rt❢♦❧✐♦✿

g =< 1, 2 >

❛♥❞

b =< −2, 1 > .

❙♦✱ ✇❡ ♥❡❡❞ t♦ r❡♣r❡s❡♥t ❡✈❡r② ♣♦rt❢♦❧✐♦ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦✳ ❋♦r ❡①❛♠♣❧❡✿

< 5, 6 >= p < 1, 2 > +q < −2, 1 >

❛♥❞

< 7, 4 >= u < 1, 2 > +v < −2, 1 > .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t✇♦ s②st❡♠s ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿

(

p − 2q = 5 2p + q = 6

❛♥❞

(

u − 2v = 7 2u + v = 4

Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

✶✳✶✶✳

✶✵✹

❚❤❡s❡ ❛r❡ t❤❡ s♦❧✉t✐♦♥s✿

p = 17/5, q = −4/5

❛♥❞

u = 4, v = −2 .

❚❤❡ s❡❝♦♥❞ ♥✉♠❜❡rs ✐♥ t❤❡s❡ ♣❛✐rs ✐♥❞✐❝❛t❡ ❤♦✇ ❢❛r ✐t ✐s ❢r♦♠ t❤❡ ✐❞❡❛❧✳ ❚❤❡ s❡❝♦♥❞ ♣♦rt❢♦❧✐♦ ✐s ❢❛rt❤❡r✳ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡ ♠❛♥② ✐♥✈❡st♠❡♥t ✈❡❤✐❝❧❡s ✐♥ ❡❛❝❤ ♣♦rt❢♦❧✐♦❄ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠❛② ❛ss✉♠❡ t❤❛t t❤❡s❡ 7 ♣♦rt❢♦❧✐♦s ❧✐✈❡ ✐♥ R ✿

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ s❤♦rt❝✉t ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠ ✐s ♥♦t ❛✈❛✐❧❛❜❧❡ ❛♥②♠♦r❡✳

❲❡ ✇✐❧❧ ♥❡❡❞ ❛ ❢✉rt❤❡r

❛♥❛❧②s✐s✳

❚❤✐s ✐s ✇❤❛t ❛ ♣r♦❥❡❝t✐♦♥ ♦♥ ❛ ❧✐♥❡ ❞❡✜♥❡❞ ❜② ❛ ✈❡❝t♦r ❧♦♦❦s ❧✐❦❡✿

❚❤❡ q✉❡st✐♦♥ ❜❡❝♦♠❡s✿ ❍♦✇ ♠✉❝❤ ❞♦❡s

◮

❱❡❝t♦r

A

❙♦✱ ❡✈❡r② ✈❡❝t♦r t♦

■❢

A

✐s ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢

A

✏♣r♦tr✉❞❡✑ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢

V

❜② ✜♥❞✐♥❣

A✬s

♣r♦❥❡❝t✐♦♥

♥❡❡❞s t♦ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢

V

V✿

P

✐s t❤❡ ♣r♦❥❡❝t✐♦♥✱ ✇❤❛t✬s t❤❡ ♦t❤❡r ✈❡❝t♦r❄ ❲❡ s✐♠♣❧② s✉❜tr❛❝t✿

A = P + (A − P ) .

P

V❄

▼♦r❡ ♣r❡❝✐s❡❧②✿

♦♥ t❤❡ ❧✐♥❡ ❝r❡❛t❡❞ ❜②

V✳

❛♥❞ s♦♠❡ ♦t❤❡r ✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r

✶✳✶✶✳

Pr♦❥❡❝t✐♦♥s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✈❡❝t♦rs

✶✵✺

❚❤❡ ✈❡❝t♦r ✇❡ ❛r❡ ❛❢t❡r ✐s ❞❡s❝r✐❜❡❞ ✐♥❞✐r❡❝t❧②✿

❉❡✜♥✐t✐♦♥ ✶✳✶✶✳✸✿ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ A ❛♥❞ V 6= 0 ❛r❡ t✇♦ ✈❡❝t♦rs ✐♥ Rn ✳ ❚❤❡♥ A ♦♥t♦ V ✐s ❛ ✈❡❝t♦r P t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ P ✐s ♣❛r❛❧❧❡❧ t♦ V ✳ ✷✳ A − P ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ V ✳

❙✉♣♣♦s❡

♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢

t❤❡

▲❡t✬s ✜♥❞ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛✳ ❋✐rst✱ ✏♣❛r❛❧❧❡❧✑ s✐♠♣❧② ♠❡❛♥s ❛ ♠✉❧t✐♣❧❡✦ ❚❤❡r❡❢♦r❡✱ t❤❡ ✜rst ♣r♦♣❡rt② ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ♥✉♠❜❡r

k

✕

t❤✐s ✐s t❤❡ ♦♥❡ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ✕ s✉❝❤ t❤❛t✿

P = kV . ❚❤❡ s❡❝♦♥❞ ♣r♦♣❡rt② ✐s ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t✿

V · (A − P ) = 0 . ❲❡ s✉❜st✐t✉t❡✿

❛♥❞ ✉s❡

❉✐str✐❜✉t✐✈✐t② ❛♥❞ ❆ss♦❝✐❛t✐✈✐t② ✿

◆❡①t ✇❡ ✉s❡

V · (A − kV ) = 0 , V · A − kV · V = 0 .

◆♦r♠❛❧✐③❛t✐♦♥ ✿

V · A = k||V ||2 .

❚❤❡♥✱

k= ❚❤✐s ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢

V

t❤❛t ❣✐✈❡s ✉s

P✳

V ·A . ||V ||2

❚❤✉s✱ ✇❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿

❚❤❡♦r❡♠ ✶✳✶✶✳✹✿ Pr♦❥❡❝t✐♦♥ ❱✐❛ ❉♦t Pr♦❞✉❝t ❚❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r

A

♦♥t♦ ❛ ✈❡❝t♦r

❣✐✈❡♥ ❜②✿

P =

◆♦t✐❝❡ t❤❛t t❤❡ ❢♦r♠✉❧❛ ✕ ❛s ❡①♣❡❝t❡❞ ✕ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡

P =



V 6= 0

✐s t❤❡ ✈❡❝t♦r

P

V ·A V ||V ||2

❞✐r❡❝t✐♦♥ ♦❢ V ❀ ♦♥❧② ✉♥✐t ✈❡❝t♦rs ❛r❡ ✐♥✈♦❧✈❡❞✿

 V V ·A . ||V || ||V ||

❊①❡r❝✐s❡ ✶✳✶✶✳✺ Pr♦✈❡ t❤❡ ❧❛st ❢♦r♠✉❧❛✳

❊①❛♠♣❧❡ ✶✳✶✶✳✻✿ ✐♥✈❡st✐♥❣ ❛❞✈✐❝❡ ❝♦♥t✐♥✉❡❞ ❲❡ ❤❛✈❡

7

✐♥str✉♠❡♥ts ✐♥ ❡❛❝❤ ♣♦rt❢♦❧✐♦✳ ❚❤❡ ♣♦rt❢♦❧✐♦

❝♦♠♣❡t✐♥❣ ♣♦rt❢♦❧✐♦s

A1

❛♥❞

A2

✐♥ ❝♦❧✉♠♥s

8

✶✳ ❲❡ ✜♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ ♣r♦❞✉❝ts ♦❢

❛♥❞

A1

9✳

❛❞✈✐❝❡

V

✐s s❤♦✇♥ ✐♥ ❝♦❧✉♠♥

4

❛♥❞ t❤❡ t✇♦

❲❡ ❝❛rr② ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥s✿

❛♥❞

A2

✇✐t❤

V

✭❝♦❧✉♠♥s

❛❞❞✐♥❣ t❤♦s❡ ♦❜t❛✐♥ t❤❡ t✇♦ ❞♦t ♣r♦❞✉❝ts ✭❜♦tt♦♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✮✳

10

❛♥❞

11✮

❛♥❞ t❤❡♥ ❜②

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn

✶✵✻

✷✳ ❋r♦♠ t❤♦s❡ t✇♦✱ ✇❡ ✜♥❞ t❤❡ ♠✉❧t✐♣❧❡s c1 ❛♥❞ c2 ❢♦r t❤❡ ♣r♦❥❡❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠ ✭t♦♣s ♦❢ ❝♦❧✉♠♥s 12 ❛♥❞ 13✮✳ ✸✳ ❲❡ ✉s❡ t❤♦s❡ t✇♦ t♦ ♠✉❧t✐♣❧② V ❝♦♠♣♦♥❡♥t✇✐s❡ t♦ ♦❜t❛✐♥ t❤❡ ♣r♦❥❡❝t✐♦♥s P1 ❛♥❞ P2 ♦❢ A1 ❛♥❞ A2 ✭❝♦❧✉♠♥s 12 ❛♥❞ 13✮✳ ✹✳ ❚❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ A1 ❛♥❞ A2 ❢r♦♠ V ❛r❡ ❢♦✉♥❞ ✭❝♦❧✉♠♥s 14 ❛♥❞ 15✮✳

❋✐♥❛❧❧②✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ t❤♦s❡ t✇♦ ✭t♦♣s ♦❢ ❝♦❧✉♠♥s 14 ❛♥❞ 15✮✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ s❡❝♦♥❞ ♣♦rt❢♦❧✐♦ ✐s ❝❧♦s❡r t♦ t❤❡ ❧✐♥❡ t❤❛t r❡♣r❡s❡♥ts t❤❡ ✐❞❡❛❧✿

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡s❡ ❞✐st❛♥❝❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥✈❡st♠❡♥t✱ ✇❤✐❝❤ s❦❡✇s t❤❡ ❝♦♥❝❧✉s✐♦♥s✳ ❲❡ t❤❡♥ t✉r♥ t♦ t❤❡ ❛♥❣❧❡s ✐♥st❡❛❞✳ ❚❤❡✐r ❝♦s✐♥❡s ❛r❡ ❝♦♠♣✉t❡❞ ✭t♦♣s ♦❢ ❝♦❧✉♠♥s 10 ❛♥❞ 11✮ ✇✐t❤ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ✜rst ♣♦rt❢♦❧✐♦ ✐s ❝❧♦s❡r t♦ t❤❡ ❧✐♥❡ t❤❛t r❡♣r❡s❡♥ts t❤❡ ✐❞❡❛❧✳ ❊①❡r❝✐s❡ ✶✳✶✶✳✼

❲❤❛t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦rt❢♦❧✐♦s ✇✐t❤ t❤❡ t♦t❛❧ ✐♥✈❡st♠❡♥t ♦❢ ✶❄

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥

Rn

❘❡❝❛❧❧ ❢r♦♠ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶P❈✲✶✮ t❤❡ ✐♠❛❣❡ ♦❢ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ❜♦✉♥❝✐♥❣ ♦✛ t❤❡ ✢♦♦r✳ ❘❡❝♦r❞✐♥❣ ✐ts ❤❡✐❣❤t ❡✈❡r② t✐♠❡ ❣✐✈❡s ✉s ❛ s❡q✉❡♥❝❡ ✿

■t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✦ ■♠❛❣✐♥❡ ♥♦✇ ✇❛t❝❤✐♥❣ ❛ ❜❛❧❧ ❜♦✉♥❝✐♥❣ ♦♥ ❛♥ ✉♥❡✈❡♥ s✉r❢❛❝❡ ✇✐t❤ ✐ts ❧♦❝❛t✐♦♥s r❡❝♦r❞❡❞ ❛t ❡q✉❛❧ ♣❡r✐♦❞s ♦❢

✶✳✶✷✳

❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn

✶✵✼

t✐♠❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡✿

❲❡ ✇✐❧❧ st✉❞② ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❡s♣❡❝✐❛❧❧② t❤❡✐r s♦♠❡t✐♠❡s ✏❛❝❝✉♠✉❧❛t✐♥❣✑ ❛r♦✉♥❞ ❛ s✐♥❣❧❡ ❧♦❝❛t✐♦♥✳

tr❡♥❞s✳

❆♥

✐♥✜♥✐t❡

s❡q✉❡♥❝❡ ✇✐❧❧ ❜❡

❚❤❡ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ ❜❛❧❧ ❛♥❞ t❤❡ ❞r❛✐♥ ❜❡❝♦♠❡s

✐♥✈✐s✐❜❧❡✦

❉❡✜♥✐t✐♦♥ ✶✳✶✷✳✶✿ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛ r❛② ✐♥ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱

{p, p + 1, ...}✱

✐♥✜♥✐t❡ s❡q✉❡♥❝❡✱ ♦r s✐♠♣❧② s❡q✉❡♥❝❡ ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥✿

✐s ❝❛❧❧❡❞ ❛♥

Ak : k = p, p + 1, p + 2, ... , ♦r✱ ❛❜❜r❡✈✐❛t❡❞✱

Ak . ❊✈❡r② ❢✉♥❝t✐♦♥

X = f (t)

✇✐t❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥ ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡✿

Ak = f (k) . ❲❡ ✈✐s✉❛❧✐③❡ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s ❛s

s❡q✉❡♥❝❡s ♦❢ ♣♦✐♥ts

♦♥ t❤❡

xy ✲♣❧❛♥❡

✈✐❛ t❤❡✐r ❣r❛♣❤s✳

❊①❛♠♣❧❡ ✶✳✶✷✳✷✿ r❡❝✐♣r♦❝❛❧s ❚❤❡ ❣♦✲t♦ ❡①❛♠♣❧❡ ✐s t❤❛t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♠❛❞❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿

Ak = ■t



cos k sin k , k k



.

t❡♥❞s t♦ 0 ✇❤✐❧❡ s♣✐r❛❧✐♥❣ ❛r♦✉♥❞ ✐t✳

❚❤✐s ❢❛❝t ✐s ❡❛s✐❧② ❝♦♥✜r♠❡❞ ♥✉♠❡r✐❝❛❧❧②✳

❯♥❢♦rt✉♥❛t❡❧②✱ ♥♦t ❛❧❧ s❡q✉❡♥❝❡s ❛r❡ ❛s s✐♠♣❧❡ ❛s t❤❛t✳

❚❤❡② ♠❛② ❛♣♣r♦❛❝❤ t❤❡✐r r❡s♣❡❝t✐✈❡ ❧✐♠✐ts ✐♥ ❛♥

✐♥✜♥✐t❡ ✈❛r✐❡t② ♦❢ ✇❛②s✳ ❆♥❞ t❤❡♥ t❤❡r❡ ❛r❡ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦ ❧✐♠✐ts✳ ❲❡ ♥❡❡❞ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤✳

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn

✶✵✽

▲❡t✬s r❡❝❛❧❧ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢ r❡❛❧ ♥✉♠❜❡rs✿

❚❤❡ ✐❞❡❛ ✐s t❤❛t t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ k t❤ ♣♦✐♥t t♦ t❤❡ ❧✐♠✐t ✐s ❣❡tt✐♥❣ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✿ |xk − a| → 0 ❛s k → ∞ .

■♥ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ t❤❡ ✐❞❡❛ r❡♠❛✐♥s t❤❡ s❛♠❡✿ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ✐♥ Rn ✐s ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ ✐ts ❧✐♠✐t✱ ✇❤✐❝❤ ✐s ❛❧s♦ ❛ ♣♦✐♥t ✐♥ Rn ✿

❲❡ r❡✇r✐t❡ ✇❤❛t ✇❡ ✇❛♥t t♦ s❛② ❛❜♦✉t t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧✐♠✐ts ✐♥ ♣r♦❣r❡ss✐✈❡❧② ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡ t❡r♠s✳ k

X = Ak

❆s k ❛♣♣r♦❛❝❤❡s ∞,

X ❛♣♣r♦❛❝❤❡s A.

❆s k ✐s ❣❡tt✐♥❣ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r,

t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ X t♦ A ❛♣♣r♦❛❝❤❡s 0.

❇② ♠❛❦✐♥❣ k ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r,

✇❡ ♠❛❦❡ d(X, A) ❛s s♠❛❧❧ ❛s ♥❡❡❞❡❞.

❆s k → ∞,

✇❡ ❤❛✈❡ X → A.

❇② ♠❛❦✐♥❣ k ❧❛r❣❡r t❤❛♥ s♦♠❡ N > 0, ✇❡ ♠❛❦❡ d(X, A) s♠❛❧❧❡r t❤❛♥ ❛♥② ❣✐✈❡♥ ε > 0. ❚❤❡♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ◮ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ε > 0✱ t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r N s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r k > N ✱ ✇❡ ❤❛✈❡ d(Ak , A) < ε .

❖✉r st✉❞② ❜❡❝♦♠❡ ♠✉❝❤ ❡❛s✐❡r ♦♥❝❡ ✇❡ r❡❛❧✐③❡ t❤❛t ❞✐st❛♥❝❡s ❛r❡ ♥✉♠❜❡rs ❛♥❞ d(Ak , A) ✐s ❥✉st ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✦ ❯♥❞❡rst❛♥❞✐♥❣ ❞✐st❛♥❝❡s ✐♥ Rn ❛♥❞ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s ❛❧❧♦✇s ✉s ❡❛s✐❧② t♦ s♦rt t❤✐s ♦✉t✳

❉❡✜♥✐t✐♦♥ ✶✳✶✷✳✸✿ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❛♥❞ ❧✐♠✐t ❙✉♣♣♦s❡ Ak : k = 1, 2, 3... ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ✐♥ Rn ✳ ❲❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ ❛♥♦t❤❡r ♣♦✐♥t A ✐♥ Rn ✱ ❝❛❧❧❡❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢✿ d(An , A) → 0 ❛s k → ∞ ,

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

Ak → A ❛s k → ∞ ,

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn

✶✵✾

♦r

A = lim Ak . k→∞

■❢ ❛ s❡q✉❡♥❝❡ ❤❛s ❛ ❧✐♠✐t✱ t❤❡♥ ✇❡ ❝❛❧❧ t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡♥t ❛♥❞ s❛② t❤❛t ✐t ❝♦♥✈❡r❣❡s ❀ ♦t❤❡r✇✐s❡ ✐t ✐s ❞✐✈❡r❣❡♥t ❛♥❞ ✇❡ s❛② ✐t ❞✐✈❡r❣❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦✐♥ts st❛rt t♦ ❛❝❝✉♠✉❧❛t❡ ✐♥ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❝✐r❝❧❡s ❛r♦✉♥❞ A✳ ❆ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ tr❡♥❞ ✐♥ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s t♦ ❡♥❝❧♦s❡ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐♥ ❛ ❞✐s❦ ✿

■t s❤♦✉❧❞ ❜❡✱ ✐♥ ❢❛❝t✱ ❛ ♥❛rr♦✇❡r ❛♥❞ ♥❛rr♦✇❡r ❜❛♥❞❀ ✐ts ✇✐❞t❤ ✐s 2ε✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ t❤❡ ❜❛♥❞ ♠♦✈❡s t♦ t❤❡ r✐❣❤t❀ t❤❛t✬s N ✳ ❊①❛♠♣❧❡s ♦❢ ❞✐✈❡r❣❡♥❝❡ ❛r❡ ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✶✳✶✷✳✹✿ ❣♦ t♦ ✐♥✜♥✐t② ❆ s❡q✉❡♥❝❡ ♠❛② t❡♥❞ t♦ ✐♥✜♥✐t②✱ s✉❝❤ ❛s Ak = (k, k) ❛t t❤❡ s✐♠♣❧❡st✿

❚❤❡♥ ♥♦ ❞✐s❦ ✕ ♥♦ ♠❛tt❡r ❤♦✇ ❧❛r❣❡ ✕ ✇✐❧❧ ❝♦♥t❛✐♥ t❤❡ s❡q✉❡♥❝❡✬s t❛✐❧✳ ❚❤✐s ❜❡❤❛✈✐♦r ❤♦✇❡✈❡r ❤❛s ❛ ♠❡❛♥✐♥❣❢✉❧ ♣❛tt❡r♥✳

❉❡✜♥✐t✐♦♥ ✶✳✶✷✳✺✿ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t② ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ An t❡♥❞s t♦ ✐♥✜♥✐t② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ◮ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r R✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r k > N ✱ ✇❡ ❤❛✈❡

d(0, An ) > R . ❲❡ ❞❡s❝r✐❜❡ s✉❝❤ ❛ ❜❡❤❛✈✐♦r ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥

Rn

✶✶✵

▲✐♠✐t

Ak → ∞

❛s

k→∞

❲❡ ♥❡❡❞ t♦ ❥✉st✐❢② ✏t❤❡✑ ✐♥ ✏t❤❡ ❧✐♠✐t✑✳ ❚❤❡♦r❡♠ ✶✳✶✷✳✻✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❙❡q✉❡♥❝❡

❆ s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ A ❛♥❞ B ❛r❡ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ s❡q✉❡♥❝❡✱ t❤❡♥ A = B ✳ Pr♦♦❢✳

❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦♦❢ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✇❡ ✇❛♥t t♦ s❡♣❛r❛t❡ t❤❡s❡ t✇♦ ♣♦✐♥ts ❜② t✇♦ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❞✐s❦s✳ ❚❤❡♥ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✜t ♦♥❡ ♦r t❤❡ ♦t❤❡r✱ ❜✉t ♥♦t ❜♦t❤✳ ■♥ ♦r❞❡r ❢♦r t❤❡♠ t♦ ❜❡ ❞✐s❥♦✐♥t✱ t❤❡✐r r❛❞✐✐ ✭t❤❛t✬s

ε✦✮

s❤♦✉❧❞ ❜❡ ❧❡ss t❤❛♥ ❤❛❧❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣♦✐♥ts✳

❚❤❡ ♣r♦♦❢ ✐s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙✉♣♣♦s❡

A ❛♥❞ B ε=

❛r❡ t✇♦ ❧✐♠✐ts✱ ✐✳❡✳✱ ❡✐t❤❡r s❛t✐s✜❡s t❤❡ ❞❡✜♥✐t✐♦♥✳ ▲❡t

d(A, B) . 2

◆♦✇✱ ✇❡ ✇r✐t❡ t❤❡ ❞❡✜♥✐t✐♦♥ t✇✐❝❡✿

•

t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r

L

s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

k > L✱

✇❡ ❤❛✈❡

d(Ak , A) < ε , ❛♥❞

•

t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r

M

s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

k > M✱

✇❡ ❤❛✈❡

d(Ak , B) < ε . ■♥ ♦r❞❡r t♦ ❝♦♠❜✐♥❡ t❤❡ t✇♦ st❛t❡♠❡♥ts ✇❡ ♥❡❡❞ t❤❡♠ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r t❤❡ s❛♠❡ ✈❛❧✉❡s ♦❢

N = min{L, M } . ❚❤❡♥✱ ❢♦r ❡✈❡r② ♥✉♠❜❡r

k > N✱

✇❡ ❤❛✈❡

d(Ak , A) < ε ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r ❡✈❡r②

k > N✱

❛♥❞

d(Ak , B) < ε .

✇❡ ❤❛✈❡ ❜② t❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② ✿

d(A, B) ≤ d(A, Ak ) + d(Ak , B) < ε + ε < 2ε . ❆ ❝♦♥tr❛❞✐❝t✐♦♥✳

k✳

▲❡t

✶✳✶✷✳ ❙❡q✉❡♥❝❡s ❛♥❞ t♦♣♦❧♦❣② ✐♥ Rn

✶✶✶

❊①❡r❝✐s❡ ✶✳✶✷✳✼ ❋♦❧❧♦✇ t❤❡ ♣r♦♦❢ ❛♥❞ ❞❡♠♦♥str❛t❡ t❤❛t t❤❛t ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❢♦r ❛ s❡q✉❡♥❝❡ t♦ ❤❛✈❡ ❛s ❧✐♠✐t✿ ❛ ♣♦✐♥t ❛♥❞ ✐♥✜♥✐t②✳ ❚❤✉s✱ t❤❡r❡ ❝❛♥ ❜❡ ♥♦ t✇♦ ❧✐♠✐ts ❛♥❞ ✇❡ ❛r❡ ❥✉st✐✜❡❞ t♦ s♣❡❛❦ ♦❢ t❤❡ ❧✐♠✐t✳

❲❤❛t ✐s t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣ ♦❢ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧❄

❈♦♠♣❛r❡ t❤❡ ❞✐s❦ ❛♥❞ t❤❡ ❞✐s❦ ✇✐t❤ ✐ts ❜♦✉♥❞❛r② ✭t❤❡ ❝✐r❝❧❡✮ r❡♠♦✈❡❞✿

{(x, y) : x2 + y 2 ≤ 1} ✈s✳ {(x, y) : x2 + y 2 < 1} . ❖r t❤✐♥❦ ♦❢ t❤❡ ❜❛❧❧ ❛♥❞ t❤❡ ❜❛❧❧ ✇✐t❤ ✐ts ❜♦✉♥❞❛r② ✭t❤❡ s♣❤❡r❡✮ r❡♠♦✈❡❞✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✐♥ t❤❡ ❧❛tt❡r ♦♥❡ ❝❛♥ r❡❛❝❤ t❤❡ ❜♦✉♥❞❛r② ✕ ❛♥❞ t❤❡ ♦✉ts✐❞❡ ♦❢ t❤❡ s❡t ✕ ❜② ❢♦❧❧♦✇✐♥❣ ❛ s❡q✉❡♥❝❡ t❤❛t ❧✐❡s ❡♥t✐r❡❧② ✐♥s✐❞❡ t❤❡ s❡t✦

❉❡✜♥✐t✐♦♥ ✶✳✶✷✳✽✿ ❝❧♦s❡❞ s❡t ❆ s❡t ✐♥ Rn ✐s ❝❛❧❧❡❞ ❝❧♦s❡❞ ✐❢ ✐t ❝♦♥t❛✐♥s t❤❡ ❧✐♠✐ts ♦❢ ❛❧❧ ♦❢ ✐ts ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✳

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✷

❉❡✜♥✐t✐♦♥ ✶✳✶✷✳✾✿ ❜♦✉♥❞❡❞ s❡t ❆ s❡t S ✐♥ Rn ✐s ❜♦✉♥❞❡❞ ✐❢ ✐t ✜ts ✐♥ ❛ s♣❤❡r❡ ✭♦r ❛ ❜♦①✮ ♦❢ ❛ ❧❛r❣❡ ❡♥♦✉❣❤ s✐③❡✿ d(x, 0) < Q ❢♦r ❛❧❧ x ✐♥ S .

✶✳✶✸✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

❍❡r❡ ✇❡ ❣♦ ❜❛❝❦ ❢r♦♠ t❤❡ tr❡❛t♠❡♥t ♦❢ t❤❡ s♣❛❝❡ t♦ t❤❡ s♣r❡❛❞✲♦✉t ❝♦♦r❞✐♥❛t❡✇✐s❡ ❛♣♣r♦❛❝❤✳ ❙✉♣♣♦s❡ s♣❛❝❡ Rn ✐s s✉♣♣❧✐❡❞ ✇✐t❤ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✳ ▲❡t✬s ✜rst ❧♦♦❦ ❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ Xk ✐♥ Rn ✳ ❚❤❡ ❧✐♠✐t ✐s ❞❡✜♥❡❞ t♦ ❜❡ s✉❝❤ ❛ ♣♦✐♥t A ✐♥ Rn t❤❛t d(Xk , A) → 0 ❛s k → ∞ . ❙✉♣♣♦s❡ ✇❡ ✉s❡ t❤❡ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝s ✐s ♦✉r s♣❛❝❡ R3 ✱ ✇❤❛t ❞♦❡s t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ ♠❡❛♥❄ ❙✉♣♣♦s❡ X k = x k , yk , z k

❚❤❡♥ ✇❡ ❤❛✈❡✿



❛♥❞ A = (a, b, c) .

p (xk − a)2 + (yk − b)2 + (zk − c)2 → 0 .

❚❤✐s ❧✐♠✐t ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ (xk − a)2 + (yk − b)2 + (zk − c)2 → 0 ,

❜❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥ u2 ✐s ❝♦♥t✐♥✉♦✉s ❛t 0✳ ❙✐♥❝❡ t❤❡s❡ t❤r❡❡ t❡r♠s ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡✱ ❛❧❧ t❤r❡❡ ❤❛✈❡ t♦ ❛♣♣r♦❛❝❤ 0✦ ❚❤❡♥ ✇❡ ❤❛✈❡✿ (xk − a)2 → 0, (yk − b)2 → 0, (zk − c)2 → 0 .

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✸

❚❤❡s❡ ❧✐♠✐ts ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❜❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥

√

|xk − a| → 0, |yk − b| → 0, |zk − c| → 0 , u

✐s ❝♦♥t✐♥✉♦✉s ❢r♦♠ t❤❡ r✐❣❤t ❛t

0✳

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡✿

xk → a, yk → b, zk → c . ❆❧❧ ❝♦♦r❞✐♥❛t❡ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡✦ ❋♦r t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡✱ ❜✉✐❧❞ ❛ t❛❜❧❡✿

1

2

...

n

Xk x1k x2k ... xnk ↓

↓

↓

...

↓

❛s

a1 a2 ... an

A

k→∞

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ s✉♠♠❛r②✳ ❚❤❡♦r❡♠ ✶✳✶✸✳✶✿ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ❆ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ❝♦♦r❞✐♥❛t❡ ♦❢

Xk

Xk → A

Xk

✐♥

Rn

❝♦♥✈❡r❣❡ t♦ ❛♥♦t❤❡r ♣♦✐♥t

A

✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r②

❝♦♥✈❡r❣❡s t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♦r❞✐♥❛t❡ ♦❢

❛s

k → ∞ ⇐⇒ xik → ai

❛s

k→∞

❢♦r ❛❧❧

A❀

✐✳❡✳✱

i = 1, 2, ..., n ,

✇❤❡r❡

Xk = (x1k , x2k , ..., xnk )

❛♥❞

A = (a1 , a2 , ..., an ) .

❊①❡r❝✐s❡ ✶✳✶✸✳✷

Pr♦✈❡ t❤❡ ✏✐❢ ✑ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✶✳✶✸✳✸✿ ❛ ❝♦♠♣✉t❛t✐♦♥

❲❡ ❝♦♠♣✉t❡ ✉s✐♥❣ t❤❡

❈♦♥t✐♥✉✐t② ❘✉❧❡ ❢♦r ◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ✿ lim

k→∞



1 1 cos , sin k k





1 1 = lim cos , lim sin k→∞ k t→∞ k = (cos 0, sin 0)



= (1, 0), ❜❡❝❛✉s❡

sin t

❛♥❞

cos t

❛r❡ ❝♦♥t✐♥✉♦✉s✳

❚❤✐s t❤❡♦r❡♠ ❛❧s♦ ♠❛❦❡s ✐t ❡❛s② t♦ ♣r♦✈❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ❢r♦♠ t❤♦s❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ◆♦✇✱ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs ❛♥❞ t❤❡✐r ❧✐♠✐ts✳ ❱❡❝t♦rs ❝♦rr❡s♣♦♥❞ t♦ ♣♦✐♥ts✿

OP ←→ P

❲❡ ❛r❡ t❤❡♥ ❛❜❧❡ t♦ ❞✐s❝✉ss

❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs✳

❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❥✉st r❡st❛t❡ t❤❡

❞❡✜♥✐t✐♦♥ ❣✐✈❡♥ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r r❡♣❧❛❝✐♥❣ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ♣♦✐♥ts ✇✐t❤ ♠❛❣♥✐t✉❞❡s ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ ✈❡❝t♦rs✿

d(P Q) = ||OQ − OP || . ❙✉♣♣♦s❡

{Ak : k = 1, 2, 3...}

❢♦❧❧♦✇✐♥❣ r❡q✉✐r❡♠❡♥t✿

✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✈❡❝t♦rs ✐♥

Rn ✳

❋✐rst✱ ♦✉r ❞❡✜♥✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✹

◮ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ε > 0✱ t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r N s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r k > N ✱ ✇❡ ❤❛✈❡ ||Ak − A|| < ε .

❉❡✜♥✐t✐♦♥ ✶✳✶✸✳✹✿ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ♦❢ {Ak : k = 1, 2, 3...} ♦❢ ✈❡❝t♦rs ✐♥ Rn ❝♦♥✈❡r❣❡s t♦ ❛♥♦t❤❡r ✈❡❝t♦r A ✐♥ Rn ✱ ❝❛❧❧❡❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✿ ||Ak − A|| → 0 ❛s k → ∞ . ❚❤✐s ❧✐♠✐t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ♦r

Ak → A ❛s k → ∞ , A = lim Ak . k→∞

❲❡ ❝♦♥s✐❞❡r ♥♦✇ t❤❡ ❛❧❣❡❜r❛ ♦❢ s❡q✉❡♥❝❡s ❛♥❞ ❧✐♠✐ts✳ ▲✐♠✐ts ❜❡❤❛✈❡ ✇❡❧❧ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❡❝t♦r ♦♣❡r❛✲ t✐♦♥s✳ ❇❡❧♦✇ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ s❡q✉❡♥❝❡s ❛r❡ ❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ s❡t ♦❢ ✐♥t❡❣❡rs✳ ❲❡ st❛rt ✇✐t❤ ❛❞❞✐t✐♦♥✳

❚♦ ❣r❛♣❤✐❝❛❧❧② ❛❞❞ t✇♦ s❡q✉❡♥❝❡s✱ ✇❡ ♣❧♦t ♣❛r❛❧❧❡❧♦❣r❛♠s✳ ❚❤❡♥✱ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ t❤❡s❡ ♣❛r❛❧❧❡❧♦❣r❛♠s ❢♦r♠ t❤❡ ♥❡✇ s❡q✉❡♥❝❡✳ ◆♦✇✱ ✐❢ ❡✐t❤❡r s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ 0✱ t❤❡♥ s♦ ❞♦ t❤❡s❡ ❞✐❛❣♦♥❛❧s✳

❚❤❡♦r❡♠ ✶✳✶✸✳✺✿ ❙✉♠ ❘✉❧❡ Ak → 0 ❛♥❞ Bk → 0 =⇒ Ak + Bk → 0

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✺

Pr♦♦❢✳

❋r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ✐t ❢♦❧❧♦✇s✿ ||Ak || → 0 ❛♥❞ ||Bk || → 0 .

❙✐♥❝❡ t❤❡s❡ t✇♦ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡ t♦ 0✱ t❤❡♥ s♦ ❞♦❡s t❤❡✐r s✉♠✿ ||Ak || + ||Bk || → 0 ,

❛s ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② ✇❡ ❝♦♥❝❧✉❞❡✿

✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✳ ❋r♦♠ t❤❡

||Ak + Bk || ≤ ||Ak || + ||Bk || → 0 .

❚❤❡r❡❢♦r❡✱ ❜② ❞❡✜♥✐t✐♦♥ Ak + Bk → 0✳ ▼✉❧t✐♣❧②✐♥❣ ❛ s❡q✉❡♥❝❡ ❜② ❛ s❝❛❧❛r s✐♠♣❧② str❡t❝❤❡s t❤❡ ✇❤♦❧❡ ♣✐❝t✉r❡ ✉♥✐❢♦r♠❧②✳

❚❤❡♦r❡♠ ✶✳✶✸✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡

Ak → 0 =⇒ cAk → 0 ❢♦r ❛♥② r❡❛❧ c Pr♦♦❢✳

❋r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ✐t ❢♦❧❧♦✇s✿ ||Ak || → 0 .

❙✐♥❝❡ t❤✐s ✐s ❛ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡ t❤❛t ❝♦♥✈❡r❣❡ t♦ 0✱ t❤❡♥ s♦ ❞♦❡s ✐ts ♠✉❧t✐♣❧❡✿ |c| ||Ak || → 0 ,

❛s ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❚❤❡♥✿

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s

❚❤❡r❡❢♦r❡✱ ❜② ❞❡✜♥✐t✐♦♥ cAk → 0✳

✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✳

||cAk || = |c| ||Ak || → 0 .

❋♦r ♠♦r❡ ❝♦♠♣❧❡① s✐t✉❛t✐♦♥s ✇❡ ♥❡❡❞ t♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s ❛r❡ ❜♦✉♥❞❡❞✳ ❚❤❡♦r❡♠ ✶✳✶✸✳✼✿ ❇♦✉♥❞❡❞♥❡ss

Ak → A =⇒ ||Ak || < Q ❢♦r s♦♠❡ r❡❛❧ Q .

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✻

Pr♦♦❢✳

❚❤❡ ✐❞❡❛ ✐s t❤❛t t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇✐❧❧ ✜t ✐♥t♦ s♦♠❡ ❜❛❧❧ ❛r♦✉♥❞ t❤❡ ❧✐♠✐t❀ ♠❡❛♥✇❤✐❧❡✱ t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② t❡r♠s ❧❡❢t✳✳✳ ❈❤♦♦s❡ ε = 1✳ ❚❤❡♥ ❜② ❞❡✜♥✐t✐♦♥✱ t❤❡r❡ ✐s s✉❝❤ N t❤❛t ❢♦r ❛❧❧ k > N ✇❡ ❤❛✈❡✿ ||Ak − A|| < 1 .

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

||Ak || = ||(Ak − A) + A||

❇② ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳

≤ ||Ak − A|| + ||A|| ❇② ✐♥❡q✉❛❧✐t② ❛❜♦✈❡✳

❚♦ ✜♥✐s❤ t❤❡ ♣r♦♦❢✱ ✇❡ ❝❤♦♦s❡✿

< 1 + ||A||.

Q = max{||A1 ||, ..., ||Ak ||, 1 + ||A||} .

❚❤❡ ♣r♦♦❢ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✿ ◆♦t ❡✈❡r② ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐s ❝♦♥✈❡r❣❡♥t✳ ❲❡ ✇✐❧❧ s❤♦✇ ❧❛t❡r t❤❛t✱ ✇✐t❤ ❛♥ ❡①tr❛ ❝♦♥❞✐t✐♦♥✱ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡s ❞♦ ❤❛✈❡ t♦ ❝♦♥✈❡r❣❡✳✳✳ ❲❡ ❛r❡ ♥♦✇ r❡❛❞② ❢♦r t❤❡ ❣❡♥❡r❛❧ r❡s✉❧ts ♦♥ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❧✐♠✐ts✳

❚❤❡♦r❡♠ ✶✳✶✸✳✽✿ ❙✉♠ ❘✉❧❡ ■❢ s❡q✉❡♥❝❡s

Ak , Bk

❝♦♥✈❡r❣❡ t❤❡♥ s♦ ❞♦❡s

Ak + Bk ✱

❛♥❞

lim (Ak + Bk ) = lim Ak + lim Bk .

k→∞

k→∞

k→∞

✶✳✶✸✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ♦❢ s❡q✉❡♥❝❡s

✶✶✼

Pr♦♦❢✳

❙✉♣♣♦s❡

Ak → A ❛♥❞ Bk → B .

❚❤❡♥✱

||Ak − A|| → 0 ❛♥❞ ||Bk − B|| → 0 .

❲❡ ❝♦♠♣✉t❡✿

||(Ak + Bk ) − (A + B)|| = ||(Ak − A) + (Bk − B)|| ❇② ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳ ≤ ||Ak − A|| + ||Bk − B||

❇② ❙❘ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s✳

→0+0

= 0.

❚❤❡♥✱ ❜② t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿ ||(Ak + Bk ) − (A + B)|| → 0 .

❚❤❡♥✱ ❜② t❤❡ ✜rst t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿ Ak + Bk → A + B .

❚❤❡♦r❡♠ ✶✳✶✸✳✾✿ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ■❢ s❡q✉❡♥❝❡s

Ak , Bk

❝♦♥✈❡r❣❡ t❤❡♥ s♦ ❞♦❡s

lim (Ak · Bk ) =

k→∞

Ak · Bk ✱

 lim Ak ·

k→∞

❛♥❞ ✇❡ ❤❛✈❡✿

lim Bk

k→∞



❊①❡r❝✐s❡ ✶✳✶✸✳✶✵

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ❝♦r♦❧❧❛r②✳ ❚❤❡♦r❡♠ ✶✳✶✸✳✶✶✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ■❢ s❡q✉❡♥❝❡

Ak

❝♦♥✈❡r❣❡s t❤❡♥ s♦ ❞♦❡s

cAk

❢♦r ❛♥② r❡❛❧

lim c Ak = c · lim Ak

k→∞

k→∞

c✱

❛♥❞ ✇❡ ❤❛✈❡✿

✶✳✶✹✳

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

✶✶✽

❲❛r♥✐♥❣✦

■t ✐s ❝♦♥s✐❞❡r❡❞ ❛ s❡r✐♦✉s ❡rr♦r ✐❢ ②♦✉ ✉s❡ t❤❡ ❝♦♥❝❧✉✲ s✐♦♥ ✭t❤❡ ❢♦r♠✉❧❛✮ ♦♥❡ ♦❢ t❤❡s❡ r✉❧❡s ✇✐t❤♦✉t ✈❡r✲ ✐❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ✭t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡✲ q✉❡♥❝❡s ✐♥✈♦❧✈❡❞✮✳

❲❡ r❡♣r❡s❡♥t t❤❡

❙✉♠ ❘✉❧❡

❛s ❛ ❞✐❛❣r❛♠✿

Ak , Bk   + y

lim

−−−−−→

A, B   + y

SR lim

Ak + Bk −−−−−→ lim(Ak + Bk ) = A + B ■♥ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

•

r✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✐t❤❡r✱ t❤❡♥ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts❀ ♦r

•

❞♦✇♥✿ ❛❞❞ t❤❡♠✱ t❤❡♥ r✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❡s✉❧t✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

✶✳✶✹✳ P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

■♥ ❱♦❧✉♠❡s ✷ ❛♥❞ ✸✱ ✇❡ ✉s❡❞ ♣❛rt✐t✐♦♥s ♦❢ ✐♥t❡r✈❛❧s✱ ❛s ✇❡❧❧ ❛s t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡✱ ✐♥ ♦r❞❡r t♦ st✉❞②

❝❤❛♥❣❡✳

❚❤✐s t✐♠❡✱ ✇❡ ♥❡❡❞ ♣❛rt✐t✐♦♥s ♦❢ t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

❢r♦♠ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ❛①❡s✳ ❋♦r ❞✐♠❡♥s✐♦♥

2✱

t❤❡s❡ ❛r❡

r❡❝t❛♥❣❧❡s✳

❆♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡

x✲❛①✐s✿

[a, b] = {x : a ≤ x ≤ b} , ❛♥❞ ❛♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡

y ✲❛①✐s✿

♠❛❦❡ ❛ r❡❝t❛♥❣❧❡ ✐♥ t❤❡

xy ✲♣❧❛♥❡✿

✐♥❝r❡♠❡♥t❛❧

❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳ ❚❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ✇✐❧❧ ❝♦♠❡

[c, d] = {y : c ≤ y ≤ d} ,

[a, b] × [c, d] = {(x, y) : a ≤ x ≤ b, c ≤ y ≤ d} .

✶✳✶✹✳

❆

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

✶✶✾

♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d] ✐s ♠❛❞❡ ♦❢ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❛❜♦✈❡✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s

❲❡ st❛rt ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

[a, b]

[a, b]

✐♥ t❤❡

✐♥ t❤❡

x✲❛①✐s

x✲❛①✐s

❛♥❞

✐♥t♦

n

[c, d]

✐♥ t❤❡

y ✲❛①✐s✿

✐♥t❡r✈❛❧s✿

[x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] , ✇✐t❤

x0 = a, xn = b✳

❚❤❡♥ ✇❡ ❞♦ t❤❡ s❛♠❡ ❢♦r

y✳

❲❡ ♣❛rt✐t✐♦♥ ❛♥ ✐♥t❡r✈❛❧

[c, d]

✐♥ t❤❡

y ✲❛①✐s

✐♥t♦

m

✐♥t❡r✈❛❧s✿

[y0 , y1 ], [y1 , y2 ], ..., [ym−1 , ym ] , ✇✐t❤

y0 = c, yn = d✳

❚❤❡ ❧✐♥❡s

y = yj

[yj , yj+1 ]✳

❚❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ ❧✐♥❡s✱

❛♥❞

x = xi

❝r❡❛t❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡

[a, b] × [c, d] ✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s [xi , xi+1 ] ×

Xij = (xi , yj ), i = 1, 2, ..., n, j = 1, 2, ..., m , ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡

♥♦❞❡s

♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❙♦✱ t❤❡r❡ ❛r❡ ♥♦❞❡s ❛♥❞ t❤❡r❡ ❛r❡ r❡❝t❛♥❣❧❡s ✭t✐❧❡s✮❀ ✐s t❤❛t ✐t❄

❚❤✐s ✐s ❤♦✇ ❛♥ ♦❜❥❡❝t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✇✐t❤ t✐❧❡s✱ ♦r ♣✐①❡❧s✿

✶✳✶✹✳

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

◆♦✇✱ ❛r❡

❝✉r✈❡s

✶✷✵

❛❧s♦ ♠❛❞❡ ♦❢ t✐❧❡s❄ ❙✉❝❤ ❛ ❝✉r✈❡ ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡ t❤✐s✿

■❢ ✇❡ ❧♦♦❦ ❝❧♦s❡r✱ ❤♦✇❡✈❡r✱ t❤✐s ✏❝✉r✈❡✑ ✐s♥✬t ❛ ❝✉r✈❡ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡❀ ✐t✬s t❤✐❝❦✦ ❚❤❡ ❝♦rr❡❝t ❛♥s✇❡r ✐s✿

❝✉r✈❡s ❛r❡ ♠❛❞❡ ♦❢ ❡❞❣❡s

♦❢ t❤❡ ❣r✐❞✿

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❛t ✇❡ ♥❡❡❞ t♦ ✐♥❝❧✉❞❡✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ sq✉❛r❡s✱ t❤❡ ✏t❤✐♥♥❡r✑ ❝❡❧❧s ❛s ❛❞❞✐t✐♦♥❛❧ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s✳ ❚❤❡ ❝♦♠♣❧❡t❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣✐①❡❧ ✐s s❤♦✇♥ ❜❡❧♦✇❀ t❤❡ ❡❞❣❡s ❛♥❞ ✈❡rt✐❝❡s ❛r❡ s❤❛r❡❞ ✇✐t❤ ❛❞❥❛❝❡♥t ♣✐①❡❧s✿

❊①❛♠♣❧❡ ✶✳✶✹✳✶✿ ❞✐♠❡♥s✐♦♥ ✶

❲❡ st❛rt ✇✐t❤ ❞✐♠❡♥s✐♦♥

n = 1✿

■♥ t❤✐s s✐♠♣❧❡st ♦❢ ♣❛rt✐t✐♦♥s✱ t❤❡ ❝❡❧❧s ❛r❡✿

• • •

0✲❝❡❧❧✱ ✐s {k} ✇✐t❤ k = ... − 2, −1, 0, 1, 2, 3, ...✳ 1✲❝❡❧❧✱ ✐s [k, k + 1] ✇✐t❤ k = ... − 2, −1, 0, 1, 2, 3, ...✳ ❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ 0✲❝❡❧❧s✳

❆ ♥♦❞❡✱ ♦r ❛

❆♥ ❡❞❣❡✱ ♦r ❛ ❆♥❞✱

1✲❝❡❧❧s

✶✳✶✹✳

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

✶✷✶

❊①❛♠♣❧❡ ✶✳✶✹✳✷✿ ❞✐♠❡♥s✐♦♥ ✷

❋♦r t❤❡ ❞✐♠❡♥s✐♦♥ n = 2 ❣r✐❞✱ ✇❡ ❞❡✜♥❡ ❝❡❧❧s ❢♦r ❛❧❧ ✐♥t❡❣❡rs k, m ❛s ♣r♦❞✉❝ts✿ • ❆ ♥♦❞❡✱ ♦r ❛ 0✲❝❡❧❧✱ ✐s {k} × {m}✳ • ❆♥ ❡❞❣❡✱ ♦r ❛ 1✲❝❡❧❧✱ ✐s {k} × [m, m + 1] ♦r [k, k + 1] × {m}✳ • ❆ sq✉❛r❡✱ ♦r ❛ 2✲❝❡❧❧✱ ✐s [k, k + 1] × [m, m + 1]✳ ❲❡ ❛❧s♦ ❤❛✈❡✿ • ❚❤❡ 2✲❝❡❧❧s ❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ 1✲❝❡❧❧s✳ • ❆♥❞✱ st✐❧❧✱ t❤❡ 1✲❝❡❧❧s ❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ 0✲❝❡❧❧s✳

❈❡❧❧s s❤♦✇♥ ❛❜♦✈❡ ❛r❡✿ • 0✲❝❡❧❧ {3} × {3}✱ • 1✲❝❡❧❧s [2, 3] × {1} ❛♥❞ {2} × [2, 3]✱ • 2✲❝❡❧❧ [1, 2] × [1, 2]✳ ❙✐♠✐❧❛r❧② ❢♦r ❞✐♠❡♥s✐♦♥ 3✱ ✇❡ ❤❛✈❡ ❜♦①❡s✳ ■♥t❡r✈❛❧s ✐♥ t❤❡ x✲✱ y ✲✱ ❛♥❞ z ✲❛①❡s✿ [a, b] = {x : a ≤ x ≤ b}, [c, d] = {y : c ≤ y ≤ d}, [p, q] = {z : p ≤ z ≤ q} ,

♠❛❦❡ ❛ ❜♦① ✐♥ t❤❡ xyz ✲s♣❛❝❡✿ [a, b] × [c, d] × [p, q] = {(x, y) : a ≤ x ≤ b, c ≤ y ≤ d, p ≤ z ≤ q} .

■♥ ❞✐♠❡♥s✐♦♥ 3✱ s✉r❢❛❝❡s

❛r❡ ♠❛❞❡ ♦❢ ❢❛❝❡s ♦❢ ♦✉r ❜♦①❡s❀ ✐✳❡✳✱ t❤❡s❡ ❛r❡ t✐❧❡s✿

❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❜♦① ❢♦❧❧♦✇s ❛♥❞ ❤❡r❡✱ ♦♥❝❡ ❛❣❛✐♥✱ t❤❡ ❢❛❝❡s✱ ❡❞❣❡s✱ ❛♥❞ ✈❡rt✐❝❡s ❛r❡ s❤❛r❡❞✿

✶✳✶✹✳

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

✶✷✷

❊①❛♠♣❧❡ ✶✳✶✹✳✸✿ ❞✐♠❡♥s✐♦♥ ✸ i, m, k ✱ ✇❡ ❤❛✈❡✿ 0✲❝❡❧❧✱ ✐s {i} × {m} × {k}✳ ❆♥ ❡❞❣❡✱ ♦r ❛ 1✲❝❡❧❧✱ ✐s {i} × [m, m + 1] × {k}✱ ❡t❝✳ ❆ sq✉❛r❡✱ ♦r ❛ 2✲❝❡❧❧✱ ✐s [i, i + 1] × [m, m + 1] × {k}✱ ❡t❝✳ ❆ ❝✉❜❡✱ ♦r ❛ 3✲❝❡❧❧✱ ✐s [i, i + 1] × [m, m + 1] × [k, k + 1]✳

❋♦r ❛❧❧ ✐♥t❡❣❡rs

• • • •

❆ ♥♦❞❡✱ ♦r ❛

❚❤✉s✱ ♦✉r ❛♣♣r♦❛❝❤ t♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ s♣❛❝❡✱ ✐♥ ❛♥② ❞✐♠❡♥s✐♦♥✱ ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮

❚❤❡

n✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✐s ❝♦♠♣♦s❡❞ (k − 1)✲❝❡❧❧s✱ k = 1, 2, ..., n✳

♦❢ ❝❡❧❧s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t

k ✲❝❡❧❧s

❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤

♦t❤❡r ❛❧♦♥❣

❚❤❡ ❡①❛♠♣❧❡s s❤♦✇ ❤♦✇ t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦

0✲✱ 1✲✱

✳✳✳✱

n✲❝❡❧❧s

✐♥ s✉❝❤ ❛

✇❛② t❤❛t

• n✲❝❡❧❧s

❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣

• (n − 1)✲❝❡❧❧s •

(n − 1)✲❝❡❧❧s✳

❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣

(n − 2)✲❝❡❧❧s✳

✳✳✳

• 1✲❝❡❧❧s

❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣

0✲❝❡❧❧s✳

❲❤❛t ❛r❡ t❤♦s❡ ❝❡❧❧s ❡①❛❝t❧②❄

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✹✿ ❝❡❧❧ ■♥ t❤❡

n

n✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✱

Rn ✱

❛

❝❡❧❧

✐s t❤❡ s✉❜s❡t ❣✐✈❡♥ ❜② t❤❡ ♣r♦❞✉❝t ✇✐t❤

❝♦♠♣♦♥❡♥ts✿

P = I1 × ... × In , ✇✐t❤ ✐ts

• •

k t❤

❝♦♠♣♦♥❡♥t ✐s ❡✐t❤❡r

Ik = [xk , xk+1 ], ♦r Ik = {xk }. ❞✐♠❡♥s✐♦♥ ✐s ❡q✉❛❧ t♦ m✱ ❛♥❞

❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❛ ♣♦✐♥t

❚❤❡ ❝❡❧❧✬s

✐t ✐s ❛❧s♦ ❝❛❧❧❡❞ ❛♥

m✲❝❡❧❧✱

✇❤❡♥ t❤❡r❡

✶✳✶✹✳

P❛rt✐t✐♦♥s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ m

❛r❡

✶✷✸

❡❞❣❡s ❛♥❞

m−n

✈❡rt✐❝❡s ♦♥ t❤✐s ❧✐st✳ ❘❡♣❧❛❝✐♥❣ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s ✐♥ t❤❡

♣r♦❞✉❝t ✇✐t❤ ♦♥❡ ♦❢ ✐ts ❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥ ♦❢

P✳

(n − 1)✲❝❡❧❧

❝❛❧❧❡❞ ❛

❜♦✉♥❞❛r② ❝❡❧❧

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✺✿ ❢❛❝❡ ❘❡♣❧❛❝✐♥❣ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s ✐♥ t❤❡ ♣r♦❞✉❝t ✇✐t❤ ♦♥❡ ♦❢ ✐ts ❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥

(n − 1)✲❝❡❧❧

❢❛❝❡ ♦❢ P ✳ ❘❡♣❧❛❝✐♥❣ s❡✈❡r❛❧ ❡❞❣❡s ✇✐t❤ k ✲❝❡❧❧✱ k < n✱ ❝❛❧❧❡❞ ❛ ❜♦✉♥❞❛r② ❝❡❧❧ ♦❢ P ✳

❝❛❧❧❡❞ ❛

❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥

♦♥❡ ♦❢ t❤❡✐r

❚❤✉s✱ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ❛①❡s ✕ ✐♥t♦ ♥♦❞❡s ❛♥❞ ❡❞❣❡s ✕ ❝r❡❛t❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s♣❛❝❡ ✕ ✐♥t♦ ❝❡❧❧s ♦❢ ❛❧❧ ❞✐♠❡♥s✐♦♥s✳

❊①❛♠♣❧❡ ✶✳✶✹✳✻✿ ✸✲❝❡❧❧ ❇❡❧♦✇✱ ❛

3✲❝❡❧❧

✐s s❤♦✇♥ ❛s ❛ ✏r♦♦♠✑ ❛❧♦♥❣ ✇✐t❤ ❛❧❧ ♦❢ t❤❡ ❝❡❧❧s ♦❢ ❞✐♠❡♥s✐♦♥s

0, 1, 2✿

❚❤❡② ❛❧❧ ❝♦♠❡ ❢r♦♠ t❤❡ ♥♦❞❡s ❛♥❞ ❡❞❣❡s ♦♥ t❤❡ ❛①❡s✿

• 0✿ • 1✿ • 2✿

❊❛❝❤ ♦❢ t❤❡ ❥♦✐♥ts ♦❢ t❤❡ ✏❜❡❛♠s✑ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤r❡❡ ♥♦❞❡s✳ ❊❛❝❤ ♦❢ t❤❡ ✏❜❡❛♠s✑ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♥♦❞❡s ❛♥❞ ❛♥ ❡❞❣❡✳ ❊❛❝❤ ♦❢ t❤❡ ✏✇❛❧❧s✑✱ ❛s ✇❡❧❧ ❛s t❤❡ ✏✢♦♦r✑ ❛♥❞ t❤❡ ✏❝❡✐❧✐♥❣✑✱ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❡❞❣❡s ❛♥❞ ❛

♥♦❞❡✳

• 3✿ ❚❤❡ ✏r♦♦♠✑ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤r❡❡ ❡❞❣❡s✳ 2✲❝❡❧❧s ❤❡r❡ ❛r❡ t❤❡ ❢❛❝❡s ♦❢ t❤❡ 3✲❝❡❧❧✱ t❤❡ 1✲❝❡❧❧s

❚❤❡

❛r❡ t❤❡ ❢❛❝❡s ♦❢ t❤❡

1✲❝❡❧❧s✱

❡t❝✳

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✼✿ ♣r♦❞✉❝t ♦❢ t❤❡ ♣❛rt✐t✐♦♥s ❙✉♣♣♦s❡ ❡❛❝❤ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ♦❢ t❤❡ t✐♦♥✳ ❚❤❡♥✱ t❤❡ ✇✐t❤

n

n✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✱ Rn ✱ ❤❛s ❛ ♣❛rt✐✲

♣r♦❞✉❝t ♦❢ t❤❡ ♣❛rt✐t✐♦♥s ❝♦♥s✐sts ♦❢ t❤❡ ❝❡❧❧s ❣✐✈❡♥ ❜② t❤❡ ♣r♦❞✉❝t

❝♦♠♣♦♥❡♥ts✿

P = I1 × ... × In , ✇✐t❤ ✐ts

• •

k t❤

❝♦♠♣♦♥❡♥t ✐s ❡✐t❤❡r

❛♥ ❡❞❣❡ Ik = [xk , xk+1 ], ♦r ❛ ♥♦❞❡ Ik = {xk },

✐♥ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ Rn ✳

k t❤

❛①✐s✳ ❚❤✐s ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❝❡❧❧s ✐s ❝❛❧❧❡❞ ❛

♣❛rt✐t✐♦♥ ✐♥

✶✳✶✺✳

❉✐s❝r❡t❡ ❢♦r♠s

✶✷✹

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✽✿ ♣❛rt✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ✐♥ Rn ❛♥❞ s✉♣♣♦s❡ ❛ s✉❜s❡t D ♦❢ Rn ✐s t❤❡ ✉♥✐♦♥ ♦❢ s♦♠❡ ♦❢ t❤❡ ❝❡❧❧s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❡ s❛② t❤❛t t❤✐s ✐s ❛ ♣❛rt✐t✐♦♥ ♦❢ D ♣r♦✈✐❞❡❞✿ ✐❢ ❛ ❝❡❧❧ ✐s ♣r❡s❡♥t✱ t❤❡♥ s♦ ❞♦ ❛❧❧ ♦❢ ✐ts ❜♦✉♥❞❛r② ❝❡❧❧s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛♥② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡s Qi , i = 1, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❝✉r✈❡✳ ❍♦✇❡✈❡r✱ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ❛❧s♦ ✐♥❝❧✉❞❡s ❛❧❧ ♦❢ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❡❞❣❡s✳

❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ❝♦♥s✐sts ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ♦r✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡✱ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❛❞❥❛❝❡♥t ♥♦❞❡s✿

❚❤❡♥ ❛

Qi = Pi−1 Pi .

❋✉rt❤❡r♠♦r❡✱ ❢♦r ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ s✉r❢❛❝❡ ♠❛❞❡ ♦❢ ❢❛❝❡s✱ ✇❡ ♠✉st ❛❧s♦ ❤❛✈❡ ❛❧❧ t❤❡ ❡❞❣❡s ❛♥❞ t❤❡ ♥♦❞❡s ♦❢ t❤❡s❡ ❢❛❝❡s✱ ❛♥❞ s♦ ♦♥✳

❲❡ ✇✐❧❧ ❝❛rr② ♦✉t ❛❧❧ ❝❛❧❝✉❧✉s ❝♦♥str✉❝t✐♦♥s ✇✐t❤✐♥ t❤❡s❡ ♣❛rt✐t✐♦♥s✳

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s

■♥ ❱♦❧✉♠❡s ✷ ❛♥❞ ✸✱ ✇❡ ❛ss✐❣♥❡❞ ♥✉♠❜❡rs t♦ ♣♦✐♥ts ✇✐t❤✐♥ ❝❡❧❧s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ t♦ r❡♣r❡s❡♥t s✉❝❤ t❤✐♥❣s ❛s ❧♦❝❛t✐♦♥ ✕ ♥♦❞❡s ♦r 0✲❝❡❧❧s ✕ ❛♥❞ ✈❡❧♦❝✐t② ✕ s❡❝♦♥❞❛r② ♥♦❞❡s ♦r 1✲❝❡❧❧s✳ ❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ❞♦ s♦✳ ■♥ ❢❛❝t✱ ✇❡ ✇✐❧❧ st✉❞② ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t ♣♦✐♥ts ❧♦❝❛t❡❞ ❛t t❤❡ ❝❡❧❧s ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❞✐♠❡♥s✐♦♥ m ✐♥ ❛ ♣❛rt✐t✐♦♥✳ ❇❡❧♦✇ ✇❡ s❡❡ m = 0, 1, 2✿

✶✳✶✺✳

❉✐s❝r❡t❡ ❢♦r♠s

✶✷✺

❋✐rst❧②✱ t❤❡s❡ ♣♦✐♥ts ✕ s❡❝♦♥❞❛r②✱ t❡rt✐❛r②✱ ❡t❝✳ ♥♦❞❡s ✕ ♠❛② ❜❡ s♣❡❝✐✜❡❞ ❛s ❛ r❡s✉❧t ♦❢ s❛♠♣❧✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ r❡❣✐♦♥✳ ◆♦t❡ t❤❛t✱ ✐♥ t❤❛t ❝❛s❡✱ ♦♥❡ ♥♦❞❡ ♠❛② ❜❡ s❤❛r❡❞ ❜② t✇♦ ❛❞❥❛❝❡♥t ❝❡❧❧s✳ ❙❡❝♦♥❞❧②✱ t❤❡s❡ ♣♦✐♥ts ❛r❡ ✉s❡❞ ❢♦r ♠❡r❡ ❜♦♦❦❦❡❡♣✐♥❣✳ ❲❡ t❤❡♥ ❝❛♥ ❝❤♦♦s❡ t❤❡♠ t♦ ❜❡ t❤❡ ❡♥❞✲♣♦✐♥ts ♦r ❝♦r♥❡rs ♦r ♠✐❞✲♣♦✐♥ts ❡t❝✳ ■♥ tr✉t❤ t❤♦✉❣❤✱ t❤❡ q✉❛♥t✐t✐❡s ❛r❡ ❛ss✐❣♥❡❞ t♦ t❤❡ ❝❡❧❧s t❤❡♠s❡❧✈❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❡❛❝❤ ❝❡❧❧ ✐s ❛♥ ✐♥♣✉t ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✱ ❛s ❡①♣❧❛✐♥❡❞ ❜❡❧♦✇✳ ❘❡❝❛❧❧ ❤♦✇ ✇❡ ❞❡✜♥❡❞ ❞✐s❝r❡t❡ ❢♦r♠s ❢♦r ❞✐♠❡♥s✐♦♥ 1✿ ✇✐t❤✐♥ ❡❛❝❤ ♦❢ t❤❡ ♣✐❡❝❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉♥❝❤❛♥❣❡❞❀ ✐✳❡✳✱ ✐t✬s ❛ s✐♥❣❧❡ ♥✉♠❜❡r✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ 0✲ ❛♥❞ 1✲❢♦r♠s ♦✈❡r R1 ✿

❚❤❡r❡ ❛r❡ ♠♦r❡ t②♣❡s ♦❢ ❝❡❧❧s ✐♥ t❤❡ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s✱ ❜✉t t❤❡ ✐❞❡❛ r❡♠❛✐♥s✿

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✶✿ ❞✐s❝r❡t❡ ❢♦r♠

❆ ❞✐s❝r❡t❡ ❢♦r♠ ♦❢ ❞❡❣r❡❡ k ❞❡✜♥❡❞ ♦♥ k ✲❝❡❧❧s ♦❢ Rn ✳ ❆♥❞ t❤❡s❡ ❛r❡ 0✲✱ 1✲✱ ❛♥❞ 2✲❢♦r♠s ♦✈❡r R2 ✿

♦✈❡r Rn✱ ♦r s✐♠♣❧② ❛ k✲❢♦r♠✱ ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥

✶✳✶✺✳

❉✐s❝r❡t❡ ❢♦r♠s

✶✷✻ 1

❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ♥❛t✉r❡ ♦❢ ❛ ❢♦r♠ ❛s ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛♥ ✉s❡ ❛rr♦✇s ✭R ✮✿

❍❡r❡ ✇❡ ❤❛✈❡ t✇♦ ❢♦r♠s✿

•

❛

0✲❢♦r♠

✇✐t❤

0 7→ 2, 1 7→ 4, 2 7→ 3, ...❀

•

❛

1✲❢♦r♠

✇✐t❤

[0, 1] 7→ 3, [1, 2] 7→ .5, [2, 3] 7→ 1, ...✳

❛♥❞

❆ ♠♦r❡ ❝♦♠♣❛❝t ✇❛② t♦ ✈✐s✉❛❧✐③❡ ✐s t❤✐s✿

❍❡r❡ ✇❡ ❤❛✈❡ t✇♦ ❢♦r♠s✿

•

❛

0✲❢♦r♠ q

✇✐t❤

•

❛

1✲❢♦r♠ s

✇✐t❤

q(0) = 2, q(1) = 4, q(2) = 3, ...❀ ❛♥❞       s [0, 1] = 3, s [1, 2] = .5, s [2, 3] = 1, ...✳

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ❧❡tt❡rs t♦ ❧❛❜❡❧ t❤❡ ❝❡❧❧s✱ ❥✉st ❛s ❜❡❢♦r❡✳ ❊❛❝❤ ❝❡❧❧ ✐s t❤❡♥ ❛ss✐❣♥❡❞

•

♦♥❡ ✐s ✐ts ♥❛♠❡ ✭❛ ❧❛tt❡r✮ ❛♥❞

•

❚❤❡ ♦t❤❡r ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦r♠ ❛t t❤❛t ❧♦❝❛t✐♦♥ ✭❛ ♥✉♠❜❡r✮✳

❚❤✐s ✐❞❡❛ ✐s ✐❧❧✉str❛t❡❞ ❢♦r ❢♦r♠s ♦✈❡r

❲❡ ❤❛✈❡ ❛

0✲❢♦r♠ q

❛♥❞ ❛

1✲❢♦r♠ s

R1

❛♥❞

R2

r❡s♣❡❝t✐✈❡❧②✿

✐♥ t❤❡ ❢♦r♠❡r ❡①❛♠♣❧❡✿

• q(A) = 2, q(B) = 4, q(C) = 3, ... • s(AB) = 3, s(BC) = .5, s(CD) = 1, ... ❲❡ ❛❧s♦ ❤❛✈❡ ❛

2✲❢♦r♠ φ

✐♥ t❤❡ ❧❛tt❡r ❡①❛♠♣❧❡✿

• q(A) = 2, q(B) = 1, q(C) = 0, q(D) = 1 • s(a) = 1, s(b) = −1, s(c) = 2, s(d) = 0 • φ(τ ) = 4 ❲❡ ❝❛♥ s✐♠♣❧② ❧❛❜❡❧ t❤❡ ❝❡❧❧s ✇✐t❤ ♥✉♠❜❡rs✱ ❛s ❢♦❧❧♦✇s ✭✐♥

R3 ✮✿

t✇♦ s②♠❜♦❧s✿

✶✳✶✺✳

✶✷✼

❉✐s❝r❡t❡ ❢♦r♠s

❚❤❡s❡ ❢♦r♠s ♠❛② r❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛ ✢♦✇ ♦❢ ❛ ❧✐q✉✐❞✿

• ❆ 0✲❢♦r♠✿ t❤❡ ♣r❡ss✉r❡ ♦❢ t❤❡ ❧✐q✉✐❞ ❛t t❤❡ ❥♦✐♥ts ♦❢ ❛ s②st❡♠ ♦❢ ♣✐♣❡s✳ • ❆ 1✲❢♦r♠✿ t❤❡ ✢♦✇ r❛t❡ ♦❢ t❤❡ ❧✐q✉✐❞ ❛❧♦♥❣ t❤❡ ♣✐♣❡✳

• ❆ 2✲❢♦r♠✿ t❤❡ ✢♦✇ r❛t❡ ♦❢ t❤❡ ❧✐q✉✐❞ ❛❝r♦ss t❤❡ ♠❡♠❜r❛♥❡✳ • ❆ 3✲❢♦r♠✿ t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❧✐q✉✐❞ ✐♥s✐❞❡ t❤❡ ❜♦①✳

❚❤❡s❡ ❢♦r♠s ✇✐❧❧ ❜❡ ✉s❡❞ t♦ st✉❞② ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❍♦✇❡✈❡r✱ t❤❡ ❡①❛♠♣❧❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✕ ❛♥❞ ❡s♣❡❝✐❛❧❧② ♠♦t✐♦♥ ✐♥ s♣❛❝❡ ✕ s✉❣❣❡sts t❤❛t ✇❡ ♠❛② ♥❡❡❞ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s t♦ ❜❡ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✳ ❲❡ s❛✇ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞✱ ❥✉st ❧✐❦❡ ❛ 0✲❢♦r♠✱ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐♥❡✱ ❜✉t ✇✐t❤ ✈❛❧✉❡s ✐♥ R2 ✱ ✉♥❧✐❦❡ ❛ 0✲❢♦r♠✳ ❚❤✐s ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❧❛st ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✷✿ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠ ❙✉♣♣♦s❡ n ❛♥❞ m ❛r❡ ❣✐✈❡♥✳ ❚❤❡♥ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠ F ♦❢ ❞❡❣r❡❡ k ✱ ♦r s✐♠♣❧② ❛ k ✲❢♦r♠✱ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ k ✲❝❡❧❧s ♦❢ Rn ✇✐t❤ ✈❛❧✉❡s ✐♥ Rm ✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ✇✐❧❧ ❜❡ ✉s✐♥❣ ❝❛♣✐t❛❧ ❧❡tt❡rs ❢♦r ✈❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ♦✉r ❝♦♥✈❡♥t✐♦♥✳ ◆♦t❡ t❤❛t ❞✐s❝r❡t❡ ❢♦r♠s ❞♦ ♥♦t ❡①❛❝t❧② ♠❛t❝❤ ♦✉r ❧✐st ♦❢ ❢✉♥❝t✐♦♥s✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❛♥❞ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❋r♦♠ t❤❡ s❛♠❡ ❞♦♠❛✐♥✱ ✇❡ ♣✐❝❦ ❝❡❧❧s ♦❢ ❞✐✛❡r❡♥t ❞✐♠❡♥s✐♦♥s ♣r♦❞✉❝✐♥❣ ❢♦r♠s ♦❢ ❞✐✛❡r❡♥t ❞❡❣r❡❡s✳ ❚❤✐s ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s✿ ❛ 0✲❢♦r♠ ❛♥❞ ❛ 1✲❢♦r♠ ✭❢♦r t❤❡ ❧❛tt❡r✱ t❤❡ ✈❡❝t♦rs ❤❛✈❡ t♦ ❜❡ ♠♦✈❡❞ t♦ ♣✉t t❤❡ st❛rt✐♥❣ ♣♦✐♥ts ❛t t❤❡ ♦r✐❣✐♥✮❀ ✐✳❡✳✱ n = 1 ❛♥❞ m = 2✿

❚❤❡ ❢♦r♠❡r ♠❛② r❡♣r❡s❡♥t t❤❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❇♦t❤ ❝❛♥ ❜❡ s❡❡♥ ❛s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ◆❡①t ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ ❛ r❡❛❧✲✈❛❧✉❡❞ ❛♥❞ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠s❀ ✐✳❡✳✱ n = 2✱ m = 1 ❛♥❞ n = 2✱ m = 2 r❡s♣❡❝t✐✈❡❧②✿

✶✳✶✺✳

✶✷✽

❉✐s❝r❡t❡ ❢♦r♠s

❚❤❡ ❢♦r♠❡r ♠❛② r❡♣r❡s❡♥t ❛ ✢♦✇ ♦❢ ✇❛t❡r ❛❧♦♥❣ ❛ s②st❡♠ ♦❢ ♣✐♣❡s ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ s❛♠❡ ✢♦✇ ✇✐t❤ ♣♦ss✐❜❧❡ ❧❡❛❦s✳ ❇♦t❤ ❝❛♥ ❜❡ s❡❡♥ ❛s ✈❡❝t♦r ✜❡❧❞s✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ❛❧❧♦✇s ✉s t♦ r❡♣r♦❞✉❝❡ t❤❡ ❞❡✜♥✐t✐♦♥s ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✹✮ ✐♥ t❤❡ ♥❡✇✱ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ✐♥ ❜♦t❤ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t✱ ❝♦♥t❡①t✳ ❲❡ ❥✉st ❛ss✉♠❡ t❤❛t ❛ s♣❛❝❡ ♦❢ ✐♥♣✉ts Rn ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛ s♣❛❝❡ ♦❢ ♦✉t♣✉ts Rm ❛r❡ ❣✐✈❡♥✳ ❲❡ ♥❡❡❞ ♦♥❡ ♠♦r❡ ❣❡♥❡r❛❧✐③❛t✐♦♥❀ ✇❡ ✐♥tr♦❞✉❝❡ ♦r✐❡♥t❛t✐♦♥

♦❢ ❝❡❧❧s✳

❆❜♦✉t t❤❡ 0✲❝❡❧❧s✱ ✇❡ ✇✐❧❧ s✐♠♣❧② ❛❧❧♦✇ t❤❡♠ t♦ ❛♣♣❡❛r ✇✐t❤ ❜♦t❤ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ s✐❣♥s✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✸✿ ♦r✐❡♥t❡❞ ✵✲❝❡❧❧ ❆♥ ♦r✐❡♥t❡❞ 0✲❝❡❧❧ ✭♦r ♥♦❞❡✮ A ✐s ❛ 0✲❝❡❧❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✇✐t❤ ✐ts s✐❣♥ s♣❡❝✐✜❡❞✿ A ♦r −A✳ ❚❤❡ ❝❤♦✐❝❡ ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ t❤❡ ❝✐r❝✉♠st❛♥❝❡s✳ ❚❤❡ ♥❡❡❞ ❢♦r t❤✐s ✇✐❧❧ ❜❡❝♦♠❡ ❝❧❡❛r s❤♦rt❧②✳ ❲❡ ❤❛✈❡ 1✲❝❡❧❧s ❞❡✜♥❡❞ ❛s ♣r♦❞✉❝ts ♦❢ ❛ s✐♥❣❧❡ ❡❞❣❡ ❛♥❞ s❡✈❡r❛❧ ♥♦❞❡s✳ ❚❤❡ ♦r❞❡r ♦❢ ♥♦❞❡s ❞♦❡s♥✬t ♠❛tt❡r✿ E = AB = BA, A = (a1 , ..., an ), B = (b1 , ..., bn ) =⇒ ai = bi , i = 1, 2, ..., n, ❛♥❞ ak 6= bk ❢♦r s♦♠❡ k .

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❞✐st✐♥❣✉✐s❤ t❤❡ t✇♦ ✇❛②s ✇❡ ❝❛♥ ❢♦❧❧♦✇ ❛♥ ❡❞❣❡✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✹✿ ♦r✐❡♥t❡❞ ✶✲❝❡❧❧ ❆♥ ♦r✐❡♥t❡❞ 1✲❝❡❧❧ ✭♦r ❡❞❣❡✮ E ✐s ❛ 1✲❝❡❧❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✇✐t❤ t❤❡ ♦r❞❡r ♦❢ ✐ts t✇♦ ♥♦❞❡s s♣❡❝✐✜❡❞✿ AB ♦r BA✳ ❚❤❡ ❝❡❧❧ ✐s ❝❛❧❧❡❞ ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ♦❢ t❤❡ ♦r❞❡r ♦❢ t❤❡ ♥♦❞❡s ❞♦❡s ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❛①❡s✿ E = AB, A = (a1 , ..., an ), B = (b1 , ..., bn ) =⇒ ak < bk ❢♦r s♦♠❡ k .

❚❤❡ ❝❡❧❧ ✐s ❝❛❧❧❡❞ ♥❡❣❛t✐✈❡❧② ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❛①❡s✿

♦r✐❡♥t❡❞

♦❢ t❤❡ ♦r❞❡r ♦❢ t❤❡ ♥♦❞❡s ❣♦❡s ❛❣❛✐♥st t❤❡

E = AB, A = (a1 , ..., an ), B = (b1 , ..., bn ) =⇒ ak > bk ❢♦r s♦♠❡ k .

✶✳✶✺✳

✶✷✾

❉✐s❝r❡t❡ ❢♦r♠s

◆❡❣❛t✐✈❡ ❡❞❣❡ BA = −AB .

❚❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧s ✐s ❛❞❞r❡ss❡❞ ❧❛t❡r✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✿

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✺✿ ❜♦✉♥❞❛r② ♦❢ ❛ ✵✲❝❡❧❧ ❚❤❡ ❜♦✉♥❞❛r②

♦❢ ❛

0✲❝❡❧❧ ✐s 0✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∂A = 0

❚❤❡ ❜♦✉♥❞❛r②

♦❢ ❛

1✲❝❡❧❧ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ✐ts ❡♥❞✲♣♦✐♥ts ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∂AB = B − A

❲❛r♥✐♥❣✦ ❲❤✐❧❡

∂A ✐s ❛ ♥✉♠❜❡r✱ ∂AB 1✲❝❡❧❧s✳

✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐✲

♥❛t✐♦♥ ♦❢

❊✈❡♥ t❤♦✉❣❤ ❛♥② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡s Ei , i = 1, ..., n✱ ✐s s❡❡♥ ❛s ❛ ❝✉r✈❡✱ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ❝♦♥s✐sts ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ♦r✐❡♥t❡❞ ❡❞❣❡s ♦r✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡✱ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❛❞❥❛❝❡♥t ♥♦❞❡s✿ Ei = Pi−1 Pi .

❲❤❛t ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ❡❞❣❡s ♦❢ C ❄

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✻✿ ❜♦✉♥❞❛r② ♦❢ ❛ ❝✉r✈❡ ❚❤❡ ❜♦✉♥❞❛r②

♦❢ ❛ ❝✉r✈❡

C ❢r♦♠ A t♦ B ✐s B − A✳

❘❡❝❛❧❧ t❤❛t 0✲❢♦r♠s ❛r❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ 0✲❝❡❧❧s✳✳✳ ❛♥❞ ♥♦✇ t❤❡② ❛r❡ ✐♥st❛♥t❧② ❡①t❡♥❞❡❞ t♦ t❤❡ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ❝❡❧❧s✳ ❙✐♠✐❧❛r❧②✱ 1✲❢♦r♠s ❛r❡ ❞❡✜♥❡❞ ♦♥ ❜♦t❤ ♣♦s✐t✐✈❡❧② ❛♥❞ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ 1✲❝❡❧❧s✳

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s

✶✸✵

❲❡ ❛♠❡♥❞ ♦✉r ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✼✿ ❞✐s❝r❡t❡ ❢♦r♠ ❙✉♣♣♦s❡ n ❛♥❞ m ❛r❡ ❣✐✈❡♥✳ ❚❤❡♥ ❛ r❡❛❧✲✈❛❧✉❡❞ ♦r ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠ F ♦❢ ❞❡❣r❡❡ k✱ ♦r s✐♠♣❧② ❛ k✲❢♦r♠✱ k = 0, 1✱ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣♦s✐t✐✈❡❧② ❛♥❞ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ k✲❝❡❧❧s ♦❢ Rn ✇✐t❤ ✈❛❧✉❡s ✐♥ R ♦r Rm r❡s♣❡❝t✐✈❡❧② s♦ t❤❛t✿ F (−a) = −F (a) .

◆♦✇ ❝❛❧❝✉❧✉s✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✽✿ ❞✐✛❡r❡♥❝❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ F ✐s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠ ❣✐✈❡♥ ❜② ✐ts ✈❛❧✉❡s ♦♥ ❡❛❝❤ ❡❞❣❡ E = AB ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ ∆F (E) = F (B) − F (A)

❆ 1✲❢♦r♠ ✐s ❝❛❧❧❡❞ ❡①❛❝t ✇❤❡♥ ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ s♦♠❡ 0✲❢♦r♠✳ ❚❤❡ ♣✐❝t✉r❡ ❛❜♦✈❡ ♠❛② s❡r✈❡ ❛s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤✐s ❝♦♥❝❡♣t✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✾✿ s✉♠ ❚❤❡ s✉♠ ♦❢ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ F ❛❧♦♥❣ ❛ ❝♦❧❧❡❝t✐♦♥ Q ♦❢ ♦r✐❡♥t❡❞ ♥♦❞❡s N1 , N2 , ..., Nk ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ X

F = F (N1 ) + F (N2 ) + ... + F (Nk )

Q

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✶✵✿ s✉♠ ❚❤❡ s✉♠ ♦❢ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠ G ❛❧♦♥❣ ❛ ❝♦❧❧❡❝t✐♦♥ C ♦❢ ♦r✐❡♥t❡❞ ❡❞❣❡s E1 , E2 , ..., Ek ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ X

G = G(E1 ) + G(E2 ) + ... + G(Ek )

C

❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s r❡♠❛✐♥s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿ t❤❡② ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✳

✶✳✶✺✳

❉✐s❝r❡t❡ ❢♦r♠s

✶✸✶

❘❡❝❛❧❧ ✜rst t❤❛t ❛♥② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡s Qi , i = 1, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s s❡❡♥ ❛s ❛ ❝✉r✈❡✱ ✇❤✐❧❡ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ❝♦♥s✐sts ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ♦r✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡✱ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❛❞❥❛❝❡♥t ♥♦❞❡s✿ Qi = Pi−1 Pi .

❚❤❡♦r❡♠ ✶✳✶✺✳✶✶✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡✲ ❣r❡❡

1

n✲❝❡❧❧ ✐♥ Rn ✐s ❣✐✈❡♥✳ ❙✉♣♣♦s❡ F ✐s ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ♦♥ t❤✐s ♣❛rt✐t✐♦♥ ❛♥❞ s✉♣♣♦s❡ A ✐s ❛ ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♥♦❞❡ X ❛♥❞ ❛♥② ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ❢r♦♠ A t♦ X ✱ ✇❡ ❤❛✈❡✿ ❙✉♣♣♦s❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥

X C

(∆F ) = F (X) − F (A)

Pr♦♦❢✳

❲❡ ❥✉st ❛❞❞ ❛❧❧ ♦❢ t❤❡s❡ ❛♥❞ ❝❛♥❝❡❧ t❤❡ r❡♣❡❛t❡❞ ♥♦❞❡s✿ X

G = G(E1 )

+G(E2 )

+... +G(Ek )

C

= G(P0 P1 ) +G(P1 P2 ) +... +G(Pk−1 Pk )       = F (P1 ) − F (P0 ) + F (P2 ) − F (P1 ) +... + F (Pk ) − F (Pk−1 ) = −f (P0 )

+F (Pk )

= F (X) − F (A) .

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ F ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦✈❡r ❡✈❡r② ❡❞❣❡✱ ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦✈❡r ❛♥② ❝♦♥t✐♥✉♦✉s ❝✉r✈❡✳

✶✳✶✺✳

❉✐s❝r❡t❡ ❢♦r♠s

✶✸✷

❈♦r♦❧❧❛r② ✶✳✶✺✳✶✷✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡✲ ❣r❡❡

1

❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿

X

(∆F ) =

C

X

F

∂C

❇✉t ❞♦ t❤❡② ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ✐♥ ❡✐t❤❡r ♦r❞❡r❄ ❉♦❡s t❤✐s ❢♦r♠✉❧❛ ❢r♦♠ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶P❈✲✶✮ st✐❧❧ ♠❛❦❡ s❡♥s❡✿

∆

X C

G

!

= G(X) ?

❲❡ ✇✐❧❧ ❛❞❞r❡ss t❤✐s q✉❡st✐♦♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳

❈❤❛♣t❡r ✷✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❈♦♥t❡♥ts

✷✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ▲✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❈♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✳ ✳ ✳ ✷✳✻ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✷✳✶✶ ❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✹ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✺ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✻ ❚❤❡ s♣❡❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✼ ❚❤❡ ❝✉r✈❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✽ ❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✾ ❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✵ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✶ ❆r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧s✿ ✇❡✐❣❤t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✷ ❚❤❡ ❤❡❧✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸✸ ✶✹✵ ✶✹✼ ✶✺✹ ✶✺✽ ✶✻✶ ✶✻✻ ✶✻✾ ✶✼✺ ✶✼✼ ✶✽✷ ✶✽✼ ✶✾✷ ✶✾✺ ✶✾✽ ✷✵✶ ✷✵✺ ✷✶✵ ✷✶✺ ✷✶✾ ✷✷✷ ✷✷✹

✷✳✶✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❘❡❝❛❧❧ t❤❛t t❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠ ❛s❦s ❢♦r ❛ t❛♥❣❡♥t ❧✐♥❡ t♦ ❛ ❝✉r✈❡ ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t✳ ■t ❤❛s ❜❡❡♥ s♦❧✈❡❞ ❢♦r t❤❡ ❣r❛♣❤s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❍♦✇❡✈❡r✱ ♠♦st ♦❢ t❤❡ ❝✉r✈❡s ✐♥ r❡❛❧ ❧✐❢❡ ❝❛♥✬t ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❣r❛♣❤s✳ ❲❡ ❤❛✈❡ t♦ ❧♦♦❦ ❛t ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❚❤❡ ❡①❛♠♣❧❡s ❛r❡ ❢❛♠✐❧✐❛r✳ ❊①❛♠♣❧❡ ✷✳✶✳✶✿ r❛❞❛r

■♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ ❛ r❛❞❛r s✐❣♥❛❧ ✇✐❧❧ ❜♦✉♥❝❡ ♦✛ ❛ ♣❧❛♥❡ ✇❤❡♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ♣❧❛♥❡ ✐s ❝✉r✈❡❞❄

✷✳✶✳

✶✸✹

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

■♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇✐❧❧ ❧✐❣❤t ❜♦✉♥❝❡ ♦✛ ❛ ❝✉r✈❡❞ ♠✐rr♦r❄

■♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇♦✉❧❞ ❛ r♦❝❦ r❡❧❡❛s❡❞ ❢r♦♠ ❛ s❧✐♥❣ ❣♦❄

❆♥❞ s♦ ♦♥✳ ❊①❛♠♣❧❡ ✷✳✶✳✷✿ ❡❧❧✐♣t✐❝❛❧ r♦♦♠

▲❡t✬s ❝♦♥✜r♠ t❤❛t t❤❡ s♦✉♥❞ ♦r✐❣✐♥❛t✐♥❣ ❢r♦♠ ♦♥❡ ❢♦❝✉s ♦❢ ❛♥ ❡❧❧✐♣t✐❝❛❧ r♦♦♠ ✇✐❧❧ ❜♦✉♥❝❡ ♦✛ t❤❡ ✇❛❧❧ t♦ ♣❛ss t❤r♦✉❣❤ t❤❡ ♦t❤❡r ❢♦❝✉s✳ ❚❤✐s ✇❛② ♦♥❡ ❝❛♥ ❧✐st❡♥ t♦ ❛ ❝♦♥✈❡rs❛t✐♦♥ ❛t t❤❡ ♦t❤❡r ❢♦❝✉s ❡✈❡♥ ✐❢ t❤❡r❡ ❛r❡ ♦❜st❛❝❧❡s ❜❡t✇❡❡♥ t❤❡♠✳ ◆♦✇✱ ✇❡ st✉❞② ❢✉♥❝t✐♦♥s✳ ❚❤❡② ❛❧❧ ❤❛✈❡ ❛ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✐♥♣✉t ❛♥❞ ❛ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ ♦✉t♣✉t✳ ❲❤❛t ✈❛r✐❡s ✐s t❤❡ ♥❛t✉r❡ ♦❢ t❤❡s❡ t✇♦ ✈❛r✐❛❜❧❡s✳ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s s✉❝❤ ❛ ❢✉♥❝t✐♦♥✿ • ❚❤❡ s✐♥❣❧❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s ❛ • ❚❤❡ s✐♥❣❧❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s

F : t 7→ F (t) .

✱ t✳

r❡❛❧ ♥✉♠❜❡r

✱ ❛ ♣♦✐♥t ♦r ❛ ✈❡❝t♦r ✐♥ Rn ✳

♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧

❋♦r ❡①❛♠♣❧❡ ❢♦r n = 2✱ ✇❡ ♠❛② ❤❛✈❡✿

F : t 7→ X = (f (t), g(t)) ,

♦r F : t 7→ OX =< f (t), g(t) > .

❆s ✇❡ ❦♥♦✇✱ t❤❡ ❢♦r♠❡r ♣♦✐♥t✱ X ✱ ✐s t❤❡ ❡♥❞ ♦❢ t❤❡ ❧❛tt❡r ✈❡❝t♦r✱ OX ✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ t❤✐s ✐s ❥✉st ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❛♠❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡ ❣♦✲t♦ ♠❡t❛♣❤♦r ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s ♠♦t✐♦♥ ✿

✷✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

• t

✶✸✺

✐s t❤♦✉❣❤t ♦❢ ❛s t✐♠❡✱

• (f (t), g(t)) ❲❡ ♣❧♦t ❡❛❝❤

✐s t❤♦✉❣❤t ♦❢ ❛s t❤❡ ♣♦s✐t✐♦♥ ✐♥ s♣❛❝❡ ❛t t✐♠❡

(x, y)

❚❤❡s❡ ✈❛❧✉❡s ♦❢

t✳

❛♥❞ ❧❛❜❡❧ ❡❛❝❤ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢

t✿

t ♠❛② ❝♦♠❡ ❛s t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ♦r r✉♥ t❤r♦✉❣❤ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✳

❊①❛♠♣❧❡ ✷✳✶✳✸✿ str❛✐❣❤t ❧✐♥❡

▲✐♥❡❛r ♠♦t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❛s ❢♦❧❧♦✇s✿

• • • •

F (t) = V · t✱ ❛ ♠♦t✐♦♥ ❛❧♦♥❣ ❛ ✈❡❝t♦r V ✭❝♦♥st❛♥t s♣❡❡❞✮ F (t) = V · t + B ✱ s❛♠❡ ❜✉t ✇✐t❤ B t❤❡ st❛rt✐♥❣ ❧♦❝❛t✐♦♥ F (t) = −V · t = V · (−t)✱ ❜❛❝❦✇❛r❞ F (t) = V · (t2 )✱ ❛❝❝❡❧❡r❛t❡❞

❢r♦♠ t❤❡ ♦r✐❣✐♥

❊①❛♠♣❧❡ ✷✳✶✳✹✿ ❝✐r❝❧❡

❚❤❡ ❞✐st❛♥❝❡ t♦ s♦♠❡ ♣♦✐♥t r❡♠❛✐♥s t❤❡ s❛♠❡✱ s❛②✱

1✱

❛♥❞ t❤❡ ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥ ✐s

x2 + y 2 = 1 .

❚❤❡ ✏s✐♠♣❧❡✑ ❝✐r❝✉❧❛r ♠♦t✐♦♥ ❤❛♣♣❡♥s ✇❤❡♥ t❤❡

❛♥❣✉❧❛r ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✿

t❤❡ ♦❜❥❡❝t t✉r♥s t❤❡ s❛♠❡

❛♥❣❧❡ ♣❡r ✉♥✐t ♦❢ t✐♠❡✿

F (t) =< cos t, sin t > . ❲❡ s✉❜st✐t✉t❡

x = cos t

❛♥❞

y = sin t

✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✉s❡ t❤❡

t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ✉♥✐t ❝✐r❝❧❡✿

x2 + y 2 = (cos t)2 + (sin t)2 = 1 .

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠

t♦ ♣r♦✈❡

✷✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✶✸✻

❚❤❡r❡ ❛r❡ ♦t❤❡r ✇❛②s t♦ tr❛✈❡❧ ❛r♦✉♥❞ ❛ ❝✐r❝❧❡ ♦❢ ❝♦✉rs❡✳

•

❈✐r❝❧✐♥❣

0

❜✉t ♠♦✈✐♥❣ ❜❛❝❦✇❛r❞✿

F (t) =< cos(−t), sin(−t) > . •

❈✐r❝❧✐♥❣

•

❈✐r❝❧✐♥❣ ♣♦✐♥t

•

❋♦r ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥✱ ❥✉st ❝❤❛♥❣❡ ❤♦✇ ❢❛st t❤❡ t✐♠❡ ❣♦❡s✿

0

❜✉t ❛❧♦♥❣ r❛❞✐✉s

r✿ F (t) = r· < cos t, sin t > .

(p, q)✿ F (t) = (p, q) + r· < cos t, sin t > .

F (t) =< cos t2 , sin t2 > . ❊①❛♠♣❧❡ ✷✳✶✳✺✿ ❡❧❧✐♣s❡

❚❤❡ ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ ✐s

❢♦r s♦♠❡ ♥♦♥✲③❡r♦ ❝♦♥st❛♥t ♥✉♠❜❡rs

x2 y 2 + 2 = 1, a2 b

a, b✳

◆♦✇✱ t❤❡ ✏s✐♠♣❧❡✑ ❡❧❧✐♣t✐❝ ♠♦t✐♦♥ ✐s s✐♠✐❧❛r t♦ ❝✐r❝✉❧❛r ♠♦t✐♦♥

❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ❛♥❣✉❧❛r ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t ❜✉t t❤❡ ❞✐st❛♥❝❡ ✈❛r✐❡s✳

F (t) =< a cos t, b sin t > . ❲❡ s✉❜st✐t✉t❡

x = a cos t

❛♥❞

y = b sin t

✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✉s❡ t❤❡

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ t♦ ♣r♦✈❡

t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❡❧❧✐♣s❡✿

(a cos t)2 (b sin t)2 x2 y 2 + = + = cos2 t + sin2 t = 1 . a2 b2 a2 b2

❖t❤❡r ✇❛②s t♦ tr❛✈❡❧ ❛r♦✉♥❞ ❛♥ ❡❧❧✐♣s❡ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❈✐r❝❧✐♥❣

0

❜✉t ♠♦✈✐♥❣ ❜❛❝❦✇❛r❞✿

F (t) =< a cos(−t), b sin(−t) > , ❝✐r❝❧✐♥❣ ♣♦✐♥t

(p, q)✿ F (t) = (p, q) + r· < a cos t, b sin t > .

✷✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✶✸✼

❈✉r✈❡s ✕ ❣✐✈❡♥ ❛s ❣r❛♣❤s ♦r ✐♠♣❧✐❝✐t❧② ✕ ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐③❡❞✳ ■t ✐s ❛s ✐❢ ✇❡ ♥❡❡❞ t♦ ❣❡t t❤❡ s❤❛♣❡ ♦❢ t❤❡ r♦❛❞ ❛♥❞ ❛❝❤✐❡✈❡ t❤❛t ❜② ❞r✐✈✐♥❣ ❛❧♦♥❣ t❤❡ r♦❛❞ ✇❤✐❧❡ r❡❝♦r❞✐♥❣ t❤❡ t✐♠❡ ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✻✿ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❝✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ✐s ❝❛❧❧❡❞ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇❤❡♥ t❤❡ ♣❛t❤ ♦❢ X ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝✉r✈❡ C ✳

♦❢ ❛ ❝✉r✈❡

C ✐♥ Rn

■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡s ✇❡ s❤♦✇❡❞ ❛ ✈❛r✐❡t② ♦❢ ✇❛② t♦ ♣❛r❛♠❡tr✐③❡ ❧✐♥❡s✱ ❝✐r❝❧❡s✱ ❛♥❞ ❡❧❧✐♣s❡s✳

❚❤❡♦r❡♠ ✷✳✶✳✼✿ ❘❡♣❛r❛♠❡tr✐③❛t✐♦♥ ■❢ t❤❡ ❢✉♥❝t✐♦♥ t❂❣✭s✮ ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✱ t❤❡♥ t❤❡ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

X = F (t)

❛♥❞

X = F (g(s))

❛r❡ ♣❛r❛♠❡tr✐③❛t✐♦♥s ♦❢ t❤❡ s❛♠❡ ❝✉r✈❡✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✽✿ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❡❧❧✐♣s❡ ❚❤❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ t❤❛t ✉s❡s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❧✐♥❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥ ❛s t❤❡ ♣❛r❛♠❡t❡r ✐s ❝❛❧❧❡❞ t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡✳ ❚❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ❛r❡ ❡s♣❡❝✐❛❧❧② ❡❛s② t♦ ♣❛r❛♠❡tr✐③❡✳

❚❤❡♦r❡♠ ✷✳✶✳✾✿ P❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ●r❛♣❤ ❚❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

x = t, y = f (t)

✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)✳ ■t ✐s ❛s ✐❢ ✇❡ ❛r❡ ♠♦✈✐♥❣ ❡❛st ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✳✳✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✶✵✿ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❣r❛♣❤ ❋♦r n = 2✱ ✇❡ ❝❛❧❧ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡   x = t,  y = f (t),

t❤❡

st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ y = f (x)✳

❊①❛♠♣❧❡ ✷✳✶✳✶✶✿ ♣❛r❛❜♦❧❛ ❚❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ✐s ✭❧❡❢t✮✿

F (t) = (t, t2 ) .

✷✳✶✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✶✸✽

√ x = y ✐s √ G(t) = ( t, t) .

▼❡❛♥✇❤✐❧❡ t❤✐s ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛❜♦❧❛

❞✐✛❡r❡♥t ✭r✐❣❤t✮✿

❈♦♥✈❡rs❡❧②✱ ✇❡ ❝❛♥ s♦♠❡t✐♠❡s r❡♣r❡s❡♥t t❤❡ ♣❛t❤ ♦❢ ❛ ♣❧❛♥❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s

• y = f (x)

✐❢ t❤❡ ♣❛t❤ s❛t✐s✜❡s t❤❡ ❍♦r✐③♦♥t❛❧ ▲✐♥❡ ❚❡st✱ ♦r

• x = g(y)

✐❢ t❤❡ ♣❛t❤ s❛t✐s✜❡s t❤❡ ❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st✳

❚❤❡ ❝✐r❝❧❡ ❢❛✐❧s ❜♦t❤✦ ❍♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ✏❞❡✲♣❛r❛♠❡tr✐③❡❞✑ ♣✐❡❝❡✲❜②✲♣✐❡❝❡ ✭t♦♣✲❜♦tt♦♠ ♦r ❧❡❢t✲r✐❣❤t ❤❛❧✈❡s✮✳ ❚❤✐s ✇♦✉❧❞ ❜❡ ❛ ❝❤❛❧❧❡♥❣✐♥❣ t❛s❦ ❢♦r ♠♦r❡ ❝♦♠♣❧❡① ❝✉r✈❡s✿

❘❡♣r❡s❡♥t✐♥❣ ❛ ♣❧❛♥❡ ❝✉r✈❡ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ ✐s ❛ str♦♥❣❧② ♣r❡❢❡rr❡❞ ❛♣♣r♦❛❝❤✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ tr② t♦ ❡❧✐♠✐♥❛t❡ t❤❡ ♣❛r❛♠❡t❡r ❜② s♦❧✈✐♥❣ ❢♦r

  x = t3  y = sin t

=⇒ t =

√ 3

t

❢♦❧❧♦✇❡❞ ❜② s✉❜st✐t✉t✐♦♥✿

x =⇒ y = sin

 √ 3 x .

❚❤❡ s❝❤❡♠❡ ✇♦r❦s ♦♥❧② ✇❤❡♥ ♦♥❡ ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ✐s ♦♥❡✲t♦✲♦♥❡✳

✷✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❚❤❡ ♠♦t✐♦♥ ♠❛② ❛❧s♦ ❜❡ ✐♥ t❤❡

• t

✶✸✾

3✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✿

✐s t❤♦✉❣❤t ♦❢ ❛s t✐♠❡✱

• (f (t), g(t), h(t)) ❲❡ ♣❧♦t ❡❛❝❤

(x, y, z)

✐s t❤❡ ♣♦s✐t✐♦♥ ✐♥ s♣❛❝❡ ❛t t✐♠❡

t✳

❛♥❞ ❧❛❜❡❧ ❡❛❝❤ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢

t✳

❊①❛♠♣❧❡ ✷✳✶✳✶✷✿ ❝✐r❝❧❡ ✐♥ s♣❛❝❡ ❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

xy ✲♣❧❛♥❡

♣❧❛❝❡❞ ✇✐t❤✐♥ t❤❡ ♣❧❛♥❡

z = c✿

❚❤✐s ✐s ✐ts ❢♦r♠✉❧❛✿

F (t) = (cos t, sin t, c) . ❊①❛♠♣❧❡ ✷✳✶✳✶✸✿ ❤❡❧✐① ❆♥ ❛s❝❡♥❞✐♥❣ s♣✐r❛❧ ✐s ✇❤❡♥ t❤❡ r♦t❛t✐♦♥ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✐s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛ ✈❡rt✐❝❛❧ ❛s❝❡♥❞✿

❚❤✐s ✐s ✐ts ❢♦r♠✉❧❛✿

F (t) = (cos t, sin t, t) .

✷✳✷✳ ▲✐♠✐ts

✶✹✵

❊①❡r❝✐s❡ ✷✳✶✳✶✹

❙✉❣❣❡st ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r t❤❡ ❝✉r✈❡ ❜❡❧♦✇✿

❲❡ ✇✐❧❧ r❡✈✐s✐t ❡✈❡r② ✐ss✉❡ ❛❜♦✉t ❢✉♥❝t✐♦♥s ✇❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❱♦❧✉♠❡s ✷ ❛♥❞ ✸ ❛s t❤❡② ❛♣♣❧② t♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❇✉t ✜rst ✇❡ ❝♦♥s✐❞❡r t❤❡ ✐ss✉❡ t❤❛t ❧✐❡s ❛t t❤❡ ❤❡❛rt ♦❢ ❝❛❧❝✉❧✉s✿ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❲❡ t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❧♦❝❛t✐♦♥✲✈❡❧♦❝✐t②✲❛❝❝❡❧❡r❛t✐♦♥ ❝♦♥♥❡❝t✐♦♥✳

✷✳✷✳ ▲✐♠✐ts

❲❡ ❤❛✈❡ ❛❧✇❛②s ❛ss✉♠❡❞ t❤❛t t♦ ❣❡t ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B ✱ ✇❡ ❤❛✈❡ t♦ ✈✐s✐t ❡✈❡r② ❧♦❝❛t✐♦♥ ❜❡t✇❡❡♥ A ❛♥❞ B ✳

■❢ t❤❡r❡ ✐s ❛ ❥✉♠♣ ✐♥ t❤❡ ♣❛t❤ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✐t ❝❛♥✬t r❡♣r❡s❡♥t ♠♦t✐♦♥✦ ❚❤❡♥ t❤❡ q✉❡st✐♦♥ ❜❡❝♦♠❡s t❤❡ ♦♥❡ ❛❜♦✉t t❤❡ ✐♥t❡❣r✐t② ♦❢ t❤❡ ♣❛t❤✿ ✐s t❤❡r❡ ❛ ❜r❡❛❦ ♦r ❛ ❝✉t❄ ❚❤✉s✱ ✇❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦ x = f (t), y = g(t) ✇❤❡♥ t ✐s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❝❤♦s❡♥ ✐♥♣✉t ✈❛❧✉❡ t = s✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ❤❛♥❞❧❡❞ t❤❡ ♣r♦❜❧❡♠ ✐♥ ❞✐♠❡♥s✐♦♥ 1✳ ❋♦r t❤❡ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✇❡ s❛② t❤❛t f (x) ❛♣♣r♦❛❝❤❡❞ b ❛s x ❛♣♣r♦❛❝❤❡s a✿ f (x) → b ❛s x → a .

✷✳✷✳ ▲✐♠✐ts

✶✹✶

❚❤❡ ♣✐❝t✉r❡ ❝❛♥ ❛❧s♦ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✿ • x ✐s ❛♣♣r♦❛❝❤✐♥❣ a✱ ❛♥❞

• y ✐s ❛♣♣r♦❛❝❤✐♥❣ b✳

❚❤✐s ❝♦♦r❞✐♥❛t❡✇✐s❡ t❤✐♥❦✐♥❣ ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❢♦r ✉s ❛♥②♠♦r❡ ❛♥❞ t❤❛t ✐s ✇❤② ✇❡ r❡♣❤r❛s❡ ♦♥❡ ♠♦r❡ t✐♠❡✿ • ❛ ♣♦✐♥t (x, y) ♦♥ t❤❡ ❝✉r✈❡ ✐s ❛♣♣r♦❛❝❤✐♥❣ ♣♦✐♥t (a, b)✳

❚❤✐s ✇❛② t♦ ❡①♣r❡ss t❤❡ ✐❞❡❛ ♦❢ ❧✐♠✐t ✐s ❢✉❧❧② ❛♣♣❧✐❝❛❜❧❡ t♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ t❤✐s t✐♠❡ ✐s t❤❛t y ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ x ❛♥②♠♦r❡❀ ✐♥st❡❛❞✱ x = x(t) ❛♥❞ y = y(t) ❞❡♣❡♥❞ ♦♥ t✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡✿ (x, y) → (a, b) ❛s t → s ,

❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡r s✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ✐♥ R ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn ✭tr❡❛t❡❞ ❛s ❡✐t❤❡r ♣♦✐♥ts ♦r ✈❡❝t♦rs✮✿ X(t) → A ❛s t → s . ❖♥❧② t❤❡ ❢♦r♠❡r ♣❛rt ♥❡❡❞s t♦ ❜❡ ❡①♣❧❛✐♥❡❞✳

❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ♣♦✐♥ts X(t) ❛r❡ ❛♣♣r♦❛❝❤✐♥❣ ❛♥♦t❤❡r ♣♦✐♥t A❄ ❚❤✐s ♠❡❛♥s t❤❛t X(t) ✐s ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ A ♦r✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ X(t) ✐s ❣❡tt✐♥❣ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❛♥❞✱ ✐♥ ❢❛❝t✱ ❛♣♣r♦❛❝❤✐♥❣ ③❡r♦✳

❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✿ ❧✐♠✐t ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ✐♥ Rn ❛t t = s ✐s ❞❡✜♥❡❞ t♦ ❜❡ s✉❝❤ ❛ ♣♦✐♥t A ✐♥ Rn t❤❛t d(X(t), A) → 0 ❛s t → s .

❋♦r ✈❡❝t♦rs✱ t❤❡ ❛♥❛❧♦❣♦✉s ❞❡✜♥✐t✐♦♥ ✐s✿

||X(t) − A|| → 0 ❛s t → s .

✷✳✷✳ ▲✐♠✐ts

✶✹✷

■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿ ▲✐♠✐t

lim X(t) = A

♦r

t→s

X(t) → A ❛s t → s

◆♦t❡ ❤♦✇ ✈❡❝t♦rs ❝♦♥✈❡r❣❡ ✐♥ ❜♦t❤ t❤❡ ♠❛❣♥✐t✉❞❡ ❛♥❞ ❞✐r❡❝t✐♦♥✿

❲❡ ❤❛✈❡ t❤✉s ❞❡✜♥❡❞ ❛ ♥❡✇ ❝♦♥❝❡♣t✱ t❤❡ ❧✐♠✐t ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ❜② r❡❧②✐♥❣ ❡♥t✐r❡❧② ♦♥ s♦♠❡t❤✐♥❣ ❢❛♠✐❧✐❛r✿ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ✉s✉❛❧✱ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✦ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝❛♥ t❛❦❡ ❛ ❞✐r❡❝t r♦✉t❡ ❜❡❧♦✇✳ ❲❡ ♠✐♠✐❝ t❤❡ t❤❡♦r② ♦❢ ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ♣♦✐♥t✳

❊①❛♠♣❧❡ ✷✳✷✳✷✿ ❝♦♥✈❡r❣❡♥❝❡

❙✉♣♣♦s❡ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ st✉❞② t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t t = 0✿ x = cos x, y = x2 .

❚❤❡ r❡❝✐♣r♦❝❛❧ s❡q✉❡♥❝❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡✿ tn =

1 → 0. n

❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥② ❢✉♥❝t✐♦♥ ❛♥❞ ❛ s❡q✉❡♥❝❡ ❣✐✈❡s ✉s ❛ ♥❡✇ s❡q✉❡♥❝❡✳✳✳ ❊①❛♠♣❧❡ ✷✳✷✳✸✿ t❤❡ s✐❣♥

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ sign ❛r♦✉♥❞ 0✳ ■ts st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐s x = t, y = sign(t)✳

✷✳✷✳ ▲✐♠✐ts

✶✹✸

❲❡ tr② tn = −1/n ❛♥❞ sn = 1/n✿ lim sign(−1/n) = −1 ❜✉t lim sign(1/n) = 1 ,

n→∞

n→∞

❛s ✇❡ ❛♣♣r♦❛❝❤ 0 ❢r♦♠ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛t ❛ t✐♠❡✳ ❚❤❡ ❧✐♠✐ts ❞♦♥✬t ♠❛t❝❤✦ ❚❤❡ ❛❧t❡r♥❛t✐✈❡ ❞❡✜♥✐t✐♦♥ ✐s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✷✳✷✳✹✿ ❆❧t❡r♥❛t✐✈❡ ❉❡✜♥✐t✐♦♥ ♦❢ ▲✐♠✐t ❚❤❡ ❧✐♠✐t ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ s❡q✉❡♥❝❡s ♦❢ ♣♦✐♥ts ✐♥

X = X(t)

✐♥

Rn

❛t

t=s

✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡s❡

Rn ✿ lim X(tn )

n→∞ ❝♦♥s✐❞❡r❡❞ ♦✈❡r ❛❧❧ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡ t♦

s✱

{tn }

✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢

s 6= tn → s

❛s

X

❡①❝❧✉❞✐♥❣

s

t❤❛t

n → ∞,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ❢♦r n = 2✿

▲❡t✬s ❝♦♥❝❡♥tr❛t❡ ♦♥ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛s ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t✿

❙✐♥❝❡ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ ✈❡❝t♦rs✱ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❛✈❛✐❧❛❜❧❡ ❢♦r ✈❡❝t♦rs ❛r❡ ❛❧s♦ ♣♦ss✐❜❧❡ ❢♦r ❢✉♥❝t✐♦♥s ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮✿ • ✈❡❝t♦r ❛❞❞✐t✐♦♥

• s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ t❤❡ s❝❛❧❛r ♠✐❣❤t ✐ts❡❧❢ ❞❡♣❡♥❞ ♦♥ t✦

✷✳✷✳ ▲✐♠✐ts

✶✹✹

• t❤❡ ❞♦t ♣r♦❞✉❝t✿ t❤❡ ♦✉t❝♦♠❡ ✐s ❛ s❝❛❧❛r ❢✉♥❝t✐♦♥✦

❯♥❞❡r t❤❡s❡ ♦♣❡r❛t✐♦♥s✱ t❤❡ ❧✐♠✐ts ❛r❡ ♣r❡s❡r✈❡❞✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡♠ ♦♥❡ ❜② ♦♥❡✳

❚❤❡♦r❡♠ ✷✳✷✳✺✿ ❙✉♠ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s ■❢ t❤❡ ❧✐♠✐ts ❛t

t=s

X = F (t) ❛♥❞ X = G(t) ❡①✐st t❤❡♥ X = F (t) + G(t)✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ✐s ❡q✉❛❧ t♦

♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r s✉♠✱ t❤❡ s✉♠ ♦❢ t❤❡ ❧✐♠✐ts✿

 lim F (t) + G(t) = lim F (t) + lim G(x) t→s

t→s

t→s

Pr♦♦❢✳

❲❡ ❝❛♥ ♣r♦✈❡ t❤❡ r❡s✉❧t ❜② ✉s✐♥❣ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠❀ ❛s ❢♦❧❧♦✇s✿ ❢♦r ❛♥② s❡q✉❡♥❝❡ t → s✱ ✇❡ ❤❛✈❡ ❜② ❙❘ ❢♦r s❡q✉❡♥❝❡s✿ lim(F (t) + G(t)) = lim (F (tn ) + G(tn )) = lim F (tn ) + lim G(tn ) . t→s

n→∞

n→∞

n→∞

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■❢ A ❛♥❞ B ❛r❡ t❤❡ t✇♦ ❧✐♠✐ts✱ ✇❡ ❝♦♥s✐❞❡r ❛♥❞ ♠❛♥✐♣✉❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ ||(F (t) + G(t)) − (A + B)|| =

||(F (t) − A)

+

(G(t) − B)||

❘❡✲❛rr❛♥❣❡ t❡r♠s✳

↓

❚❛❦✐♥❣ t❤❡ ❧✐♠✐ts✳

≤ ||(F (t) − A)|| + ||(G(t) − B)|| ❯s❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳ = =

↓ 0

+

❩❡r♦s ❜② t❤❡ ❞❡✜♥✐t✐♦♥✳

0

0.

❇② t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t F (t) + G(t) → A + B ✳ ❚❤❡♦r❡♠ ✷✳✷✳✻✿ ❈♦♥st❛♥t

▼✉❧t✐♣❧❡

❘✉❧❡

❋♦r

▲✐♠✐ts

♦❢

P❛r❛♠❡tr✐❝

❈✉r✈❡s

t = s ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❡①✐sts t❤❡♥ s♦ ❞♦❡s t❤❛t X = cF (t)✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♠✉❧t✐♣❧❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♠✉❧t✐♣❧❡

■❢ t❤❡ ❧✐♠✐t ❛t

♦❢

✐ts ♠✉❧t✐♣❧❡✱

♦❢

t❤❡ ❧✐♠✐t✿

lim cF (t) = c · lim F (t) t→s

t→s

Pr♦♦❢✳

❲❡ ❝❛♥ ♣r♦✈❡ t❤❡ r❡s✉❧t ❜② ✉s✐♥❣ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ♣r♦♦❢✿ ❢♦r ❛♥② s❡q✉❡♥❝❡ t → s✱ ✇❡ ❤❛✈❡ ❜② ❈▼❘ ❢♦r s❡q✉❡♥❝❡s✿   lim cF (t) = lim cF (tn ) = c lim F (tn ) . t→s

n→∞

n→∞

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■❢ A ✐s t❤❡ ❧✐♠✐t✱ ✇❡ ❝♦♥s✐❞❡r ❛♥❞ ♠❛♥✐♣✉❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣

✷✳✷✳ ▲✐♠✐ts ❡①♣r❡ss✐♦♥✿

✶✹✺

  || cF (t) − cA || = ||c · (F (t) − A)||

❋❛❝t♦r✳

= |c| · ||(F (t) − A)|| ❯s❡ ❍♦♠♦❣❡♥❡✐t②✳ ||

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t✳

↓

❩❡r♦ ❜② t❤❡ ❞❡✜♥✐t✐♦♥✳

= |c| · 0 = 0.

❇② t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t cF (t) → cA✳ ❲❤❛t ✐❢ t❤❡ ♠✉❧t✐♣❧❡ ✐s♥✬t ❝♦♥st❛♥t❄ ❚❤❡♦r❡♠ ✷✳✷✳✼✿ ❱❛r✐❛❜❧❡ ▼✉❧t✐♣❧❡ ❘✉❧❡ ■❢ t❤❡ ❧✐♠✐ts ❛t

t=s

X = F (t) ❛♥❞ ❛ ❢✉♥❝t✐♦♥ r = c(t) X = c(t) · F (t)✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡

♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

❡①✐st t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r ♣r♦❞✉❝t✱

♣r♦❞✉❝t ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✿

   lim c(t) · F (t) = lim c(t) · lim F (t) t→s

t→s

t→s

❊①❡r❝✐s❡ ✷✳✷✳✽

▼♦❞✐❢② t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ t♦ ♣r♦✈❡ t❤❡ ❧❛st t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✷✳✷✳✾✿ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s ■❢ t❤❡ ❧✐♠✐ts ❛t

t=s

X = F (t) ❛♥❞ X = G(t) ❡①✐st t❤❡♥ s♦ r = F (t) · G(t)✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t

♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

❞♦❡s t❤❛t ♦❢ t❤❡✐r ❞♦t ♣r♦❞✉❝t✱

✐s ❡q✉❛❧ t♦ t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✿

   lim F (t) · G(t) = lim F (t) · lim G(t) t→s

t→s

t→s

❚❤✐s ✐s ❛ s✉♠♠❛r② ♦❢ ❤♦✇ ❧✐♠✐ts ❜❡❤❛✈❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉s✉❛❧ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✷✳✷✳✶✵✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s ❙✉♣♣♦s❡

F (t) → A

❙❘✿ ❉P❘✿

❛♥❞

G(x) → B

F (t) + G(t) → A + B F (t) · G(t) → A · B

❛s

t → s✳

❈▼❘✿ ❱▼❘✿

❚❤❡♥

cF (t) → cA

c(t)F (t) → bA

❢♦r ❛♥② r❡❛❧ ✇❤❡♥

c

c(t) → b

◆♦✇ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ s♦ t❤❡ ❧❛tt❡r ❡①❤✐❜✐t ❥✉st ❛s ♠❛♥② t②♣❡s ♦❢ ❧❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛s t❤❡ ❢♦r♠❡r✳

✷✳✷✳ ▲✐♠✐ts

✶✹✻

❲❡ ✇✐❧❧ ❤♦✇❡✈❡r ♠❡♥t✐♦♥ ❤❡r❡ ♦♥❧② t❤❡ ♣❛tt❡r♥s t❤❛t ❛r❡ ❝♦♦r❞✐♥❛t❡✲✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡② ❛r❡ • ❝♦♥✈❡r❣❡♥❝❡ t♦ ❛ ♣♦✐♥t • ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t② • ♣❡r✐♦❞✐❝✐t②

❚❤❡ ✜rst ♦♥❡ ✐s ❞❡✜♥❡❞ ❥✉st ❛s t❤❡ ❝♦♠♠♦♥ ❧✐♠✐t ❜✉t ✐♥st❡❛❞ ♦❢ t ❛♣♣r♦❛❝❤✐♥❣ s♦♠❡ s✱ t → s✱ ✇❡ ❤❛✈❡ t ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t②✱ t → ∞✳

❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✶✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♣♣r♦❛❝❤❡s ♣♦✐♥t

❲❡ s❛② t❤❛t ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ❛♣♣r♦❛❝❤❡s ♣♦✐♥t A ❛s t → ±∞ ✐❢ ||X(t) − A|| → 0 ❛s t → ±∞ .

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿ X(t) → A ❛s t → ±∞ ,

♦r lim X(t) = A .

t→±∞

❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ✐♥ Rn , n > 1, ❛♥❞ t❤❛t✬s t❤❡ r❡❛s♦♥ ✇❡ ❝♦♥s✐❞❡r ♦♥❧② ♦♥❡ ✐♥✜♥✐t② ✿ t❤❡ ❣r❛♣❤ ✐s s✐♠♣❧② r✉♥♥✐♥❣ ❛✇❛② ❢r♦♠ 0✳

❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✷✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❣♦❡s t♦ ✐♥✜♥✐t② ❲❡ s❛② t❤❛t ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ❣♦❡s t♦ ✐♥✜♥✐t② ✐❢ ||X(tn )|| → ∞ ,

❢♦r ❛♥② s❡q✉❡♥❝❡ tn → ±∞ ❛s n → ∞✳ ❚❤❡♥ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿ X(t) → ∞ ❛s t → ±∞ ,

♦r lim X(t) = ∞ .

t→±∞

❊①❛♠♣❧❡ ✷✳✷✳✶✸✿ ❣r❛♣❤ ❋r♦♠ t❤❡ ❢❛♠✐❧✐❛r ❢❛❝ts✿

lim et = 0,

t→−∞

lim et = +∞ ,

t→+∞

✷✳✸✳

❈♦♥t✐♥✉✐t②

✶✹✼

✇❡ ❞❡❞✉❝❡ t❤❛t

(t, et ) → ∞ ❛s t → ±∞ .

❊①❛♠♣❧❡ ✷✳✷✳✶✹✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t② ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ ❡①❛♠♣❧❡s ♦❢ s♦♠❡ ♦❢ t❤❡s❡ ❜❡❤❛✈✐♦rs ❡①❤✐❜✐t❡❞ ❜② ❛ ♣❧❛♥❡t ✉♥❞❡r t❤❡ ❡✛❡❝t ♦❢ ❣r❛✈✐t②✳ ❚❤❡② ♠❛② ❜❡ ❜♦✉♥❞❡❞ ♦r ✉♥❜♦✉♥❞❡❞✳ ❲❤✐❝❤ ♦♥❡ ✇❡ ❛r❡ t♦ ♦❜s❡r✈❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ▲❡t✬s ♠❛❦❡ t❤❡ ❝♦♥♥❡❝t✐♦♥ ♠♦r❡ ♣r❡❝✐s❡✳

✷✳✸✳ ❈♦♥t✐♥✉✐t②

❆ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t x = a ✐❢ lim f (x) = f (a) .

x→a

❚❤✉s✱ t❤❡ ❧✐♠✐ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✳ ❲❡ ❛♣♣r♦❛❝❤ t❤❡ ✐ss✉❡ ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ ❛♥ ✐❞❡♥t✐❝❛❧ ❢❛s❤✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✿ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s

❛t t = s ✐❢

lim X(t) = X(s) . t→s

❊①❛♠♣❧❡ ✷✳✸✳✷✿ ❞✐s❝♦♥t✐♥✉✐t② ❊①❛♠♣❧❡s ♦❢ ❝♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t② ❝♦♠❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✷✮✳

✷✳✸✳

❈♦♥t✐♥✉✐t②

✶✹✽

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝❛♥ ✐♥t❡r♣r❡t ❝♦♥t✐♥✉✐t② ❛s ❥✉st ❛♥♦t❤❡r ❛❧❣❡❜r❛✐❝ r✉❧❡ ♦❢ ❧✐♠✐ts✦ ❚❤❡♦r❡♠ ✷✳✸✳✸✿ ❙✉❜st✐t✉t✐♦♥ ❘✉❧❡

■❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

X = X(t)

✐s ❝♦♥t✐♥✉♦✉s ❛t

t=s

t❤❡♥

lim X(t) = X(s) t→s

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤✐s r✉❧❡ ✐s r❡❧❛t❡❞ t♦ ❝♦♠♣♦s✐t✐♦♥s✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ♦❢ X = X(t) ❛t t = s ❝❛♥ ❜❡ r❡✲✇r✐tt❡♥ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✿ ◮ ■❢ t ✐s ❛♣♣r♦❛❝❤✐♥❣ s✱ t❤❡♥ X(t) ✐s ❛♣♣r♦❛❝❤✐♥❣ A✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ t → s =⇒ X = X(t) → A .

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X ✐♥ Rn ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ z ♦❢ n ✈❛r✐❛❜❧❡s ✇❛s ❝♦♥s✐❞❡r❡❞ ✐♥ ❈❤❛♣t❡r ✶✿ (z ◦ X)(t) = z(X(t)) .

❆s ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❤❛s♥✬t ❜❡❡♥ tr❡❛t❡❞ ②❡t✱ ✇❡ ♣♦st♣♦♥❡ t❤✐s ❡①❛♠♣❧❡ ✉♥t✐❧ ❈❤❛♣t❡r ✸ ❛♥❞ ✐♥st❡❛❞ ❧♦♦❦ ❛t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ X ✇✐t❤ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ t = t(u) ✭♦♥ t❤❡ ♦t❤❡r ❡♥❞✮✿ (X ◦ t)(u) = X(t(u)) .

❚❤✉s ✇❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s ♣r♦❝❡ss✐♥❣ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ u 7→ t 7→ X .

❋✉rt❤❡r♠♦r❡✱ ❧❡t✬s ❧♦♦❦ ❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❛t s♦♠❡ v ✿ u → v =⇒ X = (X ◦ t)(u) → A ,

✐❢ t❤✐s A ❡①✐sts✳ ❲❡ ❜r❡❛❦ t❤✐s ✐♥t♦ t✇♦✳ ❋✐rst✿ u → v =⇒ t = t(u) → s .

❙❡❝♦♥❞✱ ❧❡t✬s s✉♣♣♦s❡ t❤❛t X ✐s ❝♦♥t✐♥✉♦✉s ❛t t❤✐s t = s✳ ❚❤❡♥✱ t → s =⇒ X = X(t) → X(s) .

❚♦❣❡t❤❡r✿ u → v =⇒ t = t(u) → s =⇒ X = X(t) → X(s) .

✷✳✸✳

❈♦♥t✐♥✉✐t②

✶✹✾

❚❤❡♦r❡♠ ✷✳✸✳✹✿ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡

❙✉♣♣♦s❡ t = t(u) ✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤✿ lim t(u) = s .

u→v

❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ✐s ❝♦♥t✐♥✉♦✉s ❛t t = s✳ ❚❤❡♥✱ lim (X ◦ t)(u) = X(s) .

u→v

❚❤✐s ✐s ❛ s❤♦rt❡♥❡❞ ✈❡rs✐♦♥✿

  lim X(t(u)) = X lim t(u) .

u→v

u→v

■t r❡✈❡❛❧s ❤♦✇ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ♣✉❧❧❡❞ ♦✉t ♦❢ ❛ ❧✐♠✐t✳ ❝♦♠♣✉t❡❞ ❜②

s✉❜st✐t✉t✐♦♥✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧✐♠✐t ✐s✱ ❛❣❛✐♥✱

❲❡ ♥♦✇ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts t♦ s❤♦✇ ❤♦✇ ❝♦♥t✐♥✉✐t② ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r t❤❡s❡ ♦♣❡r❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✷✳✸✳✺✿ ❈♦♥t✐♥✉✐t② ❛♥❞ ❆❧❣❡❜r❛

❙✉♣♣♦s❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s F ❛♥❞ G ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t t = s✳ ❚❤❡♥ s♦ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ✭❙❘✮ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ F ± G✱ • ✭❈▼❘✮ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ c · F ✱ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡r c✱ • ✭❱▼❘✮ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ c · F ✱ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ c✱ ❛♥❞ • ✭❉P❘✮ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ F · G✳ ❙♦♠❡ t❤❡♦r❡♠s ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❝♦♥t✐♥✉♦✉s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❛♥❛❧♦❣✉❡s ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❋✐rst✱ ✐♥ t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s t❤❡r❡ ❛r❡ ♥♦ ✏❧❛r❣❡r✑ ♦r ✏s♠❛❧❧❡r✑ ♣♦✐♥ts ♦r ✈❡❝t♦rs✳ ❚❤✐s ✐s ✇❤② ✇❡ ❞♦♥✬t ❝♦♠♣❛r❡ ❢✉♥❝t✐♦♥s ✭♦r t❤❡✐r ❧✐♠✐ts✮ ❛s ❡❛s✐❧② ❛s

f (x) < g(x)

❛♥②♠♦r❡✳ ◆♦ ❝♦♠♣❛r✐s♦♥ t❤❡♦r❡♠s t❤❡♥✳

❋♦r t❤❡ s❛♠❡ r❡❛s♦♥ ✇❡ ❝❛♥✬t ❡❛s✐❧② sq✉❡❡③❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❜❡t✇❡❡♥ t✇♦ ♦t❤❡r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ sq✉❡❡③❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✇✐t❤ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❞✐st❛♥❝❡ t♦ ❛♥♦t❤❡r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❋♦r ❞✐♠❡♥s✐♦♥

❋♦r ❞✐♠❡♥s✐♦♥

n = 3✱

n = 2✱

t❤✐s ✐s ❛ ♥❛rr♦✇✐♥❣ str✐♣✿

✐t✬s ❛ ❢✉♥♥❡❧✳ ❚❤❡♦r❡♠ ✷✳✸✳✻✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ✜rst✿ d(P (t), Q(t)) ≤ h(t) ,

❢♦r ❛❧❧ t ✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ ❢r♦♠ t = s❀ s❡❝♦♥❞✿ lim Q(t) = A ; t→s

✷✳✸✳

✶✺✵

❈♦♥t✐♥✉✐t②

❛♥❞ t❤✐r❞✿

lim h(t) = 0 . t→s

❚❤❡♥

lim P (t) = A . t→s

Pr♦♦❢✳ ❚❤❡ t❤✐r❞ ❝♦♥❞✐t✐♦♥ ❣✉❛r❛♥t❡❡s t❤❛t d(P (t), Q(t)) → 0 .

❋♦r t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s✱ t❤❡ ✜rst ❝♦♥❞✐t✐♦♥ ❜❡❝♦♠❡s✿ ||P (t) − Q(t)|| ≤ h(t) .

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t② ✐s ♣✉r❡❧② ❧♦❝❛❧ ✿ ♦♥❧② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t ♠❛tt❡rs✳ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ✐ts ❣❧♦❜❛❧ ❜❡❤❛✈✐♦r❄

❉❡✜♥✐t✐♦♥ ✷✳✸✳✼✿ ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P = P (t) ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧ I ✐❢ ✐t ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② s ✐♥ I ✳

❉❡✜♥✐t✐♦♥ ✷✳✸✳✽✿ ❜♦✉♥❞❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ✐s ❝❛❧❧❡❞ s✉❝❤ ❛ r❡❛❧ ♥✉♠❜❡r m t❤❛t

❜♦✉♥❞❡❞

♦♥ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐❢ t❤❡r❡ ✐s

||X(t)|| ≤ m

❢♦r ❛❧❧ t ✐♥ [a, b]✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✇✐t❤✐♥ t❤❡ s♣❤❡r❡ ♦❢ r❛❞✐✉s m✳

❚❤❡♦r❡♠ ✷✳✸✳✾✿ ❇♦✉♥❞❡❞♥❡ss ❆ ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❜♦✉♥❞❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧✳

Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ r❡♣❡❛ts t❤❡ ♦♥❡ ❢♦r t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❙✉♣♣♦s❡✱ t♦ t❤❡ ❝♦♥tr❛r②✱ t❤❛t X ✐s ✉♥❜♦✉♥❞❡❞ ♦♥ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ {tn } ✐♥ [a, b] s✉❝❤ t❤❛t X(tn ) ✐s ✉♥❜♦✉♥❞❡❞✳ ❚❤❡♥✱ ❜② t❤❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠✱ s❡q✉❡♥❝❡ {tn } ❤❛s ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ {uk }✿ uk → u .

✷✳✸✳ ❈♦♥t✐♥✉✐t②

❚❤✐s ♣♦✐♥t ❜❡❧♦♥❣s t♦

✶✺✶

[a, b]✦

❋r♦♠ t❤❡ ❝♦♥t✐♥✉✐t②✱ ✐t ❢♦❧❧♦✇s t❤❛t

X(uk ) → X(u) ❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t

||X(uk )||

♦r

||X(uk ) − X(u)|| → 0 .

✐s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❞✐✈❡r❣❡s t♦ ✐♥✜♥✐t②✳

❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❤❛s ❜❡❡♥ ❛s t❤❡ ♣r♦♣❡rt② ♦❢ ❤❛✈✐♥❣ ♥♦ ❣❛♣s ✐♥ t❤❡✐r ❣r❛♣❤s✳ ❚❤❡ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ s❛②s ❛❧s♦ t❤❛t t❤❡r❡ ❛r❡ ♥♦ ❣❛♣s ✐♥ t❤❡ r❛♥❣❡s ❡✐t❤❡r✳ ❲❤❛t ❛❜♦✉t ♣❛r❛♠❡tr✐❝ ❝✉r✈❡❄ ■t✬s ♣❛t❤ s❤♦✉❧❞ ❤❛✈❡ ♥♦ ❣❛♣s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t♦ ❣❡t t♦ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ r✐✈❡r ✇❡ ❤❛✈❡ t♦ ❝r♦ss ✐t✦

❚❤✐s ✐❞❡❛ ✐s ♠♦r❡ ♣r❡❝✐s❡❧② ❡①♣r❡ss❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✷✳✸✳✶✵✿ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ♦♥ t❤❡ ♣❧❛♥❡ ✐s ❞❡✜♥❡❞ ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧ [a, b]✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t t❤❡r❡ ✐s ❛ ❧✐♥❡ L s✉❝❤ t❤❛t X(a) ❛♥❞ X(b) ❧✐❡ ♦♥ t❤❡ ❞✐✛❡r❡♥t s✐❞❡s ♦❢ L✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ d ✐♥ [a, b] s✉❝❤ t❤❛t X(d) ❧✐❡s ♦♥ L✳

❊①❡r❝✐s❡ ✷✳✸✳✶✶ ❊①♣❧❛✐♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✏❧✐❡ ♦♥ t❤❡ ❞✐✛❡r❡♥t s✐❞❡s ♦❢ ✑✳

❚❤✐s t❤❡♦r❡♠ ♠♦t✐✈❛t❡s ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t✳

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✷✿ ♣❛t❤✲❝♦♥♥❡❝t❡❞ s✉❜s❡t Q ♦❢ Rn ✐s ❝❛❧❧❡❞ ♣❛t❤✲❝♦♥♥❡❝t❡❞ ✐❢ ❛♥② t✇♦ ♣♦✐♥ts ✐♥ Q ❝❛♥ ❜❡ ❝♦♥♥❡❝t❡❞ ❜② ❛ ♣❛t❤✱ ✐✳❡✳✱ ✐❢ A ❛♥❞ B ❛r❡ t✇♦ ♣♦✐♥ts ✐♥ Q t❤❡♥ t❤❡r❡ ✐s s✉❝❤ ❛ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❞❡✜♥❡❞ ♦♥ [a, b] t❤❛t F (a) = A ❛♥❞ F (b) = B ✳ ❆ s✉❜s❡t

❚❤❡♥ t❤❡ t❤❡♦r❡♠ t❡❧❧s ✉s t❤❛t t❤❡ ♣❧❛♥❡ ✇✐t❤ ❛ ❧✐♥❡ r❡♠♦✈❡❞✱ ✐✳❡✳✱

R2 \L

✐s✱ t❤❡ ❝✐r❝❧❡ t♦♦✱ ❛ ♣♦✐♥t ❜✉t ♥♦t t✇♦✱ ❛ ❧✐♥❡✱ ❛ s♣❤❡r❡ ❜✉t ♥♦t t✇♦✱ ❡t❝✳✿

✐s ♥♦t ♣❛t❤✲❝♦♥♥❡❝t❡❞✳ ❚❤❡ ♣❧❛♥❡

✷✳✸✳ ❈♦♥t✐♥✉✐t②

✶✺✷

◆❡①t✱ ♦♥❝❡ ❛❣❛✐♥✱ t❤❡r❡ ❛r❡ ♥♦ ✏❧❛r❣❡r✑ ♦r ✏s♠❛❧❧❡r✑ ♣♦✐♥ts ♦r ✈❡❝t♦rs✳ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥✬t s♣❡❛❦ ♦❢ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts✳ ❚❤❡♦r❡♠ ✷✳✸✳✶✸✿ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❆ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦♥ ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❤❛s ❛ ♣♦✐♥t ♦❢ ♠❛①✲ ✐♠❛❧ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥✱ ✐✳❡✳✱ ✐❢ ✐s

c

✐♥

[a, b]

X = X(t) ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✱ t❤❡♥ t❤❡r❡

s✉❝❤ t❤❛t

||X(c)|| ≥ ||X(t)|| , ❢♦r ❛❧❧

t

✐♥

[a, b]✳

Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠✳

❚❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ tr❡❛t♠❡♥t ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❢♦❧❧♦✇s ❢r♦♠ t❤❛t ❢♦r s❡q✉❡♥❝❡s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶✳ ❚❤❡♦r❡♠ ✷✳✸✳✶✹✿ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ❋♦r P❛r❛♠❡tr✐❝ ❈✉r✈❡ ❆s

♦r

t → s✱

✇❡ ❤❛✈❡✿

X(t) → A ,  X(t) = p1 (t), p2 (t), ..., pn (t) → A = (a1 , a2 , ..., an ) ,

✐❢ ❛♥❞ ♦♥❧② ✐❢

p1 (t) → a1 , p2 (t) → a2 , ..., pn (t) → an .

✷✳✸✳

❈♦♥t✐♥✉✐t②

✶✺✸

❊①❛♠♣❧❡ ✷✳✸✳✶✺✿ ❛ ❧✐♠✐t ❲❡ ❝♦♠♣✉t❡✿

lim (cos t, sin t) =

t→π/2



lim cos t, lim sin t

t→π/2

t→π/2

= (cos π/2, sin π/2)



= (1, 0) , ❜❡❝❛✉s❡

sin t

❛♥❞

cos t

❛r❡ ❝♦♥t✐♥✉♦✉s✳

❚❤✐s t❤❡♦r❡♠ ❛❧s♦ ♠❛❦❡s ✐t ❡❛s② t♦ ♣r♦✈❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❢r♦♠ t❤♦s❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡

❉♦t Pr♦❞✉❝t ❘✉❧❡

lim < x, y > · < u, v > t→s



✐s ♣r♦✈❡♥ ❜❡❧♦✇✿

= lim(xu + yv) t→s

= lim(xu) + lim(yv) t→s

t→s

= lim x · lim u + lim y · lim v t→s

t→s

t→s

t→s

=< lim x, lim y > · < lim u, lim v > . t→s

◆♦✇ ❝♦♥t✐♥✉✐t②✳ ❆t

t = s✱

t→s

t = s✳

t→s

s✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

X(t) = x(t), y(t), z(t) ✐s ❝♦♥t✐♥✉♦✉s ❛t

t→s

❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ❛s



t → s✱

✇❡ ❤❛✈❡✿

X(t) → X(s) ⇐⇒ x(t) → x(s), y(t) → y(s), z(t) → z(s) . ❚❤❡ ❧❛tt❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ t❤r❡❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s

x(t), y(t), z(t)

❜❡✐♥❣ ❝♦♥t✐♥✉♦✉s ❛t

t = s✳

✷✳✹✳

▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥

✶✺✹

❚❤❡♦r❡♠ ✷✳✸✳✶✻✿ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥t✐♥✉✐t② ❖❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱

 X(t) = p1 (t), ..., pn (t) ,

t = s ✐❢ ❛♥❞ ❛t t = s✳

✐s ❝♦♥t✐♥✉♦✉s ❛t ❛r❡ ❝♦♥t✐♥✉♦✉s

♦♥❧② ✐❢ ❛❧❧ ✐ts ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱

p1 (t), ..., pn (t)✱

❊①❛♠♣❧❡ ✷✳✸✳✶✼✿ ❝✐r❝❧❡

❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❝✐r❝❧❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡❄ ❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ ✐ts ♣❛r❛♠❡tr✐③❛t✐♦♥

sin t

cos t✱

❛r❡ ❝♦♥t✐♥✉♦✉s✳

✷✳✹✳ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ s♦♠❡ ✏♣r❡✲❧✐♠✐t✑ ❝❛❧❝✉❧✉s❀ ✇❡ ✇✐❧❧ r❡✈✐s✐t t❤❡ ❧♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥ ✐ss✉❡ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s❀ t❤❡ ♠♦t✐♦♥ ✐s ♦♥ t❤❡ ♣❧❛♥❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r♦✈❡ ❛r♦✉♥❞ ✇❤✐❧❡ ♣❛②✐♥❣ ❛tt❡♥t✐♦♥ ❜♦t❤ t♦ t❤❡ ❝❧♦❝❦ ❛♥❞ t♦ t❤❡ ♠✐❧❡♣♦sts✳ ❚❤❡ r❡s✉❧t ✐s t❤✐s s✐♠♣❧❡ t❛❜❧❡ ✇✐t❤

✜✈❡

❝♦❧✉♠♥s✿

0

t✐♠❡ ✭❤♦✉rs✮✿ ❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

2

4

6

8

(0, 0) (60, 0) (120, 20) (160, 60) (160, 100)

❙♦✱ ✇❡ ❞r♦✈❡ ❡❛st ❛♥❞ t❤❡♥ ♥♦rt❤✳ ❲❤❛t ✇❛s t❤❡ ✈❡❧♦❝✐t② ♦✈❡r t❤❡s❡

❢♦✉r

♣❡r✐♦❞s ♦❢ t✐♠❡❄ ❲❡ ❡st✐♠❛t❡ ✐t ✇✐t❤ t❤❡ ❢❛♠✐❧✐❛r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

❛✈❡r❛❣❡ ✈❡❧♦❝✐t②

=

❝❤❛♥❣❡ ♦❢ ❧♦❝❛t✐♦♥ ❝❤❛♥❣❡ ♦❢ t✐♠❡

❡①❝❡♣t t❤✐s t✐♠❡ t❤❡ ♥✉♠❡r❛t♦r✱ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ✐ts❡❧❢✱ ✐s ❛ t✐♠❡ ✭❤♦✉rs✮✿ ❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿ ✈❡❧♦❝✐t② ✭♠✴❤✮✿

✈❡❝t♦r✳

2

4

6

8

(0, 0)

(60, 0)

(120, 20)

(160, 60)

(160, 100)

(60,0)−(0,0) 2−0

=

< 30, 0 > (120,20)−(60,0) 4−2

=

< 30, 10 > (160,60)−(120,20) 6−4

✈❡❧♦❝✐t② ✭♠✴❤✮✿

=

< 20, 20 > (160,100)−(160,60) 8−6

✈❡❧♦❝✐t② ✭♠✴❤✮✿ ❚❤❡ ❢♦✉r ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t✐❡s ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣

[6, 8]✱

❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿

0

✈❡❧♦❝✐t② ✭♠✴❤✮✿

❛♥❞

,

r❡s♣❡❝t✐✈❡❧②✳ ❚❤✐s ✐s✱ ♦❢ ❝♦✉rs❡✱ ❛

♣❛rt✐t✐♦♥✳

✈❡❧♦❝✐t② ✭♠✴❤✮✿

♦❢ t✐♠❡✿

[0, 2]✱ [2, 4]✱ [4, 6]✱

❚❤❡ s✉♠♠❛r② ✐s ✐♥ t❤✐s t❛❜❧❡✿

[0, 2]

t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿

✐♥t❡r✈❛❧s

= < 0, 20 >

[2, 4]

[4, 6]

[6, 8]

< 30, 0 > < 30, 10 > < 20, 20 > < 0, 20 >

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ♠❛② ❝❤♦♦s❡ t♦ ❛ss✐❣♥ t❤❡ ❢♦✉r ✈❛❧✉❡s t♦ t❤❡ ♠✐❞❞❧❡ ♣♦✐♥ts ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✱ ❛s t❤❡

♥♦❞❡s

♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✴❤✮✿

1

3

5

7

< 30, 0 > < 30, 10 > < 20, 20 > < 0, 20 >

s❡❝♦♥❞❛r②

✷✳✹✳ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥

✶✺✺

❚❤✐s ❛♠♦✉♥t t♦ s✐♠♣❧② ❝❤♦♦s✐♥❣ ❞✐✛❡r❡♥t s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤✐s ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✿

❲❡ ❛❧s♦ ❝♦♠♣✉t❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

❏✉st ❛s ✇❡ ✉s❡❞ t❤❡ ✏❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✑ ❢♦r♠✉❧❛ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t②

❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ♥♦✇ ✉s❡ ✐t t♦ ✜♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✱ ❛s ✈❡❝t♦rs✿ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥

=

❝❤❛♥❣❡ ♦❢ ✈❡❧♦❝✐t② ❝❤❛♥❣❡ ♦❢ t✐♠❡

.

❲❡ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛ t♦ t❤r❡❡ ♣❡r✐♦❞s ♦❢ t✐♠❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿

[0, 2]

[2, 4]

[4, 6]

[6, 8]

1

3

5

7

< 30, 0 >

< 30, 10 >

< 20, 20 >

< 0, 20 >

t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

− 3−1

❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

=

< 0, 5 > − 5−3

=

< −5, 5 >

− 7−5

❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

= < −10, 0 >

❚❤❡ t❤r❡❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥s ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✿ ❛♥❞

[4, 8]✱

[0, 4]✱ [2, 6]✱

r❡s♣❡❝t✐✈❡❧②✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

[0, 4]

[2, 6]

[4, 8]

< 0, 5 > < −5, 5 > < −10, 0 >

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ♠❛② ❝❤♦♦s❡ t♦ ❛ss✐❣♥ t❤❡ t❤r❡❡ ✈❛❧✉❡s t♦ t❤❡ ♠✐❞❞❧❡ ♣♦✐♥ts ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✱ ❛s ❢♦❧❧♦✇s✿ t✐♠❡ ✭❤♦✉rs✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✿

2

4

6

< 0, 5 > < −5, 5 > < −10, 0 >

✷✳✹✳ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥

✶✺✻

◆❡①t ✇❡ r❡✈❡rs❡ t❤❡ ♣r♦❜❧❡♠✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❦♥♦✇♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❛r❡ t♦ ❜❡ ❢♦✉♥❞✳ ❙✉♣♣♦s❡ ✇❡ st❛rt ♦✉r ♠♦t✐♦♥ ❛t t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥

(0, 0)

♠✐❧❡s ✇✐t❤ ❛♥ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②

< 10, 0 >

♠✐❧❡s ♣❡r

❤♦✉r ❛♥❞ ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❣✐✈❡♥ ❜❡❧♦✇✿

[0, 4]

t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

[2, 6]

[4, 8]

< −10, 10 > < 0, 10 > < 10, 10 >

❲❡ ♥♦✇ ❛♣♣❧② t❤❡ s❛♠❡ ❢♦r♠✉❧❛s ❛s t❤❡ ♦♥❡ ✇❡ ✉s❡❞ ✇❤❡♥ t❤❡ ♠♦t✐♦♥ ✇❛s ❞❡s❝r✐❜❡❞ ❜② ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✭♦r t✇♦✮✿ ♥❡①t ✈❡❧♦❝✐t②

=

❝✉rr❡♥t ✈❡❧♦❝✐t②

+

❛❝❝❡❧❡r❛t✐♦♥

·

t✐♠❡ ✐♥❝r❡♠❡♥t ,

❡①❝❡♣t t❤✐s t✐♠❡ t❤❡s❡ q✉❛♥t✐t✐❡s✱ ❡①❝❡♣t ❢♦r t✐♠❡✱ ❛r❡ ✈❡❝t♦rs✳ ❲❡ ❝❤♦♦s❡ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t t♦ ❜❡

2✳

[0, 2]

[2, 4]

[4, 6]

< −10, 10 >

< 0, 10 >

< 10, 10 >

t✐♠❡ ✭❤♦✉rs✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✴❤✴❤✮✿

❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿

✈❡❧♦❝✐t② ✭♠✴❤✮✿

< 10, 0 >

✈❡❧♦❝✐t② ✭♠✴❤✮✿

+2 < −10, 10 >=

✈❡❧♦❝✐t② ✭♠✴❤✮✿

< −10, 20 >

[6, 8]

< −10, 40 >

+2 < 0, 10 >=

+2 < 10, 10 >= < 10, 60 >

✈❡❧♦❝✐t② ✭♠✴❤✮✿

❚❤❡ ❢♦✉r ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t✐❡s ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿

[0, 2]

[2, 4]

[4, 6]

[6, 8]

< 10, 0 > < −10, 20 > < −10, 40 > < 10, 60 >

✈❡❧♦❝✐t② ✭♠✴❤✮✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ♣♦s✐t✐♦♥ ♥❡①t ✉s✐♥❣ t❤❡ s❛♠❡✱ ✈❡❝t♦r✱ ❢♦r♠✉❧❛✿ ♥❡①t ♣♦s✐t✐♦♥

=

❝✉rr❡♥t ♣♦s✐t✐♦♥ + ✈❡❧♦❝✐t②

·

t✐♠❡ ✐♥❝r❡♠❡♥t .

✷✳✹✳ ▲♦❝❛t✐♦♥ ✲ ✈❡❧♦❝✐t② ✲ ❛❝❝❡❧❡r❛t✐♦♥

✶✺✼

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿

0

2

4

6

< 10, 0 >

< −10, 20 >

< −10, 40 >

< 10, 60 >

t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✴❤✮✿ ♣♦s✐t✐♦♥ ✭♠✮✿

(0, 0)

♣♦s✐t✐♦♥ ✭♠✮✿

+2 < 10, 0 >=

(20, 0) +2 < −10, 20 >=

♣♦s✐t✐♦♥ ✭♠✮✿

8

♣♦s✐t✐♦♥ ✭♠✮✿

(0, 40) +2 < −10, 40 >=

♣♦s✐t✐♦♥ ✭♠✮✿

(−20, 120) +2 < 10, 60 >= (0, 240)

❚❤✐s ✐s t❤❡ ♣❛t❤✿ t✐♠❡ ✭❤♦✉rs✮✿ ❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

0

2

4

6

8

(0, 0) (20, 0) (0, 40) (−20, 120) (0, 240)

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤✐s ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡q✉❡♥❝❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ s♣❛❝❡✿

• tn

❢♦r t❤❡ t✐♠❡

• pn

❢♦r t❤❡ ♣♦s✐t✐♦♥

• vn

❢♦r t❤❡ ✈❡❧♦❝✐t②

• an

❢♦r t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

❚❤❡ ❧❛st t❤r❡❡ ❛r❡ ❧✐♥❦❡❞ ❜② t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r ❡❛❝❤ ❝♦♠♣♦♥❡♥t✿

pn+1 − pn tn+1 − tn

❛♥❞

an+1 =

vn+1 = vn + an+1 (tn+1 − tn )

❛♥❞

pn+1 = pn + vn+1 (tn+1 − tn ) .

vn+1 =

vn+1 − vn . tn+1 − tn

❖r✱ ❢♦r t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✿

❲✐t❤ ♠♦r❡ ❞❛t❛✱ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥s ❛r❡ ✉s❡❞ t♦ tr❛❝❡ ♦✉t ❝✉r✈❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ ❧♦❝❛t✐♦♥ ❧♦♦❦s ❧✐❦❡✿

✷✳✺✳

❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✶✺✽

◆♦t❡ t❤❛t t❤❡ ✈❡❧♦❝✐t② ❝♦❧✉♠♥ ❤❛s ♦♥❡ ❢❡✇❡r ❞❛t❛ ♣♦✐♥t ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦♥❡ ❢❡✇❡r ②❡t✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ③♦♦♠❡❞ ♦✉t✱ t❤❡ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡ ❛❝t✉❛❧ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡s ❛♥❞ ❣✐✈❡ ❛♥ ✐♠♣r❡ss✐♦♥ t❤❛t t❤❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

✷✳✺✳ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❝♦♥str✉❝t t❤❡ s❡❝❛♥t ♦r t❛♥❣❡♥t ❧✐♥❡s ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

❛t ❛❧❧ ♣♦✐♥ts

❛t t❤❡ s❛♠❡ t✐♠❡✿

❚❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ ❛ ❢✉♥❝t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ ♦♥❧②

t✇♦

✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥✿

F (a) = A a 6= b✳ ❝❤❛♥❣❡ ♦❢ t✳ ✇✐t❤

❛♥❞

F (b) = B ,

❚❤❡♥✱ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ✐ts r❛t❡ ♦❢ ❝❤❛♥❣❡❄ ❚❤❡ ❢♦r♠❡r ✐s t❤❡

❚❤❡ ❧❛tt❡r ✐s t❤❡

✐♥❝r❡♠❡♥t

❞✐✛❡r❡♥❝❡

♦❢

f✱

■t ✐s t❤❡ ❝❤❛♥❣❡ ♦❢

X

✇✐t❤ r❡s♣❡❝t t♦ t❤❡

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

∆f = B − A = F (b) − F (a) . ♦❢

t✱

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

∆t = b − a . ❚❤❡✐r r❛t✐♦ ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢

F✱

✇❤✐❝❤ ✇❡ ✇✐❧❧ ❝❛❧❧ t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

F (b) − F (a) ∆F = . ∆t b−a

♦❢

F✱

❣✐✈❡♥ ❜②✿

✷✳✺✳ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✶✺✾

▲❡t✬s r❡✈✐❡✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ X ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ♣❧❛❝❡ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ✭♦r s✐♠♣❧② ♥♦❞❡s✮ ♦♥ t❤❡ ✐♥t❡r✈❛❧✿ a = t0 ≤ t1 ≤ t2 ≤ ... ≤ tn−1 ≤ tn = b ,

♣r♦❞✉❝✐♥❣ n s♠❛❧❧❡r ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [t0 , t1 ], [t1 , t2 ], ..., [tn−1 , tn ] ,

✇✐t❤ t0 = a, tn = b✳ ❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡ t❤❡ ✐♥❝r❡♠❡♥ts ♦❢ t✿ ∆ti = ti − ti−1 , i = 1, 2, ..., n .

❲❡ ❛r❡ ❛❧s♦ ❣✐✈❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ c1 ✐♥ [t0 , t1 ], c2 ✐♥ [t1 , t2 ], ..., cn ✐♥ [tn−1 , tn ] .

❚❤✉s ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ✐s s✐♠♣❧② ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦✐♥ts✿ a = t0 ≤ c1 ≤ t1 ≤ c2 ≤ t2 ≤ ... ≤ tn−1 ≤ cn ≤ tn = b .

■♥ t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇❡ ❤❛❞ ❛ s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥✿ • ❚❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✇❡r❡ ❡q✉❛❧ ✐♥ ❧❡♥❣t❤✳

• ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✇❡r❡ ♣❧❛❝❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ✉t✐❧✐③❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✿

✷✳✺✳ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✶✻✵

❙✉♣♣♦s❡ X = F (t) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s tk , k = 0, 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✺✳✶✿ ❞✐✛❡r❡♥❝❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ F ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆F (ck ) = F (tk ) − F (tk−1 ), k = 1, 2, ..., n

❉❡✜♥✐t✐♦♥ ✷✳✺✳✷✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ F ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ F (tk ) − F (tk−1 ) ∆F (ck ) ∆F (ck ) = = , k = 1, 2, ..., n ∆t tk − tk−1 ∆tk

◆♦t❡ t❤❛t ✇❤❡♥ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡♥✬t ♣r♦✈✐❞❡❞✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ✐♥t❡r✈❛❧s t❤❡♠s❡❧✈❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✭❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✮✿ ck = [tk−1 , tk ]✳ ■t ✐s t❤❡♥ ❛ 1✲❢♦r♠✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ 0✲❢♦r♠ F ✳ ❲❤❡♥ t❤❡ ❝♦♥t❡①t ✐s ❝❧❡❛r✱ ✇❡ ✉s❡ t❤❡ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥ ✇✐t❤♦✉t t❤❡ s✉❜s❝r✐♣ts✿

❉✐✛❡r❡♥❝❡ ❛♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆F (c) ❛♥❞

∆F (c) ∆t

❊①❛♠♣❧❡ ✷✳✺✳✸✿ ❝✐r❝❧❡ ▲❡t✬s ✜♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❝✐r❝❧❡ ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉s✉❛❧ ✇❛②✿ X(t) = (cos t, sin t) .

❲❡ ✉s❡ t❤❡ ❚r✐❣♦♥♦♠❡tr✐❝ ❋♦r♠✉❧❛s ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥ ❢♦r t❤❡ ✐♥t❡r✈❛❧✱ s❛②✱ [−π/2, π/2]✱ ✐♥ t❤❡ t✲❛①✐s✿ • ❚❤❡ ♥♦❞❡s ❛r❡ x = a, a + h, ... ❛♥❞ • ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ c = a + h/2, ...✳

✷✳✻✳

✶✻✶

❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ sin x ❛♥❞ cos x ❛r❡ ❣✐✈❡♥ ❜② ❛t c✿ sin(h/2) ∆ sin(h/2) ∆ (sin x) = · cos c; (cos x) = − · sin c . ∆x h/2 ∆x h/2

❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❝♦♦r❞✐♥❛t❡✇✐s❡✿ ∆X (c) = ∆t



sin(h/2) sin(h/2) − · sin c, · cos c h/2 h/2



=

sin(h/2) < − sin c, cos c > . h/2

✷✳✻✳ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

▲❡t✬s r❡✈✐❡✇ ❛ ❢❡✇ ❡①❛♠♣❧❡s ♦❢ ✇❤② ✇❡ st✉❞② ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ♣r♦❜❧❡♠s ✇❡ ❢❛❝❡ ❛r❡ ❢❛♠✐❧✐❛r ❜✉t t❤✐s t✐♠❡ t❤❡② ❛r❡ tr❡❛t❡❞ ✇✐t❤ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✦ ❊①❛♠♣❧❡ ✷✳✻✳✶✿ ❧✐❣❤t ❜♦✉♥❝❡

▲✐❣❤t ❜♦✉♥❝❡s ♦✛ ❛ ❝✉r✈❡❞ ♠✐rr♦r ❛s ✐❢ ♦✛ ❛ str❛✐❣❤t ♠✐rr♦r t❤❛t ✐s t❛♥❣❡♥t t♦ t❤❡ ♠✐rr♦r ❛t t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t✳

❊①❛♠♣❧❡ ✷✳✻✳✷✿ ❝❛r

❚❤❡ ❤❡❛❞ ❧✐❣❤ts ♦❢ ❛ ❝❛r tr❛✈❡❧✐♥❣ ♦♥ ❛ ❝✉r✈② r♦❛❞ ♣♦✐♥t ✐♥ t❤❡ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥ t♦ t❤❡ r♦❛❞ ❛t t❤❛t ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥✳

❊①❛♠♣❧❡ ✷✳✻✳✸✿ s❧✐♥❣

❚❤❡ ❞✐r❡❝t✐♦♥ ❛ r♦❝❦ ✇✐❧❧ ❣♦ ✇❤❡♥ r❡❧❡❛s❡❞ ❢r♦♠ ❛ s❧✐♥❣ ✐s t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛t t❤❡ ♣♦✐♥t ♦❢ r❡❧❡❛s❡✳

✷✳✻✳ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

✶✻✷

❘❡❝❛❧❧ ❤♦✇ ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t ❛♥❞ r❡❛❧✐③❡ t❤❛t ✐t ✐s ✈✐rt✉❛❧❧② ❛ str❛✐❣❤t ❧✐♥❡✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t

A✿

❚❤❡ ♠❡t❤♦❞ ♦❢ ❝♦♥str✉❝t✐♥❣ t❤❡ t❛♥❣❡♥t ✐s t♦ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❧✐♥❡s t❤❛t ❝✉t t❤r♦✉❣❤ t❤❡ ❣r❛♣❤✳

❊❛❝❤

♣❛ss❡s t❤r♦✉❣❤ t✇♦ ♣♦✐♥ts✱

A

❛♥❞ s♦♠❡

B

✇✐t❤

A 6= B ✳

✷✳✻✳

✶✻✸

❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ t❤✐s t✐♠❡❄ ❋✐rst ✇❡ ❛r❡♥✬t ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✳

s❧♦♣❡

♦❢ t❤✐s ❧✐♥❡ ❛♥②♠♦r❡ ❜✉t r❛t❤❡r ✐ts

◆❡①t✱ ✇✐t❤ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ t❤❡ ❝❤❛♥❣❡ ♦❢ x ❜r✐♥❣s ❛❜♦✉t t❤❡ ❝❤❛♥❣❡ ♦❢ y ❛♥❞ t❤❡✐r r❛t✐♦ ✐s t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❛♥❞✱ ❛❢t❡r t❤❡ ❧✐♠✐t✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲✐t❤ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✇❡ ❞♦ t❤❡ s❛♠❡ ❢♦r ❡✐t❤❡r ♦❢ ✐ts t✇♦ ❝♦♠♣♦♥❡♥ts✿ t❤❡ ❝❤❛♥❣❡ ♦❢ t ❜r✐♥❣s ❛❜♦✉t t❤❡ ❝❤❛♥❣❡ ♦❢ x ❛♥❞ y ✿

❚❤❡s❡ t✇♦ r❛t✐♦s ❛r❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ❛♥❞✱ ❛❢t❡r t❤❡ ❧✐♠✐t✱ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝✉r✈❡✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡✲❢r❡❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s✿ ◮ ❚❤❡ ❝❤❛♥❣❡ ♦❢ t ❜r✐♥❣s ❛❜♦✉t t❤❡ ❝❤❛♥❣❡ ♦❢ ♣♦✐♥t X ❛♥❞ t❤✐s r❛t✐♦ ✭♦❢ ❛ ✈❡❝t♦r ❛♥❞ ❛ ♥✉♠❜❡r✮ ✐s t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❛♥❞✱ ❛❢t❡r t❤❡ ❧✐♠✐t✱ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝✉r✈❡✳

❚❤✐s ❞❡s❝r✐♣t✐♦♥ ❛♣♣❧✐❡s t♦ ❛♥② ❞✐♠❡♥s✐♦♥✿ • t ❝❤❛♥❣❡s ❢r♦♠ s t♦ s + h✱ ❛♥❞

• X ❝❤❛♥❣❡s ❢r♦♠ P = F (s) t♦ Q = F (s + h)✳

❍❡r❡✱ h ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥♣✉t✳ ■t ✐s ♦❢t❡♥ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

✷✳✻✳

❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

✶✻✹

■♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥♣✉t

h = ∆t ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ♦✉t♣✉t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

■♥❝r❡♠❡♥t ♦❢ t❤❡ ♦✉t♣✉t

∆X ❚❤❡♥✱

•

❚❤❡ ❝❤❛♥❣❡ ♦❢

t

•

❚❤❡ ❝❤❛♥❣❡ ♦❢

X

✐s t❤❡ ♥✉♠❜❡r ✐s t❤❡ ✈❡❝t♦r

h✳ XQ✳

■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ t❤❡s❡ ❛r❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡ ❛♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t

◮

❚❤❡

r❛t❡ ♦❢ ❝❤❛♥❣❡

♦❢

X

✇✐t❤ r❡s♣❡❝t t♦

t

✐s ✈❡❝t♦r

1 XQ h ■t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

♦r

∆X . ∆t

♦r✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦t✐♦♥✱ t❤❡

❛✈❡r❛❣❡ ✈❡❧♦❝✐t②✳

❲❛r♥✐♥❣✦

❲❤❡♥ ✉♥✐ts ❛r❡ ✐♥✈♦❧✈❡❞✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♠♦t✐♦♥✱ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❧✐❡s ✐♥ ❛ s♣❛❝❡ ❞✐❢✲ ❢❡r❡♥t ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ✭❛♥❞ t❤❛t ♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢

❚❤❡ ♥❡①t st❡♣ ✐s t♦ ♠❛❦❡ ❛♣♣r♦❛❝❤✐♥❣ ✈❡❝t♦r

0

h s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳

X ✮✳

❚❤✐s ✇✐❧❧ ♠❛❦❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r s❤♦rt❡r ❛♥❞ s❤♦rt❡r

✭✉♥❧❡ss t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✦✮ ❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② t❤❡ ✈❡❝t♦r ♦❢ t❤❡

r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❚❤❡ ❧❛tt❡r ♠✐❣❤t ❤❛✈❡ ❛ ❧✐♠✐t✦

✷✳✻✳

❚❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡✿ ❞❡r✐✈❛t✐✈❡

✶✻✺

❉❡✜♥✐t✐♦♥ ✷✳✻✳✹✿ ❞❡r✐✈❛t✐✈❡ X = F (t) ❞❡r✐✈❛t✐✈❡ ♦❢ F

❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❝♦♥t❛✐♥s

t = s✳

❚❤❡♥ t❤❡

✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❛t

t=s

I

t❤❛t

✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t✱ ✐❢ ❡①✐sts✱

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

 1 F (s + h) − F (s) h→0 h

F ′ (s) = lim ♦r

 1 dF (s) = lim F (s + h) − F (s) h→0 h dt ❲❤❡♥ t❤✐s ❧✐♠✐t ❡①✐sts✱ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

F

✐s ❝❛❧❧❡❞

❞✐✛❡r❡♥t✐❛❜❧❡

❛t

t = s✳

◆♦t❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ✐s t❤❡ s❛♠❡ ✐❢ ✇❡ ❝❤♦♦s❡ t♦ t❤✐♥❦ ♦❢ ♦✉r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛s ✈❡❝t♦r✲✈❛❧✉❡❞✳ ❚❤❡ r❡s✉❧t ✐s ❛ ♥❡✇ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥✱ ✐✳❡✳✱ ❛ ♥❡✇

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✦

❊①❛♠♣❧❡ ✷✳✻✳✺✿ ❧✐♥❡ ▲❡t✬s t❡st t❤❡ ❞❡✜♥✐t✐♦♥ ✐♥ ❛ ❢❛♠✐❧✐❛r t❡rr✐t♦r②✳ ❋♦r ❛♥② t✇♦ ✈❡❝t♦rs

(At + B)

′

A

B✱ 

❛♥❞

 1 = lim (A(t + h) + B) − (At + B) h→0 h  1 = lim A(t + h) − At h→0 h 1  = lim Ah h→0 h

✇❡ ❤❛✈❡✿

= lim A h→0

= A. ❚❤❡♥✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ❛ ❧✐♥❡ ✐s ✐ts ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✦

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛ ❢♦r ✏✈❡❝t♦r ♣♦❧②♥♦♠✐❛❧s✑✳

❚❤❡♦r❡♠ ✷✳✻✳✻✿ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧s ❋♦r ❛♥② ✈❡❝t♦rs



m

Am , Am−1 , ..., A1 , A0

Am t + Am−1 t

m−1

✐♥

+ ... + A1 t + A0

Rn ✱

′

✇❡ ❤❛✈❡✿

= Am tm−1 + Am−1 tm−2 + ... + A1

❊①❡r❝✐s❡ ✷✳✻✳✼ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✱ t❤❡r❡ ✐s ❛ ❣❛♣ ✐♥ t❤❡ ♣❛t❤✳ ❚❤❡♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♠✐❣❤t ♥♦t ❝♦♥✈❡r❣❡ t♦

0✦

■♥ t❤❛t ❝❛s❡✱ t❤❡ ❧✐♠✐t ❛❜♦✈❡ ❝❛♥✬t ❡①✐st✳

✷✳✼✳

❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✶✻✻

❚❤❡♦r❡♠ ✷✳✻✳✽✿ ❉✐✛ ❂❃ ❈♦♥t ❊✈❡r② ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s✳

❊①❡r❝✐s❡ ✷✳✻✳✾ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✷✳✻✳✶✵✿ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜❧❡ ❊①❛♠♣❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❝♦♠❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳

■❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✐s ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉s✉❛❧ ✇❛②✱ x = t, y = |t|✱ t❤❡ r❡s✉❧t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❝♦♥t✐♥✉♦✉s ❜✉t ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✳

❊①❡r❝✐s❡ ✷✳✻✳✶✶ ✭❛✮ Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ✭❜✮ ❋✐♥❞ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ y = |x|✳

✷✳✼✳ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❛ ❧✐♠✐t ❛♥❞ ✇❡ ❝❛♥ ✉s❡ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ✇❡ ❤❛✈❡ ❞✐s❝✉ss❡❞✳ ❲❡ ❝♦♠♣✉t❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❝♦♠♣♦♥❡♥t✇✐s❡✳

✷✳✼✳

❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✶✻✼

❊①❛♠♣❧❡ ✷✳✼✳✶✿ ❝✐r❝✉❧❛r ♠♦t✐♦♥

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❝✐r❝❧❡ ✉♥❞❡r st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥✿

F (t) =< cos t, sin t > . ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤✐s ❛s ✐❢ ❛♥ ♦❜❥❡❝t ✐s ♠♦✈✐♥❣ ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ ✕ ❛t ❛ ❝♦♥st❛♥t

❛♥❣✉❧❛r ✈❡❧♦❝✐t②✳

❲❡ ❝♦♠♣✉t❡ ✐ts ❞❡r✐✈❛t✐✈❡ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ❞❡✜♥✐t✐♦♥✿

1 (F (t + h) h→0 h  1 < cos(t + h), sin(t + h) > = lim h→0 h  1 = lim cos(t + h) − cos t, h→0 h  1 = lim (cos(t + h) − cos t), h→0 h 1 = lim (cos(t + h) − cos t), h→0 h  1 = lim (cos(t + h) − cos t), h→0 h = < (cos t)′ ,

F ′ (t) = lim

−F (t))



− < cos t, sin t >  sin(t + h) − sin t

❙✉❜st✐t✉t❡✳ ❱❡❝t♦r ❛❞❞✐t✐♦♥✳



1 (sin(t + h) − sin t) h  1 (sin(t + h) − sin t) h  1 lim (sin(t + h) − sin t) h→0 h (sin t)′ >

❙❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ▲✐♠✐t ❝♦♠♣♦♥❡♥t✇✐s❡✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡s ✳✳✳♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✳

< − sin t, cos t >

= ▲❡t✬s ❝♦♥✜r♠ t❤❡ r❡s✉❧ts✿

√

√  2 2 F (0) =< 0, 1 > ✈❡rt✐❝❛❧✱ F (π/4) = − ❞✐❛❣♦♥❛❧✱ , 2√   √2 2 2 ❞✐❛❣♦♥❛❧✳ , − F ′ (π/2) =< −1, 0 > ❤♦r✐③♦♥t❛❧✱ F ′ (3π/4) = − 2 2 ... ... ′

′

❚❤❡s❡ ❛r❡ t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦rs ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s✿



✷✳✼✳ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✶✻✽

❋✉rt❤❡r ❡①❛♠✐♥❛t✐♦♥ r❡✈❡❛❧s t❤❛t t❤❡ ✈❡❝t♦r F ′ (t) ♦❢ t❤❡ ✈❡❧♦❝✐t② r♦t❛t❡s ❛s t ✐♥❝r❡❛s❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ F ′ (t) ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ F (t)✿ F (t) · F ′ (t) =< cos t, sin t > · < − sin t, cos t >= − cos t sin t + sin t cos t = 0 .

❲❡ ❝♦♥✜r♠ ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✿ ❛ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛❞✐✉s✳ ❋✉rt❤❡r✱ t❤❡ ♦❜❥❡❝t t✉r♥s t❤❡ s❛♠❡ ❛♥❣❧❡ ♣❡r ✉♥✐t ♦❢ t✐♠❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦✈❡rs t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ♦♥ t❤❡ ❝✐r❝❧❡✳ ❲❡ ❝♦♥✜r♠ t❤❛t ❢❛❝t ❜② ❞✐s❝♦✈❡r✐♥❣ t❤❛t t❤❡ ♦❜❥❡❝t X = F (t) ♠♦✈❡s ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✿ ||F ′ (t)|| = || < − sin t, cos t > || =

p

(− sin t)2 + (cos t)2 = 1 ,

❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳ ❚❤✐s ❞♦❡s♥✬t ♠❡❛♥ ❤♦✇❡✈❡r t❤❛t t❤❡r❡ ✐s ♥♦ ❛❝❝❡❧❡r❛t✐♦♥✦ ❚❤❡ ❛❝❝❡❧❡r❛✲ t✐♦♥ ✐s ✇❤❛t t✉r♥s t❤❡ ✈❡❝t♦r F ′ ✳ ❲❡ ❝♦♠♣✉t❡ ✐t ❤❡r❡✿ F ′′ (t) = (F ′ (t))′ = (< − sin t, cos t >)′ =< − cos t, − sin t > .

❲❡ ❝❛♥ s❛② t❤❛t F s❛t✐s✜❡s t❤✐s ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✿ F ′′ = −F .

❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ t❤❡ ❢♦r❝❡✱ ✐s ❝♦♥st❛♥t✿ ||F ′′ (t)|| = 1✱ ❛♥❞ ✐t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❧♦❝✐t② F ′ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♣♦✐♥ts ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡✿

❚❤❡♥✱ t❤❡r❡ ♠✉st ❜❡ s♦♠❡t❤✐♥❣ ❛t t❤❡ ♦r✐❣✐♥ ♣✉❧❧✐♥❣ t❤❡ ♦❜❥❡❝t t♦✇❛r❞s ✐t✦ ■♥❞❡❡❞✱ t❤❡s❡ r❡s✉❧ts ♠❛t❝❤ t❤♦s❡ ❛❜♦✉t ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✭❛♥❞ ❈❤❛♣t❡r ✺❉❊✲✹✮✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❣❡♥❡r❛❧ r❡s✉❧t ✉s❡❢✉❧ ❢♦r ❝♦♠♣✉t❛t✐♦♥s✳

❚❤❡♦r❡♠ ✷✳✼✳✷✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❊❛❝❤ ❝♦♠♣♦♥❡♥t ♦❢

F′

✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♠♣♦♥❡♥t ♦❢

F✳

❊①❡r❝✐s❡ ✷✳✼✳✸ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❡r❝✐s❡ ✷✳✼✳✹ ❆♣♣❧② t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s t♦ ❛ ❝✐r❝❧❡ ♦❢ ❛r❜✐tr❛r② r❛❞✐✉s✳

❊①❡r❝✐s❡ ✷✳✼✳✺ ✭❛✮ ❆♣♣❧② t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s t♦ t❤❡ ❡❧❧✐♣s❡✳ ✭❜✮ ❈♦♠♣❛r❡ t❤❡ r❡s✉❧ts t♦ t❤♦s❡ ❛❜♦✉t ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳

✷✳✽✳ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

✶✻✾

✷✳✽✳ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s ▼♦r❡ r❡s✉❧ts ❢♦❧❧♦✇ ❢r♦♠ t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts✳ ❆❧❧ t❤❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s r❡✲❛♣♣❡❛r ✐♥ t❤✐s ❝♦♥t❡①t✳ ❚❤❡ ✐❞❡❛ ♦❢ ❛❞❞✐t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❍❡r❡✱ t❤❡ ✈❡❝t♦rs ❛r❡ ❛❞❞❡❞ ❛♥❞ s♦ ❛r❡ t❤❡ ✈❡❝t♦r ❞✐✛❡r❡♥❝❡s✳

❚❤❡♦r❡♠ ✷✳✽✳✶✿ ❙✉♠ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡s❀ ✐✳❡✳✱ ❢♦r ❛♥② t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s X = F (t) ❛♥❞ X = G(t) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢②✿ ∆ (F + G)(c) = ∆F (c) + ∆G (c)

❚❤❡♦r❡♠ ✷✳✽✳✷✿ ❙✉♠ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts❀ ✐✳❡✳✱ ❢♦r ❛♥② t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s X = F (t) ❛♥❞ X = G(t) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s x ❛♥❞ x+∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢②✿ ∆(F + G) ∆F ∆G (c) = (c) + (c) ∆t ∆t ∆t

❚❤❡♦r❡♠ ✷✳✽✳✸✿ ❙✉♠ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ❚❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s❀ ✐✳❡✳✱ ✐❢ X = F (t) ❛♥❞ X = G(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s t❤❡♥ s♦ ✐s X = F (t)+G(t) ❛♥❞ ✇❡ ❤❛✈❡✿ (F + G)′ (s) = F ′ (s) + G′ (s)

✷✳✽✳ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

✶✼✵

Pr♦♦❢✳ ❲❡ ❝❛♥ ✉s❡ ❝♦♠♣♦♥❡♥t✇✐s❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝✲ t✐♦♥s✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ❧✐♠✐ts ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ ✐❢ t✇♦ r✉♥♥❡rs ❛r❡ r✉♥♥✐♥❣ st❛rt✐♥❣ ❢r♦♠ ❛ ❝♦♠♠♦♥ ❧♦❝❛t✐♦♥✱ t❤❡♥ t❤❡ ✈❡❝t♦r ❜❡t✇❡❡♥ t❤❡♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❧♦❝❛t✐♦♥ ✈❡❝t♦rs t❤❡② ❤❛✈❡ ❝♦✈❡r❡❞✳

❊①❡r❝✐s❡ ✷✳✽✳✹ Pr♦✈✐❞❡ t❤❡ t✇♦ ♣r♦♦❢s✳ ❚❤❡ ✐❞❡❛ ♦❢ ♣r♦♣♦rt✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❍❡r❡✱ ✐❢ t❤❡ ❧♦❝❛t✐♦♥ ✈❡❝t♦rs tr✐♣❧❡ t❤❡♥ s♦ ❞♦ t❤❡✐r ❞✐✛❡r❡♥❝❡s✳

❚❤❡♦r❡♠ ✷✳✽✳✺✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❝✉r✈❡✬s ❞✐✛❡r❡♥❝❡❀ ✐✳❡✳✱ ❢♦r ❛♥② ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢② ❢♦r ❛♥② r❡❛❧ k✿ ∆(kf ) (c) = k ∆f (c)

❚❤❡♦r❡♠ ✷✳✽✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❝✉r✈❡✬s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t❀ ✐✳❡✳✱ ❢♦r ❛♥② ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢② ❢♦r ❛♥② r❡❛❧ k✿ ∆(kf ) ∆f (c) = k (c) ∆t ∆t

❚❤❡♦r❡♠ ✷✳✽✳✼✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ❆ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❞❡r✐✈❛t✐✈❡❀ ✐✳❡✳✱ ✐❢ c ✐s ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞ X = F (t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❡♥

✷✳✽✳

Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

✶✼✶

s♦ ✐s X = cF (t) ❛♥❞ ✇❡ ❤❛✈❡✿ (cF )′ (s) = cF ′ (s)

■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ ✐❢ t❤❡ ❧♦❝❛t✐♦♥ ✈❡❝t♦rs ❛r❡ r❡✲s❝❛❧❡❞✱ s✉❝❤ ❛s ❢r♦♠ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs✱ t❤❡♥ s♦ ✐s t❤❡ ✈❡❧♦❝✐t② ✕ ❛t t❤❡ s❛♠❡ ♣r♦♣♦rt✐♦♥✳

❊①❡r❝✐s❡ ✷✳✽✳✽ Pr♦✈✐❞❡ t❤❡ t✇♦ ♣r♦♦❢s✳

❚❤❡♦r❡♠ ✷✳✽✳✾✿ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ✭s❝❛❧❛r✮ ♣r♦❞✉❝t ♦❢ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐✛❡r❡♥❝❡s ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ y = g(t) ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❜♦t❤ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢②✿ ∆(gF ) (c) = g(t + ∆t) ∆F (c) + ∆g (c) F (t)

❚❤❡♦r❡♠ ✷✳✽✳✶✵✿ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ✭s❝❛❧❛r✮ ♣r♦❞✉❝t ♦❢ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ y = g(t) ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❜♦t❤ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢②✿ ∆(gF ) ∆F ∆g (c) = g(t + ∆t) (c) + (c)F (t) ∆t ∆t ∆t

❚❤❡♦r❡♠ ✷✳✽✳✶✶✿ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ■❢ y = g(t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ X = F (t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❡♥ s♦ ✐s X = g(t)F (t) ❛♥❞ ✇❡ ❤❛✈❡✿ (gF )′ (s) = g ′ (s)F (s) + g(s)F ′ (s)

Pr♦♦❢✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ❛♥❞ t❤❡♥ ✇♦r❦ ♦✉r ✇❛② t♦✇❛r❞ t❤❡

✷✳✽✳

Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

✶✼✷

❡①♣r❡ss✐♦♥ ♦♥ t❤❡ r✐❣❤t ✭h = ∆t✮✿   1 g(s + h)F (s + h) − g(s)F (s) = h   1 g(s + h)F (s + h) − g(s)F (s + h) +g(s)F (s + h) − g(s)F (s) = h    1 1 g(s + h)F (s + h) − g(s)F (s + h) + g(s)F (s + h) − g(s)F (s) = h h g(s + h) − g(s) F (s + h) − F (s) · F (s + h) +g(s) · = h h ↓ ↓ ↓

g ′ (s)

=

❞✐✛❡r❡♥t✐❛❜✐❧✐t②✱ ❛♥❞ t❤❡

· F (s)

❝♦♥t✐♥✉✐t②✱

+g(s) ·

F ′ (s),

❚✇♦ ❡①tr❛ t❡r♠s✳ ❈♦♠❜✐♥❡ t❡r♠s✳ ❋❛❝t♦r ❜♦t❤✳

h→0

▲✐♠✐ts ❛❝❝♦r❞✐♥❣ t♦✳✳✳

❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✱

❙✉♠ ❛♥❞ Pr♦❞✉❝t ▲❛✇s ❢♦r ❧✐♠✐ts✳

❘❡❝❛❧❧ t❤❛t t❤❡ r✉❧❡s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ ❛❧❧♦✇ ✉s t♦ ❝❛rr② ♦✉t ❛❧❣❡❜r❛✐❝ s✐♠♣❧✐✜❝❛t✐♦♥ ♦r ♠❛♥✐♣✉❧❛t✐♦♥ ✇✐t❤ ✈❡❝t♦rs ❛s ✐❢ t❤❡② ✇❡r❡ ♥✉♠❜❡rs ✕ ❛s ❧♦♥❣ t❤❡ ❡①♣r❡ss✐♦♥s t❤❡♠s❡❧✈❡s ♠❛❦❡ s❡♥s❡✳ ❙✐♠✐❧❛r❧②✱ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛❧❧♦✇ ✉s t♦ ❝❛rr② ♦✉t ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ✐❢ t❤❡s❡ ❝✉r✈❡s ✇❡r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✕ ❛s ❧♦♥❣ t❤❡ ❡①♣r❡ss✐♦♥s t❤❡♠s❡❧✈❡s ♠❛❦❡ s❡♥s❡✳

❊①❛♠♣❧❡ ✷✳✽✳✶✷✿ s♣✐r❛❧ ❈♦♥s✐❞❡r✿

F (t) =< t cos t, t sin t >, t ≥ 0 .

❲❡ ❛♣♣❧② t❤❡

❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ✿ F ′ (t) = (< t cos t, t sin t >)′ = (t < cos t, sin t >)′ = (t)′ < cos t, sin t > +t(< cos t, sin t >)′ =< cos t, sin t > +t < − sin t, cos t > .

❚❤❡♦r❡♠ ✷✳✽✳✶✸✿ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛✲ t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐✛❡r❡♥❝❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s X = F (t) ❛♥❞ X = G(t) ❜♦t❤ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡

✷✳✽✳

Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

✶✼✸

c s❛t✐s❢②✿ ∆(F · D) (c) = F (t + ∆t) · ∆G (c) + ∆F (c) · G(t)

❚❤❡♦r❡♠ ✷✳✽✳✶✹✿ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s X = F (t) ❛♥❞ X = G(t) ❜♦t❤ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c s❛t✐s❢②✿ ∆G ∆F ∆(F · D) (c) = F (t + ∆t) · (c) + (c) · G(t) ∆t ∆t ∆t

❚❤❡♦r❡♠ ✷✳✽✳✶✺✿ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ■❢ X = F (t) ❛♥❞ X = G(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s t❤❡♥ q = F (t) · G(t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ✇❡ ❤❛✈❡✿ (F · G)′ (s) = F ′ (s) · G(s) + F (s) · G′ (s)

❊①❡r❝✐s❡ ✷✳✽✳✶✻ Pr♦✈✐❞❡ t❤❡ t✇♦ ♣r♦♦❢s✳

❊①❛♠♣❧❡ ✷✳✽✳✶✼✿ ❝♦♥st❛♥t ❡❧❡✈❛t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ♠♦✈✐♥❣ ❛❧♦♥❣ ❛

❝✐r❝❧❡✳

■ts ❝❡♥t❡r ❛♥❞ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s ❢♦r♠ ❛♥ ✐s♦s❝❡❧❡s✳ ❇✉t

t❤❡ ♠❡❞✐❛♥ ♦❢ ❛♥ ✐s♦s❝❡❧❡s ✐s ❛❧s♦ ✐ts ❤❡✐❣❤t✦

■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳ ❚❤✐s ♠✐♠✐❝s t❤❡ ❢❛♠✐❧✐❛r ❢❛❝t t❤❛t t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐ts r❛❞✐✉s✳ ◆♦✇✱ ✇❤❛t ❛❜♦✉t ❛

s♣❤❡r❡ ❄

❙✉♣♣♦s❡ t❤❛t ♦✉r ♠♦t✐♦♥ ❣✐✈❡♥ ❜②

X = F (t)

✐s r❡str✐❝t❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♦r✐❣✐♥

r❡♠❛✐♥s t❤❡ s❛♠❡✿

||F (t)|| = r . ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❛r❡ ♠♦✈✐♥❣ ❛❧♦♥❣ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ✭❤②♣❡r✮s♣❤❡r❡ ♦❢ r❛❞✐✉s

n=2

♦r ❛ s♣❤❡r❡ ❢♦r

n=3

✭s✉❝❤ ❛s t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✮✿

r

✐♥

Rn ✱

s✉❝❤ ❛ ❝✐r❝❧❡ ❢♦r

✷✳✽✳ Pr♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ ❞❡r✐✈❛t✐✈❡s

■t ❢♦❧❧♦✇s

∆||F ||2 = 0. ∆t  ∆ F ·F = 0. ∆t

❲❡ r❡✇r✐t❡ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♥✱ ❜② t❤❡ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ✱ ✇❡ ❤❛✈❡✿ F (t + ∆t) ·

❚❤❡r❡❢♦r❡✱

✶✼✹

∆F ∆F (t) + (t) · F (t) = 0 . ∆t ∆t

  ∆F (t) · F (t + ∆t) + F (t) = 0 . ∆t ∆F

✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆t ❧♦❝❛t✐♦♥s✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✿ F (t) =< cos t, sin t > ❛♥❞ F ′ (t) =< − sin t, cos t > .

❆♥❞ ✇❡ ❦♥♦✇ t❤✐s✿ ❲❤❛t ❛❜♦✉t t❤❡ s♣❤❡r❡❄

F (t) ⊥ F ′ (t) .

❚❤❡ ❛r❣✉♠❡♥t ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿ d ||F ||2 = 0 . dt

❲❡ r❡✇r✐t❡ ❛s ❢♦❧❧♦✇s✿ ❛♥❞ ❛♣♣❧② t❤❡ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ✿ ■t ❢♦❧❧♦✇s t❤❛t

  d F · F = 0, dt F′ · F + F · F′ = 0. F′ · F = 0.

✷✳✾✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✶✼✺

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✈❡❝t♦rs F (t) ❛♥❞ F ′ (t) ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❢♦r ❡✈❡r② t✳ ❚❤❡ r❡s✉❧t ✐s t♦ ❜❡ ❡①♣❡❝t❡❞✿ ❖✉r ✈❡❧♦❝✐t② ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ r❛❞✐✉s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s t❛♥❣❡♥t✐❛❧ t♦ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤ ✭♣♦✐♥t❡❞ ♥❡✐t❤❡r ✐♥ ♥♦r ♦✉t✮✿

✷✳✾✳ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐♥ Rn , n > 2✱ t 7→ X .

❝❛♥ ❜❡ ❛ ♣❛rt ♦❢ ❛ ❝♦♠♣♦s✐t✐♦♥ ✐♥ t✇♦ ✇❛②s✿ ✇✐t❤ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ t = g(u)✱ ❜❡❢♦r❡ ✐t✿ ♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱ z = q(X)✱ ❛❢t❡r ✐t✿

u 7→ t 7→ X . t 7→ X 7→ z .

❍♦✇❡✈❡r✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❧❛tt❡r ✇✐❧❧ ♦♥❧② ❛♣♣❡❛r ✐♥ ❈❤❛♣t❡r ✸✳ ❍❡r❡✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s✐♠♣❧❡r ❝❛s❡✿ X = (F ◦ g)(u) ,

✇❤❡r❡ g ✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t✱ ❛♥❞ u✱ ✐s t✐♠❡✱ t❤❡ ❢✉♥❝t✐♦♥ g ♠❛② ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts s✉❝❤ ❛s ❢r♦♠ ❤♦✉rs t♦ ♠✐♥✉t❡s✳ ❏✉st ❛s t❤❡ r❡st ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ r✉❧❡✱ t❤✐s ♦♥❡ ✐s ❛❧s♦ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♠❜✐♥❡❞✱ ✇❡ st❛rt ✇✐t❤ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ✐❢ ✇❡ tr❛✈❡❧ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ✇❤✐❧❡ ❛❧s♦ ❡①❡❝✉t✐♥❣ ❛ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✐♥ ❛ ❧✐♥❡❛r ❢❛s❤✐♦♥❄ ❆❢t❡r t❤✐s s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❜② ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥✿ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

t = g(u) ◦

❧✐♥❡❛r ❢✉♥❝t✐♦♥

✐ts ❞❡r✐✈❛t✐✈❡

= a + m · (u − v)

m

✐♥ R

D

✐♥ Rn

Dm

✐♥ Rn

X = F (t) = A + D(t − a) F (g(u))

= A + D((a + m · (u − v)) − a)

= A + Dm · (u − v)

❚❤✉s✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✿

✷✳✾✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✶✼✻

❚❤❡♦r❡♠ ✷✳✾✳✶✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❧❛tt❡r❀ ✐✳❡✳✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ t = g(u) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s u ❛♥❞ u + ∆u ♦❢ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛♥② ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s t = g(u) ❛♥❞ t + ∆t = g(u + ∆u) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s v ❛♥❞ a ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿ ∆(F ◦ g) (v) = ∆F (a)

❚❤❡♦r❡♠ ✷✳✾✳✷✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ ♣r♦❞✲ ✉❝t ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts❀ ✐✳❡✳✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ t = g(u) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s u ❛♥❞ u + ∆u ♦❢ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛♥② ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s t = g(u) ❛♥❞ t + ∆t = g(u + ∆u) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s v ❛♥❞ a ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿ ∆F ∆g ∆(F ◦ g) (v) = (a) · (v) ∆u ∆t ∆u

❚❤❡♦r❡♠ ✷✳✾✳✸✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❛t ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s❀ ✐✳❡✳✱ ✐❢ t = g(u) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t u = v ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ X = F (t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = g(v) ♣❛r❛✲ ♠❡tr✐❝ ❝✉r✈❡ t❤❡♥ X = F (g(u)) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t u = v ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ✇❡ ❤❛✈❡✿ dF dg d(F ◦ g) (v) = (a) · (v) dt dt du

❖♥ t❤❡ ❧❡❢t✱ ✇❡ s❡❡ s✐♠♣❧② ❛ ♥❡✇ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✇❤✐❧❡ ♦♥ t❤❡ r✐❣❤t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ✈❛r✐❛❜❧❡ s❝❛❧❛r✳

Pr♦♦❢✳ ■t ✐s ♣r♦✈❡♥ ❝♦♠♣♦♥❡♥t✇✐s❡ ❢r♦♠ t❤❡

❈❤❛✐♥ ❘✉❧❡

❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳

❊①❛♠♣❧❡ ✷✳✾✳✹✿ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ❆♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤✐s ❝♦♥❝❧✉s✐♦♥ ♠❛② ❜❡ ❛ s✐t✉❛t✐♦♥ ✇❤❡♥ ✇❡ s✇✐t❝❤ t❤❡ ✉♥✐ts ♦❢ t✐♠❡ ❢r♦♠ ♠✐♥✉t❡s t♦ ❤♦✉rs✱

g(u) = u/60, g ′ (u) = 1/60 , ❛♥❞ t❤❡♥ r❡❛❧✐③❡ t❤❛t t❤❡ ✈❡❧♦❝✐t②

F′

✐♥ ♠✐❧❡s ♣❡r ❤♦✉r ✇✐❧❧ ❜❡

1/60 ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐♥ ♠✐❧❡s ♣❡r ♠✐♥✉t❡✳

❊①❛♠♣❧❡ ✷✳✾✳✺✿ ❛❝❝❡❧❡r❛t❡❞ r♦t❛t✐♦♥ ❚❤✐s ♠❛② ❧♦♦❦ ❧✐❦❡ s♣❡❡❞✐♥❣ ✉♣ ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡✿

G(u) =< cos u2 , sin u2 > ,

✷✳✶✵✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✶✼✼

❜✉t ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s ♠♦t✐♦♥ ✇✐t❤ ❛❝❝❡❧❡r❛t❡❞ t✐♠❡✿ G = F ◦g,

✇❤❡r❡

F (t) =< cos t, sin t >, g(u) = u2 .

✷✳✶✵✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❲❡ ❦♥♦✇ t❤❛t ✐❢ X = F (t) ❢♦❧❧♦✇s ❛ ❝✐r❝❧❡ ♦r ❛ s♣❤❡r❡✱ t❤❡ ✈❡❝t♦rs F (t) ❛♥❞ F ′ (t) ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❢♦r ❡✈❡r② t✳ ❲❤❛t ❛❜♦✉t ♦t❤❡r ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s❄ ■♥ ❣❡♥❡r❛❧✱ t❤✐s ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ t❤❡ ❝❛s❡✱ ❜✉t ♦♥❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡ tr✐♣ t❤❛t r❡q✉✐r❡s ✉s t♦ t✉r♥ ❛r♦✉♥❞ ✇✐❧❧ ❤❛✈❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ t❤❡ tr✐♣✳ ❲❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t r♦✉♥❞ tr✐♣s✳ ■♥❞❡❡❞✱ ✇❤❡♥❡✈❡r ✇❡ ❛r❡ t❤❡ ❢❛rt❤❡st ❢r♦♠ ❤♦♠❡ ♦r ❛♥② ❧♦❝❛t✐♦♥✱ ✇❡ ❛r❡ ♠♦✈✐♥❣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐r❡❝t✐♦♥ ❤♦♠❡✱ ❛t ❧❡❛st ❢♦r ❛♥ ✐♥st❛♥t✳

▲❡t✬s ♠❛❦❡ t❤✐s ♦❜s❡r✈❛t✐♦♥ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❞ t✉r♥ ✐t ✐♥t♦ ❛ t❤❡♦r❡♠✳ ❲❡ ❛❣❛✐♥ ❛ss✉♠❡ t❤❛t x ✐s t✐♠❡✱ ❧✐♠✐t❡❞ t♦ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♦r❡♠ ✷✳✶✵✳✶✿ ❘♦❧❧❡✬s ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) s❛t✐s✜❡s✿ • ✶✳ F ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ • ✷✳ F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ • ✸✳ F (a) = F (b) = 0✳ ❚❤❡♥ F ′ (c) · F (c) = 0 ❢♦r s♦♠❡ c ✐♥ (a, b)✳ Pr♦♦❢✳

❲❡ ✇✐❧❧ ❜❡ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❢❛rt❤❡st ❧♦❝❛t✐♦♥✳ ❚♦ ✜♥❞ ✐t✱ ✇❡ ❢♦❧❧♦✇ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ❢♦r t❤❡ ❝✐r❝❧❡✿ ❝♦♥s✐❞❡r t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥✱ ♦r ❜❡tt❡r t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✳ ❲❡ ❞❡✜♥❡ ❛ ♥❡✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✿ g(t) = ||F (t)||2 .

■t ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b] ❛♥❞ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ ❆❧s♦ g(a) = g(b) = 0 .

❚❤❡♥✱ ❜② t❤❡ ♦r✐❣✐♥❛❧ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✺✮✱ t❤❡r❡ ✐s s✉❝❤ ❛ c ✐♥ (a, b) t❤❛t g ′ (c) = 0 .

✷✳✶✵✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✶✼✽

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛t c✿  d d F · F = 0 =⇒ F ′ · F + F · F ′ = 0 , ||F ||2 = 0 =⇒ dt dt

❜② t❤❡ ❉♦t Pr♦❞✉❝t ❘✉❧❡✳ ■t ❢♦❧❧♦✇s t❤❛t

F′ · F = 0.

❚❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ s✐♠♣❧② st❛t❡s t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ③❡r♦✱ ∆F = 0, ∆t

✇❤❡♥ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✐s tr✐✈✐❛❧✿ n = 1, x0 = a, x1 = b✳ ❚❤❡ ♣r♦♦❢ ✐♥❞✐❝❛t❡s t❤❛t ✐❢ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ③❡r♦ t❤❡♥ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ❜❡❣✐♥♥✐♥❣ ✐s ③❡r♦ t♦♦ ❛t s♦♠❡ ♣♦✐♥t✳ ❊①❡r❝✐s❡ ✷✳✶✵✳✷

❉❡r✐✈❡ t❤❡ ♦r✐❣✐♥❛❧ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❢r♦♠ t❤✐s t❤❡♦r❡♠✳ ❋✉rt❤❡r♠♦r❡✱ ✇❤❛t ✐❢ t❤✐s ✐s♥✬t ❛ r♦✉♥❞ tr✐♣❄

❚❤❡ ♣✐❝t✉r❡ s✉❣❣❡sts ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❡♥t✐t✐❡s ✇❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❘♦❧❧❡✬s t❤❡♦r❡♠✿ t❤❡r❡ ✐s ♥♦✇ ❛ ❧✐♥❡ t❤❛t ❝♦♥♥❡❝ts t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡ ❛♥❞ t❤✐s ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ t❛♥❣❡♥t ❛t s♦♠❡ ♣♦✐♥t✦ ❚❤❡♦r❡♠ ✷✳✶✵✳✸✿ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ✶✳ F ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✱ ✷✳ F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ ❚❤❡♥ ❢♦r s♦♠❡ c ✐♥ (a, b)✳

F (b) − F (a) = F ′ (c) , b−a

✷✳✶✵✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✶✼✾

Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ r❡♣❡❛ts t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✺✮✳ ▲❡t✬s r❡♥❛♠❡ F ✐♥ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❛s H t♦ ✉s❡ ✐t ❧❛t❡r✳ ❚❤❡♥ ✐ts ❝♦♥❞✐t✐♦♥s t❛❦❡ t❤✐s ❢♦r♠✿ ✶✳ H ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ ✷✳ H ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ [a, b]✳ ✸✳ H(a) = H(b)✳

❙✉♣♣♦s❡ X = L(t) ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡ ❢r♦♠ F (a) t♦ F (b) ✇✐t❤ L(a) = F (a) ❛♥❞ L(b) = F (b)✳ ❚❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s s✐♠♣❧② ✐ts ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✿ L′ (t) =

F (b) − F (a) . b−a

❚❤❡ ❦❡② st❡♣ ✐s t♦ ❞❡✜♥❡ h ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✿ H(x) = F (x) − L(x) .

❲❡ ♥♦✇ ✈❡r✐❢② t❤❡ ❝♦♥❞✐t✐♦♥s ❛❜♦✈❡✳ ✶✳ H ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b] ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭❙❘✮✳ ✷✳ H ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b) ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✭❙❘✮✿ H ′ (x) = F ′ (x) −

F (b) − F (a) . b−a

✸✳ ❲❡ ❛❧s♦ ❤❛✈❡✿ F (a) = L(a), F (b) = L(b) =⇒ H(a) = 0, H(b) = 0 =⇒ H(a) = H(b) .

❚❤✉s✱ H s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❘♦❧❧❡✬s ❚❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s s❛t✐s✜❡❞ t♦♦✿ H ′ (c) = 0

❢♦r s♦♠❡ c ✐♥ (a, b)❀ ✐✳❡✳✱ F ′ (c) −

F (b) − F (a) = 0. b−a

●❡♦♠❡tr✐❝❛❧❧②✱ c ✐s ❢♦✉♥❞ ❜② s❤✐❢t✐♥❣ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r ✉♥t✐❧ ✐t t♦✉❝❤❡s t❤❡ ❣r❛♣❤✿

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t❀ ♥♦✇ ✇❡ ❛❧s♦ ❤❛✈❡ ❛ ❜❛❝❦ ❧✐♥❦✳ ■♥❞❡❡❞✱ ✇❤❛t

✷✳✶✵✳

❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✐❢ ✇❡ t❛❦❡ t❤❡ ♣❛rt✐t✐♦♥ ♦❢

[a, b]

t♦ tr✐✈✐❛❧✿

n = 1, x0 = a, x1 = b❄

✶✽✵

❚❤❡ t❤❡♦r❡♠ s✐♠♣❧② st❛t❡s t❤❛t t❤❡ t✇♦

❛r❡ ❡q✉❛❧✱ ♣r♦✈✐❞❡❞ ✇❡ ❝❤♦♦s❡ t❤❡ r✐❣❤t ♣♦✐♥t ✇❤❡r❡ t♦ s❛♠♣❧❡ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❋♦r ❛♥ ❛r❜✐tr❛r② ♣❛rt✐t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ ❝♦♥str✉❝t✐♦♥ ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ❛s ❢♦❧❧♦✇s✳

❈♦r♦❧❧❛r② ✷✳✶✵✳✹✿ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❋♦r P❛rt✐t✐♦♥ ❙✉♣♣♦s❡✿ ✶✳ F ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ ✷✳ F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ ❚❤❡♥ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b]✱ t❤❡r❡ ❛r❡ s✉❝❤ s❡❝♦♥❞❛r② ♥♦❞❡s c1 , ..., cn t❤❛t ∆F (ck ) = F ′ (ck ), k = 1, 2, ..., n . ∆x

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❧✐♥❡s ❝♦♥♥❡❝t✐♥❣ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ♣❛t❤ ♦❢

F

❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥

❛r❡ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❜❡t✇❡❡♥ t❤❡♠✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✇✐❧❧ ❤❡❧♣ ✉s t♦ ❞❡r✐✈❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ ❢❛❝ts ❛❜♦✉t ✐ts ❞❡r✐✈❛t✐✈❡✳ ❇❡❢♦r❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ♦♥❧② ❜❡❡♥ ❛❜❧❡ t♦ ✜♥❞ ❢❛❝ts ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ t❤❡ ❢❛❝ts ❛❜♦✉t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❚❤✐s ✐s ❛ s❤♦rt ❧✐st ♦❢ ❢❛♠✐❧✐❛r ❢❛❝ts✿ ✐♥❢♦ ❛❜♦✉t

F

F

✐♥❢♦ ❛❜♦✉t

✐s ❝♦♥st❛♥t

=⇒ F ′

F′

✐s ③❡r♦

?

⇐=

F

=⇒ F ′

✐s ❧✐♥❡❛r

✐s ❝♦♥st❛♥t

?

⇐=

F

✐s q✉❛❞r❛t✐❝

=⇒ F ′

✐s ❧✐♥❡❛r

?

⇐= ❆r❡ t❤❡s❡ ❛rr♦✇s r❡✈❡rs✐❜❧❡❄

■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s

❝♦♥st❛♥t❄ ❈♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿

◮ ■❢

✏■❢ ♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✱ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧✑✳

X = F (t)

r❡♣r❡s❡♥t t❤❡ ♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❚❤❡ t❤❡♦r❡♠ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡

✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✱ ❜✉t t❤❡ ❝♦♥st❛♥t ✐s ❛

✈❡❝t♦r

t❤✐s t✐♠❡✦

❚❤❡♦r❡♠ ✷✳✶✵✳✺✿ ❈♦♥st❛♥t ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ■❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r ❛❧❧ ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡♥ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ [a, b]❀ ✐✳❡✳✱ ∆F (c) = 0 =⇒ F = ❝♦♥st❛♥t ∆t

Pr♦♦❢✳ ∆F (ci ) = 0 =⇒ F (ti ) − F (ti−1 ) = 0 =⇒ F (ti ) = F (ti−1 ) . ∆x

✷✳✶✵✳

❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✶✽✶

❚❤❡♦r❡♠ ✷✳✶✵✳✻✿ ❈♦♥st❛♥t ❋♦r ❉❡r✐✈❛t✐✈❡s ■❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ❢✉♥❝t✐♦♥ ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❢♦r ❛❧❧ t ✐♥ I ✱ t❤❡♥ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦♥ I ❀ ✐✳❡✳✱ F ′ = 0 =⇒ F = ❝♦♥st❛♥t

Pr♦♦❢✳ ❚♦ ♣r♦✈❡ t❤❛t

f

✐s ❝♦♥st❛♥t✱ ✐t s✉✣❝❡s t♦ s❤♦✇ t❤❛t

F (a) = F (b) , ❢♦r ❛❧❧

a, b ✐♥ I ✳

❆ss✉♠❡

a < b ❛♥❞ ✉s❡ t❤❡ ▼❡❛♥

I✿

❢♦r s♦♠❡

c (a, b)✳

❚❤✐s ✐s

0

❱❛❧✉❡ ❚❤❡♦r❡♠

✇✐t❤ ✐♥t❡r✈❛❧

F (b) − F (a) = F ′ (c) , b−a

❜② ❛ss✉♠♣t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❢♦r ❛❧❧ ♣❛✐rs

(a, b) ✐♥s✐❞❡ t❤❡ ✐♥t❡r✈❛❧

a, b

✐♥

I✿

F (b) − F (a) = 0 =⇒ F (b) − F (a) = 0 =⇒ F (a) = F (b) . b−a

❊①❡r❝✐s❡ ✷✳✶✵✳✼ ❲❤❛t ✐❢

F′ = 0

♦♥ ❛ ✉♥✐♦♥ ♦❢ t✇♦ ✐♥t❡r✈❛❧s❄

❚❤❡ ♣r♦❜❧❡♠ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ r❡❝♦✈❡r✐♥❣ t❤❡ ❢✉♥❝t✐♦♥

❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

G=F +C

❢r♦♠ ✐t ❞❡r✐✈❛t✐✈❡

F ′ ✱ t❤❡ ♣r♦❝❡ss ✇❡ ❤❛✈❡ ❝❛❧❧❡❞

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ r❡❝♦♥str✉❝t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ✏✜❡❧❞ ♦❢ ✈❡❝t♦rs✑ ✭✐✳❡✳✱ ❛ ✈❡❝t♦r ✜❡❧❞✮✿

◆♦✇✱ ❡✈❡♥ ✐❢ ✇❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ s✉❝❤ ❛s

F

F

❢r♦♠ ✐t ❞❡r✐✈❛t✐✈❡

❢♦r ❛♥② ❝♦♥st❛♥t r❡❛❧ ♥✉♠❜❡r

C✳

F ′ ✱ t❤❡r❡ ♠❛♥② ♦t❤❡rs ✇✐t❤ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡✱

❆r❡ t❤❡r❡ ♦t❤❡rs❄ ◆♦✳

❚❤❡♦r❡♠ ✷✳✶✵✳✽✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ■❢ t✇♦ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ✈❡❝t♦r❀ ✐✳❡✳✱ ∆G ∆F (c) = (c) =⇒ F (x) − G(x) = ❝♦♥st❛♥t ∆x ∆x

❚❤❡♦r❡♠ ✷✳✶✵✳✾✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ■❢ t✇♦ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❤❛✈❡ t❤❡ s❛♠❡ ❞❡r✐✈❛✲ t✐✈❡✱ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ✈❡❝t♦r❀ ✐✳❡✳✱ F ′ (x) = G′ (x) =⇒ F (x) − G(x) = ❝♦♥st❛♥t

✷✳✶✶✳

❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✽✷

Pr♦♦❢✳

❉❡✜♥❡

H(t) = F (t) − G(t) . ❚❤❡♥✱ ❜② ❙❘✱ ✇❡ ❤❛✈❡✿

❢♦r ❛❧❧

x✳

❚❤❡♥

H

H ′ (t) = (F (t) − G(t))′ = F ′ (t) − G′ (t) = 0 ,

✐s ❝♦♥st❛♥t✱ ❜② t❤❡

❈♦♥st❛♥t ❚❤❡♦r❡♠✳

●❡♦♠❡tr✐❝❛❧❧②✱ t❤✐s ♠❡❛♥s t❤❛t s❤✐❢t✐♥❣ t❤❡ ❣r❛♣❤ ♦❢

F

❣✐✈❡s ✉s t❤❡ ❣r❛♣❤ ♦❢

G✳

❚❤❡ t❤❡♦r❡♠ r❡♣r❡s❡♥ts ❛ ❧❡ss ♦❜✈✐♦✉s ❢❛❝t ❛❜♦✉t ♠♦t✐♦♥✿

◮

✏■❢ t✇♦ r✉♥♥❡rs r✉♥ ✇✐t❤ t❤❡ s❛♠❡ ✈❡❧♦❝✐t②✱ t❤❡✐r r❡❧❛t✐✈❡ ❧♦❝❛t✐♦♥ ✐s♥✬t ❝❤❛♥❣✐♥❣✑✳

✷✳✶✶✳ ❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

▲❡t✬s ✈✐s✉❛❧✐③❡ ✇❤❛t ✐t ♠❡❛♥s t♦ ❞✐✛❡r❡♥t✐❛t❡ ❛♥❞ ✐♥t❡❣r❛t❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❙✉♣♣♦s❡ ♦✉r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

ti+1 = ti + h, i = 1, 2, ..., n✳

X = F (t), a ≤ t ≤ b✱

✐s ❞❡✜♥❡❞ ❛t t❤❡s❡ ♣♦✐♥ts✿

❇❡t✇❡❡♥ t❤❡s❡ ❧♦❝❛t✐♦♥s ✇❡ ❞r❛✇ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦rs✱

Xi = F (ti )✱ ✇❤❡r❡ Di = Xi Xi+1 ✿

❲❡ t❤❡♥ ♠♦✈❡ t❤❡♠ s♦ t❤❛t t❤❡ st❛rt✐♥❣ ♣♦✐♥ts ❛r❡ ❛❧❧ t❤❡ s❛♠❡✱ ❛t t❤❡ ♦r✐❣✐♥✳ ❲❡ ❤❛✈❡ ❝r❡❛t❡❞ t❤❡ ✈❡❝t♦rs ♦❢ ❞✐s♣❧❛❝❡♠❡♥t✳ ❇② tr❛❝✐♥❣ t❤❡ ❡♥❞s ♦❢ t❤❡s❡ ✈❡❝t♦rs ✇❡ ❛❝q✉✐r❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✏❧♦❝❛t✐♦♥s✑ ✐♥ t❤✐s ♥❡✇ s♣❛❝❡✳ ❚❤❡ r❡s✉❧t ✐s t❤❡

❞✐✛❡r❡♥❝❡

♦❢

X = F (t)✳ Vi = h1 Di , i = 1, 2, ..., n✱ ♦❢ X = F (t)✳

❲❡ ❝❛♥ ❛❧s♦ r❡✲s❝❛❧❡ t❤❡s❡ ✈❡❝t♦rs ❛❧❧ ❛t ♦♥❝❡✿ ✈❡❧♦❝✐t②✳ ❚❤❡ r❡s✉❧t ✐s t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

♣r♦❞✉❝✐♥❣ t❤❡ ✈❡❝t♦rs ♦❢ ❛✈❡r❛❣❡

✷✳✶✶✳

❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

■❢ t❤❡ ♣♦✐♥ts ❝❛♠❡ ❢r♦♠ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱

✶✽✸

s❛♠♣❧❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ t❤❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ X = F (t)✳

h → 0

❧✐♠✐t

Q = G(t)✱

✜♥✐s❤❡s t❤❡ ❥♦❜✿ ❲❡ ❤❛✈❡ ❛ ♥❡✇

◆♦✇ ✐♥ r❡✈❡rs❡✳ ❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

ti + h, i = 1, 2, ..., n✳

Q = G(t), a ≤ t ≤ b✱

✐s ❞❡✜♥❡❞ ❛t t❤❡s❡ ♣♦✐♥ts✿

❲❡ r❡♣r❡s❡♥t t❤❡s❡ ✏❧♦❝❛t✐♦♥s✑ ❛s ✈❡❝t♦rs

Di = OQi

Qi = G(ti )✱

✇❤❡r❡

ti+1 =

st❛rt✐♥❣ ❛t t❤❡ s❛♠❡ ♣♦✐♥t✱ t❤❡

♦r✐❣✐♥✿

❲❡ t❤✐♥❦ ♦❢ t❤❡♠ ❛s ❞✐s♣❧❛❝❡♠❡♥ts✳ ❲❡ ❛rr❛♥❣❡ t❤❡♠ ❤❡❛❞✲t♦✲t❛✐❧ st❛rt✐♥❣ ❛t s♦♠❡ ❧♦❝❛t✐♦♥ ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s✿

Xi+1 = Xi + Di , i = 1, 2, ..., n✳

❚❤❡ r❡s✉❧t ✐s t❤❡

s✉♠ ♦❢ Q = G(t)✳

X0

♣r♦❞✉❝✐♥❣

❚❤❡ r❡s✉❧t ✐s t❤❡

❘✐❡♠❛♥♥ s✉♠ ♦❢

Di = Li h✱ X0 ♣r♦❞✉❝✐♥❣ Q = G(t)✳

h → 0

✜♥✐s❤❡s t❤❡ ❥♦❜✿

✇❡ ❤❛✈❡ ❛ ♥❡✇

■❢ ✇❡ ✐♥st❡❛❞ t❤✐♥❦ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✈❡❝t♦rs ❛s ✈❡❧♦❝✐t✐❡s✱ ✇❡ r❡✲s❝❛❧❡ t❤❡s❡ ✈❡❝t♦rs ❛❧❧ ❛t ♦♥❝❡ ✜rst✿ ♣r♦❞✉❝✐♥❣ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦rs✳ ❲❡ ❛rr❛♥❣❡ t❤❡♠ ❤❡❛❞✲t♦✲t❛✐❧ st❛rt✐♥❣ ❛t s♦♠❡ ❧♦❝❛t✐♦♥ ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s✿

Xi+1 = Xi + Di , i = 1, 2, ..., n✳

s❛♠♣❧❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t)✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢ Q = G(t)✳ ■❢ t❤❡ ♣♦✐♥ts ❝❛♠❡ ❢r♦♠ ❛

t❤❡ ❧✐♠✐t

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ❢♦r s✉♠s ❛♥❞ ❘✐❡♠❛♥♥ s✉♠s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ st❛rt ✇✐t❤ ❛♥

n

X = F (t)✱

♠❛②❜❡ ❝♦♥t✐♥✉♦✉s ♠❛②❜❡ ♥♦t✱ ❞❡✜♥❡❞ ♦♥

❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧✳

●✐✈❡♥ ❛♥ ✐♥t❡❣❡r

✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿

[t0 , t1 ], [t1 , t2 ], ..., [tn−1 , tn ] , ✇✐t❤

t0 = a, tn = b✳

❚❤❡ ✐♥t❡r✈❛❧s ❝❛♥ ❜❡ ♣♦ss✐❜❧② r❡✈❡rs❡❞✦

n ≥ 1✱

✇❡ ❤❛✈❡ ❛

[a, b], a < b✳

❲❡

♣❛rt✐t✐♦♥ ♦❢ [a, b] ✐♥t♦

✷✳✶✶✳

❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✽✹

❚❤❡ ♣♦✐♥ts✱ t0 , t1 , t2 , ..., tn−1 , tn ,

✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡✿ ❲❡ ❛r❡ ❛❧s♦ ❣✐✈❡♥ t❤❡ s❡❝♦♥❞❛r②

∆ti = ti − ti−1 , i = 1, 2, ..., n .

♥♦❞❡s ✿

c1 ✐♥ [t0 , t1 ], c2 ✐♥ [t1 , t2 ], ..., cn ✐♥ [tn−1 , tn ] .

❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ❧❡❢t✲ ♦r t❤❡ r✐❣❤t✲❡♥❞ ♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s✱ ♦r t❤❡ ♠✐❞✲♣♦✐♥ts✱ ❡t❝✳

❉❡✜♥✐t✐♦♥ ✷✳✶✶✳✶✿ s✉♠ ❚❤❡ s✉♠ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Q = G(t) ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡✜♥❡❞ t♦ ❜❡ b X

G = G(c1 ) + G(c2 ) + ... + G(cn ) =

a

n X

G(ci )

i=1

❉❡✜♥✐t✐♦♥ ✷✳✶✶✳✷✿ ❘✐❡♠❛♥♥ s✉♠ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Q = F (t) ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

✷✳✶✶✳

❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✽✺

♦❢ ❛♥ ✐♥t❡r✈❛❧

b X

[a, b]

✐s ❞❡✜♥❡❞ t♦ ❜❡

F ∆t = F (c1 ) ∆t1 + F (c2 ) ∆t2 + ... + F (cn ) ∆tn =

a

n X

F (ci ) ∆ti

i=1

❚❤❡ s✉♠ ♠❛② ❜❡ ✐❧❧✉str❛t❡❞ ❛s ❜❡❧♦✇ ❢♦r t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡✿

■♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ✇✐t❤ ❜❛s❡s ❛♥❞ ❤❡✐❣❤ts ♣r♦❞✉❝❡❞ ❜② t❤❡

❝♦♠♣♦♥❡♥ts

♦❢

[ti , ti+1 ]

F (ci )✳

❚❤❡♦r❡♠ ✷✳✶✶✳✸✿ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❘✉❧❡ ❙✉♣♣♦s❡ ✈❡❝t♦r

Q = F (t)

C✳

✐s ❝♦♥st❛♥t ♦♥

[a, b]❀

✐✳❡✳✱

F (t) = C

❢♦r ❛❧❧

t

✐♥

[a, b]

❛♥❞ s♦♠❡

❚❤❡♥

b X a

◆♦✇✱ ✐♥ ♦r❞❡r ✐♠♣r♦✈❡ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡

r❡✜♥❡

F ∆t = C(b − a)

t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ❦❡❡♣ r❡✜♥✐♥❣❀ ✐✳❡✳✱ ✇❡ ❤❛✈❡ s✐♠✉❧✲

t❛♥❡♦✉s❧②✿

n→∞

❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❞❡✜♥❡ t❤❡

♠❡s❤ ♦❢ ❛ ♣❛rt✐t✐♦♥ P

❛♥❞

∆ti → 0 .

❛s ❢♦❧❧♦✇s✿

|P | = max ∆ti . i

■t ✐s ❛ ♠❡❛s✉r❡ ♦❢ ✏r❡✜♥❡♠❡♥t✑ ♦❢

P✳

❉❡✜♥✐t✐♦♥ ✷✳✶✶✳✹✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❚❤❡

❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

F

♦✈❡r ✐♥t❡r✈❛❧

[a, b]

✐s ❞❡✜♥❡❞ t♦

❜❡ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ✐ts ❘✐❡♠❛♥♥ s✉♠s ♦✈❡r ❛ s❡q✉❡♥❝❡ ♣❛rt✐t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡s❤ ❛♣♣r♦❛❝❤✐♥❣ ❛❧❧ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✱

F

0

❛s

✐s ❝❛❧❧❡❞

k → ∞✳

✐♥t❡❣r❛❜❧❡

♦✈❡r

[a, b]

❛s ❢♦❧❧♦✇s✿

Z

b

F dt = lim a

k→∞

Pk

❲❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡

b X a

Fk ∆t

❛♥❞ t❤❡ ❧✐♠✐t ✐s ❞❡♥♦t❡❞

✷✳✶✶✳ ❙✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✽✻

✇❤❡r❡ Fk ✐s F s❛♠♣❧❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ Pk ✳ ■t ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ♦❢ F ✳ ❚❤❡ ✐♥t❡r✈❛❧ [a, b] ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❚❤❡♦r❡♠ ✷✳✶✶✳✺✿ ❈♦♥st❛♥t ■♥t❡❣r❛❧ ❘✉❧❡

F ✐s ❝♦♥st❛♥t ♦♥ [a, b]✱ ✐✳❡✳✱ F (t) = C C ✳ ❚❤❡♥ F ✐s ✐♥t❡❣r❛❜❧❡ ♦♥ [a, b] ❛♥❞

❙✉♣♣♦s❡

Z

❢♦r ❛❧❧

t

✐♥

[a, b]

❛♥❞ s♦♠❡ ✈❡❝t♦r

b a

F dt = C(b − a)

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♣r♦✈❡s t❤❛t ♦✉r ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❢♦r ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❚❤❡♦r❡♠ ✷✳✶✶✳✻✿ ❈♦♥t✳ ❂❃ ■♥t❡❣r✳ ❆❧❧ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥

[a, b]

❛r❡ ✐♥t❡❣r❛❜❧❡ ♦♥

[a, b]✳

❊①❛♠♣❧❡s ❝❛♥ ❜❡ t❛❦❡♥ ❢r♦♠ t❤❡ ♥✉♠❡r✐❝❛❧ ❝❛s❡ ✭❱♦❧✉♠❡s ✷ ❛♥❞ ✸✮✿

❚❤❡♦r❡♠ ✷✳✶✶✳✼✿ ◆❡❣❛t✐✈❡ ■♥t❡❣r❛❧ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

♦✈❡r ✐♥t❡r✈❛❧

[a, b]✿ Z

❊①❡r❝✐s❡ ✷✳✶✶✳✽

F

a b

F dt = −

Z

b

F dt a

[b, a]

✐s ❡q✉❛❧ t♦ t❤❡

✷✳✶✷✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✽✼

✷✳✶✷✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

❲❤❛t ✐s t❤❡ ♦♣♣♦s✐t❡ ♦❢ s✉❜tr❛❝t✐♦♥❄ ❆❞❞✐t✐♦♥✳ ❚❤❡♥ t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❡ s✉♠ ✿ ❞✐✛❡r❡♥❝❡✱ ∆F : s✉♠✱

t X

F :

✈❡❝t♦r s✉❜tr❛❝t✐♦♥ ✈❡❝t♦r ❛❞❞✐t✐♦♥

a

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✦

          

♦♣♣♦s✐t❡✦

❚❤❡♦r❡♠ ✷✳✶✷✳✶✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡✲ ❣r❡❡

1

❙✉♣♣♦s❡ F ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ❛♥❞ s✉♣♣♦s❡ a ✐s ❛ ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♥♦❞❡ t✱ ✇❡ ❤❛✈❡✿ t X a

(∆F ) = F (t) − F (a)

❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ❛♥❞ t❤❡ s✉♠ ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❛♥❞ s✉♠ ♦❢ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ❛s ❞❡✜♥❡❞ ✐♥ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✹✮✳ ❚❤❡ r❡❧❛t✐♦♥ ✇❡ ♦❜s❡r✈❡❞ t❤❡♥ ♥♦✇ ❛♣♣❡❛rs s❡♣❛r❛t❡❧② ❢♦r ❡❛❝❤ ❝♦♠♣♦♥❡♥t✿

❲❡ t❛❦❡ t❤✐s ✐❞❡❛ ❢✉rt❤❡r ❛♥❞ ❡①❛♠✐♥❡ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ■t ✐s ♦♥❧② s❧✐❣❤t❧② ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✳

✷✳✶✷✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✽✽

■♥ ♦✉r ✈❡❝t♦r✲✈❛❧✉❡❞ s❡tt✐♥❣✱ ❧❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ♠♦t✐♦♥✳ ❋✐rst✱ t❤✐s ✐s ❤♦✇ ✇❡ ✉s❡ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ t♦ ❛❝q✉✐r❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❚❤❡♥ ✇❡ ❞✐s❝♦✈❡r t❤❛t ❡❛❝❤ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧❛tt❡r ✐s ❛ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❢♦r♠❡r✦

■♥❞❡❡❞✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ G ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✱ ✐✳❡✳✱ ❛ 1✲❢♦r♠ ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn ✳ ■ts ❘✐❡♠❛♥♥ s✉♠ ❞❡✜♥❡s ❛ ♥❡✇ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Γ ♦♥ t❤❡ ♥♦❞❡s✱ ✐✳❡✳✱ ❛ 0✲❢♦r♠✳ ■t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❞✐r❡❝t❧②✿ Γ(tk ) =

tk X

G ∆t ,

a

♦r r❡❝✉rs✐✈❡❧②✿

Γ(tk+1 ) = Γ(tk ) + G(ck ) ∆tk .

❙❡❝♦♥❞✱ t❤✐s ✐s ❤♦✇ ✇❡ ✉s❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ❧♦❝❛t✐♦♥ t♦ ❛❝q✉✐r❡ t❤❡ ✈❡❧♦❝✐t②✳ ❚❤❡♥ ✇❡ ❞✐s❝♦✈❡r t❤❛t ❡❛❝❤ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧❛tt❡r ✐s ❛ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢♦r♠❡r✦

■♥❞❡❡❞✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Φ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✱ ✐✳❡✳✱ ❛ 0✲❢♦r♠ ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn ✳ ■ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝♦♠♣✉t❡❞ ♦✈❡r ❡❛❝❤ s❡❣♠❡♥t ♦❢ t❤❡ ♣❛rt✐t✐♦♥ [tk−1 , tk ], k = 1, 2, ..., n✱ ❛♥❞ ❛ss✐❣♥❡❞ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ ck ✳ ❚❤✐s ❞❡✜♥❡s ❛ ♥❡✇ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ F ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ ✐✳❡✳✱ ❛ 1✲❢♦r♠✳ ■t ✐s ❝♦♠♣✉t❡❞ ❜② t❤❡ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛✿ F (ck ) =

Φ(tk ) − Φ(tk−1 ) . ∆tk

❲❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❣✐✈❡ ✉s t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡❀ ✇❡ ❥✉st s✉❜st✐t✉t❡ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧❛tt❡r ✐♥t♦ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦r♠❡r✿ Φ = Γ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ Γ(tk ) − Γ(tk−1 ) ∆tk G(ck )∆tk = ∆tk

F (ck ) =

= G(ck ) .

✷✳✶✷✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✽✾

❚❤❡ r❡s✉❧t ✐s s❡❡♥ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t ❜❡❧♦✇✿

◆♦✇✱ ✈✐❝❡ ✈❡rs❛✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❣✐✈❡ ✉s t❤❡ ♦r✐❣✐♥❛❧ ✕ ✉♣ t♦ ❛ ❝♦♥st❛♥t ✈❡❝t♦r ✕ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡❀ ✇❡ ❥✉st s✉❜st✐t✉t❡ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦r♠❡r ✐♥t♦ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧❛tt❡r✿ G(ck ) = F (ck )✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t Γ(tk ) = Φ(tk ) + C ❢♦r ❛❧❧ k ✱ t❤❡ ✇❡ ❝♦♥❝❧✉❞❡✿

Γ(tk ) = Γ(tk−1 ) + F (ck )∆tk = Γ(tk−1 ) +

Φ(tk ) − Φ(tk−1 ) ∆tk ∆tk

= Γ(tk−1 ) + Φ(tk ) − F (tk−1 ) = Φ(tk ) + C . ▲❡t✬s s✉♠♠❛r✐③❡ ✇❤❛t ✇❡ ❤❛✈❡ ♣r♦✈❡♥ ✕ ❢♦r ❞✐s❝r❡t❡❧② ❞❡✜♥❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ✐✳❡✳✱ ❛ 0✲ ❛♥❞ 1✲❢♦r♠s ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn ✳

❉❡✜♥✐t✐♦♥ ✷✳✶✷✳✷✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❙✉♣♣♦s❡ Φ ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s tk , k = 0, 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t F ♦❢ Φ ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿

F (ck ) =

Φ(tk ) − Φ(tk−1 ) ∆tk

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s

F =

∆Φ ∆t

❉❡✜♥✐t✐♦♥ ✷✳✶✷✳✸✿ ❘✐❡♠❛♥♥ s✉♠ ❙✉♣♣♦s❡ G ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ck , k = 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ G ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿

Γ(tk ) = Γ(tk−1 ) + G(ck )∆tk

✷✳✶✷✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✾✵

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

Γ=

t X

G ∆t

a

❚❤❡♦r❡♠ ✷✳✶✷✳✹✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ✭✶✮ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ G ✐s G✿ ∆

 G ∆t =G ∆t

Pt

a

✭✷✮ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ Φ ✐s Φ + C ✱ ✇❤❡r❡ C ✐s ❛ ❝♦♥st❛♥t✿  t  X ∆Φ a

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s

∆t = Φ(t) + C

❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✦ ❚❤❡ r❡s✉❧t s❤♦✉❧❞♥✬t ❜❡ s✉r♣r✐s✐♥❣ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♦♣❡r❛t✐♦♥s ✐♥✈♦❧✈❡❞✿

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱

❘✐❡♠❛♥♥ s✉♠✱

t X

∆F : ∆t F ∆t :

a

◆♦✇ t❤❡

∆t

❝♦♥t✐♥✉♦✉s ❝❛s❡✳

✈❡❝t♦r s✉❜tr❛❝t✐♦♥

✈❡❝t♦r ❛❞❞✐t✐♦♥

→ ←

s❝❛❧❛r ❞✐✈✐s✐♦♥

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥

❲❡ ❥✉st t❛❦❡ t❤❡ ❛❜♦✈❡ r❡❧❛t✐♦♥s ❛♥❞ ❧❡t

∆t

❛♣♣r♦❛❝❤

0✳

      

♦♣♣♦s✐t❡✦

     

❲❡ ✏③♦♦♠ ♦✉t✑ ♦♥ t❤❡

❣r❛♣❤✿

❉❡✜♥✐t✐♦♥ ✷✳✶✷✳✺✿ ❛♥t✐❞❡r✐✈❛t✐✈❡

❛♥t✐❞❡r✐✈❛t✐✈❡

❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐s ❝❛❧❧❡❞ ❛♥ ′ ✐❢ F = G✳ ■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

Q = G(t)

G=

Z

F dt

❲❡ ✉s❡ ✏❛♥✑ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s ❢♦r ❡❛❝❤ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳

♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

✷✳✶✷✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✾✶

❚❤❡♦r❡♠ ✷✳✶✷✳✻✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✶✳ ●✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Q = F (t) ♦♥ [a, b]✱ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✲ ✜♥❡❞ ❜② Φ(x) =

Z

x

F dt a

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ F ♦♥ (a, b)✳ ✷✳ ❋♦r ❛ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ Q = F (t) ♦♥ [a, b] ❛♥❞ ❛♥② ♦❢ ✐ts ❛♥✲ t✐❞❡r✐✈❛t✐✈❡s Φ✱ ✇❡ ❤❛✈❡ Z

b a

F dt = Φ(b) − Φ(a)

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠✱ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✿ F → Φ →

Z

( ) dt d ( ) dt

d ( ) dt

→ Φ → → F →

Z

( ) dt

→

F.

→ Φ + C.

❈♦r♦❧❧❛r② ✷✳✶✷✳✼✿ ❆♥t✐❞❡r✐✈❛t✐✈❡ P❧✉s ❈♦♥st❛♥t

■❢ G ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ F t❤❡♥ s♦ ✐s G + C ✱ ✇❤❡r❡ C ✐s ❛♥② ❝♦♥st❛♥t ✈❡❝t♦r✳

❚❤✉s✱ ♦❢ X = F (t) ❛♥❞ X = G(t) ❛r❡ t✇♦ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ s❛♠❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ t❤❡♥ ♥♦t ♦♥❧② t❤❡✐r ♣❛t❤s ❛r❡ t❤❡ s❛♠❡✱ s❤✐❢t❡❞ ❜② ❛ ✜①❡❞ ✈❡❝t♦r✱ ❜✉t ❛❧s♦ t❤❡ t✇♦ ❧♦❝❛t✐♦♥s F (t) ❛♥❞ G(t) ❛t t❤❡ ❡①❛❝t ♠♦♠❡♥t ♦❢ t✐♠❡ ❛r❡ ❛❧s♦ s❡♣❛r❛t❡❞ ❜② t❤❛t ✈❡❝t♦r✳

✷✳✶✸✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✾✷

✷✳✶✸✳ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

❏✉st ❛s ✐♥ ❞✐♠❡♥s✐♦♥

1

♣r❡s❡♥t❡❞ ✐♥ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶P❈✲✶✮✱ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s ♣r♦❞✉❝❡

❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ t❤❡✐r s✉♠s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ❢♦r ✐♥t❡❣r❛❧s ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s✉♠s ❛♥❞ t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts✳ ■❢ ✇❡ ♣r♦❝❡❡❞ t♦ t❤❡ ❛❞❥❛❝❡♥t ✐♥t❡r✈❛❧✱ ✇❡ ❝❛♥ ❥✉st ❝♦♥t✐♥✉❡ t♦ ❛❞❞ t❡r♠s ♦❢ t❤❡ s✉♠ ✭♦♥❡ ❝♦♠♣♦♥❡♥t s❤♦✇♥✮✿

❚❤❡♦r❡♠ ✷✳✶✸✳✶✿ ❆❞❞✐t✐✈✐t② ❘✉❧❡

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b, c✱ ✇❡ ❤❛✈❡✿ b X

F+

a

c X b

F =

c X

F

a

♦✈❡r ♣❛rt✐t✐♦♥s ♦❢ [a, b]✱ [b, c]✱ ❛♥❞ [a, c] r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♦r❡♠ ✷✳✶✸✳✷✿ ❆❞❞✐t✐✈✐t② ❘✉❧❡

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b, c✱ ✇❡ ❤❛✈❡✿ b X

F ∆t +

a

c X

F ∆t =

c X

F ∆t

a

b

♦✈❡r ♣❛rt✐t✐♦♥s ♦❢ [a, b]✱ [b, c]✱ ❛♥❞ [a, c] r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♦r❡♠ ✷✳✶✸✳✸✿ ❆❞❞✐t✐✈✐t② ❘✉❧❡

❙✉♣♣♦s❡ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ F ✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r [a, b] ❛♥❞ ♦✈❡r [b, c]✳ ❚❤❡♥ F ✐s ✐♥t❡❣r❛❜❧❡ [a, c] ❛♥❞ ✇❡ ❤❛✈❡✿ Z

b

F dt + a

Z

❚❤❡ ❛r❡❛ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛❞❞✐t✐✈✐t② ♦❢ ✐♥t❡❣r❛❧s ✐s t❤❡ s❛♠❡✿

c

F dt = b

Z

c

F dt a

✷✳✶✸✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✾✸

❲✐t❤ t❤❡ ♠♦t✐♦♥ ✐♥t❡r♣r❡t❛t✐♦♥✱ ✇❡ ❤❛✈❡✿

❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ t❤❡ ✶st ❤♦✉r

+

❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ t❤❡ ✷♥❞ ❤♦✉r

=

❞✐s♣❧❛❝❡♠❡♥t t❤❡ t✇♦ ❤♦✉rs

.

❚❤❡♦r❡♠ ✷✳✶✸✳✹✿ ❊st✐♠❛t❡ ❘✉❧❡

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b ✇✐t❤ a < b✱ ✐❢ ||F (x)|| ≤ M ,

❢♦r ❛❧❧ t ✇✐t❤ a ≤ t ≤ b✱ t❤❡♥

b

X

F ∆t ≤ M (b − a) .

a

❚❤❡♦r❡♠ ✷✳✶✸✳✺✿ ❊st✐♠❛t❡ ❘✉❧❡

❙✉♣♣♦s❡ F ✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ♦✈❡r [a, b]✳ ❚❤❡♥✱ ✐❢ a < b ❛♥❞ ❢♦r ❛❧❧ t ✇✐t❤ a ≤ t ≤ b✱ ✇❡ ❤❛✈❡✿

||F (x)|| ≤ M ,

Z b

F dt

≤ M (b − a) .

a

◆♦t❡ t❤❛t t❤❡ st❛t❡♠❡♥t ♦❢ t❤❡ t❤❡♦r❡♠ st✐❧❧ ❤♦❧❞s ❡✈❡♥ ✐❢

F

✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r

[b, a]

✇✐t❤

b < a✱

❡t❝✳

❚❤✐s

✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧t✳

❚❤❡♦r❡♠ ✷✳✶✸✳✻✿ ▲♦❝❛❧ ■♥t❡❣r❛❜✐❧✐t②

■❢ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ F ✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r [a, b] t❤❡♥ ✐t ✐s ❛❧s♦ ✐♥t❡❣r❛❜❧❡ ♦✈❡r ❛♥② [a′ , b′ ] ✇✐t❤ a ≤ a′ < b′ ≤ b✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥♦t❤❡r ✐♠♣♦rt❛♥t ❝♦r♦❧❧❛r②✳

❚❤❡♦r❡♠ ✷✳✶✸✳✼✿ ❈♦♥t✳ ❂❃ ■♥t❡❣r✳

❆❧❧ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛r❡ ✐♥t❡❣r❛❜❧❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s✳

✷✳✶✸✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s

✶✾✹

❚❤❡♦r❡♠ ✷✳✶✸✳✽✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❙✉♠s

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b ❛♥❞ ❛♥② r❡❛❧ c✱ ✇❡ ❤❛✈❡✿ b X a

(c · F ) = c ·

b X

F

a

❚❤❡♦r❡♠ ✷✳✶✸✳✾✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b ❛♥❞ ❛♥② r❡❛❧ c✱ ✇❡ ❤❛✈❡✿ b X a

(c · F ) ∆t = c ·

b X

F ∆t

a

❚❤❡♦r❡♠ ✷✳✶✸✳✶✵✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s

❙✉♣♣♦s❡ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ F ✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r [a, b]✳ ❚❤❡♥ s♦ ✐s c · f ❢♦r ❛♥② r❡❛❧ c ❛♥❞ ✇❡ ❤❛✈❡✿ Z

b a

(c · F ) dt = c ·

Z

b

F dt a

❚❤❡♦r❡♠ ✷✳✶✸✳✶✶✿ ❙✉♠ ❘✉❧❡ ❋♦r ❙✉♠s

❙✉♣♣♦s❡ F ❛♥❞ G ❛r❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b✱ ✇❡ ❤❛✈❡✿ b X

(F + G) =

a

b X

F+

a

b X

G

a

❚❤❡♦r❡♠ ✷✳✶✸✳✶✷✿ ❙✉♠ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s

❙✉♣♣♦s❡ F ❛♥❞ G ❛r❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛♥② ♥♦❞❡s a, b✱ ✇❡ ❤❛✈❡✿ b X a

(F + G) ∆t =

b X

F ∆t +

a

b X

G ∆t

a

❚❤❡♦r❡♠ ✷✳✶✸✳✶✸✿ ❙✉♠ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s

❙✉♣♣♦s❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮ F ❛♥❞ G ❛r❡ ✐♥t❡❣r❛❜❧❡ ♦✈❡r

✷✳✶✹✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ [a, b]✳

❚❤❡♥ s♦ ✐s

✶✾✺

F +G Z

❛♥❞ ✇❡ ❤❛✈❡✿

b

(F + G) dt = a

Z

b

F dt + a

Z

b

G dt a

✷✳✶✹✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

❆s ✇❡ ❦♥♦✇✱ t❤✐s ✐s ❛❜♦✉t t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳ ❲❤✐❧❡ t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✇❛s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✐ts ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛r❡ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ✐❞❡❛s ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✹✮✱ ✇❡ tr❡❛t t❤❡ ❧❛tt❡r ❛s ❢✉♥❝t✐♦♥s t❤❛t ❛❧s♦ ❤❛✈❡ t❤❡✐r ♦✇♥ ❞✐✛❡r❡♥❝❡ ❛♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❲❤❛t ✐s t❤❡ ♣❛rt✐t✐♦♥ t❤❡♥❄ ❲❡ s❛✇ ❡❛r❧✐❡r ✐♥ t❤❡ ❝❤❛♣t❡r ❤♦✇ t❤✐s ✐❞❡❛ ✐s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ♦r❞❡r t♦ ❞❡r✐✈❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ ❋✐rst✱ ❛❧s♦ ❢♦❧❧♦✇✐♥❣ ❈❤❛♣t❡r ✷❉❈✲✹✱ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡ t❤❛t s❦✐♣s ❛ ♥♦❞❡✳ ❚❤❛t✬s ✇❤② t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❞♦❡s♥✬t ♣r♦✈✐❞❡ ✉s ✕ ✐♥ t❤✐s ✇✐t❤ ❛♥② ♠❡❛♥✐♥❣❢✉❧ ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ✇✐❧❧ ❧✐♠✐t ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ ❲❡ ♥❡❡❞

t❤r❡❡

✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

F

1✲❞✐♠❡♥s✐♦♥❛❧

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳

s❡tt✐♥❣ ✕

❛t t❤❡ ♥♦❞❡s✳ ❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛❧♦♥❣ t❤❡

t✇♦ ✐♥t❡r✈❛❧s ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ✇✐t❤✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✳ ❋✐♥❛❧❧②✱ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ✐s ❝❛rr✐❡❞ ♦✉t ❢♦r t❤❡s❡ t✇♦ ✈❛❧✉❡s✿

− F (t1 ) − − − ∆F − −•− ∆t2 − −•− −−− t1 ❚❤✐s ✐s ❛❧❧

F (t2 ) −•−

∆F ∆t3

−

∆F ∆t2

c3 − c2 t2

c2

− − − F (t3 ) − ∆F −•− − ∆t3 −−− −•− − c3

t3

✈❡❝t♦r ❛❧❣❡❜r❛ ✦

❚❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛❜♦✈❡ r❡♣r❡s❡♥ts t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❧✐♥❡s ❝♦♥♥❡❝t✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢

F

❛t ❝♦♥s❡❝✉t✐✈❡

♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥ ❦♥♦✇♥ ♦♥❧② ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡ ✭❛✈❡r❛❣❡✮ ✈❡❧♦❝✐t② ✐s t❤❡♥ ❢♦✉♥❞ ✐♥ t❤✐s ♠❛♥♥❡r✳ ■t ✐s ♥♦✇ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t t❤❛t ✇❡ ❤❛✈❡ ✉t✐❧✐③❡❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✳ ■♥❞❡❡❞✱ ✇❡ ❝❛♥ ♥♦✇ ❝❛rr② ♦✉t ❛ s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥ ✇✐t❤ t❤✐s ❢✉♥❝t✐♦♥ ❛♥❞ ✜♥❞ t❤❡ ✭❛✈❡r❛❣❡✮ ❲❡ ❤❛✈❡ ♥♦✇ ❛ ♥❡✇

❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥✱ ♦❢ ✇❤❛t❄ [p, q],

❲❡ ♣❛rt✐t✐♦♥ ✐t ✐♥t♦ ❧❛st ♣❛rt✐t✐♦♥✿

n−1

✇✐t❤

❛❝❝❡❧❡r❛t✐♦♥ ✦

❚❤❡ ✐♥t❡r✈❛❧ ✐s

p = c0

❛♥❞

q = cn .

✐♥t❡r✈❛❧s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s t❤❛t ✉s❡❞ t♦ ❜❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡

p = c1 , c2 , c3 , ..., cn−1 , cn = b . ❲❤❡♥

ck = ck+1 ✱

✇❡ ❡①❝❧✉❞❡ t❤✐s ✏✐♥t❡r✈❛❧✑ ❢r♦♠ t❤❡ ♥❡✇ ♣❛rt✐t✐♦♥✳ ◆♦✇✱ ✇❤❛t ❛r❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s❄

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ❧❛st ♣❛rt✐t✐♦♥ ♦❢ ❝♦✉rs❡✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿

t1

✐♥

[c1 , c2 ], t2

✐♥

[c2 , c3 ], ..., tn−1

✐♥

[cn−1 , cn ] .

✷✳✶✹✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✶✾✻

✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ❲❡ ❛♣♣❧② t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥ t♦ t❤✐s ♣❛rt✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ G = ∆F ∆t ♦❢ F ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♥❡✇ ♣❛rt✐t✐♦♥ ✭t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♦❧❞ ♣❛rt✐t✐♦♥✮ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✿ ∆G G(ck+1 ) − G(ck ) (tk ) = . ∆t ck+1 − ck

❉❡✜♥✐t✐♦♥ ✷✳✶✹✳✶✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ X = F (t) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆2 F (tk ) = ∆t2

− ∆F (ck ) ∆t , k = 1, 2, ..., n − 1 ck+1 − ck

∆F (ck+1 ) ∆t

■t ✐s t❤❡♥ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ 0✲❢♦r♠✳

❊①❛♠♣❧❡ ✷✳✶✹✳✷✿ ❝✐r❝❧❡ ▲❡t✬s ✜♥❞ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✱ X(t) =< cos t, sin t > .

■t ✇❛s ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧② ✐♥ t❤✐s ❝❤❛♣t❡r✿ sin(h/2) ∆X (c) = < − sin c, cos c > . ∆t h/2

❍❡r❡ ✇❡ ❤❛✈❡ ❛ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥ ❢♦r t❤❡ ✐♥t❡r✈❛❧✱ s❛②✱ [−π/2, π/2]✱ ✐♥ t❤❡ t✲❛①✐s✿ • ❚❤❡ ♥♦❞❡s ❛r❡ x = a, a + h, ... ❛♥❞ • ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ c = a + h/2, ...✳ ❖✈❡r t❤❡ s❛♠❡ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ∆2 X (a) = − ∆t2



sin(h/2) h/2

❚❤✐s ✈❡❝t♦r ♣♦✐♥ts t♦✇❛r❞ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡✦

2

< cos a, sin a > .

✷✳✶✹✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✶✾✼

❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s r❡♣❡❛t❡❞❧② ✉s❡❞ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ s✐♠✉❧❛t✐♦♥s ✇❤✐❝❤ ✐s ❢♦❧❧♦✇❡❞✱ ✇❤❡♥ ♥❡❝❡ss❛r②✱ ❜② t❛❦✐♥❣ ✐ts ❧✐♠✐t✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛❧s♦ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ❝❛♥ ❛❧s♦ ❜❡ ❞✐✛❡r❡♥t✐❛t❡❞✳ ❚❤❡ ♥♦t❛t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿

❘❡♣❡❛t❡❞ ❞❡r✐✈❛t✐✈❡s ❢✉♥❝t✐♦♥

F

F (0)

✜rst ❞❡r✐✈❛t✐✈❡

F′

F (1)

s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

F ′′ = (F ′ )′

F (2) = F (1)

t❤✐r❞ ❞❡r✐✈❛t✐✈❡

F ′′ = (F ′′ )′ F (3) = F (2)

′

...

...

nt❤ ❞❡r✐✈❛t✐✈❡

F

...

...

(n)

= F

dF dt   d2 F d dF = dt2 dt  dt  d d2 F d3 F = dt3 dt dt2

′ ′

...

 (n−1) ′

dn F d = n dt dt



dn−1 F dtn−1



...

❚❤✉s✱ ❛ ❣✐✈❡♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♠❛② ♣r♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✿ d dt

F →

→ F′ →

→ F ′′ → ... → F (n) → ...

d dt

♣r♦✈✐❞❡❞ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❡❛❝❤ st❡♣ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛s ✇❡❧❧✳ ■♥ t❤❡ ❛❜❜r❡✈✐❛t❡❞ ❢♦r♠ t❤❡ s❡q✉❡♥❝❡ ✐s✿ d

d

d

d

d

F −−−dt−−→ F ′ −−−dt−−→ F ′′ −−−dt−−→ ... −−−dt−−→ F (n) −−−dt−−→ ...

❲❛r♥✐♥❣✦ ❲❡ ✇✐❧❧ s❡❡ ✐♥ ❈❤❛♣t❡r ✸ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❛r❡ t✇♦ ❛♥✐♠❛❧s ♦❢ ✈❡r② ❞✐✛❡r❡♥t ❜r❡❡❞s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✹✳✸✿ ♠✉❧t✐♣❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ F ✐s ❝❛❧❧❡❞ t✇✐❝❡✱ t❤r✐❝❡✱ ✳✳✳✱ n t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ✇❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡r✐✈❛t✐✈❡s✱ F ′ , F ′′ , F ′′′ , ..., F (n) ,

❡①✐st✳ ❲❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡s ❡①✐st ❢♦r ❛❧❧ n✱ ✇❡ ❝❛❧❧ t❤❡ ❢✉♥❝t✐♦♥ s♠♦♦t❤✳

❊①❛♠♣❧❡ ✷✳✶✹✳✹✿ r❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❊①❛♠♣❧❡s ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❜✉t ♥♦t t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❝♦♠❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✿   −x2 ✐❢ x < 0, f (x) =  x2 ✐❢ x ≥ 0.

✷✳✶✺✳

❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s

✶✾✽

❲❤❛t ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡s❡ ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s❄ ❚❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ r❡♣r❡s❡♥ts t❤❡ ❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡

✈❡❝t♦r✲✈❛❧✉❡❞

❢✉♥❝t✐♦♥✳

❚❤❡♥ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ r❡♣r❡s❡♥ts t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡s❡ ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡s❡ ♠❛❣♥✐t✉❞❡s✳ ◆♦t✐❝❡ ❤♦✇ ❝❤❛♥❣✐♥❣ s❧♦♣❡s ❛r❡ s❡❡♥ ❛s r♦t❛t✐♥❣ t❛♥❣❡♥ts✿

■t ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s t❤❡

❛❝❝❡❧❡r❛t✐♦♥✳

✷✳✶✺✳ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s ◆♦✇ ✐♥ t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ❡♥✈✐r♦♥♠❡♥t✱ ✇❡ ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ s❛♠❡ q✉❡st✐♦♥ ❛s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✿ ✇❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t② ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥❄ ❍♦✇ ❞♦ ✇❡ ✏r❡✈❡rs❡✑ t❤❡ ❡✛❡❝t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦♥ ❛ ❢✉♥❝t✐♦♥❄

❋♦r ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

X = G(t)

❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱

t❤❡ s❛♠❡❀ ✇❡ ❤❛✈❡ ❛♥ ❛❧♠♦st ✐❞❡♥t✐❝❛❧ ♥♦❞❡s✱

tk ✱

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛

ck ✱

♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❛♥s✇❡r r❡♠❛✐♥s

❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

F (tk+1 ) = F (tk + ∆tk ) = F (tk ) + G(ck )∆tk .

X = F (t)

❞❡✜♥❡❞ ❛t t❤❡

✷✳✶✺✳

❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s

❚❤❡♥✱

✶✾✾

∆F = G(ck ) . ∆tk

❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢

F

✐s

G✳

❲❤❛t ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡❄ ❲✐t❤ t❤❡ ❝♦♠♣❧❡①✐t② ❛❞❞❡❞ ❜② t❤❡ ❧✐♠✐t✱ t❤❡r❡ ✐s ♥♦ ❢♦r♠✉❧❛✱ ❡✈❡♥ r❡❝✉rs✐✈❡✳ ❋✐rst✱ ❧❡t✬s r❡✈✐❡✇ ❤♦✇ t❤❡ ❙✉♠ ❘✉❧❡✱ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ❞✐❛❣r❛♠s✿ d

F, G   + y

d

F ′ , G′   + y

−−−dt−−→ SR d

d

F + G −−−dt−−→ (F + G)′ = F ′ + G′

cF

d

F′   ·c y

F −−−dt−−→   ·c CM R y

−−−dt−−→ (cF )′ = cF ′

F, G   ◦ y

−−−dt−−→ CR d

F ′ , G′   ◦ y

F ◦ G −−−dt−−→ (F ◦ G)′ = F ′ ◦ G′

■♥ t❤❡ ✜rst ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

•

❘✐❣❤t✿ ❞✐✛❡r❡♥t✐❛t❡✳ ❉♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts✳

•

❉♦✇♥✿ ❛❞❞ t❤❡♠✳ ❘✐❣❤t✿ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡s✉❧t✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

❲❛r♥✐♥❣✦ ◆❡✐t❤❡r t❤❡ Pr♦❞✉❝t ❘✉❧❡ ♥♦r t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❤❛s s✉❝❤ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥✳

◆♦✇✱ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ♠❡❛♥t t♦ ✏r❡✈❡rs❡✑ t❤❡ ❡✛❡❝t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦♥ ❛ ❢✉♥❝t✐♦♥ ❥✉st ❛s ❜❡❢♦r❡✳ ■t ✐s

✐♥✈❡rs❡

s✐♠✐❧❛r t♦ t❤❡

♦❢ ❛ ❢✉♥❝t✐♦♥✳

❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ✏✐♥✈❡rs❡✑ ♣r♦❜❧❡♠ st❡♠s ❢r♦♠ t❤❡ ♥❡❡❞ t♦ ✜♥❞ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ♦r ✈❡❧♦❝✐t② ❢r♦♠ ❛❝❝❡❧❡r❛t✐♦♥✳

❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ✇❡ ❞❡r✐✈❡ ❢r♦♠ ♦✉r ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ♣❛r❛♠❡tr✐❝

❝✉r✈❡s✿

•

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✱

•

❚❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱

•

❚❤❡ ❧♦❝❛t✐♦♥ ✐s ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✱

❍❡r❡

B

❛♥❞

C

A =⇒ V (t) = At + B =⇒ X(t) = At2 /2 + Bt + C ✳

❛r❡ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ r❡s♣❡❝t✐✈❡❧②✳

❲❡ ✐❧❧✉str❛t❡ t❤❡ ✐❞❡❛ ✇✐t❤ ❛ ❞✐❛❣r❛♠✿

< t2 , t 3 >

→

< 2t, 3t2 > → ❆s ❛ ❢✉♥❝t✐♦♥✱

d ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✦ dt

❲❡ ✇✐❧❧ ♥❡❡❞ t❤❡

d dt

→

 d −1 dt

< 2t, 3t2 >

→ < t2 + C, t3 + K >

r✉❧❡s ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

❋✐rst✱ ❝♦♥s✐❞❡r ❙❘✿

(F + G)′ = F ′ + G′ ✳

▲❡t✬s r❡❛❞ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✳

❚❤❡♦r❡♠ ✷✳✶✺✳✶✿ ❙✉♠ ❘✉❧❡ ■❢

• F • G

t❤❡♥

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

P ❛♥❞ Q✱

✷✳✶✺✳ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ❛♥t✐❞❡r✐✈❛t✐✈❡s • F +G

✷✵✵

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

P + Q✳

Pr♦♦❢✳

❲❡ ❛♣♣❧② ❙❘✿

(F (x) + G(x))′ = F ′ (x) + G′ (x) = P (x) + Q(x) .

❙✐♠✐❧❛r❧②✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✷✳✶✺✳✷✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ■❢

• F ✐s ❛♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡ • c ✐s ❛ ❝♦♥st❛♥t✱

P

♦❢

❛♥❞

t❤❡♥

• cF

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

cP ✳

Pr♦♦❢✳

❲❡ ❛♣♣❧② ❈▼❘✿

(cF (x))′ = cF ′ (x) = cP (x) . ❚❤❡♦r❡♠ ✷✳✶✺✳✸✿ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ■❢

• F ✐s ❛♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ P ❛♥❞ • a= 6 0, b ❛r❡ ❝♦♥st❛♥ts✳ • ❚❤❡♥ a1 F (ax + b) ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢

P (ax + b)✳

Pr♦♦❢✳

❲❡ ❛♣♣❧② ❈▼❘ ❛♥❞ ❈❘✿ 1 F (ax a

′ + b) =

1 a

(F (ax + b))′ = a1 aF ′ (ax + b) = F ′ (ax + b) = P (ax + b) .

❚❤❡s❡ ❞✐❛❣r❛♠s ✐❧❧✉str❛t❡ ❤♦✇ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞♦ ❡❛❝❤ ♦t❤❡r✿  d −1 dt

F →

d dt

F →

d dt

→ F →

 d −1

→ F →

dt

→

F

→ F +C

❏✉st ❛s ✇✐t❤ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜✉t t❤✐s t✐♠❡ ✐t ✐s ❛ ✈❡❝t♦r✳ ❲❡ r❡st❛t❡ t❤❡ r✉❧❡s✳

❙✉♠ ❘✉❧❡✿

Z

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✿ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡✿

(F + G)dt = Z

Z

Z

F dt +

(cF )dt = c

F (mt + b)dt =

1 m

Z

Z

Z

G dt .

F dt .

F (u) du

u=mt+b

.

✷✳✶✻✳

❚❤❡ s♣❡❡❞

✷✵✶

❆t ❧❡❛st✱ ✇❡ ❤❛✈❡ t❤❡s❡ t✇♦ ❞✐❛❣r❛♠s t♦ ✐❧❧✉str❛t❡ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡s ✇✐t❤ ❛❧❣❡❜r❛✿ P, Q   + y

R

←−−−− R

F ′ , G′   + y ′

P + Q ←−−−− F + G

R

P ←−−−−   ·c y

′

R

F′   ·c y

←−−−− cF ′

cP

❲❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t♦♣ r✐❣❤t ❛♥❞ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿ • ❧❡❢t✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t❡ t❤❡♠✱ t❤❡♥ ❣♦ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts❀ ♦r • ❞♦✇♥✿ ❛❞❞ t❤❡♠✱ t❤❡♥ ❣♦ ❧❡❢t✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡s✉❧ts✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ s♣❡❝✐✜❝✱ ❛ ❜r✉t❡ ❢♦r❝❡ ❛♣♣r♦❛❝❤ ♠✐❣❤t ❜❡ ❜❡st✿

❚❤❡♦r❡♠ ✷✳✶✺✳✹✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥ts✿

< f1 (t), ..., fn (t) >′ =< f1′ (t), ..., fn′ (t) >

❚❤❡♦r❡♠ ✷✳✶✺✳✺✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ■♥t❡❣r❛t✐♦♥ ❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ❛r❡ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥ts✿

Z

< f1 (t), ..., fn (t) > dt =


❊①❛♠♣❧❡ ✷✳✶✺✳✻✿ ❝✐r❝✉❧❛r ✈❡❧♦❝✐t② ▲❡t✬s ✐♥t❡❣r❛t❡✿

Z

< cos t, sin t > dt =


=< sin t + C, cos t + K > =< sin t, cos t > + < C, K > .

❍❡r❡ C ❛♥❞ K ❛r❡ ❥✉st ❛r❜✐tr❛r② ❝♦♥st❛♥ts ✇❤✐❝❤ ♠❛❦❡s < C, K > ❥✉st ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r✳

✷✳✶✻✳ ❚❤❡ s♣❡❡❞

❉r✐✈✐♥❣ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ❛❧❧♦✇s ♦♥❡ t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ s❝❡♥❡r②✳✳✳ ❛♥❞ t♦ st✉❞② t❤❡ s❤❛♣❡ ♦❢ t❤❡ r♦❛❞✦ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ✐❢ ✇❡ ❛r❡ ♠♦✈✐♥❣ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♦r✐❣✐♥ r❡♠❛✐♥s t❤❡ s❛♠❡✱ t❤❡♥ ♦✉r ✈❡❧♦❝✐t② ✐s ❛❧✇❛②s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❝t♦r ♦❢ ❧♦❝❛t✐♦♥✳

✷✳✶✻✳

❚❤❡ s♣❡❡❞

❋♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✿

✷✵✷

  ∆F (t) · F (t + ∆t) + F (t) = 0 . ||F (t)|| = r =⇒ ∆t

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❋♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✿

∆F ∆t

✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳

||F (t)|| = r =⇒ F ′ · F = 0 .

❲❤❛t ❛❜♦✉t t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡

✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❄

❙✉♣♣♦s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❤❛s ❛ ❝♦♥st❛♥t ♠❛❣♥✐t✉❞❡✿

❚❤❡♥ ✇❡ ♠❛❦❡ ❛♥ ✐♠♣♦rt❛♥t ❣❡♦♠❡tr✐❝ ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡ ♠❡❞✐❛♥ ♦❢ ❛♥ ✐s♦s❝❡❧❡s ✐s ❛❧s♦ ✐ts ❤❡✐❣❤t✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝♦♥❝❧✉❞❡✿ ◮ ❚❤❡ ✈❡❧♦❝✐t② ✭♦✈❡r ❛ ❞♦✉❜❧❡ ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✮ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

◆♦✇✱ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❲❡ s❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡✿ F ′ (t) =< − sin t, cos t > ❛♥❞ F ′′ (t) =< − cos t, − sin t > .

❚❤❡r❡❢♦r❡✿

F ′ (t) ⊥ F ′′ (t) .

❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ s✉♣♣♦s❡ t❤❛t ♦✉r ♠♦t✐♦♥ ❣✐✈❡♥ ❜② X = F (t) ✐s ❝♦♥❞✉❝t❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ s♣❡❡❞ r❡♠❛✐♥s t❤❡ s❛♠❡✿ ❚❤❡♥✱

||F ′ (t)|| = s .

 d d F ′ · F ′ = 0 =⇒ F ′′ · F ′ + F ′ · F ′′ = 0 =⇒ F ′′ · F ′ = 0 . ||F ′ ||2 = 0 =⇒ dt dt

✷✳✶✻✳

❚❤❡ s♣❡❡❞

✷✵✸

❙♦✱ ✇❡ ❤❛✈❡✿

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❛❧✇❛②s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❧♦❝✐t②✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s r❡❢♦r♠✉❧❛t✐♦♥ ✐s t❤❛t ✐s ❛♣♣❧✐❝❛❜❧❡ t♦ ❛❧❧ ❞✐♠❡♥s✐♦♥s ✿ ◮

❏✉st ❜❡❝❛✉s❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t ❞♦❡s♥✬t ♠❡❛♥ t❤❛t t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ③❡r♦ ✭♦r ❡✈❡♥ ❝♦♥st❛♥t✮✳ ❚❤❡ ❧❛tt❡r ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ t❤❡ s❤❛r♣♥❡ss ♦❢ t❤❡ t✉r♥✱ ✐✳❡✳✱ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡✿

■❢ ✇❡ ❝♦♠♣❛r❡ t❤❡ r❡s✉❧t t♦ t❤❡ ❡①❛♠♣❧❡ ✇❡ st❛rt❡❞ ✇✐t❤✱ ✇❡ r❡❛❧✐③❡ t❤❛t ✐t ✐s ❛s ✐❢ ✕ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ ✕ ✇❡ ❛r❡ ♠♦✈✐♥❣ ❛❧♦♥❣ ❛ ❝✐r❝❧❡ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♣♦✐♥ts t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤✐s ❝✐r❝❧❡✳✳✳ ❜✉t t❤❡ ❝✐r❝❧❡s ❛r❡ ❝♦♥st❛♥t❧② ❝❤❛♥❣✐♥❣✿

✷✳✶✻✳

❚❤❡ s♣❡❡❞

✷✵✹

❲❡ ♥♦✇ r❡❛❧✐③❡ t❤❛t ♥♦t ♦♥❧② t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ♣❛✐rs ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s t❤❡ s❛♠❡ ✭❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✮ ❜✉t ❛❧s♦ t❤❛t t❤❡ t✇♦ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❡①❛❝t❧② t❤❡ s❛♠❡✦

❚❤❡♦r❡♠ ✷✳✶✻✳✶✿ ❉✐s❝r❡t❡ ❈✉r✈❡ ♦♥ ❙♣❤❡r❡ ❋♦r ❛♥② ♥♦♥✲❝♦♥st❛♥t ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ U = G(t) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡✿ ||G|| = r > 0 =⇒

G(t) + G(t + ∆t) ∆G (c) ⊥ , ∆t 2

✇❤❡r❡ c ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡✱ ♣r♦✈✐❞❡❞ ∆G(c) 6= 0✳

❚❤❡♦r❡♠ ✷✳✶✻✳✷✿ ❈✉r✈❡ ♦♥ ❙♣❤❡r❡ ❋♦r ❛♥② ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ U = G(t)✱ ✇❡ ❤❛✈❡✿ ||G|| = r > 0 =⇒

♣r♦✈✐❞❡❞

dG dt

dG ⊥ G, dt

6= 0✳

❉r✐✈✐♥❣ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✱ ✇❡ ♥❡✈❡r ❢❡❡❧ ♣r❡ss❡❞ ✐♥t♦ ♦✉r s❡❛t ♦r t❤❡ s❡❛t ❜❡❧t ❜✉t ♦♥❧② ❢❡❡❧ t❤❡ s✐❞❡✲t♦✲s✐❞❡ s✇✐♥❣✳ ❆ ❝♦♥st❛♥t s♣❡❡❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ t❛❦❡s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✕ ❤♦✇ ❢❛st ✇❡ ♣r♦❣r❡ss ❢♦r✇❛r❞ ✕ ♦✉t ♦❢ ❝♦♥s✐❞❡r❛t✐♦♥ ❛♥❞ ❛❧❧♦✇s ✉s t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❝✉r✈❛t✉r❡ ✕ ❤♦✇ ❢❛st t❤❡ ❞✐r❡❝t✐♦♥ ✐s ❝❤❛♥❣✐♥❣✳ ❚❤❡② ✇✐❧❧ ❜❡ ❛ ♠❛❥♦r t♦♦❧ ♦❢ ♦✉r st✉❞② ♦❢ t❤❡ s❤❛♣❡s ♦❢ ❝✉r✈❡s✳ ❈✉r✈❡s ✈s✳ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✿ ✇❡ ❛r❡ ♥♦✇ tr❛♥s✐t✐♦♥✐♥❣ ❢r♦♠ st✉❞②✐♥❣ ❞r✐✈✐♥❣ t♦ st✉❞②✐♥❣ r♦❛❞s✳ ■♥ ♠❛t❤❡♠❛t✐❝❛❧ t❡r♠s✱ ✇❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s t♦ ❝✉r✈❡s✳ ❇✉t ✇❤❛t ✐s ❛ ❝✉r✈❡❄

❉❡✜♥✐t✐♦♥ ✷✳✶✻✳✸✿ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❝✉r✈❡

❝✉r✈❡ C ✐♥ Rn ♣❛r❛♠❡tr✐③❛t✐♦♥✳ ❆

✐s t❤❡ ♣❛t❤ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ❝❛❧❧❡❞ ✐ts

❚❤❡♥ ❛ ❝✉r✈❡ ✐s t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛❧❧ ♦❢ ✐ts ♣❛r❛♠❡tr✐③❛t✐♦♥s ❛♥❞ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✇❤❛t ❞♦ ❛❧❧ t❤❡s❡ ♣❛r❛♠❡tr✐③❛t✐♦♥s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ♣❛r❛♠❡tr✐③❛t✐♦♥s X = G(s) ❛♥❞ X = F (t) ♦❢ ❝✉r✈❡ C ✳ ❍♦✇ ❛r❡ t❤❡② r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r❄

✷✳✶✼✳

✷✵✺

❚❤❡ ❝✉r✈❛t✉r❡

❆ ❝❤❛♥❣❡

♦❢ ✈❛r✐❛❜❧❡s

✐s ❛ ❢✉♥❝t✐♦♥ ✉s❡❞ ❢♦r s✉❜st✐t✉t✐♦♥✿ t = g(s) ,

t❤❛t t✉r♥s ♦♥❡ ✐♥t♦ t❤❡ ♦t❤❡r✿ G(s) = F (g(s)) .

❚❤✐s ❝♦♠♣♦s✐t✐♦♥ ✐s s❡❡♥ ✐♥ t❤✐s ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠✿ F

[a, b] −−−−→ Rn x  g րG  [c, d]

❚❤❡♥✱ ✐❢ F ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ C ✱ G ✐s ❝❛❧❧❡❞ ❛ r❡✲♣❛r❛♠❡tr✐③❛t✐♦♥✳ ❲❡ ❦♥♦✇ t❤❛t ❛s ❧♦♥❣ ❛s t❤❡ ❢✉♥❝t✐♦♥ t = g(s) ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✱ t❤❡ ♥❡✇ ♣❛t❤ ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ❛s t❤❡ ♦❧❞✳ ■t✬s ❛s ✐❢ ❛ ❞r✐✈❡r ♠❛❞❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❤✐s ❞r✐✈❡ ❛❧♦♥❣ t❤❡ r♦❛❞ ❛♥❞ ✇❡ ❥✉st r✉♥ ✐t ❛t ❛ ❞✐✛❡r❡♥t ✭♣♦ss✐❜❧❡ ✈❛r✐❛❜❧❡✮ s♣❡❡❞✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♦♥❧② ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ ❞r✐✈❡r ❞✐❞♥✬t st♦♣ ♦r t✉r♥❡❞ ❛r♦✉♥❞✳

❉❡✜♥✐t✐♦♥ ✷✳✶✻✳✹✿ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥ ❙✉♣♣♦s❡ X = F (t) ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ♦♥ [a, b]✳ ■❢ • F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✱ • F ′ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ (a, b)✱ ❛♥❞ • F ′ 6= 0 ♦♥ (a, b), t❤❡ ❝✉r✈❡ ✐s ❝❛❧❧❡❞ r❡❣✉❧❛r✳ ■t ✐s ❝❛❧❧❡❞ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ♣❛t❤✳ ◆♦✇✱ ✇❤❛t ✐❢ t❤❡ t✇♦ ❞r✐✈❡rs ❞r♦✈❡ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s❄

✷✳✶✼✳ ❚❤❡ ❝✉r✈❛t✉r❡

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❞❡✈✐s❡ ❛ ✇❛② t♦ ❡✈❛❧✉❛t❡ s❤❛r♣♥❡ss ♦❢ t✉r♥s ♦❢ t❤❡ r♦❛❞✿

✷✳✶✼✳

❚❤❡ ❝✉r✈❛t✉r❡

✷✵✻

❙✐♥❝❡✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝❛r✱ ✇❡ ♠❛② ♥❡❡❞ s♠❛❧❧❡r ♦r ❧❛r❣❡r t✉r♥✐♥❣ ❝✐r❝❧❡s✱ ✇❡ ✇✐❧❧ ❦♥♦✇ ✐❢ t❤✐s ✐s ♣♦ss✐❜❧❡✳ ■❢ ✇❡ ♣r♦❣r❡ss ✐♥❝r❡♠❡♥t❛❧❧②✱ t❤❡ ❝✉r✈❛t✉r❡ ✐s s✐♠♣❧② t❤❡ t✉r♥ t❤❛t ✇❡ ♠❛❦❡ ❛t ❡❛❝❤ st❡♣✳ ❆t ✐ts s✐♠♣❧❡st✱ t❤❡ r♦❛❞ ✐s ❛ ❝✐t② str❡❡t✳ ❲❡ t❤❡♥ ♠♦✈❡ ❢r♦♠ ✐♥t❡rs❡❝t✐♦♥ t♦ ✐♥t❡rs❡❝t✐♦♥✳

❚❤❡s❡ ❛r❡ t❤❡ ♦♥❧② ♦♣t✐♦♥s✿ t✉r♥

r♦t❛t✐♦♥

♥♦ t✉r♥

0 ❞❡❣r❡❡s

0

❧❡❢t t✉r♥

90 ❞❡❣r❡❡s

π/2

r✐❣❤t t✉r♥ 90 ❞❡❣r❡❡s

π/2

❯✲t✉r♥

180 ❞❡❣r❡❡s

π

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ❛♥❣❧❡❄ ❲❡ ❝♦♣② ❡❛❝❤ ✈❡❝t♦r✱ ❛tt❛❝❤ t❤❡ ❝♦♣② t♦ ✐ts ❡♥❞ ♣♦✐♥t✱ ❛♥❞ ❝♦♠♣❛r❡ t♦ t❤❡ ♥❡①t ✈❡❝t♦r✳ ❉♦❡s t❤✐s ♠❡❛♥ t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ❛ 90✲❞❡❣r❡❡ t✉r♥ ✐s π/2❄ ◆♦✱ ❜❡❝❛✉s❡✱ ❛s ✇❡ ❦♥♦✇✱ t❤❡ ❝✉r✈❛t✉r❡ s❝❛❧❡s ❞♦✇♥ ❛s t❤❡ ❝✉r✈❡ ✐s s❝❛❧❡❞ ✉♣✱ ♣r♦♣♦rt✐♦♥❛❧❧②✳

■♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ t❤❡ ❝✉r✈❛t✉r❡ ✐♥ ❛ s❝❛❧❡ ✐♥❞❡♣❡♥❞❡♥t ♠❛♥♥❡r✱ ✇❡ ✉s❡ t❤❡ ❛♥❣❧❡ ♦❢ t✉r♥ ♣❡r ❞✐st❛♥❝❡ ❝♦✈❡r❡❞✱ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ √ t❤❡✐r ❝❡♥t❡rs ♦❢ t❤❡ t✇♦ ❡❞❣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥ t❤❡ st❛♥❞❛r❞ ❣r✐❞ ✇✐t❤ ❡❞❣❡s ♦❢ ❧❡♥❣t❤ 1✱ t❤✐s ❞✐st❛♥❝❡ ✐s 22 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ 90✲❞❡❣r❡❡ t✉r♥ ✐s √π2 ✳ ■t ✐s ✇❤❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ❣r✐❞ ✐s 1✱ ✇❡ ❤❛✈❡ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ s✉❝❤ ❛ t✉r♥ ❡q✉❛❧ t♦ π/2✳

✷✳✶✼✳

❚❤❡ ❝✉r✈❛t✉r❡

✷✵✼

❊①❛♠♣❧❡ ✷✳✶✼✳✶✿ sq✉❛r❡ ❛♥❞ ❝✐r❝❧❡

▲❡t✬s ❝♦♥✜r♠ t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ♠❛❦❡s s❡♥s❡ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛ sq✉❛r❡ ❛♥❞ ❝✐r❝❧❡✳

❲❡ ♣❧♦t t❤❡ ✉♥✐t

❝✐r❝❧❡✱

F (t) =< cos t, sin t > , ✇✐t❤

n

♣♦✐♥ts✳ ❋♦r

n = 4✱

✇❡ ❤❛✈❡ ❛ sq✉❛r❡ ✇✐t❤ ❛ ✉♥✐t ❞✐❛❣♦♥❛❧✳

❍❡r❡✱ ✇❡ s❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦❧✉♠♥s✳

❲❡ ❝♦♠♣✉t❡ t❤❡ ❧♦❝❛t✐♦♥s✱ t❤❡♥ ❞✐s♣❧❛❝❡♠❡♥t ❢♦r ❡❛❝❤ t✇♦

❧♦❝❛t✐♦♥s✱ t❤❡♥ ♥♦r♠❛❧✐③❡ t❤❡♠✱ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠ ✭❛s t❤❡ ❛r❝❝♦s✐♥❡ ♦❢ t❤❡✐r ❞♦t ♣r♦❞✉❝t✮✱ t❤❡♥ ✜♥❛❧❧② ❞✐✈✐❞❡ ❜② t❤❡ ❤❛❧❢ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ❞✐s♣❧❛❝❡♠❡♥ts✳ ❚❤❡ r❡s✉❧t ✐s t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ✐s ◆♦✇✱ ❢♦r

n = 100✱ 1✿

π/2✳

✇❡ ❤❛✈❡ s♦♠❡t❤✐♥❣ ✈❡r② ❝❧♦s❡ t♦ ❛ ✉♥✐t ❝✐r❝❧❡✳ ❚❤r♦✉❣❤ t❤❡ s❛♠❡ ❝♦♠♣✉t❛t✐♦♥s✱ t❤❡

❝✉r✈❛t✉r❡ ✐s

❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❤❛✈❡ t♦ t✉r♥

1

t♦

2

π

r❛❞✐❛♥s ♣❡r

♠❛❦❡s t❤❡ ❝✉r✈❛t✉r❡ ❜❡❝♦♠❡

π

✉♥✐ts ♦❢ ❧❡♥❣t❤s tr❛✈❡❧❡❞✳ ❈❤❛♥❣✐♥❣ t❤❡ r❛❞✐✉s ❢r♦♠

1/2✳ P1 , P2 , P3 ✐♥ Rn ✳ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝✉r✈❛t✉r❡ P\ 1 P2 P3 ✱ ❜✉t t❤❡ ❛♥❣❧❡ t❤❡ ✈❡❝t♦rs P1 P2 ❛♥❞ P1 P2 ❞✐✈✐❞❡❞

◆♦✇✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤r❡❡ ❞✐st✐♥❝t ♣♦✐♥ts ✏❝✉r✈❡✑❄ ■t✬s ♥♦t ❥✉st ✐ts ❛♥❣❧❡ ❛t

P2 ✱

✐✳❡✳✱

♦❢ t❤✐s ❜② t❤❡

❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ❝❡♥t❡rs✳ ❲❡ ❝❛♥ ❝❤♦♦s❡ t♦ ♠❡❛s✉r❡ t❤❡ t✉r♥ ♥♦t ❛s ❛♥ ❛♥❣❧❡ ❜✉t r❛t❤❡r t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ✈❡❝t♦rs✳ ❙✉♣♣♦s❡ ❛ s❡q✉❡♥❝❡

C

♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ✐♥

Rn

✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

Xk = F (tk )

t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ▲❡t✬s tr② t❤✐s ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

❊①❛♠♣❧❡ ✷✳✶✼✳✷✿ ❝✐r❝❧❡

F (t ) − F (t ) − F (t ) − F (t ) k k−1 k+1 k κ(ck ) = . (F (tk+1 ) − F (tk−1 ))/2

❲❡ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛s t♦ t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥✿

F (t) =< 2 cos t, 2 sin t > .

❞❡✜♥❡❞ ❛t

✷✳✶✼✳

❚❤❡ ❝✉r✈❛t✉r❡

✷✵✽

❚❤❡ ❝✉r✈❛t✉r❡ ✐s 1/2 ❛s ❡①♣❡❝t❡❞✳ ❲❡ r❡❝♦❣♥✐③❡ s♦♠❡ ♦❢ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s ❛s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳✳✳ ▲❡t✬s tr② t❤✐s ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿   ∆ κ = ∆t

✇❤❡r❡ T ✐s t❤❡ ♥♦r♠❛❧✐③❡❞ ✈❡❝t♦r ♦❢ t❤❡ ✈❡❧♦❝✐t②✳

∆T . ∆t

❊①❛♠♣❧❡ ✷✳✶✼✳✸✿ ❝✐r❝❧❡

❚❤❡ ❝✉r✈❡ X = F (t) ✐s s❛♠♣❧❡❞ ✐♥ t❤❡ t✇♦ ❝♦❧✉♠♥s ♠❛r❦❡❞ x ❛♥❞ y ❛♥❞ t❤❡♥ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t t♦ ❣❡t t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ∆F ✐♥ t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ x′ ❛♥❞ y ′ ✳ ❚❤✐s t✐♠❡✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ∆t s♣❡❡❞ s ❛s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s ✈❡❝t♦r ❛♥❞ t❤❡♥ ♥♦r♠❛❧✐③❡ t❤❡ ✈❡❧♦❝✐t② t♦ ❣❡t t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r T✿ ❂❘❈❬✲✸❪✴❘❈✽

■t ✐s ♣❧♦tt❡❞ t♦ ❝♦♥✜r♠ t❤❛t t❤❡ ✈❡❝t♦rs ❛r❡ ♦❢ ❧❡♥❣t❤ 1✿

❲❡ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛❣❛✐♥ t♦ ❣❡t t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ T ❛♥❞ ♣✉t ✐t ✐♥t♦ t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ T ′ x ❛♥❞ T ′ y ✳ ■t ✐s ❛❧s♦ ♣❧♦tt❡❞✳ ❋✐♥❛❧❧②✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s ✈❡❝t♦r ✐s ❝♦♠♣✉t❡❞ ❞✐✈✐❞❡❞ ❜② t❤❡ s♣❡❡❞ s❀ t❤❛t✬s t❤❡ ❝✉r✈❛t✉r❡✳ ■t ✐s ❝♦♥st❛♥t ❛t 1/2 ❛s ❡①♣❡❝t❡❞✳ ❆ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ ❛ ♥♦♥✲❝♦♥st❛♥t s♣❡❡❞ ♣r♦❞✉❝❡s t❤❡ s❛♠❡ r❡s✉❧t✿

✷✳✶✼✳

❚❤❡ ❝✉r✈❛t✉r❡

✷✵✾

❊①❛♠♣❧❡ ✷✳✶✼✳✹✿ ❡❧❧✐♣s❡

❲❡ ✉s❡ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t ❢♦r ❡❧❧✐♣s❡s✳✳✳ ❲❡ ❣❡t t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ♦❢ T ❛♥❞ ♣❧♦t ✐t✿

❲❡ r❡❝♦❣♥✐③❡ t❤❛t ❛t t❤❡ st❛rt ✕ ❛t (1, 0) ✕ ✇❡ ❤❛✈❡ ❛ ✈❡rt✐❝❛❧ t❛♥❣❡♥t ✈❡❝t♦r ❛♥❞ t❤❡♥ ✐t t✉r♥s ❝♦✉♥✲ t❡r❝❧♦❝❦✇✐s❡✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s ✈❡❝t♦r ✐s ❝♦♠♣✉t❡❞ ❞✐✈✐❞❡❞ ❜② t❤❡ s♣❡❡❞ s❀ t❤❛t✬s t❤❡ ❝✉r✈❛t✉r❡✳ ■t ✐s ❛❧s♦ ♣❧♦tt❡❞✿

❲❡ ❝❛♥ s❡❡ ❤♦✇ ✐t ✐s ❧♦✇ ❛t ✜rst ❛t t❤❡ ✢❛tt❡r ❡♥❞ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛♥❞ t❤❡♥ ❣r♦✇s t♦ ❛ ❧❛r❣❡r ✈❛❧✉❡ ❛s ✇❡ r❡❛❝❤ t❤❡ s❤❛r♣❡r ❡♥❞✳ ■t r✉♥s ❜❡t✇❡❡♥ 1/4 ❛♥❞ 2✳ ❚✇♦ ❝✐r❝❧❡s ✇✐t❤ r❛❞✐✐ 4 ❛♥❞ 1/2 ❛r❡ s❡❡♥ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❡❧❧✐♣s❡ ❛t ✐ts t✇♦ ❡♥❞s✿

✷✳✶✽✳

✷✶✵

❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

✷✳✶✽✳ ❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

❚❤❡ ♠❛✐♥ ✐♥t❡r❡st ❛❜♦✉t t❤❡s❡ r♦❛❞s ✐s t❤❡✐r ❝✉r✈❛t✉r❡ ✿ ■s t❤❡ t✉r♥ t♦♦ s❤❛r♣ s♦ t❤❛t t❤❡ ♠♦t♦r✐sts ♠❛② ❤❛✈❡ tr♦✉❜❧❡ t♦ st❛②✐♥❣ ♦♥ t❤❡ r♦❛❞ ❛t ❛ ♣❛rt✐❝✉❧❛r s♣❡❡❞❄ ❍♦✇ ❝❛♥ ✇❡ ✜♥❞ ♦✉t❄ ❚❤❡r❡ ✐s ♥♦ ♦t❤❡r ✇❛② t♦ ❧❡❛r♥ t❤❡ s❤❛♣❡ ♦❢ t❤❡ r♦❛❞ ❜✉t t♦ ❞r✐✈❡ ♦✈❡r ✐t✦ ❲❡ ♥❡❡❞ ❞r✐✈❡rs✱ ❜✉t ✇❤❛t ❦✐♥❞s ♦❢ ❞r✐✈❡rs ❛r❡ ❜❡st ❢♦r ♦✉r ♣✉r♣♦s❡s❄ ■♠❛❣✐♥❡ ②♦✉rs❡❧❢ ❞r✐✈✐♥❣✳ ❆t ✜rst t❤❡ r♦❛❞ ✐s str❛✐❣❤t ✇❤✐❝❤ ✐s r❡❝♦❣♥✐③❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ②♦✉ s❡❡ ❛ tr❡❡ ❛❤❡❛❞ ❛♥❞ ❝♦♥t✐♥✉❡ t♦ s❡❡ ✐t t♦ r❡♠❛✐♥ str❛✐❣❤t ❛❤❡❛❞✳ ❚❤❡♥✱ ❛s ②♦✉ st❛rt t♦ t✉r♥✱ t❤❡ tr❡❡s st❛rt t♦ ♣❛ss ②♦✉r ✜❡❧❞ ♦❢ ✈✐s✐♦♥ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿

❚❤❡ tr❡❡s ♠❛② ♣❛ss ❢❛st❡r ♦r s❧♦✇❡r✳ ❲❤❡♥ ✐t ✐s ❢❛st❡r✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ r♦❛❞ ✐s ❤✐❣❤❡r❄ ◆♦t ♥❡❝❡ss❛r✐❧②✦ ▼❛②❜❡ ②♦✉ ❛r❡ ❥✉st ❞r✐✈✐♥❣ ❢❛st❡r✳✳✳ ❙♦✱ ✇❡ ❞♦♥✬t ✇❛♥t ❞r✐✈❡rs ✇❤♦ ❞r✐✈❡ ❡rr❛t✐❝❛❧❧②✿ s♣❡❡❞ ✉♣ ❛♥❞ s❧♦✇ ❞♦✇♥✱ ♦r st♦♣✱ ♦r ❡✈❡♥ t✉r♥ ❛r♦✉♥❞✳ ❚❤❡ ♣❡r❢❡❝t ❞r✐✈❡r ❢♦r t❤❡ ❥♦❜ ❞r✐✈❡s ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ✦ ❊①❛♠♣❧❡ ✷✳✶✽✳✶✿ ❝✐r❝❧❡

❚❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 2 ✐s✿

F (t) =< 2 cos t, 2 sin t > . ❍♦✇❡✈❡r✱ ✇❡ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ❝✐r❝❧❡ ✕ ❧❡♥❣t❤ 4π ✕ ✐♥ 2π s❡❝♦♥❞s✳ ▲❡t✬s s❧♦✇ ❞♦✇♥ ✕ ❜② ❛ ❢❛❝t♦r ♦❢ 2 ✕

✷✳✶✽✳

✷✶✶

❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

❛♥❞ ❞r✐✈❡ ✇✐t❤ s♣❡❡❞ 1✦ ❚❤✐s ✐s ❤♦✇✿ G(t) =< 2 cos(t/2), 2 sin(t/2) > .

❚❤❡♥ ✐♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ t❤❡ ♥♦r♠❛❧✲ ✐③❛t✐♦♥ st❡♣ ❞✐s❛♣♣❡❛rs ❛♥❞ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♠❡r❣❡ ✐♥t♦ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

2

∆ G

κ=

∆t2 .

❋♦r ❝♦♠♣✉t❛t✐♦♥s✱ ✇❡ ❛♣♣❧② t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t t✇✐❝❡ ❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ✈❡❝t♦r✿

❚❤❡ ❝✉r✈❛t✉r❡ ✐s 1/2 ❛s ❡①♣❡❝t❡❞✳ ❆t ✐ts s✐♠♣❧❡st✱ t❤❡ ♠♦t✐♦♥ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❝❡♣ts✿ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦rs ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s t✐♠❡ ||❞✐s♣❧❛❝❡♠❡♥t|| ❞✐st❛♥❝❡ = ♥✉♠❜❡rs ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s = t✐♠❡ t✐♠❡

✈❡❧♦❝✐t② = s♣❡❡❞

❚❤❡ ❢❛♠✐❧✐❛r ✇❛② t♦ ❧♦♦❦ ❛t t❤❡ ❝✉r✈❛t✉r❡ ✐s t♦ ♠❡❛s✉r❡ ❤♦✇ ♠✉❝❤ t❤❡ ❝✉r✈❡ ❞❡✈✐❛t❡s ❢r♦♠ ❛ str❛✐❣❤t ❧✐♥❡✳ ❇✉t ✇❤❛t ❧✐♥❡ ✐s t❤❛t❄ ❚❤❡r❡ ✐s ♥♦ s✐♥❣❧❡ ❧✐♥❡✳ ❆t ❡✈❡r② ❧♦❝❛t✐♦♥ ✇❡ ✇✐❧❧ ❧♦♦❦ ❛t ❤♦✇ ♠✉❝❤ ♦✉r ♠♦t✐♦♥ t❛❦❡s ✉s ❛✇❛② ❢r♦♠ t❤❡ ❝✉rr❡♥t t❛♥❣❡♥t

❚❤❡ ❝✉r✈❛t✉r❡ ✐s t❤❡♥ ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ❢❛st t❤❡ ❞✐r❡❝t✐♦♥ ✐s ❝❤❛♥❣✐♥❣✳ ■t ✐s t❤❡ ♠❡❛s✉r❡ ♦❢ t❛♥❣❡♥t ❧✐♥❡ ✐s t✉r♥✐♥❣✳

❧✐♥❡

✿

❤♦✇ ❢❛st t❤❡

❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ♥♦t t❤❡ s❛♠❡ ❛s t♦ s❛② t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ✐s ❤♦✇ ❢❛st t❤❡ t❛♥❣❡♥t ✈❡❝t♦r✱ F ′ ✱ ✐s ❝❤❛♥❣✐♥❣✱ ✇❤✐❝❤ ✐s F ′′ ✱ ❛s t❤✐s ✇♦✉❧❞ ♠❛❦❡ t❤❡ ❝♦♥❝❡♣t ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ s♣❡❡❞✱ ||F ′ ||✳ ❯♥❧❡ss✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s♣❡❡❞ ✐s ✜①❡❞ ❛t✱ s❛②✱ 1✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ❛r❡ ♠❡❛s✉r✐♥❣ t❤❡ s♣❡❡❞ ♦❢ t✉r♥✐♥❣ ♦❢ ❛ ✉♥✐t ✈❡❝t♦r✿

✷✳✶✽✳ ❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

✷✶✷

❚❤✐s s♣❡❡❞ ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ✈❡❝t♦r✳ ❚❤✉s✱ ❢♦r ❛ ✏✉♥✐t✲s♣❡❡❞✑ ♣❛r❛♠❡tr✐③❛t✐♦♥ X = G(s)✱ ✐✳❡✳✱ ||G′ || = 1✱ t❤❡ ❝✉r✈❛t✉r❡ ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r G′ ✱ ✐✳❡✳✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✶✽✳✷✿ ❝✉r✈❛t✉r❡ ❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥

X = G(s)

♦❢ ❛ ❝✉r✈❡

C

s❛t✐s✜❡s

||G′ || = 1✱

t❤❡♥ t❤❡

❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡ ✐s t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ t❤❛t ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❝t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿

κ(t) = ||G′′ (t)||

❊①❛♠♣❧❡ ✷✳✶✽✳✸✿ ❝✉r✈❛t✉r❡s ♦❢ ❝✐r❝❧❡s ▲❡t✬s ❝♦♠♣❛r❡ t✇♦ ❝✐r❝❧❡s ♦❢ r❛❞✐✉s

2

❛♥❞ r❛❞✐✉s

1✳

◆♦✇✱ ♦✈❡r t❤❡ s❛♠❡ ❞✐st❛♥❝❡✱

♦✈❡r t❤❡ s❛♠❡ t✐♠❡✱ ❝♦✈❡r❡❞ ❜② ❛ ♣♦✐♥t ♠♦✈✐♥❣ ❛t s♣❡❡❞

π/2

❛♥❞

1

π✱

❛♥❞✱ t❤❡r❡❢♦r❡✱

t❤❡ r♦t❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ✐s r❡s♣❡❝t✐✈❡❧②

π✿

❚❤❡ s♠❛❧❧❡r ❝✐r❝❧❡ ❤❛s t✇✐❝❡ ❛s ❧❛r❣❡ ❝✉r✈❛t✉r❡✦ ■t s❡❡♠s t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ❝✐r❝❧❡ ✐s t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ ✐ts r❛❞✐✉s✳✳✳ ❆t t❤❡ ❡①tr❡♠❡✱ t❤❡ ❧❛r❣❡ ❝✐r❝❧❡ t✉r♥s ✐♥t♦ ❛ str❛✐❣❤t ❧✐♥❡ ✇✐t❤ ③❡r♦ ❝✉r✈❛t✉r❡✦

❉❡✜♥✐t✐♦♥ ✷✳✶✽✳✹✿ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ ❆ ♣❛r❛♠❡tr✐③❛t✐♦♥

X = G(s) ✐s ❝❛❧❧❡❞ ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

1✳

✐❢

||G′ (s)|| =

❊①❛♠♣❧❡ ✷✳✶✽✳✺✿ ❝✐r❝❧❡ ❚❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

R

✭❛ ❝♦♥st❛♥t ❛♥❣✉❧❛r ✈❡❧♦❝✐t②✮✱

F (t) =< R cos t, R sin t > , ✐s♥✬t ❛r❝✲❧❡♥❣t❤ ✉♥❧❡ss

R=1

❜❡❝❛✉s❡

F ′ (t) =< −R sin t, R cos t > =⇒ ||F ′ (t)|| =

p p (−R sin t)2 + (R cos t)2 = R (sin t)2 + (cos t)2 = R .

❍♦✇ ❝❛♥ ✇❡ ♠❛❦❡ t❤✐s ✐♥t♦ ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥❄ ❲❡ ❤❛✈❡ t♦ ❜❡ ❝❛r❡❢✉❧ s♦ t❤❛t t❤❡ ♠♦❞✐✜❡❞ ❝✉r✈❡ ✐s st✐❧❧ t❤❡ ❝✐r❝❧❡✳ ❚❤❡ s❛❢❡st ✇❛② t♦ ❡♥s✉r❡ t❤❛t ✐s t♦ r❡✲s❝❛❧❡ t❤❡ t✐♠❡

t✳

❋r♦♠ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

✷✳✶✽✳

✷✶✸

❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

✇❡ ❤❛✈❡ t♦ ♠♦✈❡ s❧♦✇❡r ✐❢ R < 1 ❛♥❞ ❢❛st❡r ✐❢ R > 1✳ ❲❤❛t s❤♦✉❧❞ ✇❡ r❡♣❧❛❝❡ t ✇✐t❤ s♦ t❤❛t R ✐s ❝❛♥❝❡❧❧❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t✐❛t✐♦♥❄ ❏✉st ♣✉t✿ t = s/R . ❚❤❡♥✱ ✐♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿

G(s) =< R cos(s/R), R sin(s/R) > =⇒ G′ (s) =< − sin(s/R), cos(s/R) > =⇒ ||G′ (s)|| = 1 , ❜② t❤❡

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✷✳✶✽✳✻✿ ❧✐♥❡

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡✿

F (t) =< 3, 4 > t =⇒ F ′ (t) =< 3, 4 > =⇒ ||F ′ (t)|| =

√

32 + 42 = 5 .

❍♦✇ ❝❛♥ ✇❡ ♠❛❦❡ t❤✐s ✐♥t♦ ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥❄ ❲❡ r❡✲s❝❛❧❡ t❤❡ t✐♠❡ t ❛❣❛✐♥ t♦ s❧♦✇ ✐t ❞♦✇♥✿ t = s/5 . ❚❤✉s✱ t❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✿

G(s) =< 3/5, 4/5 > s . ❍❡r❡ t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ✐s ❛ ✉♥✐t ✈❡❝t♦r✦ ❚❤❡♦r❡♠ ✷✳✶✽✳✼✿ ❈♦♥st❛♥t ❙♣❡❡❞ P❛r❛♠❡tr✐③❛t✐♦♥ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ❛ ❧✐♥❡✱

F (t) = At + B , ✐s ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢

❚❤✉s✱ ❛ ✇❡❧❧✲❝❤♦s❡♥

❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱

||A|| = 1✳

✐✳❡✳✱ ❛ s✉❜st✐t✉t✐♦♥✿

t = g(s) , ❝❛♥ ♠❛❦❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥t♦ ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥✿

G(s) = F (g(s)) . ❙✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ t = g(s) ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✱ t❤❡ ♥❡✇ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❤❛s t❤❡ s❛♠❡ ♣❛t❤ ❛s t❤❡ ♦❧❞✳ ■t✬s ❛s ✐❢ ❛ r❛♥❞♦♠ ❞r✐✈❡r ♠❛❞❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❤✐s ❞r✐✈❡ ❛❧♦♥❣ t❤❡ r♦❛❞ ✇❡ st✉❞②✱ ❛♥❞ ♥♦✇ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♣r♦❝❡ss t❤❡ ✈✐❞❡♦ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❞r✐✈❡ ❛♣♣❡❛rs t♦ ❜❡ ❛t 1 ♠✐❧❡ ♣❡r ❤♦✉r✳ ❙✉❝❤ ❛ ♥❡✇ ♣❛r❛♠❡t❡r s ✐s ❝❛❧❧❡❞ ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r ♦❢ t❤❡ ❝✉r✈❡✳ ❚❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② t✇♦ ♦❢ t❤♦s❡ ❢♦r ❛ ❣✐✈❡♥ ❝✉r✈❡ ✕ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ✕ t❤❡ r❡st ❛r❡ ❥✉st t❤❡ s❤✐❢ts ♦❢ t❤♦s❡ t✇♦✳ ■♥ t❤❡ ✜rst ❡①❛♠♣❧❡ ❛❜♦✈❡✱ t❤❡s❡ ❛r❡✿

t = s/R ❛♥❞ t = −s/R , ❛♥❞ ✐♥ t❤❡ s❡❝♦♥❞✿

t = s/5 ❛♥❞ t = −s/5 .

❲❡ ❝❛♥ s♣❡❛❦ ♦❢ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇❤❡♥ t❤❡ ❝✉r✈❡ ✐s ♦r✐❡♥t❡❞✱ ✐✳❡✳✱ ✐ts ❞✐r❡❝t✐♦♥ ✐s ✐♥❞✐❝❛t❡❞✱ ❛♥❞ t = g(s) ✐s ✐♥❝r❡❛s✐♥❣✳ ❚❤❡s❡ s✐♠♣❧❡ ✭❧✐♥❡❛r✮ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s ❤❛✈❡ ✇♦r❦❡❞ ♦♥❧② ❜❡❝❛✉s❡ t❤❡ s♣❡❡❞ ❤❛s ❜❡❡♥ ❝♦♥st❛♥t✳

✷✳✶✽✳

✷✶✹

❚❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥

❊①❛♠♣❧❡ ✷✳✶✽✳✽✿ str❛✐❣❤t ❛❝❝❡❧❡r❛t✐♦♥

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❛❝❝❡❧❡r❛t✐♥❣ ❛❧♦♥❣ t❤❡ str❛✐❣❤t ❧✐♥❡ y = x✿ F (t) =< t2 , t2 >, t ≥ 0 .

❚❤❡ ❞❡♥s❡r ❞♦ts ✐♥❞✐❝❛t❡ s❧♦✇❡r ♠♦t✐♦♥✿

❲❡ ✇✐❧❧ ♥❡❡❞ ❛ ♥♦♥✲❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ❲❡ ❤❛✈❡✿ F ′ (t) =< 2t, 2t >=< 1, 1 > 2t =⇒ ||F ′ (t)|| =

√

2t .

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ t = g(s) t❤❛t G(s) = F (g(s)) =⇒ ||G′ (s)|| = 1 .

❲❡ s✉❜st✐t✉t❡ ❛♥❞ ✉s❡ t❤❡ ❈❤❛✐♥

❘✉❧❡ ✿

√ ||G′ (s)|| = ||F ′ (g(s))g ′ (s)|| = || < 1, 1 > 2g(s)g ′ (s)|| = 2 2g(s)g ′ (s) = 1 .

❚❤❡♥✱

1 . g ′ (s) = √ 2 2g(s)

❚❤❡ ❛♥s✇❡r ✐s♥✬t ❡❛s② t♦ ❣✉❡ss✱ ❜✉t ❤❡r❡ ✐t ✐s g(s) =

❋r♦♠ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣❛r❛♠❡tr✐③❛t✐♦♥s✱

p s/2 .

G(s) = F (g(s)) ,

✇❡ ❞❡r✐✈❡ ❜② t❤❡

❈❤❛✐♥ ❘✉❧❡ ✿

G′ = F ′ g ′ .

❚❤❡ t❤❡♦r❡♠ ❜❡❧♦✇ t❤❡♥ ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✷✳✶✽✳✾✿ ❆r❝✲❧❡♥❣t❤ P❛r❛♠❡tr✐③❛t✐♦♥

❙✉♣♣♦s❡ X = F (t) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❚❤❡♥✱ t = g(s) ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s t❤❛t ♣r♦❞✉❝❡s t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ X = G(s) = F (g(s)) ✐❢ ❛♥❞ ♦♥❧② ✐❢ g ′ (s) =

1 ||F ′ (g(s))||

❚❤❡ ♥♦♥✲③❡r♦ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡♥ ❛ ♣r❡r❡q✉✐s✐t❡ ❢♦r ❛r❝✲❧❡♥❣t❤ r❡✲♣❛r❛♠❡tr✐③❛t✐♦♥✳

✷✳✶✾✳

✷✶✺

❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥

❈♦r♦❧❧❛r② ✷✳✶✽✳✶✵✿ ▲❡♥❣t❤ ■s ■♥t❡❣r❛❧ ♦❢ ❙♣❡❡❞

❙✉♣♣♦s❡ X = F (t) ✐s ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ♦♥ [a, b]✳ ❚❤❡♥✱ s=

Z

t a

||F ′ (u)|| du, a ≤ t ≤ b

✐s t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❊①❡r❝✐s❡ ✷✳✶✽✳✶✶

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❋♦r ❡❛❝❤ t✱ t❤❡ ✐♥t❡❣r❛❧ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ✉♣ t♦ F (t)✳ ❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

✷✳✶✾✳ ❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥

▲❡t✬s r❡❝❛❧❧ t❤❛t ✐❢ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ X = G(s) ♦❢ ❛ ❝✉r✈❡ C s❛t✐s✜❡s ||G′ (s)|| = 1✱ ✐✳❡✳✱ s ✐s ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✱ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡ ✇❛s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❝t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿ κ(s) = ||G′′ (s)|| .

❊①❛♠♣❧❡ ✷✳✶✾✳✶✿ ❝✐r❝❧❡

❋♦r ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R✱ ✇❡ ✉s❡ t❤❡ ❛r❝✲❧❡♥❣t❤

♣❛r❛♠❡tr✐③❛t✐♦♥

✿

G(s) =< R cos(s/R), sin(s/R) > .

❲❡ st❛rt ✇✐t❤ R = 2✳ ❚❤❡ ❝✉r✈❡ ✐s s❛♠♣❧❡❞ ✐♥ t❤❡ t✇♦ ❝♦❧✉♠♥s ♠❛r❦❡❞ x ❛♥❞ y ❛♥❞ t❤❡s❡ ♣♦✐♥ts ❛r❡ ♣❧♦tt❡❞✳ ❲❡ t❤❡♥ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❂✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✴❘✻❈✶

❲❡ ❣❡t t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❛♥❞ ♣✉t ✐t ✐♥t♦ t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ x′ ❛♥❞ y ′ ✳ ❲❡ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛❣❛✐♥ t♦ ❣❡t t❤❡ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ♣✉t ✐t ✐♥t♦ t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ x′′ ❛♥❞ y ′′ ✿

❋✐♥❛❧❧②✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s ✈❡❝t♦r ✐s ❝♦♠♣✉t❡❞❀ t❤❛t✬s t❤❡ ❝✉r✈❛t✉r❡✳ ❚❤❡ r❡s✉❧t ✐s✱ ❛s ❡①♣❡❝t❡❞✱ 1/2✳ ◆♦✇ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ tr② t♦ ❡①♣r❡ss t❤❡ ❝✉r✈❛t✉r❡ ✐♥ t❡r♠s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛♠❡t❡r t✳

✷✳✶✾✳

✷✶✻

❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥

❚❤❡ ❝✉r✈❛t✉r❡ ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ✕ ❣✐✈❡♥ ❜② t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✕ ♦❢ t❤❡ ❝✉r✈❡✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ F ′ ✐s t❛♥❣❡♥t ❜✉t ✐t ❛❧s♦ ❝❤❛♥❣❡s ✐s ♠❛❣♥✐t✉❞❡✳ ❚♦ r❡❝t✐❢② t❤❛t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r ❛t t❤❡ ♣♦✐♥t F (t)✿ F′ . T (t) = kF ′ k ❚❤✐s ✐s ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② t♦ st❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✶✾✳✷✿ ❝✉r✈❛t✉r❡ ❙✉♣♣♦s❡ X = F (t) ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ C ✇✐t❤ ♥♦♥✲③❡r♦ ❞❡r✐✈❛t✐✈❡✱ F ′ 6= 0✳ ❚❤❡♥ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡ ✐s ❞❡✜♥❡❞ t♦ t❤❡ s♣❡❡❞ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✱ ✐✳❡✳✱



dT

κ=

ds ❘❡❝❛❧❧ t❤❛t ♦✉r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✿

t 7→ s ,

❝r❡❛t❡s ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥✿

F

t −−−−−→ X

l s

րG

❆s t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s t 7→ s ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✱ ✐t ✐s ❛rr♦✇ ✐♥ ❡✐t❤❡r ❞✐r❡❝t✐♦♥✳

✐♥✈❡rt✐❜❧❡✳

❚❤❛t ✐s ✇❤② ✇❡ ❝❛♥ ❣♦ ❛❧♦♥❣ t❤✐s

❲❡ ✇✐❧❧ tr❡❛t t❤❡ ❞✐❛❣r❛♠ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤r❡❡ q✉❛♥t✐t✐❡s✳ ❚❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ t❤r❡❡ ❛r❡ t❤❡♥ t❤❡ ❢❛♠✐❧✐❛r r❡❧❛t❡❞ r❛t❡s ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✹✮✳ ❏✉st ❛s ❛❜♦✈❡ ❜✉t ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ ✇❡ ❤❛✈❡ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✿ dX ds dX = . dt ds dt ❲❡ ♥♦✇ ✉s❡ t❤❡ ❢❛❝t t❤❛t s ✐s ❛♥ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✿

dX

ds = 1 . ❚❤❡♥✱ ✇❡ ❤❛✈❡✿





dX dX ds ds

=



dt ds dt = dt ,

◆♦✇ ❛ss✉♠✐♥❣ t❤❛t s ✐s ✐♥❝r❡❛s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ t ✭s❛♠❡ ❞✐r❡❝t✐♦♥✦✮✱ ✇❡ ❤❛✈❡ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛② t♦ ❞❡s❝r✐❜❡ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✿ ds = kF ′ k . dt ■ts ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ s♣❡❡❞ ✦ ◆❛t✉r❛❧❧②✱ ✐❢ t❤✐s ✐s 1✱ t❤✐s ✐s t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✱ s = t✳ ◆♦✇✱ ❢r♦♠ t❤❡ ❈❤❛✐♥ t❤❡ ♦❧❞ ♣❛r❛♠❡t❡r✿

❘✉❧❡

❛♥❞ t❤❡ ❛❜♦✈❡ ✐❞❡♥t✐t②✱ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ T ✇✐t❤ r❡s♣❡❝t t♦

❚❤❡♥✱

❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳

dT dT ds = . dt ds dt





dT dT ds dT ′ ′

=



dt ds dt = ds kF k = κ kF k .

✷✳✶✾✳

✷✶✼

❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥

❚❤❡♦r❡♠ ✷✳✶✾✳✸✿ ❈✉r✈❛t✉r❡ ❙✉♣♣♦s❡

′

F 6= 0✳

X = F (t)

✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡

C

✇✐t❤ ♥♦♥✲③❡r♦ ❞❡r✐✈❛t✐✈❡✱

❚❤❡♥ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜②✿

κ(t) =

kT ′ (t)k kF ′ (t)k

❉❡✜♥✐t✐♦♥ ✷✳✶✾✳✹✿ r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡ ❚❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐s ❝❛❧❧❡❞ t❤❡ ♦❢ t❤❡ ❝✉r✈❡✿

r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡

R=

1 κ

✉♥❧❡ss t❤❡ ❢♦r♠❡r ✐s 0✱ t❤❡♥ t❤❡ r❛❞✐✉s ✐s s❛✐❞ t♦ ❜❡ ✐♥✜♥✐t❡✳ ❚❤❡ ❝✐r❝❧❡ ❝✉r✈❛t✉r❡ ❛t t❤❡ ♣♦✐♥t Q = F (a) ♦❢ t❤❡ ❝✉r✈❡ ✐s t❤❡ ❝✐r❝❧❡ • t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✱ • ✇✐t❤ ✐ts r❛❞✐✉s ❡q✉❛❧ t♦ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡✱ ❛♥❞ • ✇✐t❤ ✐ts ❝❡♥t❡r ♦♥ t❤❡ ❧✐♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝✉r✈❡ ❛t Q✳

♦❢

■♥ ❣❡♥❡r❛❧✱ t❤❡ ❝✐r❝❧❡s ♦❢ ❝✉r✈❛t✉r❡ ❛r❡ ❝♦♥t✐♥✉♦✉s❧② ❡✈♦❧✈✐♥❣ ❛s ✇❡ ♠♦✈❡ ❛❧♦♥❣ t❤❡ ❝✉r✈❡✿

❚❤❡ ❝✉r✈❛t✉r❡ ✐s ❛♥ ✐♥tr✐♥s✐❝ ♣r♦♣❡rt② ♦❢ t❤❡ ❝✉r✈❡✿ ❊✈❡♥ ✐❢ ✇❡ ❛r❡ r✐❞✐♥❣ ❛ ❝❛r ❜❧✐♥❞✲❢♦❧❞❡❞✱ ✇❡ ❝❛♥ ✜❣✉r❡ ♦✉t t❤❡ s❤❛♣❡ ♦❢ t❤❡ r♦❛❞ ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ♣✉❧❧✳ ■♥❞❡❡❞✱ t❤❡ ❝✉r✈❛t✉r❡ ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡ ❧❛tt❡r ❢❡❡❧s ❧✐❦❡ ❛ ❢♦r❝❡✦ ❚❤❡ ❥♦❜ ♦❢ ✜♥❞✐♥❣ t❤❡ ❝✉r✈❛t✉r❡ ✐s ♠❛❞❡ ❡❛s✐❡r ✐❢ t❤❡ ❝❛r ✐s ❞r✐✈❡♥ ❛t ❛ ❝♦♥st❛♥t✱ ❦♥♦✇♥ s♣❡❡❞✳ ❲❡ ❞♦♥✬t ♥❡❡❞ t♦ ✇❛❧❦ t❤❡ s✉rr♦✉♥❞✐♥❣ ❛r❡❛ ❛♥❞ ♠❛❦❡ ♠❡❛s✉r❡♠❡♥ts✳✳✳ ■♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r✱ ❛ ❜✉❣ tr❛✈❡❧✐♥❣ ❛❧♦♥❣ ❛ t❤✐♥ t✉❜❡ ♠✐❣❤t ❜❡ ❛❜❧❡ t♦ ♠❛♣ ✐ts s❤❛♣❡✳ ❋♦❧❧♦✇✐♥❣ t❤✐s ✐❞❡❛✱ ✇❡ ❤❛✈❡ t♦ tr❛✈❡❧ t❤❡ ✉♥✐✈❡rs❡ t♦ ♠❛♣ ✐ts

✷✳✶✾✳

❘❡✲♣❛r❛♠❡tr✐③❛t✐♦♥

✷✶✽

❝✉r✈❛t✉r❡ ❛s ✇❡ ❛r❡ ✉♥❛❜❧❡ t♦ st❡♣ ♦✉ts✐❞❡ ❛♥❞ ♠❡❛s✉r❡ ✐t✳

❊①❛♠♣❧❡ ✷✳✶✾✳✺✿ ■s t❤❡ ❊❛rt❤ ✢❛t❄ ❍♦✇ ❞♦❡s ♦♥❡ ♣r♦✈❡ t❤❛t t❤❡ ❊❛rt❤ ✐s r♦✉♥❞❄ ❖♥❡ ❦♥♦✇♥ ♠❡t❤♦❞ ✐s t♦ ❝❧✐♠❜ ❛ t❛❧❧❡r ❜✉✐❧❞✐♥❣ ❛♥❞ s❡❡ t❤❛t ②♦✉r ✜❡❧❞ ♦❢ ✈✐s✐♦♥ ❤❛s ❡①t❡♥❞❡❞✿

❚❤❡ s❛♠❡ ✐❞❡❛ ✐s t♦ ♠♦✈❡ ❛✇❛② ❢r♦♠ ❛ t❛❧❧❡r ❜✉✐❧❞✐♥❣ ❛♥❞ s❡❡ t❤❛t ✐t ❞✐s❛♣♣❡❛rs ❜❡②♦♥❞ t❤❡ ❤♦r✐③♦♥✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤✐s ♠❡t❤♦❞ ✐s t❤❛t ♦♥❡ ♥❡❡❞s ❛ ♣❡r❢❡❝t❧② ✢❛t ✭✦❄✮ ❛r❡❛ ❢♦r t❤❡ ❡①♣❡r✐♠❡♥t✳ ❆♥♦t❤❡r ♠❡t❤♦❞ ✐s t♦ ♠❡❛s✉r❡ t❤❡ s❤❛❞♦✇s ♦❢ t✇♦ ✐❞❡♥t✐❝❛❧ st✐❝❦s ✐♥ t✇♦ ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✿

❚❤❡ ♠❡t❤♦❞ ❛ss✉♠❡s ❤♦✇❡✈❡r t❤❛t t❤❡ ❙✉♥ ✐s s♦ ❧❛r❣❡ t❤❛t ✐ts r❛②s ❛r❡ ❡ss❡♥t✐❛❧❧② ♣❛r❛❧❧❡❧✳ ❆❧❧ t❤❡s❡ ♠❡t❤♦❞s ❤❛✈❡ ✢❛✇s ❜✉t t❤❡ ♠❛✐♥ ♦♥❡ ✐s t❤❛t t❤❡② ♦♥❧② ♣r♦✈❡ t❤❛t t❤❡ ❊❛rt❤ ❤❛s ❝✉r✈❛t✉r❡ ✕ ❛t t❤❛t ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥✦ ❨♦✉✬❞ ❤❛✈❡ t♦ ✈✐s✐t ❡✈❡r② ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤ ❛♥❞ r❡♣❡❛t t❤✐s ❡①♣❡r✐♠❡♥t✳ ❖♥❝❡ ②♦✉ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ✐s t❤❡ s❛♠❡ ❡✈❡r②✇❤❡r❡✱ ②♦✉✬❞ st✐❧❧ ♥❡❡❞ s♦♠❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ♣r♦✈❡s r♦✉♥❞♥❡ss ♦❢ t❤❡ ✇❤♦❧❡ s✉r❢❛❝❡✳ ❇✉t✱ ✐❢ ②♦✉✬✈❡ ✈✐s✐t❡❞ ❡✈❡r② ♣❧❛❝❡ ♦♥ ❊❛rt❤✱ ✇❤② ♥♦t ♠❛❦❡ ❛ ♠❛♣ ♦❢ ✐t ❛♥❞ ♣r♦✈❡ ✐ts s❤❛♣❡ t❤✐s ✇❛②❄✦

❊①❡r❝✐s❡ ✷✳✶✾✳✻ ❙❤♦✇ t❤❛t ✐❢ ❛ ♣❧❛♥❡ ❝✉r✈❡ ❤❛s ❛ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡✱ ✐t ✐s ❡✐t❤❡r ❛ str❛✐❣❤t ❧✐♥❡ ♦r ❛ ❝✐r❝❧❡✳ ❲❤❛t ✐❢ t❤❡ ❝✉r✈❡ ✐s♥✬t ♥❡❝❡ss❛r✐❧② ♣❧❛♥❡❄

✷✳✷✵✳

▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✶✾

✷✳✷✵✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

❊①❛♠♣❧❡ ✷✳✷✵✳✶✿ ❝✐r❝❧❡

■♥ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❛s ✐t ✐s r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥✳ ❚❤✐s t✐♠❡✱ ❧❡t✬s ❝♦♠♣✉t❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✐r❝❧❡ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡✿ ✶✳ P❧❛❝❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡✳ ✷✳ ❈♦♥♥❡❝t t❤❡♠ ❝♦♥s❡❝✉t✐✈❡❧② ❜② ❡❞❣❡s✳ ✸✳ ❆♣♣r♦①✐♠❛t❡ t❤❡ ❝✉r✈❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ♠❛❞❡ ♦❢ t❤❡s❡ ❡❞❣❡s✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❡❛❝❤ ❡❞❣❡ ✐s ❝♦♠♣✉t❡❞ ✈✐❛ t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ✿ ❂❙◗❘❚✭✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✂✷✰✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✂✷✮

❆s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ s❡❣♠❡♥ts✱ t❤❡ r❡s✉❧t t❤❛t ✇❡ ❦♥♦✇ t♦ ❜❡ ❝♦rr❡❝t✱ 2π ✱ ✐s ❜❡✐♥❣ ❛♣♣r♦❛❝❤❡❞✳ ❊❛❝❤ ♦❢ t❤❡s❡ ❝✉r✈❡s ✐s ❛ 0✲❢♦r♠ ♦✈❡r R ✇✐t❤ ✈❛❧✉❡s ✐♥ R2 ✳ ❲❡ ♥♦✇ r❡✈✐❡✇ ❛♥❞ ❣❡♥❡r❛❧✐③❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✸■❈✲✸ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❧❡♥❣t❤s ♦❢ ❝✉r✈❡s✳ ❙✉♣♣♦s❡ X = F (t) ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ t❤❡ m✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛ 0✲❢♦r♠ ♦✈❡r R ✇✐t❤ ✈❛❧✉❡s ✐♥ Rm ✳

❲❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✇✐t❤ n ✐♥t❡r✈❛❧s✳ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡❛❝❤ ✐♥t❡r✈❛❧ [tk−1 , tk ], k = 1, 2, ..., n ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ t❤❡ ❝✉r✈❡ ❧❡❛♣s ❢r♦♠ F (tk−1 ) t♦ F (tk )✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s s❡❣♠❡♥t ✐s✿ ||F (tk ) − F (tk−1 )|| .

✷✳✷✵✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✷✵

❚❤✉s✱ t❤❡ ❢✉❧❧ ❧❡♥❣t❤ ♦❢ t❤❡s❡ s❡❣♠❡♥ts ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ ❛❧❧ n ♦❢ t❤♦s❡✱ ❛s ❢♦❧❧♦✇s✿ t♦t❛❧ ❧❡♥❣t❤ =

n X k=1

||F (tk ) − F (tk−1 )|| .

❚❤✐s ❢♦r♠✉❧❛ ✐s t♦ ❜❡ ✉s❡❞ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s✱ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❋♦r ❡①❛❝t ❛♥s✇❡rs✱ ✇❡ ♠❛❦❡ t❤❡ ✐♥t❡r✈❛❧s s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳ ❲❡ ❛❧s♦ ❛♥t✐❝✐♣❛t❡ t❤❛t t❤✐s ♣r♦❝❡ss ✇✐❧❧ ❡♥❞s ✇✐t❤ ❛ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✇❡ ❤❛✈❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t), a ≤ t ≤ b✳ ❲❡ ✇✐❧❧ ❞❡✜♥❡ ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❛t❤ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ ✐♥t❡r✈❛❧ t♦ s❛♠♣❧❡ t❤❡ ❝✉r✈❡ ❛♥❞ t❤❡♥ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♣✐❡❝❡s ♦❢ t❤❡ ❝✉r✈❡ ✇✐t❤ str❛✐❣❤t s❡❣♠❡♥ts ✇✐t❤ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✳

❖✈❡r ❡❛❝❤ ✐♥t❡r✈❛❧ [tk−1 , tk ], k = 1, 2, ..., n ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡ ♣❛t❤ ♦❢ F ❣♦❡s ❢r♦♠ F (tk−1 ) t♦ F (tk )✳ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ❛ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣♦✐♥ts✳ ❚❤❡♥✱ t❤❡ ❢✉❧❧ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❛t❤ ♦❢ F ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜②✿ ❧❡♥❣t❤ ♦❢ ♣❛t❤ ≈

n X k=1

||F (tk ) − F (tk−1 )|| .

❙✐♥❝❡ t❤✐s ❞♦❡s♥✬t ❧♦♦❦ ❧✐❦❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ❝r❡❛t❡ t❤❡ ♠✐ss✐♥❣ ∆t ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❲❡ ❛❧s♦ ✉s❡ t❤❡ ❡❛r❧✐❡r ✐♥s✐❣❤t✿ t❤❡r❡ ♠✉st ❜❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ F ♣r❡s❡♥t✳ ❲❡ ❝r❡❛t❡ t❤❡ ♠✐ss✐♥❣ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❜② ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❢♦r♠✉❧❛✳ ❚❤❡ t✇♦ ❣♦❛❧s ♠❛t❝❤ ✉♣✿ ✇❡ ❞✐✈✐❞❡ ❛♥❞ ♠✉❧t✐♣❧② ❡❛❝❤ t❡r♠ ❜② ∆t✱ ❛s ❢♦❧❧♦✇s✿ ❙✉♠ ♦❢ ❧❡♥❣t❤s

=

n X k=1

||F (tk ) − F (tk−1 )||

n X

∆t ||F (tk ) − F (tk−1 )|| · ∆t k=1

n X 1

· ∆t .

(F (t ) − F (t )) = k k−1

∆t k=1 =

❲❡ t❤❡♥ ❤❛✈❡ ❜♦t❤ ∆t ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❇✉t t❤✐s ✐s st✐❧❧ ♥♦t ❛ ❘✐❡♠❛♥♥ s✉♠❀ t❤❡ ❡①♣r❡ss✐♦♥ t❤❛t ♣r❡❝❡❞❡s ∆t s❤♦✉❧❞ ❜❡ t❤❡ ✈❛❧✉❡ ♦❢ s♦♠❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❡✈❛❧✉❛t❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❡ ❤❛✈❡♥✬t s♣❡❝✐✜❡❞ t❤♦s❡ ❛♥❞ t❤✐s ✐s t❤❡ t✐♠❡ t♦ ❞♦ t❤❛t✳ ❲❡ ❛♣♣❧②✱ ❛s ✇❡✬✈❡ ❞♦♥❡ ❜❡❢♦r❡✱ t❤❡

✷✳✷✵✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✷✶

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✿ ❚❤❡r❡ ✐s s♦♠❡ ck ✐♥ t❤❡ ✐♥t❡r✈❛❧ [tk−1 , tk ] s✉❝❤ t❤❛t 1 (F (tk ) − F (tk−1 )) = F ′ (ck ) . ∆t

❚❤❡r❡❢♦r❡✱ ❙✉♠ ♦❢ ❧❡♥❣t❤s =

n X k=1

||F ′ (ck )|| ∆t .

❚❤✐s ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ g(t) = ||F ′ (t)|| ♦✈❡r t❤❡ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✇✐t❤ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c1 , ..., cn ✳ ❚❤❡ ❛♥❛❧②s✐s ❛❜♦✈❡ r❡✈❡❛❧s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣t✳

❉❡✜♥✐t✐♦♥ ✷✳✷✵✳✷✿ ❧❡♥❣t❤ ♦❢ ❝✉r✈❡ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❝✉r✈❡ ❣✐✈❡♥ ❜② t❤❡ ♣❛t❤ ♦❢ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ♦✈❡r ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥t❡❣r❛❧ L=

Z

b a

||F ′ (t)|| dt

✐❢ ✐t ❡①✐sts✳ ❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ❢✉♥❝t✐♦♥ F ✐ts❡❧❢ ✐s ❛❜s❡♥t ❢r♦♠ t❤❡ ❢♦r♠✉❧❛ ❜❡❝❛✉s❡ ♦♥❧② t❤❡ s❤❛♣❡ ✭❣✐✈❡♥ ❜② t❤❡ ❞❡r✐✈❛t✐✈❡✮ ❛♥❞ ♥♦t ✐ts ❧♦❝❛t✐♦♥ ♠❛tt❡rs ❢♦r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✳

❚❤❡♦r❡♠ ✷✳✷✵✳✸✿ ▲❡♥❣t❤ ♦❢ ❈✉r✈❡ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❝✉r✈❡ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡✜♥❡❞ ✉♥✐q✉❡❧② ✐❢ ✐t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ♣❛t❤ ♦❢ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✭✐✳❡✳✱ ✐t ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✮✳

Pr♦♦❢✳ ❲❡ ♥❡❡❞ t❤❡ ❡①tr❛ ❝♦♥❞✐t✐♦♥ t♦ ❡♥s✉r❡ t❤❛t t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❛♣♣❧✐❡s ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡✳

❊①❛♠♣❧❡ ✷✳✷✵✳✹✿ ❝✐r❝✉♠❢❡r❡♥❝❡ ▲❡t✬s r❡✲♣r♦✈❡ t❤❛t t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ✐s 2πR✳ ❲❡ ❦♥♦✇ t❤❡ r❡s✉❧t ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✱ ❜✉t t❤✐s t✐♠❡ ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ r❡♣r❡s❡♥t t❤❡ ❝✉r✈❡ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✳

❲❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ X = F (t) =< R cos t, R sin t >, 0 ≤ t ≤ 2π .

❚❤❡♥✱

F ′ (x) =< −R sin t, R cos t > .

✷✳✷✶✳

✷✷✷

❆r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧s✿ ✇❡✐❣❤t

❲❡ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛✿ ❚❤❡ ❧❡♥❣t❤

= = =

Z

2π

Z0 2π

Z

0

||F ′ || dt || < −R sin t, R cos t > || dt

2π

R dt 0

= 2πR .

▼✉❝❤ ❡❛s✐❡r✦ ❙✐♥❝❡ ♦r✐❣✐♥❛❧❧② ✇❡ ♦❜t❛✐♥❡❞ t❤✐s r❡s✉❧t ✈✐❛ tr✐❣ s✉❜st✐t✉t✐♦♥✱ t❤❡ ♥❡✇ ❝♦♠♣✉t❛t✐♦♥ r❡✈❡❛❧s ✐ts tr✉❡ ♠❡❛♥✐♥❣✿ ♣❛r❛♠❡tr✐③❛t✐♦♥✳ ❊①❡r❝✐s❡ ✷✳✷✵✳✺

❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ❢r♦♠ (0, 0) t♦ (1, 1)✳ ❊①❡r❝✐s❡ ✷✳✷✵✳✻

❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ♦♥❡ ❛r❝ ♦❢ t❤❡ ❝②❝❧♦✐❞✿ x = R(1 − sin t), y = R(1 − cos t)✳

✷✳✷✶✳ ❆r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧s✿ ✇❡✐❣❤t

❲❤❛t ✐❢ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ✇❡✐❣❤t ♦❢ ❛ ❝✉r✈❡ ♦❢ ✈❛r✐❛❜❧❡ ❞❡♥s✐t② ❄ ❲❡ ❦♥♦✇ ✭❢r♦♠ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✸✮ t❤❡ ❛♥s✇❡r ✇❤❡♥ t❤✐s ✐s ❛ str❛✐❣❤t s❡❣♠❡♥t ✇✐t❤ t❤❡ ❞❡♥s✐t② ❣✐✈❡♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦♥ t❤✐s s❡❣♠❡♥t s✉❝❤ ❛s t❤✐s ♠❡t❛❧ r♦❞✿

❋♦r ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ f ♦♥ s❡❣♠❡♥t [a, b] ✭❧✐♥❡❛r ❞❡♥s✐t②✮✱ ✇❡ ❞❡✜♥❡❞ t❤❡ ✇❡✐❣❤t ❛s ✐ts ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ Z b

f dx✳

a

❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤✐s ❛♥❛❧②s✐s ❛♥❞ t❤❡ ❛♥❛❧②s✐s ❛❜♦✈❡ ♣r♦✈✐❞❡s t❤❡ s♦❧✉t✐♦♥✳

❙✉♣♣♦s❡ t❤❡ ❝✉r✈❡ ✐s ✉s❡❞ t♦ ❝✉t ❛ ✈❡r② t❤✐♥ str✐♣✱ ❛ ✇✐r❡✱ ❢r♦♠ ❛ s❤❡❡t ♦❢ ♠❡t❛❧ ✇✐t❤ ✈❛r✐❛❜❧❡ ❞❡♥s✐t②✳ ❲❡ ✇✐❧❧ ❞❡✜♥❡ ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ♠❛ss ♦❢ t❤❡ ❝✉r✈❡✳ ❙✉♣♣♦s❡ X = X(t), a ≤ t ≤ b✱ ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ Rn ❛♥❞ t❤❡ ❞❡♥s✐t② ✐s ❣✐✈❡♥ ❜② ❛ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡ z = f (X)✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ ✐♥t❡r✈❛❧ t♦ s❛♠♣❧❡ t❤❡ ❝✉r✈❡ ❛♥❞ t❤❡♥ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♣✐❡❝❡s ♦❢ t❤❡ ❝✉r✈❡ ✇✐t❤ str❛✐❣❤t s❡❣♠❡♥ts ❛♥❞ ❝♦♥st❛♥t ❞❡♥s✐t✐❡s✳ ❲❡ st❛rt ✇✐t❤ ❛ s❛♠♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✇✐t❤ n ✐♥t❡r✈❛❧s✿ ck ✐♥ [tk−1 , tk ], k = 1, 2, ..., n✳ ❲✐t❤ t❤❡ ♣❛t❤ ♦❢ X = X(t) ❣♦✐♥❣ ❢r♦♠ X(tk−1 ) t♦ X(tk )✱ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣♦✐♥ts ✇❡✐❣❤s ❛♣♣r♦①✐♠❛t❡❧②✿ f (ck )||X(tk ) − X(tk−1 )|| .

✷✳✷✶✳

✷✷✸

❆r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧s✿ ✇❡✐❣❤t

❚❤✉s✱ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡s❡ s❡❣♠❡♥ts ✐s✿ t♦t❛❧ ♠❛ss =

n X k=1

f (ck )||X(tk ) − X(tk−1 )|| .

❚♦ ❡①tr❛❝t ❛ ❘✐❡♠❛♥♥ s✉♠ ❢r♦♠ t❤✐s s✉♠✱ ✇❡ ❞✐✈✐❞❡ ❛♥❞ ♠✉❧t✐♣❧② ❡❛❝❤ t❡r♠ ❜② h✱ ❛s ❢♦❧❧♦✇s✿ ❙✉♠ ♦❢ ❧❡♥❣t❤s

=

n X k=1

f (ck )||X(tk ) − X(tk−1 )||

n X

h f (ck )||X(tk ) − X(tk−1 )|| · h k=1

n X

1

· h. = f (ck ) (X(t ) − X(t )) k k−1

h

k=1

=

❉❡✜♥✐t✐♦♥ ✷✳✷✶✳✶✿ ✇❡✐❣❤t ♦❢ ❝✉r✈❡

❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❝✉r✈❡ ❣✐✈❡♥ ❜② t❤❡ ♣❛t❤ ♦❢ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ♦✈❡r ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ ❞❡♥s✐t② ❣✐✈❡♥ ❜② ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ z = f (X) ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭♥✉♠❡r✐❝❛❧✮ ✐♥t❡❣r❛❧✿ Z

f ds = C

Z

b

f (X(t))||X ′ (t)|| dt a

✐❢ ✐t ❡①✐sts✳ ■t ✐s ❛❧s♦ ❝❛❧❧❡❞ ❛♥ ❛r❝✲❧❡♥❣t❤

✐♥t❡❣r❛❧ ♦❢

f

❛❧♦♥❣

C✳

❚❤❡♦r❡♠ ✷✳✷✶✳✷✿ ❲❡✐❣❤t ♦❢ ❈✉r✈❡ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❝✉r✈❡ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡✜♥❡❞ ✉♥✐q✉❡❧② ✐❢ ✐t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✭♣❛r❛♠❡tr✐③❡❞✮ ❛s t❤❡ ♣❛t❤ ♦❢ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳

❊①❡r❝✐s❡ ✷✳✷✶✳✸ ❋✐♥❞ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ❢r♦♠ (0, 0) t♦ (1, 1)✳

✷✳✷✷✳

❚❤❡ ❤❡❧✐①

✷✷✹

❖❢ ❝♦✉rs❡✱ t❤❡ ✐♥t❡❣r❛❧ ❝❛♥ ❛❧s♦ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❛♠♦✉♥t ♦❢ ❧✐q✉✐❞ ✐♥ ❛ ♣✐♣❡ ❛♥❞ s♦ ♦♥✳

✷✳✷✷✳ ❚❤❡ ❤❡❧✐①

❚❤❡ t✉r♥✐♥❣ ❝✐r❝❧❡ ♦❢ ❛ ❝❛r t❡❧❧s ✉s t❤❡ ♥❛rr♦✇❡st ✇✐❞t❤ ♦❢ ❛ str❡❡t t❤❛t ❛❧❧♦✇s t❤✐s ❝❛r t♦ ❯✲t✉r♥✿

❲❤❛t ✐❢ ✐♥st❡❛❞ ♦❢ ❛ str❡❡t✱ ✇❡ ❛r❡ t♦ ❜✉✐❧❞ ❛

❚❤❡ s❤❛♣❡ ♦❢ s✉❝❤ ❛ ❝✉r✈❡ ✐s t❤❡

❤❡❧✐①✳

❝✐r❝✉❧❛r r❛♠♣ ✐♥ ❛ ♣❛r❦✐♥❣ ❣❛r❛❣❡❄

❚❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❛ ❤❡❧✐① ♦❢ r❛❞✐✉s R ✐s ❣✐✈❡♥ ❜❡❧♦✇✿

F (t) =< R cos t, R sin t, t > . ■t ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛ r♦t❛t✐♦♥ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ❛♥❞ ❛ ✈❡rt✐❝❛❧ ❛s❝❡♥❞✳ ◆♦✇✱ s✉♣♣♦s❡ t❤❡ t✉r♥✐♥❣ ❝✐r❝❧❡ ♦❢ ♦✉r ❝❛r ❤❛s r❛❞✐✉s r✳ ❆❧s♦✱ s✉♣♣♦s❡ t❤❛t ♦✉r ❤❡❧✐❝❛❧ r❛♠♣ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❝②❧✐♥❞❡r ♦❢ r❛❞✐✉s R✳ ❲❤❛t s❤♦✉❧❞ t❤❡ r❛❞✐✉s ♦❢ t❤✐s ❝②❧✐♥❞❡r ❜❡ t♦ ❛❝❝♦♠♠♦❞❛t❡ ♦✉r ❝❛r❄ ▲❛r❣❡r✱ s♠❛❧❧❡r✱ ♦r t❤❡ s❛♠❡❄ ❚❤❡ ❛♥s✇❡r ✐s s♠❛❧❧❡r✱ ❜❡❝❛✉s❡ s♦♠❡ ♦❢ t❤❡ t✉r♥ ✐s ❝❛rr✐❡❞ ♦✉t ✐♥ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥✳ ❚❤❡ s♠❛❧❧❡r ❝②❧✐♥❞❡r ❞♦❡s♥✬t ❤❛✈❡ t♦ ❝♦♥t❛✐♥ t❤✐s ❧❛r❣❡r ❝✐r❝❧❡ ❜❡❝❛✉s❡ s✉❝❤ ❛ ❝✐r❝❧❡ ❛♣♣r♦①✐♠❛t❡s t❤❡ ❝✉r✈❡ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✳ ❲❡ t❛❦❡ t❤❡ s♣r❡❛❞s❤❡❡t ❢♦r t❤❡ ❝✐r❝❧❡ ❛♥❞ s✐♠♣❧② ❛❞❞ ❛ z ✲❝♦❧✉♠♥ ✇❤❡r❡✈❡r ✇❡ ❤❛✈❡ x✲ ❛♥❞ y ✲❝♦❧✉♠♥s✳ ❚❤❡ ❝♦♠♣✉t❡❞ ♠❛❣♥✐t✉❞❡s ✕ ✐♥❝❧✉❞✐♥❣ t❤❡ ❝✉r✈❛t✉r❡ ✐ts❡❧❢ ✕ ❛r❡ t❤❡♥ ❛❞❥✉st❡❞ t♦ ✐♥❝❧✉❞❡ t❤❡ t❤✐r❞ ❝♦♠♣♦♥❡♥t✳

✷✳✷✷✳

❚❤❡ ❤❡❧✐①

✷✷✺

❚❤❡ ❝✉r✈❛t✉r❡ ✐s κ = 1/2 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡ ✐s R = 2✳ ❙♦ t❤❡ ❤❡❧✐① ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ❝✐r❝❧❡ ♦❢ t✇✐❝❡ ✐ts r❛❞✐✉s✦ ▲❡t✬s ♣r♦✈✐❞❡ ❛♥ ❛❧❣❡❜r❛✐❝ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✷✳✷✷✳✶✿ ❈✉r✈❛t✉r❡ ♦❢ ❍❡❧✐① ❚❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❤❡❧✐① ♦❢ r❛❞✐✉s

R

κ=

■t ❢♦❧❧♦✇s✱ ✜rst✱ t❤❛t

✐s ❝♦♥st❛♥t ❛♥❞ ❡q✉❛❧ t♦✿

R +1

R2

κ → 0 ❛s R → ∞ .

❚❤✐s ♠❡❛♥s t❤❛t ❛ ✇✐❞❡♥✐♥❣ ❤❡❧✐① ❧♦♦❦s ♠♦r❡ ❛♥❞ ♠♦r❡ ❧✐❦❡ ❛ ❝✐r❝❧❡✱ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡✿

κ → 0 ❛s R → 0 .

❚❤✐s ♠❡❛♥s t❤❛t ❛ ♥❛rr♦✇✐♥❣ ❤❡❧✐① ❧♦♦❦s ♠♦r❡ ❛♥❞ ♠♦r❡ ❧✐❦❡ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✳ ■♥❞❡❡❞✱ t❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ✐♥❝r❡❛s❡ t❤❡ r❛t❡ ♦❢ ❛s❝❡♥❞✿

❙♦ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ ❝✉r✈❛t✉r❡ ✐s 2 ❜✉t ✇❤❡r❡ ✐s t❤✐s ❝✐r❝❧❡ ❧♦❝❛t❡❞❄ ❖♥ t❤❡ ❧✐♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝✉r✈❡✱ ✐✳❡✳✱ t♦ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❙✐♥❝❡ ||T || = 1✱ ✇❡ ❝❛♥ ❥✉st ❝❤♦♦s❡ T ′ ✦

✷✳✷✷✳

❚❤❡ ❤❡❧✐①

✷✷✻

❉❡✜♥✐t✐♦♥ ✷✳✷✷✳✷✿ ✉♥✐t ♥♦r♠❛❧ ✈❡❝t♦r ❚❤❡ ✉♥✐t

♥♦r♠❛❧ ✈❡❝t♦r ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐s ❞❡✜♥❡❞ t♦ ❜❡ N (t) =

T ′ (t) ||T ′ (t)||

❲❡ ♥♦✇ r❡♣❡❛t ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✇✐t❤ ♦✉r ❝❛❧❝✉❧✉s t♦♦❧s✳ ❲❡ t❛❦❡ t❤❡ st❛♥❞❛r❞ ❤❡❧✐① ♦❢ r❛❞✐✉s 1 ❛♥❞ ❞✐✛❡r❡♥t✐❛t❡ ✐t✿ F (t) =< cos t, sin t, t > =⇒ F ′ (t) =< − sin t, cos t, 1 > .

❚❤❡ s♣❡❡❞ ♥❡①t✿ ❛♥❞ t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r✿

||F ′ (t)|| =

p √ (− sin t)2 + (cos t)2 + 1 = 2 ,

1 T (t) = √ < − sin t, cos t, 1 > . 2

❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❛t✿

1 T ′ (t) = √ < − cos t, − sin t, 0 > . 2

■t✬s ❤♦r✐③♦♥t❛❧✦ ■ts ♠❛❣♥✐t✉❞❡ ✐s✿ 1 p 1 ||T ′ (t)|| = √ (− cos t)2 + (− sin t)2 + 02 = √ . 2 2

■t✬s ❝♦♥st❛♥t✦ ❚❤❡♥ t❤❡ ✉♥✐t ♥♦r♠❛❧ ✈❡❝t♦r ✐s✿

1 1 N (t) = √ < − cos t, − sin t, 0 > ÷ √ =< − cos t, − sin t, 0 > . 2 2

❍♦✇ ❢❛r ❞♦ ✇❡ ❣♦❄ ❚❤❡ st❡♣ ✐s t❤❡ r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡ R = 2✳ ❚❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ✐s✿ C = F (t) + 2N (t) =< cos t, sin t, t > +2 < − cos t, − sin t, 0 >=< − cos t, − sin t, 0 >= −F (t) .

■t✬s t❤❡ ♣♦✐♥t ❡①❛❝t❧② ♦♣♣♦s✐t❡ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❧♦❝❛t✐♦♥ F (t)✦ ❋♦r t❤❡ ❤❡❧✐①✱ ✇❡ ❛r❡ t♦ ♠❛❦❡ ❛ st♦♣ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✱ F (t) ✐♥ t❤✐s ❞✐r❡❝t✐♦♥✱ N (t)✱ ♦❢ ❧❡♥❣t❤ 2✳

✷✳✷✷✳

❚❤❡ ❤❡❧✐①

✷✷✼

❚❤✐s ✐s t❤❡ ❡♥❞ r❡s✉❧t✿ t❤❡ r❛❞✐✉s ✐s t❤❡ ❞♦✉❜❧❡ ♦❢ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝②❧✐♥❞❡r ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝❡♥t❡r ❧✐❡s ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ ❝②❧✐♥❞❡r✳

❊①❡r❝✐s❡ ✷✳✷✷✳✸

❚❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❤❡❧✐① ♦❢ r❛❞✐✉s 1 ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 2✳ ❈❛♥ ✇❡ t❡❧❧ t❤❡♠ ❛♣❛rt ✕ ❢r♦♠ t❤❡ ✐♥s✐❞❡❄

❈❤❛♣t❡r ✸✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❈♦♥t❡♥ts

✸✳✶ ❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥ R2 ❛♥❞ ♣❧❛♥❡s ✐♥ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ●r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ▲✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❈♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s ✸✳✶✸ ❚❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✶✳ ❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s

▲❡t✬s r❡✈✐❡✇ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳ ❲❡ ❤❛✈❡ t✇♦ ❛①❡s✿ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ r❛♥❣❡✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷✽ ✷✸✸ ✷✸✽ ✷✹✶ ✷✹✽ ✷✺✼ ✷✻✹ ✷✼✷ ✷✼✺ ✷✽✶ ✷✽✼ ✷✾✶ ✷✾✺

✸✳✶✳ ❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s

✷✷✾

❲❡ ❝♦✈❡r❡❞ t❤❡ ✈❡r② ✜rst ❝❡❧❧ ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ♠❛❞❡ ❛ st❡♣ ✐♥ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥ ❛♥❞ ❡①♣❧♦r❡❞ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ t❤✐s t❛❜❧❡✳ ■t ✐s ♥♦✇ t✐♠❡ t♦ ♠♦✈❡ t♦ t❤❡ r✐❣❤t✳ ❲❡ r❡tr❡❛t t♦ t❤❡ ✜rst ❝❡❧❧ ❜❡❝❛✉s❡ t❤❡ ♥❡✇ ♠❛t❡r✐❛❧ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♠❛t❡r✐❛❧ ♦❢ ❈❤❛♣t❡r ✷ ✕ ♦r ✈✐❝❡ ✈❡rs❛✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡② ❞♦ ✐♥t❡r❛❝t ✈✐❛ ❝♦♠♣♦s✐t✐♦♥s✳ ❲❡ ✇✐❧❧ ♥♦t ❥✉♠♣ ❞✐❛❣♦♥❛❧❧②✦ ❲❡ ♥❡❡❞ t♦ ❛♣♣r❡❝✐❛t❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❞✐✛❡r❡♥t ❝❤❛❧❧❡♥❣❡s t❤❡s❡ t✇♦ st❡♣s ♣r❡s❡♥t✳ ❊✈❡r② t✇♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♠❛❦❡ ❛ ♣❧❛♥❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞✱ ❝♦♥✈❡rs❡❧②✱ ❡✈❡r② ♣❧❛♥❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❥✉st ❛ ♣❛✐r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ s✉r❢❛❝❡ t❤❛t ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ♣r♦❞✉❝❡s ✕ t❤r♦✉❣❤ ❝✉tt✐♥❣ ❜② ✈❡rt✐❝❛❧ ♣❧❛♥❡s ✕ ✐♥✜♥✐t❡❧② ♠❛♥② ❣r❛♣❤s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ◆♦t❡ t❤❛t t❤❡ ✜rst ❝❡❧❧ ❤❛s ❝✉r✈❡s ❜✉t ♥♦t ❛❧❧ ♦❢ t❤❡♠ ❜❡❝❛✉s❡ s♦♠❡ ♦❢ t❤❡♠ ❢❛✐❧ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ t❡st ✕ s✉❝❤ ❛s t❤❡ ❝✐r❝❧❡ ✕ ❛♥❞ ❝❛♥✬t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❣r❛♣❤s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❍❡♥❝❡ t❤❡ ♥❡❡❞ ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ✜rst ❝❡❧❧ ♦❢ t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ❤❛s s✉r❢❛❝❡s ❜✉t ♥♦t ❛❧❧ ♦❢ t❤❡♠ ❜❡❝❛✉s❡ s♦♠❡ ♦❢ t❤❡♠ ❢❛✐❧ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ t❡st ✕ s✉❝❤ ❛s t❤❡ s♣❤❡r❡ ✕ ❛♥❞ ❝❛♥✬t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❍❡♥❝❡ t❤❡ ♥❡❡❞ ❢♦r ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s✱ s❤♦✇♥ ❤✐❣❤❡r ✐♥ t❤✐s ❝♦❧✉♠♥✳ ❚❤❡② ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✹✳ ❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t ♦❢ ✇❤❛t❡✈❡r ♥❛t✉r❡✿ ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t

x

7→

f

7→

y

❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s

✸✳✶✳

✷✸✵

▲❡t✬s ❝♦♠♣❛r❡ t❤❡ t✇♦✿ ♣❛r❛♠❡tr✐❝ ✐♥♣✉t

t 7→ R ♥✉♠❜❡r

❝✉r✈❡

F

❢✉♥❝t✐♦♥ ♦❢ ♦✉t♣✉t

✐♥♣✉t

X 7→

7→ X

Rm

Rm ♣♦✐♥t ♦r ✈❡❝t♦r

t✇♦ ✈❛r✐❛❜❧❡s

♣♦✐♥t ♦r ✈❡❝t♦r

f

♦✉t♣✉t

7→ z R ♥✉♠❜❡r

❚❤❡② ❝❛♥ ❜❡ ❧✐♥❦❡❞ ✉♣ ✐♥ t❤❡ ♠✐❞❞❧❡✱ ♣r♦❞✉❝✐♥❣ ❛ ❝♦♠♣♦s✐t✐♦♥✳ ❇✉t ♦✉r ✐♥t❡r❡st ✐s t❤❡ ❧❛tt❡r st❛rt✐♥❣ ✇✐t❤ ❞✐♠❡♥s✐♦♥ m = 2✿

❚❤❡ ♠❛✐♥ ♠❡t❛♣❤♦r ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐❧❧ r❡♠❛✐♥ t♦ ❜❡

t❡rr❛✐♥ ✿

❍❡r❡✱ ❡✈❡r② ♣❛✐r X = (x, y) r❡♣r❡s❡♥ts ❛ ❧♦❝❛t✐♦♥ ♦♥ ❛ ♠❛♣ ❛♥❞ z = f (x, y) ✐s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ t❡rr❛✐♥ ❛t t❤❛t ❧♦❝❛t✐♦♥✳ ❚❤✐s ✐s ❤♦✇ ✐t ✐s ♣❧♦tt❡❞ ❜② ❛ s♣r❡❛❞s❤❡❡t✿

◆♦✇

❝❛❧❝✉❧✉s✳

❖♥❡ s✉❜❥❡❝t ♦❢ ❝❛❧❝✉❧✉s ✐s ❝❤❛♥❣❡ ❛♥❞ ♠♦t✐♦♥ ❛♥❞✱ ❛♠♦♥❣ ♦t❤❡rs✱ ✇❡ ✇✐❧❧ ❛❞❞r❡ss t❤❡ q✉❡st✐♦♥✿ ✐❢ ❛ ❞r♦♣ ♦❢ ✇❛t❡r ❧❛♥❞ ♦♥ t❤✐s s✉r❢❛❝❡✱ ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇✐❧❧ ✐t ✢♦✇❄ ❲❡ ✇✐❧❧ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ✐ss✉❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✕ ✐♥ ❛♥② ❞✐r❡❝t✐♦♥✳

✸✳✶✳

❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s

✷✸✶

❙❡❝♦♥❞❧②✱ ❝❛❧❝✉❧✉s st✉❞✐❡s t❛♥❣❡♥❝② ❛♥❞ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞✱ ❛♠♦♥❣ ♦t❤❡rs✱ ✇❡ ✇✐❧❧ ❛❞❞r❡ss t❤❡ q✉❡st✐♦♥✿ ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ♦♥ t❤❡ s✉r❢❛❝❡✱ ✇❤❛t ❞♦❡s ✐t ❧♦♦❦ ❧✐❦❡❄ ❚❤❡ s❤♦rt ❛♥s✇❡r ✐s✿ ❧✐❦❡ ❛ ♣❧❛♥❡✳ ■t ✐s ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❊①❛♠♣❧❡s ♦❢ t❤✐s ✐ss✉❡ ❤❛✈❡ ❜❡❡♥ s❡❡♥ ♣r❡✈✐♦✉s❧②✳ ■♥❞❡❡❞✱ r❡❝❛❧❧ t❤❛t t❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠ ❛s❦s ❢♦r ❛ t❛♥❣❡♥t ❧✐♥❡ t♦ ❛ ❝✉r✈❡ ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t✳ ■t ❤❛s ❜❡❡♥ s♦❧✈❡❞ ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ ❈❤❛♣t❡r ✷✳ ❍♦✇❡✈❡r✱ ✐♥ r❡❛❧ ❧✐❢❡ ✇❡ s❡❡ s✉r❢❛❝❡s r❛t❤❡r t❤❛♥ ❝✉r✈❡s✳ ❚❤❡ ❡①❛♠♣❧❡s ❛r❡ ❢❛♠✐❧✐❛r✳ ❊①❛♠♣❧❡ ✸✳✶✳✶✿ r❛❞❛r

■♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ ❛ r❛❞❛r s✐❣♥❛❧ ✇✐❧❧ ❜♦✉♥❝❡ ♦✛ ❛ ♣❧❛♥❡ ✇❤❡♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ♣❧❛♥❡ ✐s ❝✉r✈❡❞❄

■♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇✐❧❧ ❧✐❣❤t ❜♦✉♥❝❡ ♦✛ ❛ ❝✉r✈❡❞ ♠✐rr♦r❄

❲❤❛t ✐❢ ✐t ✐s ❛ ✇❤♦❧❡ ❜✉✐❧❞✐♥❣❄

❘❡❝❛❧❧ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛t x = a ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ t❤❛t ❝♦♥♥❡❝ts (a, f (a)) t♦ t❤❡ ♥❡①t ♣♦✐♥t (x, f (x))✿ ∆f f (x) − f (a) = . ∆x x−a

◆♦✇✱ ❧❡t✬s s❡❡ ❤♦✇ t❤✐s ♣❧❛♥ ❛♣♣❧✐❡s t♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (x, y)✳

✸✳✶✳

❖✈❡r✈✐❡✇ ♦❢ ❢✉♥❝t✐♦♥s

✷✸✷

■❢ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♣♦✐♥t (a, b, f (a, b)) ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ✇❡ st✐❧❧ ♣❧♦t t❤❡ ❧✐♥❡ t❤❛t ❝♦♥♥❡❝ts t❤✐s ♣♦✐♥t t♦ t❤❡ ♥❡①t ♣♦✐♥t ♦♥ t❤❡ ❣r✐❞✳ ❚❤❡r❡ ❛r❡ t✇♦ ♣♦✐♥t t❤✐s t✐♠❡❀ t❤❡② ❧✐❡ ✐♥ t❤❡ x✲ ❛♥❞ t❤❡ y ✲❞✐r❡❝t✐♦♥s ❢r♦♠ (a, b)✱ ✐✳❡✳✱ (x, b) ❛♥❞ (a, y) ✇✐t❤ x 6= a ❛♥❞ y 6= b✳

❚❤❡ t✇♦ s❧♦♣❡s ✐♥ t❤❡s❡ t✇♦ ❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦ y ✿ ∆f f (x, b) − f (a, b) ∆f f (a, y) − f (a, b) = ❛♥❞ = . ∆x

x−a

∆y

y−b

❲❤❡♥ ❞♦♥❡ ✇✐t❤ ❡✈❡r② ♣❛✐r ♦❢ ♥♦❞❡s ♦♥ t❤❡ ❣r❛♣❤✱ t❤❡ r❡s✉❧t ✐s ❛ ♠❡s❤ ♦❢ tr✐❛♥❣❧❡s✿

❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡ s✉r❢❛❝❡ ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❛♥❞ ✇❡ ③♦♦♠ ✐♥ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r ♦♥ ❛ ♣❛rt✐❝✉❧❛r ♣♦✐♥t✱ ✇❡ ♠✐❣❤t ❡①♣❡❝t t❤❡ s✉r❢❛❝❡ t♦ st❛rt t♦ ❧♦♦❦ ♠♦r❡ ❛♥❞ ♠♦r❡ str❛✐❣❤t ❧✐❦❡ ❛ ♣❧❛♥❡✳

✸✳✷✳

▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

✷✸✸

R3

✸✳✷✳ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

R3

❚❤❡ st❛♥❞❛r❞✱ s❧♦♣❡✲✐♥t❡r❝❡♣t✱ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ t❤❡ xy ✲♣❧❛♥❡ ✐s✿ y = mx + p .

❍❡r❡ m ✐s t❤❡ s❧♦♣❡ ❛♥❞ p t❤❡ y ✲✐♥t❡r❝❡♣t✳ ◆❡①t✱ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ t❤❡ xy ✲♣❧❛♥❡ ✐s✿ y − b = m(x − a) .

❚❤✐s ✐s ❤♦✇ ✇❡ ❝❛♥ ♣❧♦t t❤✐s ❧✐♥❡✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ♣♦✐♥t (a, b) ✐♥ R2 ✳ ❚❤❡♥ ✇❡ ♠❛❦❡ ❛ st❡♣ ❛❧♦♥❣ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ s❧♦♣❡ m✱ ✐✳❡✳✱ ✇❡ ❡♥❞ ✉♣ ❛t (a + 1, b + m) ♦r (a + 1/m, b + 1)✱ ❡t❝✳ ❚❤❡s❡ t✇♦ ♣♦✐♥ts ❞❡t❡r♠✐♥❡ t❤❡ ❧✐♥❡✳

❲❡ ❛❧s♦ ♥❡❡❞ t❤❡ ❣❡♥❡r❛❧ ✭✐♠♣❧✐❝✐t✮ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ R2 ✿ m(x − a) + n(y − b) = 0 .

▲❡t✬s t❛❦❡ ❛ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤✐s ❡①♣r❡ss✐♦♥✳ ❚❤❡r❡ ✐s ❛ s②♠♠❡tr② ❤❡r❡ ♦✈❡r t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡s✿     m · (x − a)+ n

❚❤✐s ✐s t❤❡ ✈❛r✐❛❜❧❡✿

❞♦t ♣r♦❞✉❝t

⇐⇒

· (y − b) = 0

x−a m  = 0.  · y−b n

✦ ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ s❡❝♦♥❞ ✈❡❝t♦r ❛s t❤❡

❚❤❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿

♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✈❡❝t♦r✱

     a x x−a  = −  . b y y−b 



< m, n > · (x, y) − (a, b)

❋✐♥❛❧❧② ❛ ❝♦♦r❞✐♥❛t❡✲❢r❡❡ ✈❡rs✐♦♥✿

✐♥❝r❡♠❡♥t



= 0.

N · (P − P0 ) = 0 ♦r N · P0 P = 0

❍❡r❡ ✇❡ ❤❛✈❡ ✐♥ R2 ✿ • P ✐s t❤❡ ✈❛r✐❛❜❧❡ ♣♦✐♥t✳ • P0 ✐s t❤❡ ✜①❡❞ ♣♦✐♥t✳

• N ✐s ❛♥② ✈❡❝t♦r t❤❛t r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳

❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✈❡❝t♦r N ✐s r❡✈❡❛❧❡❞ ♦♥❝❡ ✇❡ r❡♠❡♠❜❡r t❤❛t t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ✈❡❝t♦rs ✐s 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳

✸✳✷✳ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

R3

✷✸✹

❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✿ ❧✐♥❡ t❤r♦✉❣❤ ♣♦✐♥t ✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r ❙✉♣♣♦s❡ ❛ ♣♦✐♥t

❧✐♥❡ t❤r♦✉❣❤

P0

P0

❛♥❞ ❛ ♥♦♥✲③❡r♦ ✈❡❝t♦r

✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r

N

N

❛r❡ ❣✐✈❡♥ ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡♥ t❤❡

✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ♣♦✐♥ts

P

✕ ❛♥❞

P0

✐ts❡❧❢ ✕ t❤❛t s❛t✐s❢②✿

P0 P ⊥ N . ◆♦✇ t❤❡ ♣❧❛♥❡s ✐♥ s♣❛❝❡✳ ❲❡ ✇✐❧❧ ❛♣♣r♦❛❝❤ t❤❡ ✐ss✉❡ ✐♥ ❛ ♠❛♥♥❡r ❛♥❛❧♦❣♦✉s t♦ t❤❛t ❢♦r ❧✐♥❡s✳ ❚❤❡ s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ t❤❡

xy ✲♣❧❛♥❡✿

y = mx + p . ❤❛s ❛♥ ❛♥❛❧♦❣✉❡✳ ❆ s✐♠✐❧❛r✱ ❛❧s♦ ✐♥ s♦♠❡ s❡♥s❡ s❧♦♣❡✲✐♥t❡r❝❡♣t✱ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ✐♥

R3

✐s✿

z = mx + ny + p . ■♥❞❡❡❞✱ ✐❢ ✇❡ s✉❜st✐t✉t❡

x=y=0

❚❤❡ ✏s❧♦♣❡s✑ ❞❡♣✐❝t❡❞ ❛r❡ ■♥ ✇❤❛t s❡♥s❡ ❛r❡

r❡s♣❡❝t✐✈❡❧②

❛♥❞

n = −3/2 n

❛♥❞

z = p✳

❚❤❡♥

p

✐s t❤❡

z ✲✐♥t❡r❝❡♣t✦

m = −3/4✳ y = 0 ✜rst✳ ❲❡ ❤❛✈❡ z = mx + p✱ ❛♥ ❡q✉❛t✐♦♥ ♦❢ s✉❜st✐t✉t❡ x = 0✳ ❲❡ ❤❛✈❡ z = ny + p✱ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉t t❤❡ ♣❧❛♥❡ ✇✐t❤ ❛♥② ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✱ ♦r

t❤❡ s❧♦♣❡s❄ ▲❡t✬s s✉❜st✐t✉t❡

xz ✲♣❧❛♥❡✳ ■ts s❧♦♣❡ ✐s m✳ ◆♦✇ ✇❡ yz ✲♣❧❛♥❡✳ ■ts s❧♦♣❡ ✐s n✳ ◆♦✇✱ ✐❢ ✇❡ yz ✲♣❧❛♥❡✱ t❤❡ r❡s✉❧t✐♥❣ ❧✐♥❡ ❤❛s t❤❡ s❛♠❡

❛ ❧✐♥❡ ✕ ✐♥ t❤❡ ❧✐♥❡ ✕ ✐♥ t❤❡

m

✇❡ ❤❛✈❡

y = 1 =⇒ z = mx + n + p;

s❧♦♣❡❀ ❢♦r ❡①❛♠♣❧❡✿

x = 1 =⇒ z = m + ny + p .

✸✳✷✳

▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

✷✸✺

R3

❚❤❡r❡❢♦r❡✱ ✇❡ ❛r❡ ❥✉st✐✜❡❞ t♦ s❛② t❤❛t m ❛♥❞ n ❛r❡ t❤❡ ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✿ z= m ·x + n

x✲s❧♦♣❡

s❧♦♣❡s

y ✲s❧♦♣❡

♦❢ t❤❡ ♣❧❛♥❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❛r✐❛❜❧❡s x

·y + p

z ✲✐♥t❡r❝❡♣t

❈✉tt✐♥❣ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ❛❧s♦ ♣r♦❞✉❝❡s ❛ ❧✐♥❡✿

z = 1 =⇒ 1 = mx + ny + p . ❚❤❡s❡ t✇♦ s❧♦♣❡s ❛r❡ ❛❧s♦ ✐♥❞❡♣❡♥❞❡♥t✦ ❲❡ ❝❛♥ ✉s❡ t❤❡ ❧✐♥❡ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ❛s ❛ ❤✐♥❣❡ ✕ s♦ t❤❛t t❤✐s s❧♦♣❡ r❡♠❛✐♥s t❤❡ s❛♠❡ ✕ ❛♥❞ r♦t❛t❡ t❤❡ ♣❧❛♥❡ ❛s ❛ ❞♦♦r s♦ t❤❛t t❤❡ ♦t❤❡r s❧♦♣❡ ✇✐❧❧ ❝❤❛♥❣❡✿

❚❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ t❤❡ xy ✲♣❧❛♥❡✿

y − b = m(x − a) . ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤✐s ✐s ❛ s✐♠✐❧❛r✱ ❛❧s♦ ✐♥ s♦♠❡ s❡♥s❡ ♣♦✐♥t✲s❧♦♣❡✱ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡✳ ❚❤✐s ✐s t❤❡ st❡♣ ✇❡ ♠❛❦❡✿ ✐♥ R2 ✐♥ R3 ♣♦✐♥t (a, b) s❧♦♣❡ m

♣♦✐♥t (a, b, c) s❧♦♣❡s m, n

❚❤✐s ❛♥❛❧♦❣② ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

z − c = m(x − a) + n(y − b) . ❊①♣❛♥❞✐♥❣ ✐t t❛❦❡s ✉s ❜❛❝❦ t♦ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✉❧❛✳ ❊①❛♠♣❧❡ ✸✳✷✳✷✿ ♣❧❛♥❡

▲❡t✬s ♣❧♦t s✉❝❤ ❛ ♣❧❛♥❡ ✇✐t❤✿

(a, b, c) = (2, 4, 1), m = −1, n = −2 .

✸✳✷✳

▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

R3

✷✸✻

❋✐rst✱ ✇❡ ✜① y = 4 ❛♥❞ ❝❤❛♥❣❡ x✿ ❢r♦♠ t❤❡ ♣♦✐♥t (2, 4, 1)✱ ✇❡ ♠❛❦❡ ❛ st❡♣ ❛❧♦♥❣ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ s❧♦♣❡ −1✱ ✐✳❡✳✱ ✇❡ ❡♥❞ ✉♣ ❛t (2 + 1, 4, 1 − 1) = (3, 4, 9)✳ ❲❡ ♣❧♦t t❤✐s ❧✐♥❡✿

❙❡❝♦♥❞✱ ✇❡ ✜① x = 2 ❛♥❞ ❝❤❛♥❣❡ y ✿ ❢r♦♠ t❤❡ ♣♦✐♥t (2, 4, 1)✱ ✇❡ ♠❛❦❡ ❛ st❡♣ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ✇✐t❤ t❤❡ s❧♦♣❡ −2✱ ✐✳❡✳✱ ✇❡ ❡♥❞ ✉♣ ❛t (2, 4 + 1, 1 − 2) = (2, 5, −1)✳ ❲❡ ♣❧♦t t❤✐s ❧✐♥❡✳ ❆♥❞✱ ✜♥❛❧❧②✱ ✇❡ ♣❧♦t ❛ ♣❧❛♥❡ t❤r♦✉❣❤ t❤♦s❡ t✇♦ ❧✐♥❡s✳ ●❡♥❡r❛❧❧②✱ t❤✐s ✐s ❤♦✇ ✇❡ ♣❧♦t ❛ ♣❧❛♥❡ ❣✐✈❡♥ ❜② s✉❝❤ ❛♥ ❡q✉❛t✐♦♥✿

z − c = m(x − a) + n(y − b) . ❲❡ st❛rt ❜② ♣❧♦tt✐♥❣ t❤❡ ♣♦✐♥t (a, b, c) ✐♥ R3 ✳ ◆♦✇ ✇❡ tr❡❛t ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✳ ❋✐rst✱ ✇❡ ✜① y ❛♥❞ ❝❤❛♥❣❡ x✳ ❋r♦♠ t❤❡ ♣♦✐♥t (a, b, c)✱ ✇❡ ♠❛❦❡ ❛ st❡♣ ❛❧♦♥❣ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ x✲s❧♦♣❡✱ ✐✳❡✳✱ ✇❡ ❡♥❞ ✉♣ ❛t (a + 1, b, c + m) ♦r (a + 1/m, b, c + 1)✱ ❡t❝✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤✐s ❧✐♥❡ ✐s✿

z − c = m(x − a), y = b . ❙❡❝♦♥❞✱ ✇❡ ✜① x ❛♥❞ ❝❤❛♥❣❡ y ✳ ❲❡ ♠❛❦❡ ❛ st❡♣ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ✇✐t❤ t❤❡ y ✲s❧♦♣❡✱ ✐✳❡✳✱ ✇❡ ❡♥❞ ✉♣ ❛t (a, b + 1, c + n) ♦r (a, b + 1/n, c + 1)✱ ❡t❝✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤✐s ❧✐♥❡ ✐s✿

x = a, z − c = n(y − b) . ❚❤❡s❡ t❤r❡❡ ♣♦✐♥ts ✭♦r t❤♦s❡ t✇♦ ❧✐♥❡s t❤r♦✉❣❤ t❤❡ s❛♠❡ ♣♦✐♥t✮ ❞❡t❡r♠✐♥❡ t❤❡ ♣❧❛♥❡✳ ❲❤❛t ✐s ❛ ♣❧❛♥❡ ❛♥②✇❛②❄ ❚❤❡ ♣❧❛♥❡s ✇❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞ s♦ ❢❛r ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ ✭❧✐♥❡❛r✮ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ z = f (x, y) = mx + ny + p . ❚❤❡② ❤❛✈❡ t♦ s❛t✐s❢② t❤❡ ❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s ✕ ❡✈❡♥ t❤❡ t✇♦ xz ✲ ❛♥❞ yz ✲♣❧❛♥❡s ✕ ❛r❡ ❡①❝❧✉❞❡❞✳ ❚❤✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ s✐t✉❛t✐♦♥ ✇✐t❤ ❧✐♥❡s ❛♥❞ t❤❡ ✐♠♣♦ss✐❜✐❧✐t② t♦ r❡♣r❡s❡♥t ❛❧❧ ❧✐♥❡s ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠ y = mx + p ❛♥❞ ✇❡ ♥❡❡❞ t❤❡ ❣❡♥❡r❛❧ ✭✐♠♣❧✐❝✐t✮ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ R2 ✿

m(x − a) + n(y − b) = 0 . ❚❤❡ ❣❡♥❡r❛❧ ✭✐♠♣❧✐❝✐t✮ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✐♥ R3 ✐s✿

m(x − a) + n(y − b) + k(z − c) = 0 .

✸✳✷✳ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ❧✐♥❡s ✐♥

R2

❛♥❞ ♣❧❛♥❡s ✐♥

R3

✷✸✼

▲❡t✬s t❛❦❡ ❛ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥✿

m · (x − a)+ n k

⇐⇒

· (y − b)+

· (z − c) = 0

    x−a m      n  · y − b = 0 .     z−c k

❚❤✐s ✐s t❤❡ ❞♦t ♣r♦❞✉❝t ✦ ❚❤❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿

< m, n, k > · < x − a, y − b, z − c >= 0 , ♦r ❡✈❡♥ ❜❡tt❡r✿



< m, n, k > · (x, y, z) − (a, b, c)



= 0.

❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ❧❛st ✈❡❝t♦r ❛s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡

(x, y, z)✳

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ❝♦♦r❞✐♥❛t❡✲❢r❡❡ ✈❡rs✐♦♥ ♦❢ ♦✉r ❡q✉❛t✐♦♥✿

N · (P − P0 ) = 0

♦r

N · P0 P = 0

■t ✐s ❡①❛❝t❧② t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ❢♦r t❤❡ ❧✐♥❡✦ ❍❡r❡ ✇❡ ❤❛✈❡ ✐♥

• P

R3 ✿

✐s t❤❡ ✈❛r✐❛❜❧❡ ♣♦✐♥t✳

• P0

✐s t❤❡ ✜①❡❞ ♣♦✐♥t✳

• N

✐s ❛♥② ✈❡❝t♦r t❤❛t s♦♠❡❤♦✇ r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡ ♦❢ t❤❡ ♣❧❛♥❡✳

❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿

❚❤❡s❡ ✈❡❝t♦rs ❛r❡ ❧✐❦❡ ❜✐❝②❝❧❡✬s s♣♦❦❡s t♦ t❤❡ ❤✉❜

N✳

❚❤✐s ✐❞❡❛ ❣✐✈❡s ✉s ♦✉r ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✸✳✷✳✸✿ ♣❧❛♥❡ t❤r♦✉❣❤ ♣♦✐♥t ✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r ❙✉♣♣♦s❡ ❛ ♣♦✐♥t

P0

P0

❛♥❞ ❛ ♥♦♥✲③❡r♦ ✈❡❝t♦r

✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r

N

N

❛r❡ ❣✐✈❡♥✳ ❚❤❡♥ t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤

✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ♣♦✐♥ts

P

✕ ❛♥❞

P0

✐ts❡❧❢ ✕ t❤❛t

s❛t✐s❢②✿

P0 P ⊥ N . ❚❤✐s ❞❡✜♥✐t✐♦♥ ❣✐✈❡s ✉s ❞✐✛❡r❡♥t r❡s✉❧ts ✐♥ ❞✐✛❡r❡♥t ❞✐♠❡♥s✐♦♥s✿ ❞✐♠❡♥s✐♦♥

❛♠❜✐❡♥t s♣❛❝❡

✏❤②♣❡r♣❧❛♥❡✑

2

R2

R1

❧✐♥❡

3

R3

R2

♣❧❛♥❡

4

R4

R3

✕

...

...

...

...

❆ ❤②♣❡r♣❧❛♥❡ ✐s s♦♠❡t❤✐♥❣ ✈❡r② ✏t❤✐♥✑ r❡❧❛t✐✈❡ t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❜✉t ♥♦t ❛s t❤✐♥ ❛s✱ s❛②✱ ❛ ❝✉r✈❡✳

✸✳✸✳

✷✸✽

❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥

❚❤✐s ✇✐❞❡ ❛♣♣❧✐❝❛❜✐❧✐t② s❤♦✇s t❤❛t ❧❡❛r♥✐♥❣ t❤❡ ❞♦t ♣r♦❞✉❝t r❡❛❧❧② ♣❛②s ♦✛✦ ❲❡ ✇✐❧❧ ♥❡❡❞ t❤✐s ❢♦r♠✉❧❛ t♦ st✉❞② ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s ❧❛t❡r✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♥♦♥✲✈❡rt✐❝❛❧ ♣❧❛♥❡s✳ ❲❤❛t ♠❛❦❡s ❛ ♣❧❛♥❡ ♥♦♥✲✈❡rt✐❝❛❧❄ ❆ ♥♦♥✲③❡r♦ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ♥♦r♠❛❧ ✈❡❝t♦r✳ ❙✐♥❝❡ ❧❡♥❣t❤ ❞♦♥✬t ♠❛tt❡r ❤❡r❡ ✭♦♥❧② t❤❡ ❛♥❣❧❡s✮✱ ✇❡ ❝❛♥ s✐♠♣❧② ❛ss✉♠❡ t❤❛t t❤✐s ❝♦♠♣♦♥❡♥t ✐s ❡q✉❛❧ t♦ ♦♥❡✿

N =< m, n, 1 > . ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ s✐♠♣❧✐✜❡s✿

0 = N · (P − P0 ) =< m, n, 1 > · < x − a, y − b, z − c >= m(x − a) + n(y − b) + z − c , ♦r t❤❡ ❢❛♠✐❧✐❛r

z = c + m(x − a) + n(y − b) .

■♥ t❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥✱ ✇❡ ❤❛✈❡✿

z = c + M · (Q − Q0 ) .

■♥ ❝❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2 ✇❡ ❤❛✈❡ ❤❡r❡✿

• Q = (x, y) ✐s t❤❡ ✈❛r✐❛❜❧❡ ♣♦✐♥t✳ • Q0 = (a, b) ✐s t❤❡ ✜①❡❞ ♣♦✐♥t✳

• M =< m, n > ✐s t❤❡ ✈❡❝t♦r t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✇❤✐❝❤ ❛r❡ t❤❡ t✇♦ s❧♦♣❡s ♦❢ t❤❡ ♣❧❛♥❡✳

❚❤✐s ✐s ❛

❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✿

z = f (x, y) = c + m(x − a) + n(y − b) = p + mx + ny , ❛♥❞ M =< m, n > ✐s ❝❛❧❧❡❞ t❤❡

❣r❛❞✐❡♥t

♦❢ f ✳

✸✳✸✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥

❲❤❛t ♠❛❦❡s ❛ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r ❄ ❚❤❡ ✐♠♣❧✐❝✐t ❛♥s✇❡r ❤❛s ❜❡❡♥✿ t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ ♦♥❧② ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t ❛♥❞ ❛❞❞❡❞ t♦ ❛ ❝♦♥st❛♥t✳ ❚❤❡ ✜rst ♣❛rt ♦❢ t❤❡ ❛♥s✇❡r st✐❧❧ ❛♣♣❧✐❡s✱ ❡✈❡♥ t♦ ❢✉♥❝t✐♦♥s ♦❢ ♠❛♥② ✈❛r✐❛❜❧❡s ❛s ✐t ♣r♦❤✐❜✐ts ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✳ ❨♦✉ ❝❛♥ st✐❧❧ ❛❞❞ t❤❡♠✳ ❚❤❡ ♥♦♥✲❧✐♥❡❛r✐t② ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐s s❡❡♥ ❛s s♦♦♥ ❛s ✐t ✐s ♣❧♦tt❡❞ ✕ ✐t✬s ♥♦t ❛ ♣❧❛♥❡ ✕ ❜✉t ✐t ❛❧s♦ s✉✣❝❡s t♦ ❧✐♠✐t ✐ts ❞♦♠❛✐♥✳ ❙♦✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s♥✬t ❧✐♥❡❛r✿

f (x, y) = xy . ■♥ ❢❛❝t✱ xy = x1 y 1 ✐s s❡❡♥ ❛s q✉❛❞r❛t✐❝ ✐❢ ✇❡ ❛❞❞ t❤❡ ♣♦✇❡rs ♦❢ x ❛♥❞ y ✿ 1 + 1 = 2✳ ❲❡ ❝♦♠❡ t♦ t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ✇❤❡♥ ✇❡ ❧✐♠✐t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡ ❧✐♥❡ y = x ✐♥ t❤❡ xy ✲♣❧❛♥❡❀ ✉s✐♥❣ y = x s ❛ s✉❜st✐t✉t✐♦♥ ✇❡ ❛rr✐✈❡ t♦ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✿

g(x) = f (x, x) = x · x = x2 . ❙♦✱ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f t❤❛t ❧✐❡s ❡①❛❝t❧② ❛❜♦✈❡ t❤❡ ❧✐♥❡ y = x ✐s ❛ ❧✐❡s ❛❜♦✈❡ t❤❡ ❧✐♥❡ y = −x❀ ✐t✬s ❥✉st ♦♣❡♥ ❞♦✇♥ ✐♥st❡❛❞ ♦❢ ✉♣✿

♣❛r❛❜♦❧❛✳

❆♥❞ s♦ ✐s t❤❡ ♣❛rt t❤❛t

h(x) = f (x, −x) = x · (−x) = −x2 . ❚❤❡s❡ t✇♦ ♣❛r❛❜♦❧❛s ❤❛✈❡ ❛ s✐♥❣❧❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥ ❛♥❞ t❤❡r❡❢♦r❡ ♠❛❦❡ ✉♣ t❤❡ ❡ss❡♥t✐❛❧ ♣❛rt ♦❢ ❛ ♣♦✐♥t ✿

s❛❞❞❧❡

✸✳✸✳

✷✸✾

❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥

❚❤❡ ❢♦r♠❡r ♣❛r❛❜♦❧❛ ❣✐✈❡s r♦♦♠ ❢♦r t❤❡ ❤♦rs❡✬s ❢r♦♥t ❛♥❞ ❜❛❝❦ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ❢♦r t❤❡ ❤♦rs❡♠❛♥✬s ❧❡❣s✳ ❆ s✐♠♣❧❡r ✇❛② t♦ ❧✐♠✐t t❤❡ ❞♦♠❛✐♥ ✐s t♦ ✜① ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✳ ❲❡ ✜① y ✜rst✿ ♣❧❛♥❡

❡q✉❛t✐♦♥ ❝✉r✈❡

y=2

z = x · 2 ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 2

y=1 y=0 y = −1

y = −2

z = x · 1 ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 1

z = x · 0 = 0 ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 0 z = x · (−1) ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 1

z = x · (−2) ❧✐♥❡ ✇✐t❤ s❧♦♣❡ − 2

❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✿

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ x✲❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲❝♦❧✉♠♥s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ❡❛st✲✇❡st ❞✐r❡❝t✐♦♥✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✳ ❲❡ ✜① x s❡❝♦♥❞✿

♣❧❛♥❡

❡q✉❛t✐♦♥ ❝✉r✈❡

x=2

z = 2 · y ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 2

x=1 x=0 x = −1

x = −2

z = 1 · y ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 1

z = 0 · y = 0 ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 0 z = (−1) · y ❧✐♥❡ ✇✐t❤ s❧♦♣❡ 1

z = (−2) · y ❧✐♥❡ ✇✐t❤ s❧♦♣❡ − 2

❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ ❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✿

✸✳✸✳

✷✹✵

❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ y ✲r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲r♦✇s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ♥♦rt❤✲s♦✉t❤ ❞✐r❡❝t✐♦♥✳ ❚❤✉s t❤❡ s✉r❢❛❝❡ ♦❢ t❤✐s ❣r❛♣❤ ✐s ♠❛❞❡ ♦❢ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s✦ ■t✬s ❛ ♣♦t❛t♦ ❝❤✐♣✿

❆♥♦t❤❡r ✇❛② t♦ ❛♥❛❧②③❡ t❤❡ ❣r❛♣❤ ✐s t♦ ❧✐♠✐t t❤❡ r❛♥❣❡ ✐♥st❡❛❞ ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❲❡ ✜① t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ t❤✐s t✐♠❡✿ ❡❧❡✈❛t✐♦♥ ❡q✉❛t✐♦♥ ❝✉r✈❡ z=2 z=1 z=0 z = −1

❚❤❡ r❡s✉❧t ✐s ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s✿

z = −2

2 = x · y ❤②♣❡r❜♦❧❛ 1 = x · y ❤②♣❡r❜♦❧❛

0 = x · y t❤❡ t✇♦ ❛①❡s

−1 = x · y ❤②♣❡r❜♦❧❛

−2 = x · y ❤②♣❡r❜♦❧❛

❊❛❝❤ ✐s ❧❛❜❡❧❧❡❞ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ z ✱ t✇♦ ❜r❛♥❝❤❡s ❢♦r ❡❛❝❤✳ ❚❤❡s❡ ❧✐♥❡s ❛r❡ t❤❡ ❧✐♥❡s ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥✳

♦❢ ❡q✉❛❧

❲❡ ❝❛♥ s❡❡ ✇❤♦ t❤❡s❡ ❧✐♥❡s ❝♦♠❡ ❢r♦♠ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✭♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✮✳

✸✳✹✳

●r❛♣❤s

✷✹✶

❲❡ ❝❛♥ ✉s❡ t❤❡♠ t♦ r❡❛ss❡♠❜❧❡ t❤❡ s✉r❢❛❝❡ ❜② ❧✐❢t✐♥❣ ❡❛❝❤ t♦ t❤❡ ❡❧❡✈❛t✐♦♥ ✐♥❞✐❝❛t❡❞ ❜② ✐ts ❧❛❜❡❧✿

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❝♦❧♦r❡❞ ♣❛rts ♦❢ t❤❡ ❣r❛♣❤ ❝♦rr❡s♣♦♥❞ t♦ ✐♥t❡r✈❛❧s ♦❢ ♦✉t♣✉ts✳ ❚❤✐s s✉r❢❛❝❡ ♣r❡s❡♥t❡❞ ❤❡r❡ ✐s ❝❛❧❧❡❞ t❤❡ ❤②♣❡r❜♦❧✐❝

♣❛r❛❜♦❧♦✐❞✳

✸✳✹✳ ●r❛♣❤s

❘❡❝❛❧❧ t❤❡ ✈❡r② ❜❛s✐❝✱ ❜✉t ✈❡r② ❣❡♥❡r❛❧✱ ❞❡✜♥✐t✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✶✿ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ z = f (x) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ✐♥ R2 ♦❢ t❤❡ ❢♦r♠ (x, f (x))✳ ■♥ s♣✐t❡ ♦❢ ❛ ❢❡✇ ❡①❝❡♣t✐♦♥s✱ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❤❛✈❡ ❜❡❡♥ ❝✉r✈❡s✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✷✿ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (x, y) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ✐♥ R3 ♦❢ t❤❡ ❢♦r♠ (x, y, f (x, y))✳ ■♥ s♣✐t❡ ♦❢ ♣♦ss✐❜❧❡ ❡①❝❡♣t✐♦♥s✱ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇❡ ❡♥❝♦✉♥t❡r ✇✐❧❧ ♣r♦❜❛❜❧② ❜❡ s✉r❢❛❝❡s✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ r❡♠❡♠❜❡r t❤❛t t❤❡ t❤❡♦r② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐♥❝❧✉❞❡ t❤❛t ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✦ ❚❤❡ ❢♦r♠✉❧❛ ❢♦r f ✱ ❢♦r ❡①❛♠♣❧❡✱ ♠✐❣❤t ❤❛✈❡ ♥♦ ♠❡♥t✐♦♥ ♦❢ y s✉❝❤ ❛s ❢♦r ❡①❛♠♣❧❡ z = f (x, y) = x3 ✳ ❚❤❡ ❣r❛♣❤ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❜❡ ❢❡❛t✉r❡✲❧❡ss✱ ✐✳❡✳✱ ❝♦♥st❛♥t✱ ✐♥ t❤❡ y ✲❞✐r❡❝t✐♦♥✳ ■t ✇✐❧❧ ❧♦♦❦ ❛s ✐❢ ♠❛❞❡ ♦❢ ♣❧❛♥❦s ❧✐❦❡ ❛ ♣❛r❦ ❜❡♥❝❤✿

✸✳✹✳

●r❛♣❤s

✷✹✷

■♥ ❢❛❝t✱ t❤❡ ❣r❛♣❤ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ z = x2 ✭t❤❡ ❝✉r✈❡✮ ❜② s❤✐❢t✐♥❣ ✐t ✐♥ t❤❡ y ✲❞✐r❡❝t✐♦♥ ♣r♦❞✉❝✐♥❣ t❤❡ s✉r❢❛❝❡✿

■♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇❡ ✜①❡❞ ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡ ♠❛❦✐♥❣ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ s✉❜❥❡❝t t♦ t❤❡ ❢❛♠✐❧✐❛r ♠❡t❤♦❞s✳ ❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✸✿ ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡ z = f (x1 , ..., xn ) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✱ ✐✳❡✳✱ x1 , ..., xn ❛♥❞ z ❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡♥ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ k = 1, 2, ..., n ❛♥❞ ❛♥② ❝♦❧❧❡❝t✐♦♥ ♦❢ ♥✉♠❜❡rs xi = ai , i 6= k ✱ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜②✿ z = h(x) = f (a1 , ..., ak−1 , x, ak+1 , ..., an ) ,

✐s ❝❛❧❧❡❞ t❤❡ k t❤ ✈❛r✐❛❜❧❡ ✈❛r✐❛❜❧❡ ❝✉r✈❡✳

❢✉♥❝t✐♦♥ ♦❢ f ✳ ❲❤❡♥ n = 2✱ ✐ts ❣r❛♣❤ ✐s ❝❛❧❧❡❞ t❤❡ kt❤

❲❡ t❤✉s ✜① ❛❧❧ ✈❛r✐❛❜❧❡s ❜✉t ♦♥❡ ♠❛❦✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ✈❛r✐❛❜❧❡✳

❊①❡r❝✐s❡ ✸✳✹✳✹ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ n = 1❄ ▼❡❛♥✇❤✐❧❡✱ ✜①✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❞♦❡s♥✬t ❤❛✈❡ t♦ ♣r♦❞✉❝❡ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡s✉❧t ✐s ✐♥st❡❛❞ ❛♥ ✐♠♣❧✐❝✐t r❡❧❛t✐♦♥✳ ❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s t♦♦✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✺✿ ❧❡✈❡❧ s❡t ❙✉♣♣♦s❡ z = f (X) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✱ ✐✳❡✳✱ X ❜❡❧♦♥❣s t♦ s♦♠❡ Rn ❛♥❞ z ✐s ❛ r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ z = c✱ t❤❡ s✉❜s❡t ♦❢ Rn ✐s ❝❛❧❧❡❞ ❛ ❧❡✈❡❧

s❡t ♦❢ f ✳

{X : f (X) = c}

✸✳✹✳

●r❛♣❤s

✷✹✸

❲❤❡♥ n = 2✱ ✐t ✐s ✐♥❢♦r♠❛❧❧② ❝❛❧❧❡❞ ❛ ❧❡✈❡❧

❝✉r✈❡ ♦r ❛ ❝♦♥t♦✉r ❝✉r✈❡✳

❊①❡r❝✐s❡ ✸✳✹✳✻

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ n = 1❄ ■♥ ❣❡♥❡r❛❧✱ ❛ ❧❡✈❡❧ s❡t ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ❛ ❝✉r✈❡ ❛s t❤❡ ❡①❛♠♣❧❡ ♦❢ f (x, y) = 1 s❤♦✇s✳ ❚❤❡♦r❡♠ ✸✳✹✳✼✿ ▲❡✈❡❧ ❙❡ts ▲❡✈❡❧ s❡ts ❞♦♥✬t ✐♥t❡rs❡❝t✳

Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❱❡rt✐❝❛❧

▲✐♥❡ ❚❡st✳

❊①❡r❝✐s❡ ✸✳✹✳✽

Pr♦✈✐❞❡ t❤❡ ♣r♦♦❢✳ ❊①❛♠♣❧❡ ✸✳✹✳✾✿ ❧❡✈❡❧ ❝✉r✈❡s

▲❡t✬s ❝♦♥s✐❞❡r✿ f (x, y) =

p x2 + y 2 .

❚❤❡ ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ✐♠♣❧✐❝✐t ❝✉r✈❡s ❛♥❞ t❤❡② ❛r❡ ♣❧♦tt❡❞ ♣♦✐♥t ❜② ♣♦✐♥t✿

✳✳✳✉♥❧❡ss ✇❡ r❡❝♦❣♥✐③❡ t❤❡♠✿ c=

p x2 + y 2 =⇒ c2 = x2 + y 2 .

❚❤❡② ❛r❡ ❝✐r❝❧❡s✦ ❚❤✐s s✉r❢❛❝❡ ✐s ❛ ❤❛❧❢ ♦❢ ❛ ❝♦♥❡✳ ❍♦✇ ❞♦ ✇❡ ❦♥♦✇ ✐t✬s ❛ ❝♦♥❡ ❛♥❞ ♥♦t ❛♥♦t❤❡r s❤❛♣❡❄ ❲❡ ❧✐♠✐t t❤❡ ❞♦♠❛✐♥ t♦ t❤✐s ❧✐♥❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡✿ y = mx =⇒ z =

❚❤❡s❡ ❛r❡ ❱✲s❤❛♣❡❞ ❝✉r✈❡s✳

p

x2 + (mx)2 =

√

1 + m · |x| .

❊①❛♠♣❧❡ ✸✳✹✳✶✵✿ ❤②♣❡r❜♦❧❛s

❈♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ f (x, y) = y 2 − x2 .

■ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ❤②♣❡r❜♦❧❛s ❛❣❛✐♥ ❥✉st ❛s ✐♥ t❤❡ ✜rst ❡①❛♠♣❧❡✳

✸✳✹✳

●r❛♣❤s

✷✹✹

❚❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r s❛❞❞❧❡ ♣♦✐♥t ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳ ■ts ✈❛r✐❛❜❧❡ ❝✉r✈❡s ❛r❡ ❞✐✛❡r❡♥t s✐♠♣❧② ❜❡❝❛✉s❡ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❣r❛♣❤ r❡❧❛t✐✈❡ t♦ t❤❡ ❛①❡s ✐s ❞✐✛❡r❡♥t✳ ❊①❛♠♣❧❡ ✸✳✹✳✶✶✿ ❡①tr❡♠❡ ♣♦✐♥t

■❢ ✇❡ r❡♣❧❛❝❡ − ✇✐t❤ +✱ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❜❡❝♦♠❡s f (x, y) = y 2 + x2

✇✐t❤ ❛ ✈❡r② ❞✐✛❡r❡♥t ❣r❛♣❤❀ ✐t ❤❛s ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ✉♣✇❛r❞ t❤✐s t✐♠❡✿

❚❤❡ ❧❡✈❡❧ ❝✉r✈❡s

♣♦✐♥t✳

❚❤❡ ✈❛r✐❛❜❧❡ ❝✉r✈❡s ❛r❡ st✐❧❧ ♣❛r❛❜♦❧❛s ❜✉t ❜♦t❤

y 2 + x2 = c

❛r❡ ❝✐r❝❧❡s ❡①❝❡♣t ✇❤❡♥ c = 0 ✭❛ ♣♦✐♥t✮ ♦r c < 0 ✭❡♠♣t②✮✳ ❚❤❡② ❞♦♥✬t ❣r♦✇ ✉♥✐❢♦r♠❧②✱ ❛s ✇✐t❤ t❤❡ ❝♦♥❡✱ ✇✐t❤ c ❤♦✇❡✈❡r✳ ❚❤❡ s✉r❢❛❝❡ ✐s ❝❛❧❧❡❞ t❤❡ ♣❛r❛❜♦❧♦✐❞ ♦❢ r❡✈♦❧✉t✐♦♥✳ ❖♥❡ ♦❢ t❤❡ ♣❛r❛❜♦❧❛s t❤❛t ♣❛ss❡s t❤r♦✉❣❤ ③❡r♦ ✐s r♦t❛t❡❞ ❛r♦✉♥❞ t❤❡ z ✲❛①✐s ♣r♦❞✉❝✐♥❣ t❤✐s s✉r❢❛❝❡✳ ❚❤❡ ♠❡t❤♦❞ ♦❢ ❧❡✈❡❧ ❝✉r✈❡s ❤❛s ❜❡❡♥ ✉s❡❞ ❢♦r ❝❡♥t✉r✐❡s t♦ ❝r❡❛t❡ ❛❝t✉❛❧ ♠❛♣s✱ ✐✳❡✳✱ 2✲❞✐♠❡♥s✐♦♥❛❧ ✈✐s✉❛❧✐③❛t✐♦♥s ♦❢ 3✲❞✐♠❡♥s✐♦♥❛❧ t❡rr❛✐♥s✿

❚❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❧❡✈❡❧ ❝✉r✈❡s ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥t♦✉r ♠❛♣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t ③♦♦♠✐♥❣ ✐♥ ♦♥ ❛♥② ♣♦✐♥t ♦❢ t❤❡ ❝♦♥t♦✉r ♠❛♣ ✐s ❡✐t❤❡r ❛ ❣❡♥❡r✐❝ ♣♦✐♥t ✇✐t❤ ♣❛r❛❧❧❡❧ ❧❡✈❡❧ ❝✉r✈❡s ♦r ❛ s✐♥❣✉❧❛r ♣♦✐♥t ❡①❡♠♣❧✐✜❡❞ ❜② ❛ s❛❞❞❧❡ ♣♦✐♥t ❛♥❞ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t✳

✸✳✹✳ ●r❛♣❤s

✷✹✺

❚❤❡ s✐♥❣✉❧❛r ♣♦✐♥ts ❛r❡ ❧✐tt❧❡ ✐s❧❛♥❞s ✐♥ t❤❡ s❡❛ ♦❢ ❣❡♥❡r✐❝ ♣♦✐♥ts✳✳✳

❊①❛♠♣❧❡ ✸✳✹✳✶✷✿ s✉❜✲❧❡✈❡❧ s❡t ❙✉♣♣♦s❡

z = f (x, y)

✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤❡♥ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢

z = c✱

t❤❡ s✉❜s❡t

{(x, y) : f (x, y) ≤ c} ♦❢ t❤❡ ♣❧❛♥❡ ✐s ❝❛❧❧❡❞ ❛ s✉❜✲❧❡✈❡❧ s❡t ♦❢

f✳

❚❤❡s❡ s❡ts ❛r❡ ✉s❡❞ t♦ ❝♦♥✈❡rt ❣r❛②✲s❝❛❧❡ ✐♠❛❣❡s t♦ ❜✐♥❛r②✿

❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ✏t❤r❡s❤♦❧❞✐♥❣✑✳

❚❤✉s✱ t❤❡ ✈❛r✐❛❜❧❡ ❝✉r✈❡s ❛r❡ t❤❡ r❡s✉❧t ♦❢ r❡str✐❝t✐♥❣ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✇❤✐❧❡ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ t❤❡ r❡s✉❧t ♦❢ r❡str✐❝t✐♥❣ t❤❡ ✐♠❛❣❡✳ ❊✐t❤❡r ♠❡t❤♦❞ ✐s ❛♣♣❧✐❡❞ ✐♥ ❤♦♣❡ ♦❢ s✐♠♣❧✐❢②✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡ ❞❡❣r❡❡ t❤❛t ✇✐❧❧ ♠❛❦❡ ✐s s✉❜❥❡❝t t♦ t❤❡ t♦♦❧ ✇❡ ❛❧r❡❛❞② ❤❛✈❡✳ ❍♦✇❡✈❡r✱ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✐s ♠❛❞❡ ♦❢ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤♦s❡✳✳✳ ■♥❢♦r♠❛❧❧②✱ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ s❛♠❡✿ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✐♥♣✉ts✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✶✸✿ ♥❛t✉r❛❧ ❞♦♠❛✐♥ ❚❤❡ ♥❛t✉r❛❧ ❞♦♠❛✐♥ ✭♦r ✐♠♣❧✐❡❞ ❞♦♠❛✐♥✮ ♦❢ ❛ ❢✉♥❝t✐♦♥ X ✐♥ Rn ❢♦r ✇❤✐❝❤ f (X) ♠❛❦❡s s❡♥s❡✳

z = f (X)

✐s t❤❡ s❡t ♦❢ ❛❧❧

❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ✐ss✉❡ ✐s t❤❛t ♦❢ ❞✐✈✐s✐♦♥ ❜② ③❡r♦✱ sq✉❛r❡ r♦♦ts✱ ❡t❝✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❝♦♠❡s ❢r♦♠ t❤❡ n ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♣❛❝❡ ♦❢ ✐♥♣✉ts✿ t❤❡ ♣❧❛♥❡ ✭❛♥❞ ❢✉rt❤❡r R ✮ ✈s✳ t❤❡ ❧✐♥❡✳

❊①❛♠♣❧❡ ✸✳✹✳✶✹✿ r❡❝✐♣r♦❝❛❧ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = ✐s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ♠✐♥✉s t❤❡ ❛①❡s✿

1 xy

✸✳✹✳

●r❛♣❤s

✷✹✻

❊①❛♠♣❧❡ ✸✳✹✳✶✺✿ r♦♦t ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

√

✐s t❤❡ ❤❛❧❢ ♦❢ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜② t❤❡ ✐♥❡q✉❛❧✐t② y ≤ x✿

x−y

❊①❛♠♣❧❡ ✸✳✹✳✶✻✿ ❣r❛✈✐t② ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡ ✐s ❛❧❧ ♣♦✐♥ts ❜✉t 0✿

■t ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

f (X) =

1 . d(X, 0)

❊①❡r❝✐s❡ ✸✳✹✳✶✼ Pr♦✈✐❞❡ t❤❡ ❧❡✈❡❧ ❛♥❞ t❤❡ ✈❛r✐❛❜❧❡ ❝✉r✈❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛✈✐✲ t❛t✐♦♥❛❧ ❢♦r❝❡✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✶✽✿ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (X)✱ ✇❤❡r❡ X ❜❡❧♦♥❣s t♦ Rn ✱ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ✐♥ Rn+1 s♦ t❤❛t t❤❡ ✜rst n ❝♦♦r❞✐♥❛t❡s ❛r❡ t❤♦s❡ ♦❢ X ❛♥❞ t❤❡ ❧❛st ✐s f (X)✳ ❲❡ ❤❛✈❡ ✉s❡❞ t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ✐♠❛❣❡ ❛♥❞ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s t❤❛t ❝♦♠❡ ❢r♦♠ ✐t ❛s ❛ t♦♦❧ ♦❢ r❡❞✉❝t✐♦♥✳ ❲❡ r❡❞✉❝❡ t❤❡ st✉❞② ♦❢ ❛ ❝♦♠♣❧❡① ♦❜❥❡❝t ✕ t❤❡ ❣r❛♣❤ ♦❢ z = f (x, y) ✕ t♦ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ s✐♠♣❧❡r ♦♥❡s ✕ ✐♠♣❧✐❝✐t ❝✉r✈❡s c = f (x, y)✳ ❚❤❡ ✐❞❡❛ ✐s ❡✈❡♥ ♠♦r❡ ✉s❡❢✉❧ ✐♥ t❤❡ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥✬t s✐♠♣❧② ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s ❛♥②♠♦r❡ ✕ ✐t ✐s ❧♦❝❛t❡❞ ✐♥ R4 ✕ ❛s ✇❡ ❞✐❞ ❢♦r ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❚❤❡ ❧❡✈❡❧ s❡ts ✕ ❧❡✈❡❧ s✉r❢❛❝❡s ✕ ✐s t❤❡ ❜❡st ✇❛② t♦ ✈✐s✉❛❧✐③❡ ✐t✳ ❲❡ ❝❛♥ ❛❧s♦ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ✐♥st❡❛❞ ❜② ✜①✐♥❣ ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡ ❛♥❞ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❚❤❡ ❞♦♠❛✐♥s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥ ♦✉r ♣❤②s✐❝❛❧ s♣❛❝❡✳ ❚❤❡② ♠❛② r❡♣r❡s❡♥t✿

✸✳✹✳

●r❛♣❤s

✷✹✼

• t❤❡ t❡♠♣❡r❛t✉r❡ ♦r t❤❡ ❤✉♠✐❞✐t②✱ • t❤❡ ❛✐r ♦r ✇❛t❡r ♣r❡ss✉r❡✱

• t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ❢♦r❝❡ ✭s✉❝❤ ❛s ❣r❛✈✐t❛t✐♦♥✮✱ ❡t❝✳ ❊①❛♠♣❧❡ ✸✳✹✳✶✾✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥

❚❤❡ ❧❡✈❡❧ s❡ts ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ f (x, y, z) = A + mx + ny + kz .

❛r❡ ♣❧❛♥❡s✿ d = A + mx + ny + kz .

❚❤❡s❡ ♣❧❛♥❡s ❧♦❝❛t❡❞ ✐♥ xyz ✲s♣❛❝❡ ❛r❡ ❛❧❧ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♥♦r♠❛❧ ✈❡❝t♦r < m, n, k >✳ ❚❤❡ ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❛r❡♥✬t t❤❛t ❞✐✛❡r❡♥t❀ ❧❡t✬s ✜① z = c✿ d = f (x, y, c) = h(x, y) = A + mx + ny + kc .

❋♦r ❛❧❧ ✈❛❧✉❡s ♦❢ c✱ t❤❡s❡ ♣❧❛♥❡s ❧♦❝❛t❡❞ ✐♥ xyu✲s♣❛❝❡ ❛r❡ ❛❧s♦ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♥♦r♠❛❧ ✈❡❝t♦r < m, n, 1 >✳ ❊①❛♠♣❧❡ ✸✳✹✳✷✵✿ ❧❡✈❡❧ s❡ts

❲❡ st❛rt ✇✐t❤ ❛ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ f (x, y) = sin(xy) .

❛♥❞ ❥✉st s✉❜tr❛❝t z ❛s t❤❡ t❤✐r❞ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ h(x, y, z) = sin(xy) − z .

❚❤❡♥ ❡✈❡r② ♦❢ ✐ts ❧❡✈❡❧ s❡ts ✐s ❣✐✈❡♥ ❜②✿ d = sin(xy) − z ,

❢♦r s♦♠❡ r❡❛❧ d✳ ❲❤❛t ✐s t❤✐s❄ ❲❡ ❝❛♥ ♠❛❦❡ t❤❡ r❡❧❛t✐♦♥ ❡①♣❧✐❝✐t✿ z = sin(xy) − d ,

◆♦t❤✐♥❣ ❜✉t t❤❡ ❣r❛♣❤ ♦❢ f s❤✐❢t❡❞ ❞♦✇♥ ✭❛♥❞ ✉♣✮ ❜② d✿

■♥ t❤✐s s❡♥s❡✱ t❤❡② ❛r❡ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❣r♦✇✐♥❣ ❛s ✇❡ ♠♦✈❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ z ✳ ◆♦✇✱ ✐❢ ✇❡ ✜① ❛♥ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦❢ h✱ s❛② z = c✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ g(x, y) = sin(xy) − c .

❚❤❡ ❣r❛♣❤s ❛r❡ t❤❡ s❛♠❡✳

✸✳✺✳

▲✐♠✐ts

✷✹✽

❊①❛♠♣❧❡ ✸✳✹✳✷✶✿ t❤r❡❡ ✈❛r✐❛❜❧❡s

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✿

f (x, y, z) =

p x2 + y 2 + z 2 .

❚❤❡♥ ❡✈❡r② ♦❢ ✐ts ❧❡✈❡❧ s❡ts ✐s ❣✐✈❡♥ ❜② t❤✐s ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥✿

d 2 = x2 + y 2 + z 2 , ❢♦r s♦♠❡ r❡❛❧ d✳ ❊❛❝❤ ✐s ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ ♦❢ r❛❞✐✉s |d| ❝❡♥t❡r❡❞ ❛t 0 ✭❛♥❞ t❤❡ ♦r✐❣✐♥ ✐ts❡❧❢ ✇❤❡♥ d = 0✮✳ ❚❤❡② ❛r❡ ❝♦♥❝❡♥tr✐❝✿

❚❤❡ r❛❞✐✐ ❛❧s♦ ❣r♦✇ ✉♥✐❢♦r♠❧② ✇✐t❤ d ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡s❡ s✉r❢❛❝❡s ❛r❡ ♥♦t ❛r❡ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❙♦✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣r♦✇✐♥❣ ✕ ❛♥❞ ❛t ❛ ❝♦♥st❛♥t r❛t❡ ✕ ❛s ✇❡ ♠♦✈❡ ✐♥ ❛♥② ❞✐r❡❝t✐♦♥ ❛✇❛② ❢r♦♠ 0✳ ❲❤❛t ✐s t❤✐s ❢✉♥❝t✐♦♥❄ ■t✬s t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ✐ts❡❧❢✿ p f (X) = f (x, y, z) = x2 + y 2 + z 2 = ||X|| . ❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡ ❛❧s♦ ❤❛s ❝♦♥❝❡♥tr✐❝ s♣❤❡r❡s ❛r❡ ❧❡✈❡❧ s✉r❢❛❝❡s ❜✉t t❤❡ r❛❞✐✐ ❞♦ ♥♦t ❝❤❛♥❣❡ ✉♥✐❢♦r♠❧② ✇✐t❤ d✳ ❇♦t❤ ❛r❡ ✈✐s✉❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✿

❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡❝r❡❛s✐♥❣ ❛s ✇❡ ♠♦✈❡ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✳ ❚❤✐s ✐s ❜❡st ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✳ ❇❡②♦♥❞ 3 ✈❛r✐❛❜❧❡s✱ ✇❡ ❛r❡ ♦✉t ♦❢ ❞✐♠❡♥s✐♦♥s✳✳✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✈✐❛ ✜①✐♥❣ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ✐s ❝♦♥s✐❞❡r❡❞ ❧❛t❡r✳

✸✳✺✳ ▲✐♠✐ts

❲❡ ♥♦✇ st✉❞② s♠❛❧❧ s❝❛❧❡ ❜❡❤❛✈✐♦r ♦❢ ❢✉♥❝t✐♦♥s❀ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛ s✐♥❣❧❡ ♣♦✐♥t ♦❢ t❤❡ ❣r❛♣❤✳

✸✳✺✳

▲✐♠✐ts

✷✹✾

❏✉st ❛s ✐♥ t❤❡ ❧♦✇❡r ❞✐♠❡♥s✐♦♥s✱ ♦♥❡ ♦❢ t❤❡ ♠♦st ❝r✉❝✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ✐♥t❡❣r✐t② ♦❢ ✐ts ❣r❛♣❤✿

✐s t❤❡r❡ ❛ ❜r❡❛❦ ♦r ❛ ❝✉t ♦r ❛ ❤♦❧❡❄

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ❣r❛♣❤ ❛s ❛ t❡rr❛✐♥✱ ✐s t❤❡r❡ ❛ ✈❡rt✐❝❛❧

❞r♦♣❄

❲❡ ❛♣♣r♦❛❝❤ t❤❡ ✐ss✉❡ ✈✐❛ t❤❡

❧✐♠✐ts✳

■♥ s♣✐t❡ ♦❢ ❛❧❧ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ ✕ s✉❝❤ ❛s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✈s✳ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✕ t❤❡ ✐❞❡❛ ♦❢ ❧✐♠✐t ✐s ✐❞❡♥t✐❝❛❧✿ ❛s t❤❡ ✐♥♣✉t ❛♣♣r♦❛❝❤❡s ❛ ♣♦✐♥t✱ t❤❡ ♦✉t♣✉t ✐s ❛❧s♦ ❢♦r❝❡❞ t♦ ❛♣♣r♦❛❝❤ ❛ ♣♦✐♥t✳ ❚❤✐s ✐s t❤❡ ❝♦♥t❡①t ✇✐t❤ t❤❡ ❛rr♦✇ ✏ →✑ t♦ ❜❡ r❡❛❞ ❛s ✏❛♣♣r♦❛❝❤❡s✑✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

y = f (x) → l

❛s

x→a

♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

Y = F (t) → L

❛s

❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

t→a

z = f (X) → l

❛s

X→A

❲❡ ✉s❡ ❧♦✇❡r ❝❛s❡ ❢♦r s❝❛❧❛rs ❛♥❞ ✉♣♣❡r ❝❛s❡ ❢♦r ❛♥②t❤✐♥❣ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ✭♣♦✐♥t ♦r ✈❡❝t♦rs✮✳ ❲❡ t❤❡♥ s❡❡ ❤♦✇ t❤❡ ❝♦♠♣❧❡①✐t② s❤✐❢ts ❢r♦♠ t❤❡ ♦✉t♣✉t t♦ t❤❡ ✐♥♣✉t✳ ❆♥❞ s♦ ❞♦❡s t❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✐t②✳ ❲❡ ❛r❡ r❡❛❞② ❢♦r t❤✐s ❝❤❛❧❧❡♥❣❡❀ t❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r ♠❡❛♥✐♥❣ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐♥

Rm

t♦ ❜❡ ✉s❡❞

t❤r♦✉❣❤♦✉t✿

Xn → A ⇐⇒ d(Xn , A) → 0,

♦r

❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ ✐s ❛s ❡①♣❡❝t❡❞✿

||Xn − A|| → 0 .

❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛❧♠♦st ❛♥ ❡①❛❝t ❝♦♣② ♦❢ ✇❤❛t ✇❡ ✉s❡❞ t♦ ❤❛✈❡✿

❉❡✜♥✐t✐♦♥ ✸✳✺✳✶✿ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❚❤❡

❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (A)

❛t ❛ ♣♦✐♥t

X=A

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t

lim f (Xn )

n→∞ ❝♦♥s✐❞❡r❡❞ ❢♦r ❛❧❧ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡ t♦

A✱

{Xn }

✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢

A 6= Xn → A

❛s

f

❡①❝❧✉❞✐♥❣

A

t❤❛t

n → ∞,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ✉s❡ t❤❡

✸✳✺✳ ▲✐♠✐ts

✷✺✵ ♥♦t❛t✐♦♥✿ lim f (X) .

X→A

❖t❤❡r✇✐s❡✱ t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳ ❲❡ ✉s❡ t❤✐s ❝♦♥str✉❝t✐♦♥ t♦ st✉❞② ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦ z = f (X) ✇❤❡♥ ❛ ♣♦✐♥t X ✐s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❝❤♦s❡♥ ♣♦✐♥t X = A✱ ✇❤❡r❡ f ♠✐❣❤t ❜❡ ✉♥❞❡✜♥❡❞✳ ❲❡ st❛rt ✇✐t❤ ❛♥ ❛r❜✐tr❛r② s❡q✉❡♥❝❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡ t❤❛t ❝♦♥✈❡r❣❡s t♦ t❤✐s ♣♦✐♥t✱ Xn → A✱ t❤❡♥ ❣♦ ✈❡rt✐❝❛❧❧② ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡s❡ ♣♦✐♥t t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ (Xn , f (Xn ))✱ ❛♥❞ ✜♥❛❧❧② ♣❧♦t t❤❡ ♦✉t♣✉t ✈❛❧✉❡s ✭❥✉st ♥✉♠❜❡rs✮ ♦♥ t❤❡ z ✲❛①✐s✱ zn = f (Xn )✳

■s t❤❡r❡ ❛ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡❄ ❲❤❛t ❛❜♦✉t ♦t❤❡r s❡q✉❡♥❝❡s❄ ❆r❡ ❛❧❧ t❤❡s❡ ❧✐♠✐ts t❤❡ s❛♠❡❄ ❚❤❡ ❛❜✐❧✐t② t♦ ❛♣♣r♦❛❝❤ t❤❡ ♣♦✐♥t ❢r♦♠ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s ❤❛❞ t♦ ❜❡ ❞❡❛❧t ✇✐t❤ ❡✈❡♥ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿ sign(x) → −1 ❛s x → 0− ✱ ❜✉t sign(x) → 1 ❛s x → 0+ . ■♥ t❤❡ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ t❤✐♥❣s ❛r❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❛s ✇❡ ❝❛♥ ❛♣♣r♦❛❝❤ ❛ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❢r♦♠ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s✳

❊①❛♠♣❧❡ ✸✳✺✳✷✿ ♥♦ s✐♥❣❧❡ ♣❧❛♥❡

■t ✐s ❡❛s② t♦ ❝♦♠❡ ✉♣ ✇✐t❤ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❤❛s ❞✐✛❡r❡♥t ❧✐♠✐ts ❢r♦♠ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s✳ ❲❡ t❛❦❡ ♦♥❡ t❤❛t ✇✐❧❧ ❜❡ ✐♠♣♦rt❛♥t ✐♥ t❤❡ ❢✉t✉r❡✿ |2x + y| p . (x,y)→(0,0) x2 + y 2 lim

✸✳✺✳

▲✐♠✐ts

✷✺✶

❇② t❤❡ ✇❛②✱ t❤✐s ✐s ❤♦✇ ♦♥❡ ♠✐❣❤t tr② t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ✏s❧♦♣❡✑ ✭r✐s❡ ♦✈❡r t❤❡ r✉♥✮ ❛t

z = 2x + y ✳

❋✐rst✱ ✇❡ ❛♣♣r♦❛❝❤ ❛❧♦♥❣ t❤❡

x✲❛①✐s✱

✐✳❡✳✱

y

✐s ✜①❡❞ ❛t

0

♦❢ t❤❡ ♣❧❛♥❡

0✿

|2x + 0| |2x| lim √ = lim = 2. 2 2 x→0 x→0 |x| x +0 ❙❡❝♦♥❞✱ ✇❡ ❛♣♣r♦❛❝❤ ❛❧♦♥❣ t❤❡

y ✲❛①✐s✱

✐✳❡✳✱

x

✐s ✜①❡❞ ❛t

0✿

|2 · 0 + y| |y| = 1. lim p = lim y→0 02 + y 2 y→0 |y|

❚❤❡r❡ ❝❛♥✬t ❜❡ ❥✉st ♦♥❡ s❧♦♣❡ ❢♦r ❛ ♣❧❛♥❡✳✳✳

❚❤✐♥❣s ♠✐❣❤t ❜❡ ❜❛❞ ✐♥ ❛♥ ❡✈❡♥ ♠♦r❡ s✉❜t❧❡ ✇❛②✳ ❊①❛♠♣❧❡ ✸✳✺✳✸✿ s♠♦♦t❤

◆♦t ❛❧❧ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❤❛✈❡ ❣r❛♣❤s ❧✐❦❡ t❤❡ ♦♥❡ ❛❜♦✈❡✳✳✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥✿

f (x, y) = ❛♥❞ ✐ts ❧✐♠✐t

(x, y) → (0, 0)✳

x2 y . x4 + y 2

❉♦❡s ✐t ❡①✐st❄ ■t ❛❧❧ ❞❡♣❡♥❞s ♦♥ ❤♦✇

❚❤❡ t✇♦ ❣r❡❡♥ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ✐♥❞✐❝❛t❡ t❤❛t ✇❡ ❣❡t

y

❞✐r❡❝t✐♦♥✳

0

X

❛♣♣r♦❛❝❤❡s

✐❢ ✇❡ ❛♣♣r♦❛❝❤

0

A = 0✿

❢r♦♠ ❡✐t❤❡r t❤❡

x

♦r t❤❡

❚❤❛t✬s t❤❡ ❤♦r✐③♦♥t❛❧ ❝r♦ss ✇❡ s❡❡ ♦♥ t❤❡ s✉r❢❛❝❡ ✭❥✉st ❧✐❦❡ t❤❡ ♦♥❡ ✐♥ t❤❡ ❤②♣❡r❜♦❧✐❝

♣❛r❛❜♦❧♦✐❞✮✳ ■♥ ❢❛❝t✱ ❛♥② ✭❧✐♥❡❛r✮ ❞✐r❡❝t✐♦♥ ✐s ❖❑✿

y = mx =⇒ f (x, mx) =

x2 mx mx3 m = = →0 4 2 4 2 2 x + (mx) x +m x x + m2 /x

❛s

x → 0.

❙♦✱ t❤❡ ❧✐♠✐ts ❢r♦♠ ❛❧❧ ❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡✦ ❍♦✇❡✈❡r✱ t❤❡r❡ s❡❡♠s t♦ ❜❡ t✇♦ ❝✉r✈❡❞ ❝❧✐✛s ♦♥ ❧❡❢t ❛♥❞ r✐❣❤t ✈✐s✐❜❧❡ ❢♦r♠ ❛❜♦✈❡✳✳✳ ❲❤❛t ✐s ✇❡ ❛♣♣r♦❛❝❤ 0 ❛❧♦♥❣✱ ✐♥st❡❛❞ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡✱ ❛ ♣❛r❛❜♦❧❛ y = x2 ❄ ❚❤❡ r❡s✉❧t ✐s s✉r♣r✐s✐♥❣✿

y=x

2

x4 1 x2 x2 = 4 = . =⇒ f (x, x ) = 4 2 2 x + (x ) 2x 2 2

❚❤✐s ✐s t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❝❧✐✛s✦ ❙♦✱ t❤✐s ❧✐♠✐t ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦t❤❡r ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ❞❡✜♥✐t✐♦♥✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❡①✐st✿

x2 y DN E . (x,y)→(0,0) x4 + y 2 lim

■♥ ❢❛❝t✱ t❤❡ ✐❧❧✉str❛t✐♦♥ ❝r❡❛t❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t ❛tt❡♠♣ts t♦ ♠❛❦❡ ❛♥ ✉♥❜r♦❦❡♥ s✉r❢❛❝❡ ❢r♦♠ t❤❡s❡ ♣♦✐♥ts ✇❤✐❧❡ ✐♥ r❡❛❧✐t② t❤❡r❡ ✐s ♥♦ ♣❛ss❛❣❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝❧✐✛s ❛s ✇❡ ❥✉st ❞❡♠♦♥str❛t❡❞✳

✸✳✺✳

▲✐♠✐ts

✷✺✷

❙♦✱ ♥♦t ♦♥❧② ✇❡ ❤❛✈❡ t♦ ❛♣♣r♦❛❝❤ t❤❡ ♣♦✐♥t ❢r♦♠ ❛❧❧ ❞✐r❡❝t✐♦♥s ❛t ♦♥❝❡ ❜✉t ❛❧s♦ ❛❧♦♥❣ ❛♥② ♣♦ss✐❜❧❡ ♣❛t❤✳

❆ s✐♠♣❧❡r ❜✉t ✈❡r② ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t st✉❞②✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡ ♠✐❣❤t ❜❡ ✐♥s✉✣❝✐❡♥t ♦r ❡✈❡♥ ♠✐s❧❡❛❞✐♥❣✳ ❆ s♣❡❝✐❛❧ ♥♦t❡ ❛❜♦✉t t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t ❛ ♣♦✐♥t t❤❛t ❧✐❡s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳✳✳ ■t ❞♦❡s♥✬t ♠❛tt❡r✦ ❚❤❡ ❝❛s❡ ♦❢ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ❛t a ♦r b ♦❢ t❤❡ ❞♦♠❛✐♥ [a, b] ✐s ♥♦✇ ✐♥❝❧✉❞❡❞ ✐♥ ♦✉r ♥❡✇ ❞❡✜♥✐t✐♦♥✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ t❤❡♦r② ♦❢ ❧✐♠✐ts ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❚❤❡r❡ ❛r❡ ❛s ♠❛♥② r✉❧❡s ❜❡❝❛✉s❡ ✇❤❛t❡✈❡r ②♦✉ ❝❛♥ ❞♦ ✇✐t❤ t❤❡ ♦✉t♣✉ts ②♦✉ ❝❛♥ ❞♦ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ✕ ♦❢ ♥✉♠❜❡rs ✕ t♦ ♣r♦✈❡ ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤❡♦r❡♠ ✸✳✺✳✹✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❙✉♣♣♦s❡

an → a

❛♥❞

❙❘✿ P❘✿

bn → b✳

❚❤❡♥✿

an + bn → a + b

an · bn → ab

❈▼❘✿ ◗❘✿

c · an → ca

r❡❛❧

c

an /bn → a/b b 6= 0

❊❛❝❤ ♣r♦♣❡rt② ✐s ♠❛t❝❤❡❞ ❜② ✐ts ❛♥❛❧♦❣ ❢♦r ❢✉♥❝t✐♦♥s✳ ❚❤❡♦r❡♠ ✸✳✺✳✺✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ❙✉♣♣♦s❡

f (X) → a

❙❘✿ P❘✿

❛♥❞

g(X) → b

❛s

f (X) + g(X) → a + b f (X) · g(X) → ab

X → A✳

❚❤❡♥✿

❈▼❘✿

c · f (X) → ca

◗❘✿

r❡❛❧

c

f (X)/g(X) → a/b b 6= 0

✸✳✺✳ ▲✐♠✐ts

✷✺✸

◆♦t❡ t❤❛t t❤❡r❡ ✇❡r❡ ♥♦ P❘ ♦r ◗❘ ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳✳✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡♠ ♦♥❡ ❜② ♦♥❡✳ ◆♦✇✱ ❧✐♠✐ts ❜❡❤❛✈❡ ✇❡❧❧ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉s✉❛❧ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✸✳✺✳✻✿ ❙✉♠ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s

A

■❢ t❤❡ ❧✐♠✐ts ❛t

f (P ) + g(P )✱

♦❢ ❢✉♥❝t✐♦♥s

f (X), g(X)

❡①✐st t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r s✉♠✱

❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❧✐♠✐ts✿

 lim f (X) + g(X) = lim f (X) + lim g(X) X→A

X→A

X→A

❙✐♥❝❡ t❤❡ ♦✉t♣✉ts ❛♥❞ t❤❡ ❧✐♠✐ts ❛r❡ ❥✉st ♥✉♠❜❡rs✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❧✐♠✐ts ✇❡ ❢♦❧❧♦✇ t❤❡ s❛♠❡ r✉❧❡s ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s ❛s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✿ ♥✉♠❜❡r

+ (+∞) = +∞

♥✉♠❜❡r

+ (−∞) = −∞

+∞

+ (+∞) = +∞

−∞

+ (−∞) = −∞

❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ❛r❡ ✐❞❡♥t✐❝❛❧✳ ❚❤❡♦r❡♠ ✸✳✺✳✼✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ■❢ t❤❡ ❧✐♠✐t ❛t

cf (X)✱

X=A

♦❢ ❢✉♥❝t✐♦♥

f (X)

❡①✐sts t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ ✐ts ♠✉❧t✐♣❧❡✱

❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♠✉❧t✐♣❧❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❧✐♠✐t✿

lim cf (X) = c · lim f (X)

X→A

X→A

❚❤❡♦r❡♠ ✸✳✺✳✽✿ Pr♦❞✉❝t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐✲ ❛❜❧❡s ■❢ t❤❡ ❧✐♠✐ts ❛t

f (X) · g(X)✱

a ♦❢ ❢✉♥❝t✐♦♥s f (X), g(X) ❡①✐st t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r ♣r♦❞✉❝t✱

❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♣r♦❞✉❝t ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✿



lim f (X) · g(X) =

X→A



   lim f (X) · lim g(X)

X→A

X→A

❚❤❡♦r❡♠ ✸✳✺✳✾✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐✲ ❛❜❧❡s

X = A ♦❢ ❢✉♥❝t✐♦♥s f (X), g(X) ❡①✐st t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r f (X)/g(X)✱ ♣r♦✈✐❞❡❞ lim g(X) 6= 0✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦ ✐s ❡q✉❛❧ t♦

■❢ t❤❡ ❧✐♠✐ts ❛t r❛t✐♦✱

X→A

t❤❡ r❛t✐♦ ♦❢ t❤❡ ❧✐♠✐ts✿

lim

X→A



f (X) g(X)



lim f (X)

=

X→A

lim g(X)

X→A

✸✳✺✳

▲✐♠✐ts

✷✺✹

❏✉st ❛s ✇✐t❤ s❡q✉❡♥❝❡s✱ ✇❡ ❝❛♥ r❡♣r❡s❡♥t t❤❡s❡ r✉❧❡s ❛s ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠s✿

f, g   + y

lim

−−−−−→

l, m   + y

SR lim

f + g −−−−−→ lim(f + g) = l + m ❊①❡r❝✐s❡ ✸✳✺✳✶✵

❋✐♥✐s❤✿

x2 + y 2 =? (x,y)→(0,0) x + y lim

❲❡ ❝❛♥ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❢❛❝t t❤❛t t❤❡ ❞♦♠❛✐♥

Rm

f

♦❢

❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s ♠❛❞❡ ♦❢

✈❡❝t♦rs

❛♥❞ ✈❡❝t♦rs

❛r❡ s✉❜❥❡❝t t♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✸✳✺✳✶✶✿ ❆❧t❡r♥❛t✐✈❡ ❋♦r♠✉❧❛ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈✲ ❡r❛❧ ❱❛r✐❛❜❧❡s ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥

z = f (X)

❛t

X=A

✐s ❡q✉❛❧ t♦

l

✐❢ ❛♥❞ ♦♥❧② ✐❢

lim f (A + H) = l .

||H||→0

❚❤❡ ♥❡①t r❡s✉❧t st❛♥❞s ✈✐rt✉❛❧❧② ✉♥❝❤❛♥❣❡❞ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✷✮✳

❚❤❡♦r❡♠ ✸✳✺✳✶✷✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❋♦r ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s

■❢ ❛ ❢✉♥❝t✐♦♥ ✐s sq✉❡❡③❡❞ ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ❧✐♠✐t ❛t ❛ ♣♦✐♥t✱ ✐ts ❧✐♠✐t ❛❧s♦ ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❡ t❤❛t ♥✉♠❜❡r❀ ✐✳❡✳✱ ✐❢

f (X) ≤ h(X) ≤ g(X) , ❢♦r ❛❧❧

X

✇✐t❤✐♥ s♦♠❡ ❞✐st❛♥❝❡ ❢r♦♠

X = A✱

❛♥❞

lim f (X) = lim g(X) = l ,

X→A

X→A

t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts ❛♥❞ ❡q✉❛❧ t♦ t❤❛t ♥✉♠❜❡r✿

lim h(X) = l .

X→A

❚❤❡ ❡❛s✐❡st ✇❛② t♦ ❤❛♥❞❧❡ ❧✐♠✐ts ✐s

❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡♥ ♣♦ss✐❜❧❡✳

✸✳✺✳

▲✐♠✐ts

✷✺✺

❚❤❡♦r❡♠ ✸✳✺✳✶✸✿ ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ■❢ t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❡①✐sts t❤❡♥ ✐t ❡①✐sts ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s❀ ✐✳❡✳✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥

A = (a1 , ..., am ) f (X) → l ❢♦r ❡❛❝❤

Rm ✱

✐♥

❛s

z = f (X) = f (x1 , ..., xm )

X→A

=⇒ f (a1 , ..., ak−1 , x, ak+1 , ..., am ) → l

k = 1, 2, ..., m

❛♥❞ ❛♥②

✇❡ ❤❛✈❡✿

❛s

x → ak ,

✳

❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✦

❊①❛♠♣❧❡ ✸✳✺✳✶✹✿ ✷❞ ❧✐♠✐t ❘❡❝❛❧❧ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❤❛s ❧✐♠✐ts ❛t

(0, 0)

✇✐t❤ r❡s♣❡❝t t♦ ❡✐t❤❡r ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✿

f (x, y) = ❜✉t t❤❡ ❧✐♠✐t ❛s

(x, y) → (0, 0)

x2 y , x4 + y 2

❞♦❡s ♥♦t ❡①✐st✳

❙♦✱ ✇❡ ❤❛✈❡ t♦ ❡st❛❜❧✐s❤ t❤❛t t❤❡ ❧✐♠✐t ❡①✐sts ✕ ✐♥ t❤❡ ✏♦♠♥✐✲❞✐r❡❝t✐♦♥❛❧✑ s❡♥s❡ ✕ ✜rst ❛♥❞ ♦♥❧② t❤❡♥ ✇❡ ❝❛♥ ✉s❡ ❧✐♠✐ts ✇✐t❤ r❡s♣❡❝t t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ ✕ ✐♥ t❤❡ ✏✉♥✐✲❞✐r❡❝t✐♦♥❛❧✑ s❡♥s❡ ✕ t♦ ✜♥❞ t❤✐s ❧✐♠✐t✳ ◆♦✇✱ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r✳✳✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✶✺✿ ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ✐♥✜♥✐t② ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

✐♥✜♥✐t② ❛t A ✐❢

z = f (X)

❛♥❞ ❛ ♣♦✐♥t

A

✐♥

Rn ✱

✇❡ s❛② t❤❛t

f ❛♣♣r♦❛❝❤❡s

lim f (x) = ±∞ ,

n→∞ ❢♦r ❛♥② s❡q✉❡♥❝❡

Xn → A

❛s

n → ∞✳

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

lim f (x) = ±∞ .

X→A ❚❤❡ ❧✐♥❡

X=A

❧♦❝❛t❡❞ ✐♥

Rn+1

✐s t❤❡♥ ❝❛❧❧❡❞ ❛

✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡

♦❢

f✳

❊①❛♠♣❧❡ ✸✳✺✳✶✻✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t② z = f (x, y, z) r❡♣r❡s❡♥ts t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ♦❢ X = (x, y, z) r❡❧❛t✐✈❡ t♦ ❛♥♦t❤❡r ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✱ t❤❡♥ ✇❡ ■❢

lim f (X) = ∞ .

X→0

◆❡①t✱ ✇❡ ❝❛♥ ❛♣♣r♦❛❝❤ ✐♥✜♥✐t② ✐♥ ❛ ♥✉♠❜❡r ♦❢ ✇❛②s t♦♦✿

❛♥ ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♣♦✐♥t ❤❛✈❡✿

✸✳✺✳

▲✐♠✐ts

✷✺✻

❚❤❛t✬s ✇❤② ✇❡ ❤❛✈❡ t♦ ❧♦♦❦ ❛t t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♦r✐❣✐♥✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✶✼✿ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❣♦❡s t♦ ✐♥✜♥✐t② ❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

z = f (X)✱

✇❡ s❛② t❤❛t

f ❣♦❡s t♦ ✐♥✜♥✐t②

✐❢

f (Xn ) → ±∞ , ❢♦r ❛♥② s❡q✉❡♥❝❡

Xn

✇✐t❤

||Xn || → ∞

❛s

f (X) → ±∞

n → ∞✳ ❛s

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

X → ∞,

♦r

lim f (x) = ±∞ .

x→∞

❊①❛♠♣❧❡ ✸✳✺✳✶✽✿ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ❲❡ ♣r❡✈✐♦✉s❧② ❞❡♠♦♥str❛t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

lim ex = 0,

lim ex = +∞ .

x→−∞

x→+∞

❲❡ ✇♦♥✬t s♣❡❛❦ ♦❢ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ❜✉t t❤❡ ♥❡①t ✐❞❡❛ ✐s r❡❧❛t❡❞✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✶✾✿ ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ✈❛❧✉❡ ❛t ✐♥✜♥✐t② ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

z = f (X)✱

✇❡ s❛② t❤❛t

f ❛♣♣r♦❛❝❤❡s z = d ❛t ✐♥✜♥✐t②

✐❢

lim f (Xn ) = d ,

n→∞ ❢♦r ❛♥② s❡q✉❡♥❝❡

Xn

✇✐t❤

||Xn || → ∞

❛s

n → ∞✳

lim f (X) = d .

X→∞

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✺✼

❊①❛♠♣❧❡ ✸✳✺✳✷✵✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t② ■❢ z = f (x, y, z) r❡♣r❡s❡♥ts t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ♦❢ ❛♥ ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♣♦✐♥t X = (x, y, z) r❡❧❛t✐✈❡ t♦ ❛♥♦t❤❡r ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✱ t❤❡♥ ✇❡ ❤❛✈❡✿ lim f (X) = 0 .

X→∞

✸✳✻✳ ❈♦♥t✐♥✉✐t②

❚❤❡ ✐❞❡❛ ♦❢ ❝♦♥t✐♥✉✐t② ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦r ♣❛r❛♠❡tr✐❝ ❝✉r❡s✿ ❛s t❤❡ ✐♥♣✉t ❛♣♣r♦❛❝❤❡s ❛ ♣♦✐♥t✱ t❤❡ ♦✉t♣✉t ✐s ❢♦r❝❡❞ t♦ ❛♣♣r♦❛❝❤ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✳ ❚❤❡ ❝♦♥❝❡♣t ✢♦✇s ❢r♦♠ t❤❡ ✐❞❡❛ ♦❢ ❧✐♠✐t ❥✉st ❛s ❜❡❢♦r❡✳

❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✿ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t ❆ ❢✉♥❝t✐♦♥ z = f (X) ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s • f (X) ✐s ❞❡✜♥❡❞ ❛t X = A✳ • ❚❤❡ ❧✐♠✐t ♦❢ f ❡①✐sts ❛t A✳ • ❚❤❡ t✇♦ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿

❛t ♣♦✐♥t X = A ✇❤❡♥✿

lim f (X) = f (A) .

X→A

❋✉rt❤❡r♠♦r❡✱ ❛ ❢✉♥❝t✐♦♥ ✐s ❞♦♠❛✐♥✳

❝♦♥t✐♥✉♦✉s

✐❢ ✐t ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ✐ts

❚❤✉s✱ t❤❡ ❧✐♠✐ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥ ✿ lim f (X) = f (A)

X→A

♦r f (X) → f (A) ❛s X → A

❊q✉✐✈❛❧❡♥t❧②✱ ❛ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t a ✐❢ ✇❡ ❤❛✈❡✿ lim f (An ) = f (A) ,

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ Xn → A✳

❆ t②♣✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✺✽ ❚❤❡♦r❡♠ ✸✳✻✳✷✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♥t✐♥✉✐t②

g ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t X = A✳ ✭❙❘✮ f ± g ✱ ✭❈▼❘✮ c · f ❢♦r ❛♥② r❡❛❧ c✱ ✭P❘✮ f · g ✱ ❛♥❞ ✭◗❘✮ f /g ♣r♦✈✐❞❡❞ g(A) 6= 0✳

❙✉♣♣♦s❡ ✶✳ ✷✳ ✸✳ ✹✳

f

❛♥❞

❚❤❡♥ s♦ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿

❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ ✇❡ ❝❛♥ s❛② t❤❛t ✐❢ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣ r❡♣r❡s❡♥t❡❞ ❜② f ❛♥❞ g r❡s♣❡❝t✐✈❡❧② ♦❢ ❛ ❝❛✈❡ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧② t❤❡♥ s♦ ✐s ✐ts ❤❡✐❣❤t✱ ✇❤✐❝❤ ✐s g − f ✿

❖r✱ ✐❢ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣ ✭f ❛♥❞ −g ✮ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧② t❤❡♥ s♦ ✐s ✐ts ❤❡✐❣❤t ✭g + f ✮✳ ❆♥❞ s♦ ♦♥✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s t♦ ❝♦♠♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✳✳✳ ❚❤❡♦r❡♠ ✸✳✻✳✸✿ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■

■❢ t❤❡ ❧✐♠✐t ❛t

X=A

♦❢ ❢✉♥❝t✐♦♥

z = f (X)

❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦

t❤❛t ♦❢ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛♥② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

lim (g ◦ f )(X) = g(l)

X→A

Pr♦♦❢✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✱ Xn → A . bn = f (Xn ) .

❚❤❡ ❝♦♥❞✐t✐♦♥ f (X) → l ❛s X → A ✐s r❡st❛t❡❞ ❛s ❢♦❧❧♦✇s✿ bn → l ❛s n → ∞ .

❚❤❡r❡❢♦r❡✱ ❝♦♥t✐♥✉✐t② ♦❢ g ✐♠♣❧✐❡s✱ g(bn ) → g(l) ❛s n → ∞ .

■♥ ♦t❤❡r ✇♦r❞s✱

(g ◦ f )(Xn ) = g(f (Xn )) → g(l) ❛s n → ∞ .

❙✐♥❝❡ s❡q✉❡♥❝❡ Xn → A ✇❛s ❝❤♦s❡♥ ❛r❜✐tr❛r✐❧②✱ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s r❡st❛t❡❞ ❛s✱ (g ◦ f )(X) → g(l) ❛s X → A .

t❤❡♥ s♦ ❞♦❡s

u = g(z) ❝♦♥t✐♥✉♦✉s ❛t z = l

❛♥❞

❚❤❡♥✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡✱

l

✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✺✾

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ r❡s✉❧t ❛s ❢♦❧❧♦✇s✿

lim (g ◦ f )(X) = g(l) X→A

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿

l=limX→A f (X)

   lim g f (X) = g lim f (X)

X→A

X→A

❚❤❡♦r❡♠ ✸✳✻✳✹✿ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■■

t = a ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ L t❤❡♥ z = g(X) ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❛t X = L ❛♥❞

■❢ t❤❡ ❧✐♠✐t ❛t

s♦ ❞♦❡s t❤❛t ♦❢ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛♥② ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s

lim(g ◦ F )(t) = g(L) t→a

Pr♦♦❢✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✱

tn → a . ❚❤❡♥✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡✱

Bn = F (tn ) . ❚❤❡ ❝♦♥❞✐t✐♦♥

F (t) → L

❛s

t→a

✐s r❡st❛t❡❞ ❛s ❢♦❧❧♦✇s✿

Bn → L ❚❤❡r❡❢♦r❡✱ ❝♦♥t✐♥✉✐t② ♦❢

F

❛s

n → ∞.

✐♠♣❧✐❡s✱

g(Bn ) → g(L)

❛s

n → ∞.

■♥ ♦t❤❡r ✇♦r❞s✱

(g ◦ F )(tn ) = g(F (tn )) → g(L) ❙✐♥❝❡ s❡q✉❡♥❝❡

Xn → A

❛s

n → ∞.

✇❛s ❝❤♦s❡♥ ❛r❜✐tr❛r✐❧②✱ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s r❡st❛t❡❞ ❛s✱

(g ◦ F )(t) → g(L)

❛s

t → a.

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ r❡s✉❧t ❛s ❢♦❧❧♦✇s✿

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿

lim(g ◦ F )(t) = g(L) t→a 



L=limt→a F (t)

lim g F (t) = g lim F (t) t→a

t→a



✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✻✵

❈♦r♦❧❧❛r② ✸✳✻✳✺✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s

g◦f y = f (a)

f

❚❤❡ ❝♦♠♣♦s✐t✐♦♥

♦❢ ❛ ❢✉♥❝t✐♦♥

❝♦♥t✐♥✉♦✉s ❛t

✐s ❝♦♥t✐♥✉♦✉s ❛t

❚❤❡ ❡❛s✐❡st ✇❛② t♦ ❤❛♥❞❧❡ ❝♦♥t✐♥✉✐t② ✐s

❝♦♥t✐♥✉♦✉s ❛t

x = a

❛♥❞ ❛ ❢✉♥❝t✐♦♥

g

x = a✳

❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡♥ ♣♦ss✐❜❧❡✳

❚❤❡♦r❡♠ ✸✳✻✳✻✿ ❈♦♥t✐♥✉✐t② ♦❢ ❋✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ■❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s ❝♦♥t✐♥✉♦✉s t❤❡♥ ✐t ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✳

❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✦ ❊①❛♠♣❧❡ ✸✳✻✳✼✿ ✷❞ ❧✐♠✐t

❘❡❝❛❧❧ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛t

(0, 0)

✇✐t❤ r❡s♣❡❝t t♦ ❡✐t❤❡r ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✿

x2 y f (x, y) = 4 , x + y2 ❜✉t t❤❡ ❧✐♠✐t ❛s

(x, y) → (0, 0)

s✐♠♣❧② ❞♦❡s ♥♦t ❡①✐st✳

❙♦✱ ✇❡ ❤❛✈❡ t♦ ❡st❛❜❧✐s❤ ❝♦♥t✐♥✉✐t② ✕ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ✏♦♠♥✐✲❞✐r❡❝t✐♦♥❛❧✑ ❧✐♠✐t ✕ ✜rst ❛♥❞ ♦♥❧② t❤❡♥ ✇❡ ❝❛♥ ✉s❡ t❤✐s ❢❛❝t t♦ ✜♥❞ ❢❛❝ts ❛❜♦✉t t❤❡ ❝♦♥t✐♥✉✐t② ✇✐t❤ r❡s♣❡❝t t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ ✕ ✐♥ t❤❡ ✏✉♥✐✲❞✐r❡❝t✐♦♥❛❧✑ s❡♥s❡✳ ❆ s♣❡❝✐❛❧ ♥♦t❡ ❛❜♦✉t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t ❛ ♣♦✐♥t t❤❛t ❧✐❡s ♦♥ t❤❡ ❛❣❛✐♥✱ ✐t ❞♦❡s♥✬t ♠❛tt❡r✦ ❚❤❡ ❝❛s❡ ♦❢ ♦♥❡✲s✐❞❡❞ ❝♦♥t✐♥✉✐t② ❛t

a

♦r

b

❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳✳✳

♦❢ t❤❡ ❞♦♠❛✐♥

[a, b]

❖♥❝❡

✐s ♥♦✇ ✐♥❝❧✉❞❡❞ ✐♥

♦✉r ♥❡✇ ❞❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t② ✐s ♣✉r❡❧②

❧♦❝❛❧ ✿

♦♥❧② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱

✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t ♠❛tt❡rs✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ✇❤♦❧❡ s❡t✱ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ✐ts

❣❧♦❜❛❧

❜❡❤❛✈✐♦r❄ ❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❤❛s ❜❡❡♥ ❛s t❤❡ ♣r♦♣❡rt② ♦❢ ❤❛✈✐♥❣

❣r❛♣❤s✳

❚❤✐s ✐❞❡❛ ✐s ♠♦r❡ ♣r❡❝✐s❡❧② ❡①♣r❡ss❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❜② t❤❡

◮

■❢ ❛ ❢✉♥❝t✐♦♥

❛♥❞

f (b)✱

f

t❤❡r❡ ✐s

■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✿

✐s ❞❡✜♥❡❞ ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛♥ ✐♥t❡r✈❛❧

d

✐♥

[a, b]

s✉❝❤ t❤❛t

f (d) = c✳

♥♦ ❣❛♣s ✐♥ t❤❡✐r

[a, b]✱

t❤❡♥ ❢♦r ❛♥②

c

❜❡t✇❡❡♥

f (a)

✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✻✶

❆♥ ♦❢t❡♥ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ✇❛② t♦ st❛t❡ ✐s✿

◮

■❢ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ❛♥ ✐♥t❡r✈❛❧ t❤❡♥ s♦ ✐s ✐ts ✐♠❛❣❡✳

◆♦✇ t❤❡ ♣❧❛♥❡ ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ ❛ ❧✐♥❡ ❛♥❞ ✇❡ ❝❛♥✬t ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ ✐♥t❡r✈❛❧s✳ ❇✉t ✇❤❛t ✐s t❤❡ ❛♥❛❧♦❣ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡❄

❚❤❡♦r❡♠ ✸✳✻✳✽✿ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❢♦r ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ s❡t t❤❛t ❝♦♥t❛✐♥s t❤❡ ♣❛t❤ C ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤✐s ♣❛t❤ ✐s ❛♥ ✐♥t❡r✈❛❧✳ Pr♦♦❢✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡

❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■■

■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠

❛♥❞ t❤❡

❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳

❚❤❡ t❤❡♦r❡♠ s❛②s t❤❛t t❤❡r❡ ❛r❡ ♥♦ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ s✉❝❤ ❛ s❡t✳

❊①❡r❝✐s❡ ✸✳✻✳✾ ❙❤♦✇ t❤❛t t❤❛t t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s♥✬t tr✉❡✳

❆ ❝♦♥✈❡♥✐❡♥t r❡✲st❛t❡♠❡♥t ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ❜❡❧♦✇✳

❈♦r♦❧❧❛r② ✸✳✻✳✶✵✿ ■♠❛❣❡ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥

❚❤❡ ✐♠❛❣❡ ♦❢ ❛ ♣❛t❤✲❝♦♥♥❡❝t❡❞ s❡t ✉♥❞❡r ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ❛♥ ✐♥t❡r✈❛❧✳ ❘❡❝❛❧❧ t❤❛t ❛ ❢✉♥❝t✐♦♥ ♥✉♠❜❡r

m

f

✐s ❝❛❧❧❡❞

❜♦✉♥❞❡❞

♦♥ ❛ s❡t

S

✐♥

Rn

✐❢ ✐ts ✐♠❛❣❡ ✐s ❜♦✉♥❞❡❞✱ ✐✳❡✳✱ t❤❡r❡ ✐s s✉❝❤ ❛ r❡❛❧

t❤❛t

|f (X)| ≤ m ❢♦r ❛❧❧

X

✐♥

S✳

❚❤❡♦r❡♠ ✸✳✻✳✶✶✿ ❈♦♥t✳ ❂❃ ❇♦✉♥❞❡❞

■❢ t❤❡ ❧✐♠✐t ❛t X = A ♦❢ ❢✉♥❝t✐♦♥ z = f (X) ❡①✐sts t❤❡♥ f ✐s ❜♦✉♥❞❡❞ ♦♥ s♦♠❡

✸✳✻✳

✷✻✷

❈♦♥t✐♥✉✐t②

♦♣❡♥ ❞✐s❦ t❤❛t ❝♦♥t❛✐♥s

A✿

lim f (X)

X→A ❢♦r ❛❧❧

X

✇✐t❤

d(X, A) < δ

❡①✐sts

❢♦r s♦♠❡

=⇒ |f (X)| ≤ m

δ>0

❛♥❞ s♦♠❡

m✳

❊①❡r❝✐s❡ ✸✳✻✳✶✷

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ✉♥❞❡r ❝❡rt❛✐♥ ❝✐r❝✉♠st❛♥❝❡s✳ ❚❤❡ ✈❡rs✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐s s✐♠♣❧②✿ ❛ ❝♦♥t✐♥✉♦✉s ♦♥ [a, b] ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞✳

❇✉t ✇❤❛t ✐s t❤❡ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣ ♦❢ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧❄ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ❛ s❡t S ✐♥ Rn ✐s ❜♦✉♥❞❡❞ ✐❢ ✐t ✜ts ✐♥ ❛ s♣❤❡r❡ ✭♦r ❛ ❜♦①✮ ♦❢ ❛ ❧❛r❣❡ ❡♥♦✉❣❤ s✐③❡✿ ||x|| < Q ❢♦r ❛❧❧ x ✐♥ S ;

❛♥❞ ❛ s❡t ✐♥ Rn ✐s ❝❛❧❧❡❞ ❝❧♦s❡❞ ✐❢ ✐t ❝♦♥t❛✐♥s t❤❡ ❧✐♠✐ts ♦❢ ❛❧❧ ♦❢ ✐ts ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✳ ❚❤❡♦r❡♠ ✸✳✻✳✶✸✿ ❇♦✉♥❞❡❞♥❡ss ❆ ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ s❡t ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞✳

Pr♦♦❢✳

❙✉♣♣♦s❡✱ t♦ t❤❡ ❝♦♥tr❛r②✱ t❤❛t x = f (X) ✐s ✉♥❜♦✉♥❞❡❞ ♦♥ s❡t S ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ {Xn } ✐♥ S s✉❝❤ t❤❛t f (Xn ) → ∞✳ ❚❤❡♥✱ ❜② t❤❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠✱ s❡q✉❡♥❝❡ {Xn } ❤❛s ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ {Yk }✿ Yk → Y .

❚❤✐s ♣♦✐♥t ❜❡❧♦♥❣ t♦ S ✦ ❋r♦♠ t❤❡ ❝♦♥t✐♥✉✐t②✱ ✐t ❢♦❧❧♦✇s t❤❛t f (Yk ) → f (Y ) .

❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t {Yk } ✐s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❞✐✈❡r❣❡s t♦ ∞✳ ❊①❡r❝✐s❡ ✸✳✻✳✶✹

❲❤② ❛r❡ ✇❡ ❥✉st✐✜❡❞ t♦ ❝♦♥❝❧✉❞❡ ✐♥ t❤❡ ♣r♦♦❢ t❤❛t t❤❡ ❧✐♠✐t Y ♦❢ {Yk } ✐s ✐♥ S ❄ ■s t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❝❧♦s❡❞ s❡t ❝❧♦s❡❞❄ ■❢ ✐t ✐s✱ t❤❡ ❢✉♥❝t✐♦♥ ❜♦✉♥❞ sup ❛♥❞ t❤❡ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞ inf ✳

r❡❛❝❤❡s ✐ts ❡①tr❡♠❡ ✈❛❧✉❡s✱

✐✳❡✳✱ t❤❡ ❧❡❛st ✉♣♣❡r

✸✳✻✳

❈♦♥t✐♥✉✐t②

✷✻✸

❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✺✿ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ ♦♥ s❡t

S

z = f (X)✳

❚❤❡♥

X=D

❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t

♦❢

f

✐❢

f (D) ≥ f (X) ❆♥❞

✐s ❝❛❧❧❡❞ ❛

X=C

✐s ❝❛❧❧❡❞ ❛

❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t

f (C) ≤ f (X) ✭❚❤❡② ❛r❡ ❛❧s♦ ❝❛❧❧❡❞ ❛r❡ ❛❧❧ ❝❛❧❧❡❞

X

❢♦r ❛❧❧

f

♦❢

❢♦r ❛❧❧

✐♥

X

S.

♦♥ s❡t

✐♥

S

✐❢

S.

❛❜s♦❧✉t❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts✳✮

❈♦❧❧❡❝t✐✈❡❧② t❤❡②

❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳

❏✉st ❜❡❝❛✉s❡ s♦♠❡t❤✐♥❣ ✐s ❞❡s❝r✐❜❡❞ ❞♦❡s♥✬t ♠❡❛♥ t❤❛t ✐t ❝❛♥ ❜❡ ❢♦✉♥❞✳

❚❤❡♦r❡♠ ✸✳✻✳✶✻✿ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ s❡t ✐♥ ❛♥❞ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡s❀ ✐✳❡✳✱ ✐❢ s❡t

S✱

t❤❡♥ t❤❡r❡ ❛r❡

C, D

✐♥

S

z = f (X)

Rn ❛tt❛✐♥s ✐ts ❣❧♦❜❛❧ ♠❛①✐♠✉♠

✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞

s✉❝❤ t❤❛t

f (C) ≤ f (X) ≤ f (D) , ❢♦r ❛❧❧

X

✐♥

S✳

Pr♦♦❢✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡

❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠✳

❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✼✿ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ ♦❢

f

♦♥ s❡t

S

z = f (X)✳

❚❤❡♥

X =M

y=m

✐s ❝❛❧❧❡❞ t❤❡

✭❚❤❡② ❛r❡ ❛❧s♦ ❝❛❧❧❡❞

❚❤❡♥

t❤❡

❢♦r ❛❧❧

❢♦r ❛❧❧

❛r❡ ❢❛❝✐♥❣ ❤❛s ❛ s♦❧✉t✐♦♥✳

X

✐♥ ♦❢

✐♥

S. f

♦♥ s❡t

S

✐❢

S.

❛❜s♦❧✉t❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ✈❛❧✉❡s✳✮

❈♦❧❧❡❝t✐✈❡❧② t❤❡②

❣❧♦❜❛❧ ❡①tr❡♠❡ ✈❛❧✉❡s✳

❣❧♦❜❛❧ ♠❛① ✭♦r ♠✐♥✮ ✈❛❧✉❡ ✐s r❡❛❝❤❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❛t

◆♦t❡ t❤❛t t❤❡ r❡❛s♦♥ ✇❡ ♥❡❡❞ t❤❡

X

❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡

m ≤ f (X) ❛r❡ ❛❧❧ ❝❛❧❧❡❞

❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✈❛❧✉❡

✐❢

M ≥ f (X) ❆♥❞

✐s ❝❛❧❧❡❞ t❤❡

❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠

❛♥②

♦❢ ✐ts ❣❧♦❜❛❧ ♠❛① ✭♦r ♠✐♥✮ ♣♦✐♥ts✳

✐s t♦ ❡♥s✉r❡ t❤❛t t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇❡

✸✳✼✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✻✹

❲❡ ❝❛♥ ❞❡✜♥❡ ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✇✐t❤♦✉t ✐♥✈♦❦✐♥❣ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s✳ ▲❡t✬s r❡✇r✐t❡ ✇❤❛t ✇❡ ✇❛♥t t♦ s❛② ❛❜♦✉t t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧✐♠✐ts ✐♥ ♣r♦❣r❡ss✐✈❡❧② ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡ t❡r♠s✳ X

z = f (X)

❆s X → A, ❆s X ❛♣♣r♦❛❝❤❡s A, ❆s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ X t♦ A ❛♣♣r♦❛❝❤❡s 0, ❆s d(X, A) → 0, ❇② ♠❛❦✐♥❣ d(X, A) ❛s s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r, ❇② ♠❛❦✐♥❣ d(X, A) ❧❡ss t❤❛♥ s♦♠❡ δ > 0,

✇❡ ❤❛✈❡ y → l. y ❛♣♣r♦❛❝❤❡s l. t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ y t♦ l ❛♣♣r♦❛❝❤❡s 0. ✇❡ ❤❛✈❡ |y − l| → 0. ✇❡ ♠❛❦❡ |y − l| ❛s s♠❛❧❧ ❛s ♥❡❡❞❡❞. ✇❡ ♠❛❦❡ |y − l| s♠❛❧❧❡r t❤❛♥ ❛♥② ❣✐✈❡♥ ε > 0.

❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✽✿ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥ z = f (X) ❛t X = A ✐s ❛ ♥✉♠❜❡r l✱ ✐❢ ❡①✐sts✱ s✉❝❤ t❤❛t ❢♦r

❛♥② ε > 0 t❤❡r❡ ✐s s✉❝❤ ❛ δ > 0 t❤❛t

0 < d(X, A) < δ =⇒ |f (X) − l| < ε .

❚❤✐s ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✿ ✐❢ X ✐s ✇✐t❤✐♥ δ ❢r♦♠ A✱ t❤❡♥ f (X) ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ✇✐t❤✐♥ ε ❢r♦♠ l✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ✜ts ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣❧❛♥❡s ε ❛✇❛② ❢r♦♠ t❤❡ ♣❧❛♥❡ z = l✳

✸✳✼✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❲❡ st❛rt ✇✐t❤ ♥✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳

❊①❛♠♣❧❡ ✸✳✼✳✶✿ ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥✿

f (x, y) = −x2 + y 2 + xy .

❋♦r ❡❛❝❤ x ✐♥ t❤❡ ❧❡❢t✲♠♦st ❝♦❧✉♠♥ ❛♥❞ ❡❛❝❤ y ✐♥ t❤❡ t♦♣ r♦✇✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ t❤✐s t❛❜❧❡✱ ❥✉st ❛s ❜❡❢♦r❡✿ ❂❘✷❈✸✯✭❘✹❈✂✷✲❘❈✷✂✷✰❘✹❈✯❘❈✷✮

❲❤❡♥ ♣❧♦tt❡❞ ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛ ❢❛♠✐❧✐❛r ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞✿

✸✳✼✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✻✺

❇❡❧♦✇ ✇❡ ♦✉t❧✐♥❡ t❤❡ ♣r♦❝❡ss ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡ ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✕ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ✕ ❛r❡ s❤♦✇♥✳ ❚❤❡s❡ ❛r❡ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ✇✐❧❧ ❞✐✛❡r❡♥t✐❛t❡✳ ❋✐rst✱ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ y ❣✐✈❡♥ ✐♥ t❤❡ t♦♣ r♦✇✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ x ❜② ❣♦✐♥❣ ❞♦✇♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❧✉♠♥ ❛♥❞ t❤❡♥ ♣❧❛❝✐♥❣ t❤❡s❡ ✈❛❧✉❡s ♦♥ r✐❣❤t ✐♥ ❛ ♥❡✇ t❛❜❧❡ ✭✐t ✐s ♦♥❡ r♦✇ s❤♦rt ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧✮✿

❂✭❘❈❬✲✷✾❪✲❘❬✲✶❪❈❬✲✷✾❪✮✴❘✷❈✶ ❙❡❝♦♥❞✱ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ x ❣✐✈❡♥ ✐♥ t❤❡ ❧❡❢t✲♠♦st ❝♦❧✉♠♥✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ y ❜② ❣♦✐♥❣ r✐❣❤t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r♦✇ ❛♥❞ t❤❡♥ ♣❧❛❝✐♥❣ t❤❡s❡ ✈❛❧✉❡s ❜❡❧♦✇ ✐♥ ❛ ♥❡✇ t❛❜❧❡ ✭✐t ✐s ♦♥❡ ❝♦❧✉♠♥ s❤♦rt ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧✮✳

❂✭❘❬✲✷✾❪❈✲❘❬✲✷✾❪❈❬✲✶❪✮✴❘✷❈✶ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿

❚❤❡ r❡s✉❧ts ❛r❡ ❛❧❧ str❛✐❣❤t ❧✐♥❡s ❡q✉❛❧❧② s♣❛❝❡❞ ❛♥❞ ✐♥ ❜♦t❤ ❝❛s❡s t❤❡② ❢♦r♠ ♣❧❛♥❡s✳ ❚❤❡s❡ ♣❧❛♥❡s ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ t✇♦ ♥❡✇ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✱ t❤❡ t❛❜❧❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞✳ ❙♦♠❡ t❤✐♥❣s ❞♦♥✬t ❝❤❛♥❣❡✿ ❥✉st ❛s ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ❛r❡ ❧✐♥❡❛r✳

✸✳✼✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✻✻

❲❡ ❞❡❛❧ ✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ∆f ✱ r❡❧❛t✐✈❡ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ ✐ts ✐♥♣✉t ✈❛r✐❛❜❧❡✳ ❚❤✐s t✐♠❡✱ t❤❡r❡ ❛r❡ t✇♦✿ ∆x ❛♥❞ ∆y ✳ ■❢ ✇❡ ❦♥♦✇ ♦♥❧② ❢♦✉r ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✭❧❡❢t✮✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ∆y f ♦❢ f ❛❧♦♥❣ ❜♦t❤ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❡❞❣❡s ✭r✐❣❤t✮✿

y + ∆y : f (x, y + ∆y) − − − f (x + ∆x, y + ∆y) t: y:

|

|

|

|

|

|

f (x, y) x

−−−

−•−

∆x f (s, y + ∆y)

|

x + ∆x

|

−•−

−•− |

→ ∆y f (x, t)

f (x + ∆x, y)

s

❞✐✛❡r❡♥❝❡s ∆x f ❛♥❞

∆y f (x + ∆x, t) ∆x f (s, y)

x

s

|

−•−

x + ∆x

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ s✉❜tr❛❝t t❤❡ ✈❛❧✉❡s ❛t t❤❡ ❝♦r♥❡rs ✕ ✈❡rt✐❝❛❧❧② ❛♥❞ ❤♦r✐③♦♥t❛❧❧② ✕ ❛♥❞ ♣❧❛❝❡ t❤❡♠ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡s✳ ❲❡ t❤❡♥ ❛❝q✉✐r❡ t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❜② ❞✐✈✐❞✐♥❣ ❜② t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛r✐❛❜❧❡✿ −•−

∆f (s, y + ∆y) ∆x

|

→

∆f (x, t) ∆y | −•− x

−•− |

∆f (s, y) ∆x s

∆f (x + ∆x, t) ∆y | −•− x + ∆x

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❜✉✐❧❞ ❛ ♣❛rt✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐♥ t❤❡ xy ✲♣❧❛♥❡✳ ■t ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s [a, b] ❛♥❞ [c, d]✿

❲❡ st❛rt ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐♥ t❤❡ x✲❛①✐s ✐♥t♦ n ✐♥t❡r✈❛❧s✿

[x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] , ✇✐t❤ x0 = a, xn = b✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ x ❛r❡✿

∆xi = xi − xi−1 , i = 1, 2, ..., n . ❚❤❡♥ ✇❡ ❞♦ t❤❡ s❛♠❡ ❢♦r y ✳ ❲❡ ♣❛rt✐t✐♦♥ ❛♥ ✐♥t❡r✈❛❧ [c, d] ✐♥ t❤❡ y ✲❛①✐s ✐♥t♦ m ✐♥t❡r✈❛❧s✿

[y0 , y1 ], [y1 , y2 ], ..., [ym−1 , ym ] , ✇✐t❤ y0 = c, yn = d✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ y ❛r❡✿

∆yi = yi − yi−1 , i = 1, 2, ..., m .

✸✳✼✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

y = yj ❛♥❞ x = xi [xi , xi+1 ] × [yj , yj+1 ]✿

❚❤❡ ❧✐♥❡s

❝r❡❛t❡ ❛ ♣❛rt✐t✐♦♥

P

✷✻✼

♦❢ t❤❡ r❡❝t❛♥❣❧❡

[a, b] × [c, d]

✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s

❚❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ ❧✐♥❡s✱

Xij = (xi , yj ), i = 1, 2, ..., n, j = 1, 2, ..., m , ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ✭♣r✐♠❛r②✮ ❚❤❡

s❡❝♦♥❞❛r② ♥♦❞❡s

i = 0, 1, 2, ..., n − 1

♦❢

❛♥❞

♥♦❞❡s

♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

P ❛♣♣❡❛r ♦♥ ❡❛❝❤ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ i = 0, 1, 2, ..., m − 1✱ ✇❡ ❤❛✈❡✿

•

❛ ♣♦✐♥t

Sij

✐♥ t❤❡ s❡❣♠❡♥t

[xi , xi+1 ] × {yj }✱

•

❛ ♣♦✐♥t

Tij

✐♥ t❤❡ s❡❣♠❡♥t

{xi } × [yj , yj+1 ]✳

❛♥❞ ✈❡rt✐❝❛❧ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥❀ ❢♦r ❡❛❝❤ ♣❛✐r

❛♥❞

❚❤❡ ❧♦❝❛t✐♦♥s ✇✐t❤✐♥ t❤❡ s❡❣♠❡♥ts ❛r❡ ❛r❜✐tr❛r②✿

❲❡ ❝❛♥ ❤❛✈❡ ❧❡❢t✲ ❛♥❞ r✐❣❤t✲❡♥❞ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s ❛s ✇❡❧❧ ❛s ♠✐❞✲♣♦✐♥t ♦♥❡s✳ ❊①❛♠♣❧❡ ✸✳✼✳✷✿ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

❆s ❛ s✐♠♣❧❡ ❝❤♦✐❝❡✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s ♦❢

[c, d]✱ s❛② {tj }✿ • ❛ ♣♦✐♥t Sij = (si , yj ) ✐♥ t❤❡ s❡❣♠❡♥t [xi−1 , xi ] × {yj }✱ • ❛ ♣♦✐♥t Tij = (xi , tj ) ✐♥ t❤❡ s❡❣♠❡♥t {xi } × [yj−1 , yj ]✳

❛♥❞

[a, b]✱

s❛②

{si }✱

❛♥❞

✸✳✼✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✻✽

◆♦✇ ❝❛❧❝✉❧✉s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ❦♥♦✇♥ ♦♥❧② ❛t t❤❡ ♥♦❞❡s ✭❧❡❢t✮✿

❲❤❡♥ y = yj ✐s ✜①❡❞✱ ✐ts ❞✐✛❡r❡♥❝❡ ∆f ✐s ❝♦♠♣✉t❡❞ ♦✈❡r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ [xi , xi+1 ], i = 0, 1, 2..., n − 1 ♦❢ t❤❡ s❡❣♠❡♥t ✭r✐❣❤t✮✳ ❚❤✐s ❞❡✜♥❡s ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❙✐♠✐❧❛r ❢♦r ❡✈❡r② ✜①❡❞ x = xi ✳ ■t ✐s ❛ 1✲❢♦r♠✳ ■t ✐♥❝❧✉❞❡s ❜♦t❤ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❞✐✛❡r❡♥❝❡s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡s ✇❡ ❤❛✈❡ s❡❡♥✳ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (X) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥✳

❉❡✜♥✐t✐♦♥ ✸✳✼✳✸✿ ❞✐✛❡r❡♥❝❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ❡❞❣❡ N = [X, X + ∆X] ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆f (N ) = f (X + ∆X) − f (X)

❚❤❡ ❞❡✜♥✐t✐♦♥ ✇♦r❦ ❢♦r ❛❧❧ ❞✐♠❡♥s✐♦♥s✳ ❖❢t❡♥✱ ✇❡ ♣r❡❢❡r t♦ ❜❡ ♠♦r❡ s♣❡❝✐✜❝ ❛❜♦✉t t❤❡ ✐♥❝r❡♠❡♥ts ♦❢ X ✳ ❙✉♣♣♦s❡ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s Xij , i, j = 0, 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ ✇❡ ❢♦❧❧♦✇ t❤❡ t✇♦ ❛①❡s s❡♣❛r❛t❡❧②✿

❉❡✜♥✐t✐♦♥ ✸✳✼✳✹✿ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ❚❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆x f (Sij ) = f (Xij ) − f (Xi−1,j )

❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ y ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆y f (Tij ) = f (Xij ) − f (Xi,j−1 )

✸✳✼✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✻✾

❚❤❡ ❝♦♥♥❡❝t✐♦♥ ✐s ♦❜✈✐♦✉s✿

❚❤❡♦r❡♠ ✸✳✼✳✺✿ ❉✐✛❡r❡♥❝❡ ❆♥❞ P❛rt✐❛❧ ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ z = f (x, y) ✐s t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t ❡❛❝❤ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ♦r y ✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿   ∆ f (S ) x ij ∆f (N ) =  ∆y f (Tij )

✐❢ N = Sij , ✐❢ N = Tij .

❚❤✐s ✐s ♦✉r tr✉❡ ✐♥t❡r❡st ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✿

❉❡✜♥✐t✐♦♥ ✸✳✼✳✻✿ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆x f (Sij ) f (Xij ) − f (Xi−1,j ) ∆f (Sij ) = = ∆x ∆xi xi − xi−1

❆♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ y ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ ∆y f (Tij ) f (Xij ) − f (Xi,j−1 ) ∆f (Tij ) = = ∆y ∆yj yj − yj−1

❚❤❡s❡ t✇♦ ♥✉♠❜❡rs r❡♣r❡s❡♥t t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s ♦✈❡r t❤❡ ♥♦❞❡s r❡s♣❡❝t✐✈❡❧②✿

◆♦t❡ t❤❛t ❜♦t❤

∆f ∆x

❛♥❞

∆f ∆y

❛r❡ ❧✐t❡r❛❧❧② ❢r❛❝t✐♦♥s✳

❋♦r ❡❛❝❤ ♣❛✐r (i, j)✱ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♣♣❡❛r ❛s t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡

✸✳✼✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

✷✼✵

t❤r♦✉❣❤ t❤❡ t❤r❡❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ❛❜♦✈❡ t❤❡ t❤r❡❡ ❛❞❥❛❝❡♥t ♥♦❞❡s ✐♥ t❤❡ xy ✲♣❧❛♥❡✿ ( xi ,

yj ,

( xi+1 , yj , ( xi ,

❚❤✐s ♣❧❛♥❡ ✐s ❣✐✈❡♥ ❜②✿ y − f (xi , yj ) =

f (xi ,

yj )

)

f (xi+1 , yj )

)

yj+1 , f (xi ,

yj+1 ) )

∆f ∆f (Sij )(x − xi ) + (Tij )(y − yj ) . ∆x ∆y

❚❤✐s ♣❧❛♥❡ r❡str✐❝t❡❞ t♦ t❤❡ tr✐❛♥❣❧❡ ❢♦r♠❡❞ ❜② t❤♦s❡ t❤r❡❡ ♣♦✐♥ts ✐s ❛ tr✐❛♥❣❧❡ ✐♥ ♦✉r 3✲s♣❛❝❡✳ ❋♦r ❡❛❝❤ (i, j)✱ t❤❡r❡ ❢♦✉r s✉❝❤ tr✐❛♥❣❧❡s❀ ❥✉st t❛❦❡ (i ± 1, j ± 1)✳

❊①❡r❝✐s❡ ✸✳✼✳✼

❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s✱ ✇❤❡♥ t❛❦❡♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ s✉❝❤ ♣❛✐rs (i, j)✱ ❞♦ t❤❡s❡ tr✐❛♥❣❧❡s ❢♦r♠ ❛ ♠❡s❤ ❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ ❞♦ t❤❡② ✜t t♦❣❡t❤❡r ✇✐t❤♦✉t ❜r❡❛❦s❄

❋♦r ❛ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ ✐♥❞✐❝❡s ✿

❉✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts f (x + ∆x, y) − f (x, y) ∆f (s, y) = ∆x ∆x ∆f f (x, y + ∆y) − f (x, y) (x, t) = ∆y ∆y

❲❛r♥✐♥❣✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ♥♦t ❛ ❢r❛❝t✐♦♥✳

❊①❛♠♣❧❡ ✸✳✼✳✽✿ ❤②❞r❛✉❧✐❝ ❛♥❛❧♦❣② ❆ s✐♠♣❧❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤✐s ❞❛t❛ ✐s ✇❛t❡r ✢♦✇✱ ❛s ❢♦❧❧♦✇s✿ • ❊❛❝❤ ❡❞❣❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ r❡♣r❡s❡♥ts ❛ ♣✐♣❡✳ • ❚❤❡ ❢✉♥❝t✐♦♥ f (xi , yj ) r❡♣r❡s❡♥ts t❤❡ ✇❛t❡r ♣r❡ss✉r❡ ❛t t❤❡ ❥♦✐♥t (xi , yj )✳ • ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣r❡ss✉r❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ❛❞❥❛❝❡♥t ❥♦✐♥ts ❝❛✉s❡s t❤❡ ✇❛t❡r t♦ ✢♦✇✳ • ❚❤❡ ❞✐✛❡r❡♥❝❡s ∆x f (si , yj ) ❛♥❞ ∆y f (xi , tj ) ❛r❡ t❤❡ ✢♦✇ ❛♠♦✉♥ts ❛❧♦♥❣ t❤❡s❡ ♣✐♣❡s✳ (s , y ) ❛♥❞ ∆f (xi , tj ) ❛r❡ t❤❡ ✢♦✇ r❛t❡s ❛❧♦♥❣ t❤❡s❡ ♣✐♣❡s✳ • ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ∆f ∆x i j ∆y

✸✳✼✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts

❋♦r

❡❧❡❝tr✐❝ ❝✉rr❡♥t✱ s✉❜st✐t✉t❡ ✏❡❧❡❝tr✐❝ ♣♦t❡♥t✐❛❧✑

✷✼✶

❢♦r ✏♣r❡ss✉r❡✑✳

❲❤❡♥ t❤❡r❡ ❛r❡ ♥♦ s❡❝♦♥❞❛r② ♥♦❞❡s s♣❡❝✐✜❡❞✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢

• Sij = [xi , xi+1 ] × {yj }✱

Sij

❛♥❞

Tij

❛s st❛♥❞✐♥❣ ❢♦r t❤❡ ❡❞❣❡s t❤❡♠s❡❧✈❡s✿

❛♥❞

• Tij = {xi } × [yj , yj+1 ]✳ ❚❤❡② ❛r❡ t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ❚❛❦❡♥ ❛s ❛ ✇❤♦❧❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❲❡ ❜r✐♥❣ t❤❡ t✇♦ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts t♦❣❡t❤❡r ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡s✿

❉❡✜♥✐t✐♦♥ ✸✳✼✳✾✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r②

♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❣✐♦♥ ✐s ❞❡✜♥❡❞ ❛t s❡❝♦♥❞❛r② ♥♦❞❡s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

 ∆x f   (Sij )  ∆f ∆x (N ) = ∆y f  ∆X   (Tij ) ∆y ❚❤✐s ✐s ❛ r❡❛❧✲✈❛❧✉❡❞

✐❢

N = Sij ,

✐❢

N = Tij .

1✲❢♦r♠✳

❲❤② ❛r❡♥✬t t❤❡s❡ ✈❡❝t♦rs❄ ❋r♦♠ ♦✉r st✉❞② ♦❢ ♣❧❛♥❡s✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ✈❡❝t♦r ❢♦r♠❡❞ ❜② t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐s ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t✿

❚❤✐s ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞

1✲❢♦r♠✳

   ∆x f   (Sij ), 0    ∆x ∆ f  y  (Tij )  0, ∆y

✐❢

N = Sij ,

✐❢

N = Tij .

❚❤❡r❡ ♠❛② ❜❡ ♦t❤❡rs✱ ♠♦r❡ ❝♦♠♣❧❡①✳

✸✳✽✳ ❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡

✷✼✷

❊①❛♠♣❧❡ ✸✳✼✳✶✵✿ ❤②❞r❛✉❧✐❝ ❛♥❛❧♦❣② ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❤②❞r❛✉❧✐❝ ❛♥❛❧♦❣②✱ ♦♥❧② t❤❡ ✢♦✇ ❛❧♦♥❣ t❤❡ ♣✐♣❡ ♠❛tt❡rs ✇❤✐❧❡ ❛♥② ❧❡❛❦❛❣❡ ✐s ✐❣♥♦r❡❞✳ ❲❡ ❝❛♥ ❛❧s♦ s❡❡ ❛ ✢♦✇ ♦♥ ❛ s✉r❢❛❝❡✳ ❲❡ ❝♦♥s✐❞❡r ✈❡❝t♦r ✜❡❧❞s ✭♦r ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠s✮❀ t❤❡r❡ ✐s ❛ ✈❡❝t♦r ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ❡❞❣❡✿ F (s, y) = F (x, t) =





p(s, y) , q(s, y) r(s, y) , s(x, t)



, .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ✐s ❛❧s♦ ❛ ❝♦♠♣♦♥❡♥t ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿

❍♦✇❡✈❡r✱ ♦♥❧② t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡s❡ ✈❡❝t♦rs ♦♥ t❤❡ ❡❞❣❡s ❛✛❡❝t t❤❡ r♦t❛t✐♦♥ ♦❢ t❤❡ ❜❛❧❧✳

❉❡✜♥✐t✐♦♥ ✸✳✼✳✶✶✿ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞ ❆ ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ❛t ❡❛❝❤ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐s ❝❛❧❧❡❞ ❣r❛❞✐❡♥t ✐❢ t❤❡r❡ ✐s s✉❝❤ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ t❤❛t F (N ) · E = ∆f (N )

❢♦r ❡✈❡r② ❡❞❣❡ E ❛♥❞ ✐ts s❡❝♦♥❞❛r② ♥♦❞❡ N ✳ ■t ❢♦❧❧♦✇s t❤❛t✿ • ❚❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ F (N ) ✐s ❡q✉❛❧ t♦

• ❚❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ F (N ) ✐s ❡q✉❛❧ t♦

∆f (N ) ∆x

∆f (N ) ∆y

✇❤❡♥ N ✐s ♦♥ ❛ ❤♦r✐③♦♥t❛❧ ❡❞❣❡✳

✇❤❡♥ N ✐s ❛ ✈❡rt✐❝❛❧ ❡❞❣❡✳

❚❤❡ ♦t❤❡r ❝♦♠♣♦♥❡♥ts ❛r❡ ✐rr❡❧❡✈❛♥t✳

◆❡①t✱ ✇❡ ❝♦♥t✐♥✉❡ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ❛s ❜❡❢♦r❡✿ 



❞✐s❝r❡t❡  = ❝❛❧❝✉❧✉s lim  ∆X→0 ❝❛❧❝✉❧✉s ✸✳✽✳ ❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡

❘❡❝❛❧❧ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❆ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛t x = a ✐s ❢✉♥❝t✐♦♥ t❤❛t ❞❡✜♥❡s ❛ s❡❝❛♥t ❧✐♥❡✱ ✐✳❡✳✱ ❛ ❧✐♥❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st (a, f (a)) ❛♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ (x, f (x))✿

✸✳✽✳

❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡

✷✼✸

■ts s❧♦♣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ ∆f f (x) − f (a) = . ∆x x−a

◆♦✇✱ ❧❡t✬s s❡❡ ❤♦✇ t❤✐s ♣❧❛♥ ❛♣♣❧✐❡s t♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❆ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) ❛t (x, y) = (a, b) ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts ❛ s❡❝❛♥t

♣❧❛♥❡✱ ✐✳❡✳✱ ❛ ♣❧❛♥❡ ✐♥ t❤❡ xyz ✲s♣❛❝❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st (a, b, f (a, b)) ❛♥❞ t✇♦ ♦t❤❡r ♣♦✐♥ts ♦♥ t❤❡

❣r❛♣❤✳ ■♥ ♦r❞❡r t♦ ❡♥s✉r❡ t❤❛t t❤❡s❡ ♣♦✐♥ts ❞❡✜♥❡ ❛ ♣❧❛♥❡✱ t❤❡② s❤♦✉❧❞ ❜❡ ❝❤♦s❡♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡② ❛r❡♥✬t ♦♥ t❤❡ s❛♠❡ ❧✐♥❡✳ ❚❤❡ ❡❛s✐❡st ✇❛② t♦ ❛❝❝♦♠♣❧✐s❤ t❤❛t ✐s t♦ ❝❤♦♦s❡ t❤❡ ❧❛st t✇♦ t♦ ❧✐❡ ✐♥ t❤❡ x✲ ❛♥❞ t❤❡ y ✲❞✐r❡❝t✐♦♥s ❢r♦♠ (a, b)✱ ✐✳❡✳✱ (x, b) ❛♥❞ (a, y) ✇✐t❤ x 6= a ❛♥❞ y 6= b✳

❚❤❡ t✇♦ s❧♦♣❡s ✐♥ t❤❡s❡ t✇♦ ❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦ y ✿ ∆f f (x, b) − f (a, b) ∆f f (a, y) − f (a, b) = ❛♥❞ = . ∆x

❚❤❡ t✇♦ ❧✐♥❡s ❢♦r♠ t❤❡ s❡❝❛♥t

♣❧❛♥❡✳

x−a

∆y

y−b

✸✳✽✳

❚❤❡ ❛✈❡r❛❣❡ ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡s ♦❢ ❝❤❛♥❣❡

✷✼✹

❉❡✜♥✐t✐♦♥ ✸✳✽✳✶✿ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❚❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ❛t (x, y) = (a, b) ❛r❡ ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐ts ♦❢ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ x ❛t x = a ❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦ y ❛t y = b r❡s♣❡❝t✐✈❡❧②✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ f (x, b) − f (a, b) ∂f (a, b) = lim x→a ∂x x−a

❛♥❞

∂f f (a, y) − f (a, b) (a, b) = lim y→b ∂y y−b

❛s ✇❡❧❧ ❛s fx′ (a, b) ❛♥❞ fy′ (a, b)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ♦❜✈✐♦✉s ❝♦♥❝❧✉s✐♦♥✳

❚❤❡♦r❡♠ ✸✳✽✳✷✿ P❛rt✐❛❧ ❉❡r✐✈❛t✐✈❡s f y✿

❚❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇✐t❤ r❡s♣❡❝t t♦

x

❛♥❞ t♦

❛t

(x, y) = (a, b)

∂f d (a, b) = f (x, b) ∂x dx x=a

❛♥❞

❆s ❛ r❡s✉❧t✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ str❛✐❣❤t✲❢♦r✇❛r❞✳

❊①❛♠♣❧❡ ✸✳✽✳✸✿ ❛ ❝♦♠♣✉t❛t✐♦♥ ❋✐♥❞ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (x, y) = (x − y)ex+y

2

❛r❡ ❢♦✉♥❞ ❛s t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢

∂f d (a, b) = f (a, y) ∂y dy y=b

f

✸✳✾✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✼✺

❛t (x, y) = (0, 0)✿ d ∂f d 2 = = 1, = ex + xex (0, 0) = (x − 0)ex+0 xex ∂x dx dx x=0 x=0 x=0 ∂f d d 2 2 2 2 2 2 (0, 0) = (0 − y)e0+y − yey = = −ey − yey 2y = −ey − 2y 2 ey . ∂y dy dy y=0 y=0

❊①❡r❝✐s❡ ✸✳✽✳✹

❋✐♥❞ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (x, y) = ? ❊①❡r❝✐s❡ ✸✳✽✳✺

◆♦✇ ✐♥ r❡✈❡rs❡✳ ❋✐♥❞ ❛ ❢✉♥❝t✐♦♥ f ♦❢ t✇♦ ✈❛r✐❛❜❧❡s t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇❤✐❝❤ ❛r❡ t❤❡s❡✿ ∂f ∂f =? =? ∂x ∂y

✸✳✾✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❚❤✐s ✐s t❤❡ ❜✉✐❧❞✲✉♣ ❢♦r t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ • t❤❡ s❧♦♣❡s ♦❢ ❛ s✉r❢❛❝❡ ✐♥ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s • t❤❡ s❡❝❛♥t ❧✐♥❡s

• t❤❡ s❡❝❛♥t ♣❧❛♥❡s

• t❤❡ ♣❧❛♥❡s ❛♥❞ ✇❛②s t♦ r❡♣r❡s❡♥t t❤❡♠ • t❤❡ ❧✐♠✐ts

❉✉❡ t♦ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠ ✇❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t ❢♦r ♥♦✇✳ ▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡s s♦ ❢❛r✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✭❛t ❛ ♣♦✐♥t✮✐s ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❜❡❝❛✉s❡ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛❧❧♦✇❡❞ ❜② ✈❡❝t♦r ❛❧❣❡❜r❛✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s f (x) − f (a) x→a x−a

f ′ (a) = lim

♣❛r❛♠❡tr✐❝ ❝✉r✈❡s F (t) − F (a) F ′ (a) = lim t→a t−a

❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s f (X) − f (A) ??? X→A X −A

f ′ (A) = lim

❚❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❢❛✐❧s ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❲❡ ❝❛♥✬t ❞✐✈✐❞❡ ❜② ❛ ✈❡❝t♦r✦ ❚❤✐s ❢❛✐❧✉r❡ ✐s t❤❡ r❡❛s♦♥ ✇❤② ✇❡ st❛rt st✉❞②✐♥❣ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❝❛❧❝✉❧✉s ✇✐t❤ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ ♥♦t ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❈❛♥ ✇❡ ✜① t❤❡ ❞❡✜♥✐t✐♦♥❄

✸✳✾✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✼✻

❍♦✇ ❛❜♦✉t ✇❡ ❞✐✈✐❞❡ ❜② t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s ✈❡❝t♦r❄ ❚❤✐s ✐s ❛❧❧♦✇❡❞ ❜② ✈❡❝t♦r ❛❧❣❡❜r❛ ❛♥❞ t❤❡ r❡s✉❧t ✐s s♦♠❡t❤✐♥❣ s✐♠✐❧❛r t♦ t❤❡ r✐s❡ ♦✈❡r t❤❡ r✉♥ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❧♦♣❡✳ ❊①❛♠♣❧❡ ✸✳✾✳✶✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥

▲❡t✬s ❝❛rr② ♦✉t t❤✐s ✐❞❡❛ ❢♦r ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥✿

f (x, y) = 2x + y . ❚❤❡ ❧✐♠✐t✿

|2x + y| p , (x,y)→(0,0) x2 + y 2 lim

✐s t♦ ❜❡ ❡✈❛❧✉❛t❡❞ ❛❧♦♥❣ ❡✐t❤❡r ♦❢ t❤❡ ❛①❡s✿

|y| |2x| |2x + 0| |2 · 0 + y| = 2 6= lim p = 1. = lim = lim lim √ x→0 y→0 x2 + 02 x→0 |x| 02 + y 2 y→0 |y|

❚❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st ❜❡❝❛✉s❡ t❤❡ s❧♦♣❡s ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s✳

❚❤❡ ❧✐♥❡ ♦❢ ❛tt❛❝❦ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ t❤❡♥ s❤✐❢ts ✕ t♦ ✜♥❞✐♥❣ t❤❛t t❛♥❣❡♥t ♣❧❛♥❡✳ ❙♦♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s s❤♦✇♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❤❛✈❡ ♥♦ t❛♥❣❡♥t ♣❧❛♥❡s ❛t t❤❡✐r ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t②✳ ❇✉t s✉❝❤ ❛ ♣❧❛♥❡ ✐s ❥✉st ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✦ ❆s s✉❝❤✱ ✐t ✐s ❛ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ■♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ❛♠♦♥❣ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛t x = a ❛r❡ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❞❡✜♥❡ s❡❝❛♥t ❧✐♥❡s✱ ✐✳❡✳✱ ❧✐♥❡s ♦♥ t❤❡ xy ✲♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st (a, f (a)) ❛♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ (x, f (x))✳ ■ts s❧♦♣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

∆f f (x) − f (a) = . ∆x x−a ■♥ ❣❡♥❡r❛❧✱ t❤❡② ❛r❡ ❥✉st ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❣r❛♣❤s ❛r❡ ❧✐♥❡s t❤r♦✉❣❤ (a, f (a))❀ ✐♥ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ t❤❡② ❛r❡✿ l(x) = f (a) + m(x − a). ❲❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✕ ❜✉t ♥♦ ♦t❤❡r ❧✐♥❡ ✕ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿

❲❡ r❡❢♦r♠✉❧❛t❡ ♦✉r t❤❡♦r② ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮ s❧✐❣❤t❧②✳

✸✳✾✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✼✼

❉❡✜♥✐t✐♦♥ ✸✳✾✳✷✿ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❙✉♣♣♦s❡ y = f (x) ✐s ❛ ❞❡✜♥❡❞ ❛t x = a ❛♥❞ l(x) = f (a) + m(x − a)

✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛t t❤❛t ♣♦✐♥t✳ ❚❤❡♥✱ y = l(x) ✐s ❝❛❧❧❡❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ❛t x = a ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s❛t✐s✜❡❞✿ lim

x→a

f (x) − l(x) = 0. |x − a|

❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛❜♦✉t t❤❡ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿ ❡rr♦r = |f (x) − l(x)|.

❚❤✐s✱ ♥♦t ♦♥❧② t❤❡ ❡rr♦r ✈❛♥✐s❤❡s✱ ❜✉t ❛❧s♦ ✐t ✈❛♥✐s❤❡s r❡❧❛t✐✈❡ t♦ ❤♦✇ ❝❧♦s❡ ✇❡ ❛r❡ t♦ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ ✐✳❡✳✱ t❤❡ r✉♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✐s ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮✿

❚❤❡♦r❡♠ ✸✳✾✳✸✿ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ❋♦r ❖♥❡ ❱❛r✐❛❜❧❡ ■❢

l(x) = f (a) + m(x − a) ✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

f

❛t

x = a✱

m = f ′ (a) .

t❤❡♥

✸✳✾✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✼✽

❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ◆♦✇ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ❛t (x, y) = (a, b) ❛r❡✿

∂f ∂f f (x, b) − f (a, b) f (a, y) − f (a, b) (a, b) = lim ❛♥❞ (a, b) = lim . x→a y→b ∂x x−a ∂y y−b ❋r♦♠ ♦✉r st✉❞② ♦❢ ♣❧❛♥❡s✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ✈❡❝t♦r ❢♦r♠❡❞ ❜② t❤❡s❡ ❢✉♥❝t✐♦♥s t❤❛t ✐s ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t✿

❉❡✜♥✐t✐♦♥ ✸✳✾✳✹✿ ❣r❛❞✐❡♥t ❚❤❡ ❣r❛❞✐❡♥t ♦❢ f ❛t (x, y) = (a, b) ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛s ✇❡❧❧ ❛s t❤❡ ✈❡❝t♦r ♦❢ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

df ∇f (a, b) = (a, b) = dX



∂f ∂f (a, b), (a, b) ∂x ∂y



■♥ ❣❡♥❡r❛❧ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❥✉st ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❣r❛♣❤s ❛r❡ ♣❧❛♥❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (a, b, f (a, b))❀ ✐♥ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ t❤❡② ❛r❡✿

l(x, y) = f (a, b) + m(x − a) + n(y − b). ❲❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ ✕ ❜✉t ♥♦ ♦t❤❡r ♣❧❛♥❡ ✕ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿

✸✳✾✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✼✾

❉❡✜♥✐t✐♦♥ ✸✳✾✳✺✿ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❙✉♣♣♦s❡ y = f (x, y) ✐s ❞❡✜♥❡❞ ❛t (x, y) = (a, b) ❢✉♥❝t✐♦♥ ❛♥❞ l(x, y) = f (a, b) + m(x − a) + n(y − b)

✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛t t❤❛t ♣♦✐♥t✳ ❚❤❡♥✱ y = l(x, y) ✐s ❝❛❧❧❡❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ❛t (x, y) = (a, b) ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s❛t✐s✜❡❞✿ f (x, y) − l(x, y) = 0. (x,y)→(a,b) ||(x, y) − (a, b)|| lim

■♥ t❤❛t ❝❛s❡✱ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t (a, b) ❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ z = l(x, y) ✐s ❝❛❧❧❡❞ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ st✐❝❦ t♦ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❧♦♦❦ ❧✐❦❡ ♣❧❛♥❡ ♦♥ ❛ s♠❛❧❧ s❝❛❧❡✦ ❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛❜♦✉t t❤❡ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✿ ❡rr♦r = |f (x, y) − l(x, y)| .

❚❤❡ ❧✐♠✐t ♦❢

❡rr♦r ✐s r❡q✉✐r❡❞ t♦ ❜❡ ③❡r♦✳ r✉♥

❚❤❡♦r❡♠ ✸✳✾✳✻✿ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ❋♦r ❚✇♦ ❱❛r✐❛❜❧❡s ■❢

l(x, y) = f (a, b) + m(x − a) + n(y − b) ✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

m=

f

∂f (a, b) ∂x

❛t

x = a✱

❛♥❞

t❤❡♥

n=

∂f (a, b) . ∂y

Pr♦♦❢✳ ❚❤❡ ❧✐♠✐t ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❧❧♦✇s ✉s t♦ ❛♣♣r♦❛❝❤ (a, b) ✐♥ ❛♥② ✇❛② ✇❡ ❧✐❦❡✳ ▲❡t✬s st❛rt ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ❡rr♦r r✐s❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s ❛♥❞ s❡❡ ❤♦✇ ✐♥ ❝♦♥✈❡rt❡❞ ✐♥t♦ ✿ r✉♥ r✉♥ f (x, y) − l(x, y) x→a , y=b ||(x, y) − (a, b)|| f (x, y) − (f (a, b) + m(x − a) + n(b − b)) = lim+ x→a x−a f (x, y) − f (a, b) −m = lim+ x→a x−a ∂f (a, b) − m . = ∂x

0 =

lim +

❚❤❡ s❛♠❡ ❝♦♠♣✉t❛t✐♦♥ ❢♦r t❤❡ y ✲❛①✐s ♣r♦❞✉❝❡s t❤❡ s❡❝♦♥❞ ✐❞❡♥t✐t②✳

✸✳✾✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✽✵

❊①❛♠♣❧❡ ✸✳✾✳✼✿ ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞

▲❡t✬s ❝♦♥✜r♠ t❤❛t t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞✱ ❣✐✈❡♥ ❜② t❤❡ ❣r❛♣❤ ♦❢ f (x, y) = xy ✱ ❛t (0, 0) ✐s ❤♦r✐③♦♥t❛❧✿ ∂f d d d ∂f d (0, 0) = f (x, 0) (x · 0) (0, 0) = f (0, y) (0 · y) = = 0, = = 0. ∂x dx dx ∂y dy dx x=0 x=0 y=0 y=0

❚❤❡ t✇♦ t❛♥❣❡♥t ❧✐♥❡s ❢♦r♠ ❛ ❝r♦ss ❛♥❞ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ ✐s s♣❛♥♥❡❞ ♦♥ t❤✐s ❝r♦ss✳ ❏✉st ❛s ❜❡❢♦r❡✱ r❡♣❧❛❝✐♥❣ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❧✐♥❡❛r✐③❛t✐♦♥ ❛♥❞ ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ ✈❛❧✉❡s ♦❢ ♥❡✇ ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✾✳✽✿ ❛♣♣r♦①✐♠❛t✐♦♥

√

√

▲❡t✬s r❡✈✐❡✇ ❛ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡✿ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ 4.1✳ ❲❡ ❝❛♥✬t ❝♦♠♣✉t❡ x ❜② ❤❛♥❞ ❜❡❝❛✉s❡✱ ✐♥ ❛ √ √ s❡♥s❡✱ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x ✐s ✉♥❦♥♦✇♥✳ ❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ y = f (x) = x ✐s ❦♥♦✇♥ ❛♥❞✱ ❛s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ✐t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❤❛♥❞✿ 1 1 1 f ′ (x) = √ =⇒ f ′ (4) = √ = . 4 2 x 2 4

❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿ 1 l(x) = f (a) + f ′ (a)(x − a) = 2 + (x − 4) . 4 √ ❋✐♥❛❧❧②✱ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ 4.1 ✐s 1 1 l(4.1) = 2 + (4.1 − 4) = 2 + · 1 = 2 + 0.025 = 2.025 . 4 4 √

√

▲❡t✬s ♥♦✇ ❛♣♣r♦①✐♠❛t❡ ♦❢ 4.1 3 7.8✳ ■♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ t✇♦ t❡r♠s s❡♣❛r❛t❡❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ √ √ 3 t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♣r♦❞✉❝t f (x, y) = x y ❛t (x, y) = (4, 8)✳ ❚❤❡♥ ✇❡ ❝♦♠♣✉t❡ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✱ x✿

❆♥❞ y ✿

 ∂f 1 d d √ √ 1 3 = √ 2 (4, 8) = f (x, 8) x 8 = = . ∂x dx dx 2 x x=4 2 x=4 x=4

 d d √ √ ∂f 1 1/3 3 (4, 8) = f (4, y) . = = 4 y = 2p 3 ∂y dy dx y 2 y=8 12 y=8 y=8

❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿

∂f ∂f 1 1 (a, b)(x − a) + (a, b)(y − b) = 2 · 2 + (x − 4) + (y − 8) . ∂x ∂y 2 12 √ √ 3 ❋✐♥❛❧❧②✱ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ 4.1 7.8 ✐s 1 1 l(4.1, 7.8) = 4 + .1 + (−.2) = 4.033333... 2 12 l(x, y) = f (a, b) +

✸✳✶✵✳ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥

✷✽✶

❊①❡r❝✐s❡ ✸✳✾✳✾

❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f (x) = (xy)1/3 ❛t (1, 1)✳ ❚♦ s✉♠♠❛r✐③❡✿ • ❯♥❞❡r t❤❡ ❧✐♠✐t x → a✱ t❤❡ s❡❝❛♥t ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s t✉r♥s ✐♥t♦ ❛ t❛♥❣❡♥t ❧✐♥❡ ✇✐t❤ t❤❡ s❧♦♣❡ fx′ (a, b)✳ • ❯♥❞❡r t❤❡ ❧✐♠✐t y → b✱ t❤❡ s❡❝❛♥t ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s t✉r♥s ✐♥t♦ ❛ t❛♥❣❡♥t ❧✐♥❡s ✇✐t❤ t❤❡ s❧♦♣❡ fy′ (a, b)✱ ♣r♦✈✐❞❡❞ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡s❡ t✇♦ ✈❛r✐❛❜❧❡s ❛t t❤✐s ♣♦✐♥t✳ ▼❡❛♥✇❤✐❧❡✱ • ❚❤❡ s❡❝❛♥t ♣❧❛♥❡ t✉r♥s ✐♥t♦ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡✱ ♣r♦✈✐❞❡❞ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤✐s ♣♦✐♥t✱ ✇✐t❤ t❤❡ s❧♦♣❡s fx′ (a, b) ❛♥❞ fy′ (a, b) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

✸✳✶✵✳ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ ❲❡ ♣✐❝❦ ♦♥❡ ✈❛r✐❛❜❧❡ ❛♥❞ tr❡❛t t❤❡ r❡st ♦❢ t❤❡♠ ❛s ♣❛r❛♠❡t❡rs✳ ❚❤❡ r✉❧❡s ❛r❡ ❡①❛❝t❧② t❤❡ s❛♠❡✳ ❚❤✐s ✐s✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✿ ∂ ∂ ∂ ∂ (xy) = y (x) = y · 1 = y , (xy) = x (y) = x · 1 = x . ∂x ∂x ∂y ∂y

❚❤✐s ✐s t❤❡ ❙✉♠ ❘✉❧❡ ✿ ∂ ∂ ∂ ∂ ∂ ∂ (x + y) = (x) + (y) = 1 + 0 = 1 , (x + y) = (x) + (y) = 0 + 1 = 1 . ∂x ∂x ∂x ∂y ∂y ∂y

✸✳✶✵✳

P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥

✷✽✷

❊①❛♠♣❧❡ ✸✳✶✵✳✶✿ r❡❧❛t❡❞ r❛t❡s❄ ❚❤✐s ♥♦t✐♦♥ ✐s ♥♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ t❤❡ ✏r❡❧❛t❡❞ r❛t❡s✑ ✭❱♦❧✉♠❡ ✷✮✳ ❍❡r❡

❜② t❤❡

❈❤❛✐♥ ❘✉❧❡✳

❍❡r❡

✐s ❛ ❢✉♥❝t✐♦♥ ♦❢

x✿

 d d xy 2 = y 2 + x · (y 2 ) = y 2 + x · 2yy ′ , dx dx

y

✐s ❥✉st ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✿

❆❢t❡r ❛❧❧✱ t❤❡s❡ ✈❛r✐❛❜❧❡ ❛r❡

◆❡①t ✇❡ ❝♦♥s✐❞❡r

y

 ∂ ∂ 2 xy 2 = y 2 + x · (y ) = y 2 + x · 0 = y 2 . ∂x ∂x

✉♥ ✲r❡❧❛t❡❞❀ t❤❡② ❛r❡ ✐♥❞❡♣❡♥❞❡♥t

✈❛r✐❛❜❧❡s✦

♦♣t✐♠✐③❛t✐♦♥✱ ✇❤✐❝❤ ✐s s✐♠♣❧② ❛ ✇❛② t♦ ✜♥❞ t❤❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ✈❛❧✉❡s ♦❢ ❢✉♥❝t✐♦♥s✳

❲❡ ❛❧r❡❛❞② ❦♥♦✇ ❢r♦♠ t❤❡

❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠

t❤❛t t❤✐s ♣r♦❜❧❡♠ ❛❧✇❛②s ❤❛s ❛ s♦❧✉t✐♦♥ ♣r♦✈✐❞❡❞ t❤❡

❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ s✉❜s❡t✳ ❋✉rt❤❡r♠♦r❡✱ ❥✉st ❛s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✺✮✱ ✇❡ ✜rst ♥❛rr♦✇ ❞♦✇♥ ♦✉r s❡❛r❝❤✳ ❘❡❝❛❧❧ t❤❛t ❛ ❢✉♥❝t✐♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞

y = f (x) ❤❛s ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t ❛t a✳ ❲❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛s ✐❢ ✇❡ ❜✉✐❧❞ ❛

❛t

x=a

✐❢

f (a) ≤ f (x)

❢♦r ❛❧❧

x

✇✐t❤✐♥ s♦♠❡

r❡❝t❛♥❣❧❡ ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s ❛♥❞

✉s❡ t♦ ❝✉t ❛ ♣✐❡❝❡ ❢r♦♠ ♦✉r ❣r❛♣❤✿

❲❡ ❝❛♥✬t ✉s❡ ✐♥t❡r✈❛❧s ✐♥ ❞✐♠❡♥s✐♦♥

(a, b)✳

2✱

✇❤❛t✬s t❤❡✐r ❛♥❛❧♦❣❄

■t ✐s ❛♥ ♦♣❡♥ ❞✐s❦ ♦♥ t❤❡ ♣❧❛♥❡ ❝❡♥t❡r❡❞ ❛t

❲❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛s ✐❢ ✇❡ ❜✉✐❧❞ ❛ ❝②❧✐♥❞❡r ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ❞✐s❦s ❛♥❞ ✉s❡ t♦ ❝✉t ❛ ♣❛t❝❤ ❢r♦♠

t❤❡ s✉r❢❛❝❡ ♦❢ ♦✉r ❣r❛♣❤✿

■♥ ❢❛❝t✱ ❜♦t❤ ✐♥t❡r✈❛❧s ❛♥❞ ❞✐s❦s ✭❛♥❞ ✸❞ ❜❛❧❧s✱ ❡t❝✳✮ ❝❛♥ ❜❡ ❝♦♥✈❡♥✐❡♥t❧② ❞❡s❝r✐❜❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦✐♥t✿

d(X, A) ≤ ε✳

❙♦✱ ✇❡ r❡str✐❝t ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡s❡ ✭♣♦ss✐❜❧② s♠❛❧❧✮ ❞✐s❦s✳

❉❡✜♥✐t✐♦♥ ✸✳✶✵✳✷✿ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t

X = A ✐❢ f (A) ≤ f (X) ❢♦r ❛❧❧ X ✇✐t❤✐♥ s♦♠❡ ♣♦s✐t✐✈❡ ❞✐st❛♥❝❡ ❢r♦♠ A✱ ✐✳❡✳✱ d(X, A) ≤ ε✳ ❋✉rt❤❡r♠♦r❡✱ ❛ ❢✉♥❝t✐♦♥ f ❤❛s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t ❛t X = A ✐❢ f (A) ≥ f (X) ❢♦r ❛❧❧ X ❢♦r ❛❧❧ X ✇✐t❤✐♥ s♦♠❡ ♣♦s✐t✐✈❡ ❞✐st❛♥❝❡ ❢r♦♠ A✳ ❲❡ ❝❛❧❧ t❤❡s❡ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✱ ❆ ❢✉♥❝t✐♦♥

♦r ❡①tr❡♠❛✳

z = f (X)

❤❛s ❛

❛t

✸✳✶✵✳ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥

✷✽✸ ❲❛r♥✐♥❣✦

❚❤❡ ❞❡✜♥✐t✐♦♥ ✐♠♣❧✐❡s t❤❛t f ✐s ❞❡✜♥❡❞ ❢♦r ❛❧❧ ♦❢ t❤❡s❡ ✈❛❧✉❡s ♦❢ X ✳ ❊①❡r❝✐s❡ ✸✳✶✵✳✸

❲❤❛t ❝♦✉❧❞ ❜❡ ❛ ♣♦ss✐❜❧❡ ♠❡❛♥✐♥❣ ♦❢ ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♦♥❡✲s✐❞❡❞ ❞❡r✐✈❛t✐✈❡ ❢♦r ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ❞✐s❦ D ❛r♦✉♥❞ A s✉❝❤ t❤❛t A ✐s t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✭♦r ♠✐♥✐♠✉♠✮ ♣♦✐♥t ✇❤❡♥ f ✐s r❡str✐❝t❡❞ t♦ A✳ ◆♦✇✱ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥ts ❛r❡ ❝❛♥❞✐❞❛t❡s ❢♦r ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳ ❚♦ ❝♦♠♣❛r❡ t♦ t❤❡ ❢❛♠✐❧✐❛r✱ ♦♥❡✲ ❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ ✉s❡ t♦ ❧♦♦❦ ❛t t❤❡ t❡rr❛✐♥ ❢r♦♠ t❤❡ s✐❞❡ ❛♥❞ ♥♦✇ ✇❡ ❧♦♦❦ ❢r♦♠ ❛❜♦✈❡✿

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡♠❄ ❲❡ ❣♦ ❢r♦♠ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠ t♦ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ■♥❞❡❡❞✱ ✐❢ (a, b) ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦❢ z = f (x, y) t❤❡♥ • x = a ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ g(x) = f (x, b)✱ ❛♥❞ • y = b ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ h(y) = f (a, y)✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s✉♠♠✐t ✇✐❧❧ ❜❡ t❤❡ ❤✐❣❤❡st ♣♦✐♥t ♦❢ ♦✉r tr✐♣ ✇❤❡t❤❡r ✇❡ ❝♦♠❡ ❢r♦♠ t❤❡ s♦✉t❤ ♦r ❢r♦♠ t❤❡ ✇❡st✳ ❲❛r♥✐♥❣✦

✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤ ✇❡ ✐❣♥♦r❡ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❡✈❡♥ ❤✐❣❤❡r ❧♦❝❛t✐♦♥s ❢♦✉♥❞ ✐♥ t❤❡ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥✳ ❍♦✇❡✈❡r✱ t❤❡ ❞❛♥❣❡r ♦❢ ♠✐ss✐♥❣ t❤✐s ✐s ♦♥❧② ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ❞✐✛❡r❡♥t✐❛❜❧❡✳

❏✉st ❛s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✺✮✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ✐♥ ♦r❞❡r t♦ ❢❛❝✐❧✐t❛t❡ ♦✉r s❡❛r❝❤✳ ❘❡❝❛❧❧ ✇❤❛t ❋❡r♠❛t✬s ❚❤❡♦r❡♠ st❛t❡s✿ ✐❢ • x = a ✐s ❛ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ z = f (x) ❛♥❞ • z = f (x) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✱

✸✳✶✵✳ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ t❤❡♥ a ✐s ❛ ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✐✳❡✳✱

✷✽✹

f ′ (a) = 0 ,

♦r ✉♥❞❡✜♥❡❞✳ ❚❤❡ ❧❛st ❡q✉❛t✐♦♥ t②♣✐❝❛❧❧② ♣r♦❞✉❝❡s ❥✉st ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝❛♥❞✐❞❛t❡s ❢♦r ❡①tr❡♠❛✳ ❆s ✇❡ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ ❡✐t❤❡r ✈❛r✐❛❜❧❡✱ ✇❡ ✜♥❞ ♦✉rs❡❧✈❡s ✐♥ ❛ s✐♠✐❧❛r s✐t✉❛t✐♦♥ ❜✉t ✇✐t❤ t✇♦ ❡q✉❛t✐♦♥s✳

❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣ ♦❢ ❋❡r♠❛t✬s ❚❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✸✳✶✵✳✹✿ ❋❡r♠❛t✬s ❚❤❡♦r❡♠

■❢ X = A ✐s ❛ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t A ❢✉♥❝t✐♦♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s z = f (X) t❤❡♥ A ✐s ❛ ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✐✳❡✳✱ ∇f (A) = 0 ❊①❛♠♣❧❡ ✸✳✶✵✳✺✿ ❝r✐t✐❝❛❧ ♣♦✐♥ts

▲❡t✬s ✜♥❞ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ ❲❡ ❞✐✛❡r❡♥t✐❛t❡✿

f (x, y) = x2 + 3y 2 + xy + 7 . ∂f (x, y) = 2x + y , ∂x ∂f (x, y) = 6y + x . ∂x

❲❡ ❤♦✇ s❡t t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❡q✉❛❧ t♦ ③❡r♦✿ 2x + y = 0, 6y + x = 0.

❲❡ ❤❛✈❡ t✇♦ ❡q✉❛t✐♦♥s t♦ ❜❡ s♦❧✈❡❞✳ ●❡♦♠❡tr✐❝❛❧❧②✱ t❤❡s❡ ❛r❡ t✇♦ ❧✐♥❡s ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥✳ ❇② s✉❜st✐t✉t✐♦♥✿ 6y + x = 0 =⇒ x = −6y =⇒ 2(−6y) + y = 0 =⇒ y = 0 =⇒ x = 0 .

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ♣♦✐♥t (0, 0)✳ ■t ✐s ❥✉st ❛ ❝❛♥❞✐❞❛t❡ ❢♦r ❛♥ ❡①tr❡♠❡ ♣♦✐♥t✱ ❢♦r ♥♦✇✳ P❧♦tt✐♥❣ r❡✈❡❛❧s t❤❛t t❤✐s ✐s ❛ ♠✐♥✐♠✉♠✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤ ❧♦♦❦s ❧✐❦❡✿

✸✳✶✵✳

P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥

❚❤✐s ✐s ❛

✷✽✺

❡❧❧✐♣t✐❝ ♣❛r❛❜♦❧♦✐❞ ✭✐ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ❡❧❧✐♣s❡s✮✳

■♥st❡❛❞ ♦❢ ♦♥❡ ❡q✉❛t✐♦♥ ✲ ♦♥❡ ✈❛r✐❛❜❧❡ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❤❛✈❡ t✇♦ ✈❛r✐❛❜❧❡s ✲ t✇♦ ❡q✉❛t✐♦♥s ✭❛♥❞ t❤❡♥ t❤r❡❡ ✈❛r✐❛❜❧❡s ✲ t❤r❡❡ ❡q✉❛t✐♦♥s✱ ❡t❝✳✮✳ ❚②♣✐❝❛❧❧② t❤❡ r❡s✉❧t ✐s ✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts✳ ❊①❡r❝✐s❡ ✸✳✶✵✳✻

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛♥ ✐♥✜♥✐t❡❧② ♠❛♥② ❝r✐t✐❝❛❧ ♣♦✐♥ts✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ✕ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡ ③❡r♦ ✕ ❝❛♥ ❜❡ r❡✲st❛t❡❞ ✐♥ ❛ ✈❡❝t♦r ❢♦r♠ ✕ t❤❡ ❣r❛❞✐❡♥t ✐s ③❡r♦✿ fx (a, b) = 0 ❛♥❞ fy (a, b) = 0 ⇐⇒ ∇f (a, b) = 0 .

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ ✐s ❤♦r✐③♦♥t❛❧✳ ❙♦✱ ❛t t❤❡ t♦♣ ♦❢ t❤❡ ♠♦✉♥t❛✐♥✱ ✐❢ ✐t✬s s♠♦♦t❤ ❡♥♦✉❣❤✱ ✐t r❡s❡♠❜❧❡s ❛ ♣❧❛♥❡ ❛♥❞ t❤✐s ♣❧❛♥❡ ❝❛♥♥♦t ❜❡ t✐❧t❡❞ ❜❡❝❛✉s❡ ✐❢ ✐t s❧♦♣❡s ❞♦✇♥ ✐♥ ❛♥② ❞✐r❡❝t✐♦♥ t❤❡♥ s♦ ❞♦❡s t❤❡ s✉r❢❛❝❡✿

❲❤❛t ✇❡ ❤❛✈❡ s❡❡♥ ✐s

❧♦❝❛❧ ♦♣t✐♠✐③❛t✐♦♥ ❛♥❞ ✐t ✐s ♦♥❧② ❛ st❡♣♣✐♥❣ st♦♥❡ ❢♦r ❣❧♦❜❛❧ ♦♣t✐♠✐③❛t✐♦♥✳

❏✉st ❛s ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ t❤❡ ❜♦✉♥❞❛r② ♣♦✐♥ts ❤❛✈❡ t♦ ❜❡ ❞❡❛❧t ✇✐t❤ s❡♣❛r❛t❡❧②✳ ❚❤✐s t✐♠❡✱ ❤♦✇❡✈❡r✱ t❤❡r❡ ✇✐❧❧ ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡♠ ✐♥st❡❛❞ ♦❢ ❥✉st t✇♦ ❡♥❞✲♣♦✐♥ts✿

✸✳✶✵✳ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥

✷✽✻

❊①❛♠♣❧❡ ✸✳✶✵✳✼✿ ❡①tr❡♠❡ ♣♦✐♥ts

▲❡t✬s ✜♥❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢

f (x, y) = x2 + y 2 ,

s✉❜❥❡❝t t♦ t❤❡ r❡str✐❝t✐♦♥ ✭♦❢ t❤❡ ❞♦♠❛✐♥✮✿ |x| ≤ 1, |y| ≤ 1 .

❚❤❡ r❡str✐❝t✐♦♥ ✐s t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤✐s sq✉❛r❡✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡✱ s❡t t❤❡ ❞❡r✐✈❛t✐✈❡s ❡q✉❛❧ t♦ ③❡r♦✱ ❛♥❞ s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ fx (x, y) = 2x = 0 =⇒ x = 0 fy (x, y) = 2y = 0 =⇒ y = 0

=⇒ (a, b) = (0, 0)

❚❤✐s ✐s t❤❡ ♦♥❧② ❝r✐t✐❝❛❧ ♣♦✐♥t✳ ❲❡ t❤❡♥ ♥♦t❡ t❤❛t f (0, 0) = 0

❜❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣✳ ❚❤❡ ❜♦✉♥❞❛r② ♣♦✐♥ts ♦❢ ♦✉r sq✉❛r❡ ❞♦♠❛✐♥ r❡♠❛✐♥ t♦ ❜❡ ✐♥✈❡st✐❣❛t❡❞✿ |x| = 1, |y| = 1 .

❲❡ ❤❛✈❡ t♦ t❡st ❛❧❧ ♦❢ t❤❡♠ ❜② ❝♦♠♣❛r✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❡❛❝❤ ♦t❤❡r ❛♥❞ t♦ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❛r❡ s♦❧✈✐♥❣ s❡✈❡r❛❧ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✦ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿ |x| = 1 =⇒ f (x, y) = 1 + y 2 =⇒ max{1 + y 2 } = 2 ❢♦r y = ±1, min{1 + y 2 } = 1 ❢♦r y = 0 |y|≤1

|y|≤1

|y| = 1 =⇒ f (x, y) = x + 1 =⇒ max{x + 1} = 2 ❢♦r x = ±1, min {x2 + 1} = 1 ❢♦r x = 0 2

|x|≤1

2

|x|≤1

❈♦♠♣❛r✐♥❣ t❤❡s❡ ♦✉t♣✉ts ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t✿ • ❚❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡ ♦❢ 2 ✐s ❛tt❛✐♥❡❞ ❛t t❤❡ ❢♦✉r ♣♦✐♥ts (1, 1), (−1, 1), (1, −1), (−1, −1)✳ • ❚❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ ♦❢ 0 ✐s ❛tt❛✐♥❡❞ ❛t (0, 0)✳ ❚♦ ❝♦♥✜r♠ ♦✉r ❝♦♥❝❧✉s✐♦♥s ✇❡ ❥✉st ❧♦♦❦ ❛t t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢❛♠✐❧✐❛r ♣❛r❛❜♦❧♦✐❞ ♦❢ r❡✈♦❧✉t✐♦♥✿

❯♥❞❡r t❤❡ ❞♦♠❛✐♥ r❡str✐❝t✐♦♥s✱ t❤❡ ❣r❛♣❤ ❞♦❡s♥✬t ❧♦♦❦ ❧✐❦❡ ❛ ❝✉♣ ❛♥②♠♦r❡✳✳✳ ❲❡ ❝❛♥ s❡❡ t❤❡ ✈❡rt✐❝❛❧ ♣❛r❛❜♦❧❛s ✇❡ ❥✉st ♠❛①✐♠✐③❡❞✳ ❋♦r ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✱ t❤❡ ❞♦♠❛✐♥s ❛r❡ t②♣✐❝❛❧❧② s♦❧✐❞s✳ ❚❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡s❡ ❞♦♠❛✐♥s ❛r❡ t❤❡♥ s✉r❢❛❝❡s✦ ❚❤❡♥✱ ❛ ♣❛rt ♦❢ s✉❝❤ ❛ ✸❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐❧❧ ❜❡ ❛ ✷❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳✳✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ♣❛✐r ♦❢ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛s ♦♥❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❍♦✇❡✈❡r✱ ✐♥ s❤❛r♣ ❝♦♥tr❛st t♦ t❤❡ ♦♥❡✲✈❛r✐❛❜❧❡ ❝❛s❡✱ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥ ❤❛s ❛ ✈❡r② ❞✐✛❡r❡♥t ♥❛t✉r❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧✿ s❛♠❡ ✐♥♣✉t ❜✉t t❤❡ ♦✉t♣✉t ✐s♥✬t ❛ ♥✉♠❜❡r ❛♥②♠♦r❡ ❜✉t ❛ ✈❡❝t♦r✦ ■t✬s ❛ ✈❡❝t♦r ✜❡❧❞ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹✳

✸✳✶✶✳

❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡

✷✽✼

✸✳✶✶✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡✲ ♣❡❛t❡❞ ✈❛r✐❛❜❧❡ ❲❡ st❛rt ✇✐t❤

♥✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

❊①❛♠♣❧❡ ✸✳✶✶✳✶✿ ❛ ❝♦♠♣✉t❛t✐♦♥

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥✿

f (x, y) = −x2 + y 2 + xy .

❲❤❡♥ ♣❧♦tt❡❞✱ ✐t ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛ ❢❛♠✐❧✐❛r ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞✳

❇❡❧♦✇ ✇❡ ♦✉t❧✐♥❡ t❤❡ ♣r♦❝❡ss ♦❢ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡ ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✕ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ✕ ❛r❡ s❤♦✇♥ ❛♥❞ s♦ ❛r❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡s❡ ❛r❡ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ✇✐❧❧ ❞✐✛❡r❡♥t✐❛t❡✳ ❋✐rst✱ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ y ❣✐✈❡♥ ✐♥ t❤❡ t♦♣ r♦✇✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ x ❜② ❣♦✐♥❣ ❞♦✇♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❧✉♠♥ ❛♥❞ t❤❡♥ ♣❧❛❝✐♥❣ t❤❡s❡ ✈❛❧✉❡s ♦♥ r✐❣❤t ✐♥ ❛ ♥❡✇ t❛❜❧❡ ✭✐t ✐s ♦♥❡ r♦✇ s❤♦rt ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧✮✿

❂✭❘❈❬✲✷✾❪✲❘❬✲✶❪❈❬✲✷✾❪✮✴❘✷❈✶ ❙❡❝♦♥❞✱ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ x ❣✐✈❡♥ ✐♥ t❤❡ ❧❡❢t✲♠♦st ❝♦❧✉♠♥✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ y ❜② ❣♦✐♥❣ r✐❣❤t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r♦✇ ❛♥❞ t❤❡♥ ♣❧❛❝✐♥❣ t❤❡s❡ ✈❛❧✉❡s ❜❡❧♦✇ ✐♥ ❛ ♥❡✇ t❛❜❧❡ ✭✐t ✐s ♦♥❡ ❝♦❧✉♠♥ s❤♦rt ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧✮✳

❂✭❘❬✲✷✾❪❈✲❘❬✲✷✾❪❈❬✲✶❪✮✴❘✷❈✶ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿

✸✳✶✶✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡

✷✽✽

❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛❞❥❛❝❡♥t t♦ t❤❡ ♦r✐❣✐♥❛❧ ❛r❡ t❤❡ ❢❛♠✐❧✐❛r ✜rst ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ t♦♣ r✐❣❤t ❛♥❞ t❤❡ ❜♦tt♦♠ ❧❡❢t ❛r❡ ❝♦♠♣✉t❡❞ ❢♦r t❤❡ ✜rst t✐♠❡ ❛s ❞❡s❝r✐❜❡❞✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ♠✐❞❞❧❡ ✐s ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ r❡s✉❧ts ❛r❡ ❛❧❧ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ❡q✉❛❧❧② s♣❛❝❡❞ ❛♥❞ ✐♥ ❜♦t❤ ❝❛s❡s t❤❡② ❢♦r♠ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡s✳ ❚❤❡s❡ ♣❧❛♥❡s ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ t✇♦ ♥❡✇ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✱ t❤❡ t❛❜❧❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞✳ ❙♦♠❡ t❤✐♥❣s ❞♦♥✬t ❝❤❛♥❣❡✿ ❥✉st ❛s ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥st❛♥t✳ ❍❡r❡ ✐s ❛ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡✿ f (x, y) = −x2 + y 2 + xy + sin(x + y + 3) .

✸✳✶✶✳

❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡

✷✽✾

❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❛t✱ ❥✉st ❛s ✇✐t❤ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✹✮ ❛♥❞ ❥✉st ❛s ✇✐t❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ ❈❤❛♣t❡r ✷✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡ t❤❛t s❦✐♣s ❛ ♥♦❞❡✳ ❚❤❛t✬s ✇❤② t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❞♦❡s♥✬t ♣r♦✈✐❞❡ ✉s ✕ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡ ✕ ✇✐t❤ ❛♥② ♠❡❛♥✐♥❣❢✉❧ ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ✇✐❧❧ ❧✐♠✐t ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ❲❡ ✇✐❧❧ ❝❛rr② ♦✉t t❤❡ s❛♠❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝♦♥str✉❝t✐♦♥ ❢♦r f ✇✐t❤ r❡s♣❡❝t t♦ x ✇✐t❤ y ✜①❡❞ ❛♥❞ t❤❡♥ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s s✉♠♠❛r✐③❡❞ ✐♥ t❤✐s ❞✐❛❣r❛♠✿

− f (x1 ) − − − ∆f − −•− ∆x2 − −•− −−− x1

f (x2 ) −•−

∆f ∆x3

−

∆f ∆x2

s3 − s2 x2

s2

− − − f (x3 ) − ∆f −•− − ∆x3 −−− −•− − s3

x3

❚❤❡ t✇♦ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✐❧❧✉str❛t❡❞ ❜② ❝♦♥t✐♥✉✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧✐❛r ❞✐❛❣r❛♠ ❢♦r t❤❡ ✭✜rst✮ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

f (x, y + ∆y) − − − f (x + ∆x, y + ∆y) |

|

|

|

|

|

f (x, y)

−−−

∆f f (s, y + ∆y) ∆x

−•−

❧❡❛❞✐♥❣ t♦

f (x + ∆x, y)

|

∆f (x, t) ∆y | −•−

−•− |

∆f (s, y) ∆x

∆f (x + ∆x, t) ∆y | −•−

❚❤❡r❡ ❛r❡ t✇♦ ❢✉rt❤❡r ❞✐❛❣r❛♠s✳ ❲❡✱ r❡s♣❡❝t✐✈❡❧②✱ s✉❜tr❛❝t t❤❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ t❤❡ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s ❤♦r✐③♦♥t❛❧❧② ✭s❤♦✇♥ ❜❡❧♦✇✮ ❛♥❞ ✇❡ s✉❜tr❛❝t t❤❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ t❤❡ ✈❡rt✐❝❛❧ ❡❞❣❡s ✈❡rt✐❝❛❧❧② ❜❡❢♦r❡ ♣❧❛❝✐♥❣ t❤❡ r❡s✉❧ts ❛t t❤❡ ♥♦❞❡s ✭♥♦t s❤♦✇♥✮✿

−•−

∆f ∆f (s, y) − • − (s + ∆s, y) − • − ∆x ∆x

❧❡❛❞✐♥❣ t♦

∆ ∆f ∆x (x, y) −− − • − − • − −− ∆x

❘❡❝❛❧❧ ❢r♦♠ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r t❤❛t ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐♥ t❤❡ xy ✲♣❧❛♥❡ ❜✉✐❧t ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s [a, b] ❛♥❞ [c, d]✿

a = x0 ≤ x1 ≤ x2 ≤ ... ≤ xn−1 ≤ xn = b , ❛♥❞ ■ts ♣r✐♠❛r② ♥♦❞❡s ❛r❡✿

c = y0 ≤ y1 ≤ y2 ≤ ... ≤ ym−1 ≤ ym = d . Xij = (xi , yj ), i = 1, 2, ..., n, j = 1, 2, ..., m .

❆♥❞ ✐ts s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡✿

• ❛ ♣♦✐♥t Sij ✐♥ t❤❡ s❡❣♠❡♥t [xi−1 , xi ] × {yj }✱ ❛♥❞

• ❛ ♣♦✐♥t Tij ✐♥ t❤❡ s❡❣♠❡♥t {xi } × [yj−1 , yj ]✳

✸✳✶✶✳

❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡

✷✾✵

■❢ z = f (x, y) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s Xij , i, j = 0, 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ❛r❡ r❡s♣❡❝t✐✈❡❧②✿

f (Xij ) − f (Xi−1,j ) ∆f (Sij ) = ∆x xi − xi−1

❛♥❞

∆f f (Xij ) − f (Xi,j−1 ) (Tij ) = . ∆y yj − yj−1

■t ✐s ♥♦✇ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t t❤❛t ✇❡ ❤❛✈❡ ✉t✐❧✐③❡❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥s✳ ■♥❞❡❡❞✱ ✇❡ ❝❛♥ ♥♦✇ ❝❛rr② ♦✉t ❛ s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥ ✇✐t❤ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ ✜♥❞ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ t❤❡ ❢♦✉r ♦❢ t❤❡♠✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s ❜❛s❡❞ ♦♥ t❤❡ ♦♥❡ ✇❡ ❝❛rr✐❡❞ ♦✉t ❢♦r ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✿

❲❡ ✇✐❧❧ st❛rt ✇✐t❤ t❤❡s❡

t✇♦ ♥❡✇ q✉❛♥t✐t✐❡s✳

❋✐rst✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ∆f ✕ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ✕ ✇✐t❤ r❡s♣❡❝t t♦ ∆x x✱ ❛❣❛✐♥✳ ❋♦r ❡❛❝❤ ✜①❡❞ j ✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✐♥t❡r✈❛❧ [a, b] × {yj } ✇✐t❤ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ Sij = (sij , yj ), i = 1, 2, ..., n , ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

Xij , i = 1, 2, ..., n − 1 .

❲❡ ♥♦✇ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝♦♥str✉❝t✐♦♥✱ st✐❧❧ ✇✐t❤✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s✳ ❙✐♠✐❧❛r❧②✱ t♦ ❞❡✜♥❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ∆f ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❋♦r ❡❛❝❤ ✜①❡❞ i✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥ ♦❢ ∆y t❤❡ ✈❡rt✐❝❛❧ ✐♥t❡r✈❛❧ {xi } × [c, d] ✇✐t❤ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿

Tij = (xi , tij ), j = 1, 2, ..., m , ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

Xij , j = 1, 2, ..., m − 1 .

❲❡ ♥♦✇ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝♦♥str✉❝t✐♦♥✱ st✐❧❧ ✇✐t❤✐♥ t❤❡ ✈❡rt✐❝❛❧ ❡❞❣❡s✳

✸✳✶✷✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s

✷✾✶

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✷✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ✭r❡♣❡❛t❡❞✮ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ x ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥ P ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2 f (Xij ) = ∆x2

− ∆f (Si,j−1 ) ∆x sij − si−1,j

∆f (Sij ) ∆x

✇✐t❤ i = 1, 2, ..., n − 1, j = 0, 1, ..., m✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ y ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥ P ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2 f (Xij ) = ∆y 2

∆f (Tij ) ∆y

−

∆f (Ti,j−1 ) ∆y

tij − ti,j−1

✇✐t❤ i = 0, 1, ..., n, j = 1, 2, ..., m − 1✳ ❚❤❡s❡ ❛r❡ ❞✐s❝r❡t❡ 0✲❢♦r♠s✳ ■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ s✐❣♥ ♦❢ t❤❡s❡ ❡①♣r❡ss✐♦♥s r❡✈❡❛❧ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ❝✉r✈❡s✳ ❋♦r ❛ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ ✐♥❞✐❝❡s ✿

❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ∆2 f (x, y) = ∆x2

∆f (s ∆x

∆2 f (x, y) = ∆y 2

∆f (x, t ∆y

+ ∆s, y) − ∆s + ∆t) −

∆f (s, y) ∆x ∆f (x, t) ∆y

∆t

✸✳✶✷✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦t❤❡r ✈❛r✐❛❜❧❡✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ❝♦♥❝❛✈✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ✐s ✐♥❝r❡❛s✐♥❣ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ y ✿

✸✳✶✷✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s

✷✾✷

❲❡ ❛r❡ ♥♦t ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ s❡tt✐♥❣ ❛♥②♠♦r❡✦ ❚❤❛t ✐s ✇❤②✱ ✐♥ ❝♦♥tr❛st t♦ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✱ t❤❡ ❝❛s❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ ❈❤❛♣t❡r ✷✱ ❛♥❞ t❤❡ r❡♣❡❛t❡❞ ✈❛r✐❛❜❧❡ ❝❛s❡ ❛❜♦✈❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ♥♦t s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡ t❤❛t s❦✐♣s ❛ ♥♦❞❡✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s ❞♦❡s ♣r♦✈✐❞❡ ✉s ✇✐t❤ ♠❡❛♥✐♥❣❢✉❧ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡s✿

∆x f (Sij ) ❛♥❞ ∆y f (Tij ) ,

❛♥❞ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

∆f (Sij ) ❛♥❞ ∆x

∆f (Tij ) . ∆y

❛r❡ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❧♦❝❛t❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d]✱ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✿

❚❤❡r❡ ❛r❡ ❢♦✉r ♠♦r❡ q✉❛♥t✐t✐❡s✿ • t❤❡ ❝❤❛♥❣❡ ♦❢ ∆x f ✇✐t❤ r❡s♣❡❝t t♦ y

• t❤❡ ❝❤❛♥❣❡ ♦❢ ∆y f ✇✐t❤ r❡s♣❡❝t t♦ x

• t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢

• t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢

∆f ∆x

✇✐t❤ r❡s♣❡❝t t♦ y

∆f ∆y

✇✐t❤ r❡s♣❡❝t t♦ x

❚❤❡② ✇✐❧❧ ❜❡ ❛ss✐❣♥❡❞ t♦ t❤❡ ♥♦❞❡s ❧♦❝❛t❡❞ ❛t t❤❡ ❢❛❝❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥✳ ❚❤❡s❡ ❛r❡ t❡rt✐❛r② ♥♦❞❡s✳

✸✳✶✷✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s

✷✾✸

❚❤❡ ♥❡✇ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✐❧❧✉str❛t❡❞ ❜② ❝♦♥t✐♥✉✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧✐❛r ❞✐❛❣r❛♠ ❢♦r t❤❡ ✭✜rst✮ ❞✐✛❡r❡♥❝❡s✿ f (x, y + ∆y) − − − f (x + ∆x, y + ∆y) |

|

|

|

|

f (x, y)

−•−

❧❡❛❞✐♥❣ t♦

|

−−−

−•−

∆x f (s, y + ∆y)

|

|

∆y f (x, t)

∆y f (x + ∆x, t)

|

|

−•−

f (x + ∆x, y)

−•−

∆x f (s, y)

❚❤❡r❡ ❛r❡ t✇♦ ❢✉rt❤❡r ❞✐❛❣r❛♠s✳ ❲❡✱ r❡s♣❡❝t✐✈❡❧②✱ s✉❜tr❛❝t t❤❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ t❤❡ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s ✈❡rt✐❝❛❧❧② ❛♥❞ ✇❡ s✉❜tr❛❝t t❤❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ t❤❡ ✈❡rt✐❝❛❧ ❡❞❣❡s ❤♦r✐③♦♥t❛❧❧② ❜❡❢♦r❡ ♣❧❛❝✐♥❣ t❤❡ r❡s✉❧ts ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ sq✉❛r❡✿ ❧❡❛❞✐♥❣ t♦

−•− |

−•−

−−−

−•−

−−−

−•−

∆y ∆x f (s, t)

❧❡❛❞✐♥❣ t♦

|

−•− |

−−−

−•−

−−−

−•−

∆x ∆y f (s, t)

−•−

|

❉❡✜♥✐t✐♦♥ ✸✳✶✷✳✶✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❚❤❡ ✭♠✐①❡❞✮ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ yx ✐s ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥ P ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2yx f (Uij ) = ∆x f (Sij ) − ∆x f (Si,j−1 )

✇✐t❤ i = 1, 2, ..., n − 1, j = 1, 2, ..., m − 1✳ ❚❤❡ ♠✐①❡❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ xy ✐s ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥ P ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2xy f (Uij ) = ∆y f (Tij ) − ∆y f (Ti−1,j )

✇✐t❤ i = 1, 2, ..., n − 1, j = 1, 2, ..., m − 1✳

❉❡✜♥✐t✐♦♥ ✸✳✶✷✳✷✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ✭♠✐①❡❞✮ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ yx ✐s ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥ P ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2 f (Uij ) = ∆y∆x

∆f (Sij ) ∆x

− ∆f (Si,j−1 ) ∆x yj − yj−1

✇✐t❤ i = 1, 2, ..., n − 1, j = 1, 2, ..., m − 1✳ ❚❤❡ ♠✐①❡❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f

✸✳✶✷✳

❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✐①❡❞ ✈❛r✐❛❜❧❡s ✇✐t❤ r❡s♣❡❝t t♦ xy

✷✾✹

✐s ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥

P

❜②

t❤❡ ❢♦❧❧♦✇✐♥❣✿

∆2 f (Uij ) = ∆x∆y ✇✐t❤

∆f (Ti,j ) ∆y

−

∆f (Ti−1,j ) ∆y

xi − xi−1

i = 1, 2, ..., n − 1, j = 1, 2, ..., m − 1✳

❲❤❡♥ t❤❡ t❡rt✐❛r② ♥♦❞❡s ❛r❡ ✉♥s♣❡❝✐✜❡❞✱ t❤❡s❡ ❛r❡ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❢❛❝❡s t❤❡♠s❡❧✈❡s✱ ✐✳❡✳✱ ❞✐s❝r❡t❡ ❋♦r ❛ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✱ ✇❡ ✇✐❧❧ ♦❢t❡♥

2✲❢♦r♠s✳

♦♠✐t t❤❡ ✐♥❞✐❝❡s ✿

❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts + ∆y) − ∆y

∆f (s, y) ∆x

+ ∆x, t) −

∆f (x, t) ∆y

∆2 f (s, t) = ∆y∆x

∆f (s, y ∆x

∆2 f (s, t) = ∆x∆y

∆f (x ∆y

∆x

◆♦✇✱ t❤❡ t✇♦ ♠✐①❡❞ ❞✐✛❡r❡♥❝❡s ❛r❡ ✏♠❛❞❡ ♦❢ ✑ t❤❡ s❛♠❡ ❢♦✉r q✉❛♥t✐t✐❡s❀ ♦♥❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡② ❛r❡ ❡q✉❛❧✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ s❡❡ ✐t ❤❡r❡✿

▲❡t✬s ❝♦♥✜r♠ t❤❛t ✇✐t❤ ❛❧❣❡❜r❛✳ ❲❡ s✉❜st✐t✉t❡ ❛♥❞ s✐♠♣❧✐❢②✿

∆2yx f (s, t) = ∆x f (s, y + ∆y) − ∆x f (s, y)

  = f (x + ∆x, y + ∆y) − f (x, y + ∆y) − f (x + ∆x, y) − f (x, y)

= f (x + ∆x, y + ∆y) − f (x, y + ∆y) − f (x + ∆x, y) + f (x, y)   = f (x + ∆x, y + ∆y) − f (x + ∆x, y) − f (x, y + ∆y) − f (x, y)

= ∆y f (x + ∆x, t) − ∆y f (x, t)

= ∆2xy f (s, t) .

❚❤❡♦r❡♠ ✸✳✶✷✳✸✿ ❉✐s❝r❡t❡ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠ ❖✈❡r ❛ ♣❛rt✐t✐♦♥ ✐♥ Rn ✱ ✜rst✱ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ ∆2yx f = ∆2xy f

✸✳✶✸✳

❚❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s

✷✾✺

❛♥❞✱ s❡❝♦♥❞✱ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ ∆2 f ∆2 f = ∆y∆x ∆x∆y ❚❤✐s t❤❡♦r❡♠ ✇✐❧❧ ❤❛✈❡ ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✹✳

✸✳✶✸✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s

❚❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✐s ❦♥♦✇♥ t♦ ❤❡❧♣ t♦ ❝❧❛ss✐❢② ❡①tr❡♠❡ ♣♦✐♥ts✳ ❆t ❧❡❛st✱ ✇❡ ❝❛♥ ❞✐s♠✐ss t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t ❛ ♣♦✐♥t ✐s ❛ ♠❛①✐♠✉♠ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥❝❛✈❡ ✉♣✱ ✐✳❡✳✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✐s ♣♦s✐t✐✈❡✳ ❊①❛♠♣❧❡ ✸✳✶✸✳✶✿ ❛ ❝♦♠♣✉t❛t✐♦♥

■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = x2 + 3y 2 + xy + 7 , ✇❡ ❞✐✛❡r❡♥t✐❛t❡ ♦♥❡ ♠♦r❡ t✐♠❡ ❡❛❝❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ✕ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡✿

  ∂ ∂f ∂ ∂f = 2x + y =⇒ = (2x + y) = 2 ∂x ∂x  ∂x  ∂x ∂f ∂ ∂ ∂f = = 6y + x =⇒ (6y + x) = 6 ∂x ∂y ∂y ∂x ❇♦t❤ ♥✉♠❜❡rs ❛r❡ ♣♦s✐t✐✈❡✱ t❤❡r❡❢♦r❡✱ ❜♦t❤ ❝✉r✈❡s ❛r❡ ❝♦♥❝❛✈❡ ✉♣✦ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♣r♦❞✉❝❡s ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ♦♥❡✲✈❛r✐❛❜❧❡ ❝❛s❡✿ d

d

d

d

d

f −−−dx−−→ f ′ −−−dx−−→ f ′′ −−−dx−−→ ... −−−dx−−→ f (n) −−−dx−−→ ... ■♥ t❤❡ t✇♦✲✈❛r✐❛❜❧❡ ❝❛s❡✱ ✐t ♠❛❦❡s t❤❡ ❢✉♥❝t✐♦♥s

♠✉❧t✐♣❧② ✿ f

fx fxx ւx ...

ւx y

ց

...

ւx y

ց

ւx ...

y

fyx

fxy

ց

fy

ւx y

ց

...

y

ց

ւx ...

fyy y

ց

...

❚❤❡ ♥✉♠❜❡r ♦❢ ❞❡r✐✈❛t✐✈❡s ♠✐❣❤t ❜❡ r❡❞✉❝❡❞ ✇❤❡♥ t❤❡ ♠✐①❡❞ ❞❡r✐✈❛t✐✈❡s ❛t t❤❡ ❜♦tt♦♠ ❛r❡ ❡q✉❛❧✳ ■t ✐s ♦❢t❡♥ ♣♦ss✐❜❧❡ t❤❛♥❦s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t t❤❛t ✇❡ ❛❝❝❡♣t ✇✐t❤♦✉t ♣r♦♦❢✳ ❚❤❡♦r❡♠ ✸✳✶✸✳✷✿ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) ❤❛s ❝♦♥t✐♥✉♦✉s s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛t ❛

✸✳✶✸✳

❚❤❡ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❣✐✈❡♥ ♣♦✐♥t

✷✾✻

(x0 , y0 )

✐♥

R2 ✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

∂ 2f ∂ 2f (x0 , y0 ) = (x0 , y0 ) ∂x ∂y ∂y ∂x

❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠✱ t❤✐s ♣❛rt ♦❢ t❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠ ❜❡❝♦♠❡s ❝♦♠♠✉t❛t✐✈❡✿

f ւx

fx fxx ■♥ ❢❛❝t✱

ւx

y

ց

y

fyx = fxy

ց

fy

ւx

y

ց

fyy

t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❝♦♠♠✉t❡ ✿ ∂ ∂ ∂ ∂ = . ∂x ∂y ∂y ∂x

❚❤✐s ❝❛♥ ❛❧s♦ ❜❡ ✇r✐tt❡♥ ❛s ❛♥♦t❤❡r ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠✿ ∂

−−−∂x −−→

f   ∂ y ∂y

ց ∂

−−−∂x −−→ fxy = fyx

fy ❋♦r t❤❡ ❝❛s❡ ♦❢

t❤r❡❡

f  x  ∂ y ∂y

✈❛r✐❛❜❧❡s✱ t❤❡r❡ ❛r❡ ❡✈❡♥ ♠♦r❡ ❞❡r✐✈❛t✐✈❡s✿

f ւx

fx fxx

ւx

↓y

z

fyx

ց

↓y

z

fy

fzx

❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠✱

ւx

fxy

↓y

z

fyy

ց

fzy

fxz

ց

ւx

fz ↓y

fxz

z

ց

t❤❡ t❤r❡❡ ♦♣❡r❛t✐♦♥s ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❝♦♠♠✉t❡

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = , = , = ∂x ∂y ∂y ∂x ∂y ∂z ∂z ∂y ∂z ∂x ∂x ∂z ❚❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠ ❛❧s♦ ❜❡❝♦♠❡s ❝♦♠♠✉t❛t✐✈❡✿

fzz

ւx

fxz =

←−−− ←−−− fzx ↑z

fx fxx

ւx

fyz fzz

↑z fz

↑z

−−−→ −−−→ fzy

f

ւx

y

y

ւx

ց

fyx = fxy

ց

y

ց

= fyz

↑z

fy y

ց

fyy

✭♣❛✐r✇✐s❡✮✿

❈❤❛♣t❡r ✹✿ ❚❤❡ ❣r❛❞✐❡♥t

❈♦♥t❡♥ts

✹✳✶ ❖✈❡r✈✐❡✇ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✹✳✹ ❚❤❡ ❣r❛❞✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ✳ ✳ ✹✳✻ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✽ ❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✵ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✶ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✷ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾✼ ✷✾✾ ✸✵✻ ✸✵✽ ✸✶✹ ✸✷✵ ✸✷✺ ✸✸✵ ✸✸✻ ✸✸✾ ✸✹✷ ✸✹✻

✹✳✶✳ ❖✈❡r✈✐❡✇ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❲❤❡r❡ ❛r❡ ✇❡ ✐♥ ♦✉r st✉❞② ♦❢ ❢✉♥❝t✐♦♥s ✐♥ ❤✐❣❤ ❞✐♠❡♥s✐♦♥s❄ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❞✐❛❣r❛♠ t❤❛t ❝❛♣t✉r❡s ❛❧❧ t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ❛s ✇❡❧❧ ❛s t❤♦s❡ ❤❛✈❡♥✬t s❡❡♥ ②❡t✳ ❚❤❡② ❛r❡ ♣❧❛❝❡❞ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✇✐t❤ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s r❡♣r❡s❡♥t✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ✐♥♣✉t s♣❛❝❡ ❛♥❞ t❤❡ ♦✉t♣✉t s♣❛❝❡✳ ❚❤❡ ✜rst ❝♦❧✉♠♥ ❝♦♥s✐sts ♦❢ ❛❧❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ t❤❡ ✜rst r♦✇ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❚❤❡ t✇♦ ❤❛✈❡ ♦♥❡ ❝❡❧❧ ✐♥ ❝♦♠♠♦♥❀ t❤❛t ✐s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳

✹✳✶✳ ❖✈❡r✈✐❡✇ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✾✽

❚❤✐s t✐♠❡ ✇❡ ✇✐❧❧ s❡❡ ❤♦✇ ❡✈❡r②t❤✐♥❣ ✐s ✐♥t❡r❝♦♥♥❡❝t❡❞✳ ❲❡ s❤♦✇ ✇✐t❤ t❤❡ r❡❞ ❛rr♦✇s ❢♦r ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s ✇❤❛t t②♣❡ ♦❢ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ■♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✸✮ ❛♥❞ ❜❡②♦♥❞✱ ✇❡ ❢❛❝❡❞ ♦♥❧② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ✇❡ ✐♠♣❧✐❝✐t❧② ✉s❡❞ t❤❡ ❢❛❝t t❤❛t ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐❧❧ ♥♦t ♠❛❦❡ ✉s ❧❡❛✈❡ t❤❡ ❝♦♥✜♥❡s ♦❢ t❤✐s ❡♥✈✐r♦♥♠❡♥t✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ ❡✈❡r② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s s♦♠❡❜♦❞②✬s ❞❡r✐✈❛t✐✈❡✳ ■♥ t❤✐s s❡♥s❡✱ t❤❡ ❛rr♦✇ ❝❛♥ ❜❡ r❡✈❡rs❡❞✳ ▼♦r❡ r❡❝❡♥t❧②✱ ✇❡ ❞❡✜♥❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ Rn ❛♥❞ t❤♦s❡ ❛r❡ ❛❣❛✐♥ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ Rn ✭t❤✐♥❦ ❧♦❝❛t✐♦♥ ✈s✳ ✈❡❧♦❝✐t②✮✳ ❚❤❛t✬s ✇❤② ✇❡ ❤❛✈❡ ❛rr♦✇s t❤❛t ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ s❛♠❡ ❝❡❧❧✳ ❖♥❝❡ ❛❣❛✐♥✱ ❡✈❡r② ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s s♦♠❡❜♦❞②✬s ❞❡r✐✈❛t✐✈❡✳ ■♥ t❤✐s s❡♥s❡✱ t❤❡ ❛rr♦✇ ❝❛♥ ❜❡ r❡✈❡rs❡❞✳ ❚❤❡ st✉❞② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✇♦✉❧❞ ❜❡ ✐♥❝♦♠♣❧❡t❡ ✇✐t❤♦✉t ✉♥❞❡rst❛♥❞✐♥❣ t❤❡✐r r❛t❡s ♦❢ ❝❤❛♥❣❡✦ ❲❤❛t ✇❡ ❦♥♦✇ s♦ ❢❛r ✐s t❤❛t ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ t✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s✿

❚❤❡ r❡s✉❧t ✐s ❣✐✈❡♥ ❜② ❛ ✈❡❝t♦r ❝❛❧❧❡❞ ✐ts ❣r❛❞✐❡♥t✳

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✷✾✾

❋♦r ❡❛❝❤ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ✈❡❝t♦r ❜✉t ✇❤❛t ✐❢ ✇❡ ❝❛rr② ♦✉t t❤✐s ❝♦♠♣✉t❛t✐♦♥ ♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡❄ ❲❤❛t ✐❢✱ ✐♥ ♦r❞❡r t♦ ❦❡❡♣ tr❛❝❦ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ✇❡ ❛tt❛❝❤ t❤✐s ✈❡❝t♦r t♦ t❤❡ ♣♦✐♥t ✐t ❝❛♠❡ ❢r♦♠❄ ❚❤❡ r❡s✉❧t ✐s ❛ ✈❡❝t♦r ✜❡❧❞✳ ■t ✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ R2 t♦ R2 ❛♥❞ ✐t ✐s ♣❧❛❝❡❞ ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ♦✉r t❛❜❧❡✳ ❚❤❡ s❛♠❡ ❤❛♣♣❡♥s t♦ ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s ❛♥❞ s♦ ♦♥✳ ❊✈❡r② ❣r❛❞✐❡♥t ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❜✉t ♥♦t ❡✈❡r② ✈❡❝t♦r ✜❡❧❞ ✐s ❛ ❣r❛❞✐❡♥t✳ ■♥ t❤✐s s❡♥s❡✱ t❤❡ ❛rr♦✇ ❝❛♥♥♦t ✇❡ r❡✈❡rs❡❞✦ ❚❤❡ s✐t✉❛t✐♦♥ s❡❡♠s t♦ ♠✐♠✐❝ t❤❡ ♦♥❡ ✇✐t❤ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿ t❤❡r❡ ❛r❡ ♥♦♥✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡ ❛rr♦✇ ✐♥ t❤❡ ✜rst ❝❡❧❧ ✐s r❡✈❡rs✐❜❧❡ ✐❢ ✇❡ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ s♠♦♦t❤ ✭✐✳❡✳✱ ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡✮ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ♠♦r❡ ♣r♦❢♦✉♥❞ ✇✐t❤ ✈❡❝t♦r ✜❡❧❞s ❛s ✇❡ s❤❛❧❧ s❡❡ ❧❛t❡r✳

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s ❱❡❝t♦r ✜❡❧❞s ❛r❡ ❥✉st ❢✉♥❝t✐♦♥s ✇✐t❤✿

• t❤❡ ✐♥♣✉t ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ♥✉♠❜❡rs ❛♥❞ • t❤❡ ♦✉t♣✉t ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ♥✉♠❜❡rs✳

❲❡ ❥✉st ❝❤♦♦s❡ t♦ tr❡❛t t❤❡ ❢♦r♠❡r ❛s ❛ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❧❛tt❡r ❛s ❛ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ t❤❛t ♣♦✐♥t✳ ❚❤✐s ✐s ❥✉st ❛ ❝❧❡✈❡r ✇❛② t♦ ✈✐s✉❛❧✐③❡ s✉❝❤ ❛ ❝♦♠♣❧❡① ✕ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♦♥❡s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ✕ ❢✉♥❝t✐♦♥✳ ■t✬s ❛ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✦ ▲❡t✬s ♣❧♦t s♦♠❡ ✈❡❝t♦r ✜❡❧❞s ❜② ❤❛♥❞ ❛♥❞ t❤❡♥ ❛♥❛❧②③❡ t❤❡♠✳

❊①❛♠♣❧❡ ✹✳✷✳✶✿ ❛❝❝❡❧❡r❛t❡❞ ♦✉t✢♦✇ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ ✈❡❝t♦r ✜❡❧❞✿

V (x, y) =< x, y > . ❆ ✈❡❝t♦r ✜❡❧❞ ✐s ❥✉st t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

V (x, y) =< p(x, y), q(x, y) > ✇✐t❤ p(x, y) = x, q(x, y) = y . P❧♦tt✐♥❣ t❤♦s❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ❛s ❜❡❢♦r❡✱ ❞♦❡s ♥♦t ♣r♦❞✉❝❡ ✉s❡❢✉❧ ✈✐s✉❛❧✐③❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ✇❡ ✇✐❧❧ st✐❧❧ ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛tt❡r♥✿ ✇❡ ♣✐❝❦ ❛ ❢❡✇ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✱ ❝♦♠♣✉t❡ t❤❡ ♦✉t♣✉t ❢♦r ❡❛❝❤✱ ❛♥❞ ❛ss✐❣♥ ✐t t♦ t❤❛t ♣♦✐♥t✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✐♥st❡❛❞ ♦❢ ❛ s✐♥❣❧❡ ♥✉♠❜❡r ✇❡ ❤❛✈❡ t✇♦ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❛ ✈❡rt✐❝❛❧ ❜❛r t❤❛t ✇❡ ❡r❡❝t ❛t t❤❛t ♣♦✐♥t t♦ ✈✐s✉❛❧✐③❡ t❤✐s ♥✉♠❜❡r ✇❡ ❞r❛✇ ❛♥ ❛rr♦✇✳ ❲❡ ❝❛rr② t❤✐s ♦✉t ❢♦r t❤❡s❡ ♥✐♥❡ ♣♦✐♥ts ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✿

(−1, 1)

(0, 1)

(1, 1)

(−1, 0)

(0, 0)

(1, 0)

(−1, −1) (0, −1) (1, −1)

❧❡❛❞✐♥❣ t♦

< −1, 1 >

< −1, 0 >

< 0, 1 >

< 1, 1 >

< 0, 0 >

< 1, 0 >

< −1, −1 > < 0, −1 > < 1, −1 >

❊❛❝❤ ✈❡❝t♦r ♦♥ r✐❣❤t st❛rts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t ♦♥ ❧❡❢t✿

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✵

❲❤❛t ❛❜♦✉t t❤❡ r❡st❄ ❲❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡ ♠❛❣♥✐t✉❞❡s ✐♥❝r❡❛s❡ ❛s ✇❡ ♠♦✈❡ ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✇❤✐❧❡ t❤❡ ❞✐r❡❝t✐♦♥s r❡♠❛✐♥ t❤❡ s❛♠❡✿ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ♦r✐❣✐♥✳

❋♦r ❡❛❝❤ ♣♦✐♥t (x, y) ✇❡ ❝♦♣② t❤❡ ✈❡❝t♦r t❤❛t ❡♥❞s t❤❡r❡✱ ✐✳❡✳✱ < x, y > ❛♥❞ ♣❧❛❝❡ ✐t ❛t t❤✐s ❧♦❝❛t✐♦♥✳ ◆♦✇✱ ✐❢ t❤❡ ✈❡❝t♦rs r❡♣r❡s❡♥t ✈❡❧♦❝✐t✐❡s ♦❢ ♣❛rt✐❝❧❡s✱ ✇❤❛t ❦✐♥❞ ♦❢ ✢♦✇ ✐s t❤✐s❄ ❚❤✐s ✐s♥✬t ❛ ❢♦✉♥t❛✐♥ ♦r ❛♥ ❡①♣❧♦s✐♦♥ ✭t❤❡ ♣❛rt✐❝❧❡s ✇♦✉❧❞ ❣♦ s❧♦✇❡r ❛✇❛② ❢r♦♠ t❤❡ s♦✉r❝❡✮✳ ❚❤❡ ❛♥s✇❡r ✐s✿ t❤✐s ✐s ❛ ✢♦✇ ♦♥ ❛ s✉r❢❛❝❡ ✕ ✉♥❞❡r ❣r❛✈✐t② ✕ t❤❛t ❣❡ts st❡❡♣❡r ❛♥❞ st❡❡♣❡r ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✦ ❚❤✐s s✉r❢❛❝❡ ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡ t❤✐s ♣❛r❛❜♦❧♦✐❞✿

❈❛♥ ✇❡ ❜❡ ♠♦r❡ s♣❡❝✐✜❝❄ ❲❡❧❧✱ t❤✐s s✉r❢❛❝❡ ❧♦♦❦s ❧✐❦❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ✢♦✇ s❡❡♠s t♦ ❢♦❧❧♦✇ t❤❡ ❧✐♥❡ ❢❛st❡st ❞❡s❝❡♥t❀ ♠❛②❜❡ ♦✉r ✈❡❝t♦r ✜❡❧❞ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄ ❲❡ ✇✐❧❧ ✜♥❞ ♦✉t ❜✉t ✜rst ❧❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ ✈✐s✉❛❧✐③❡❞ ❛s ❛ s②st❡♠ ♦❢ ♣✐♣❡s✿

❲❡ r❡❝♦❣♥✐③❡ t❤✐s ❛s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳ ◆♦✇✱ t❤❡ q✉❡st✐♦♥ ❛❜♦✈❡ ❜❡❝♦♠❡s✿ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦❞✉❝❡ t❤✐s ♣❛tt❡r♥ ♦❢ ✢♦✇ ✐♥ t❤❡ ♣✐♣❡s ❜② ❝♦♥tr♦❧❧✐♥❣ t❤❡ ♣r❡ss✉r❡ ❛t t❤❡ ❥♦✐♥ts❄ ❙♦✱ ✐s t❤✐s ✈❡❝t♦r ✜❡❧❞ ✕ ✇❤❡♥ ❧✐♠✐t❡❞ t♦ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❣r✐❞ ✕ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ▲❡t✬s t❛❦❡ t❤❡ ❧❛tt❡r q✉❡st✐♦♥❀ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) t❤❛t

∆f =< x, y > . ❚❤❡ ❧❛tt❡r ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ❀ t❤❡ ❛❝t✉❛❧ ❞✐✛❡r❡♥❝❡s ❛r❡ x ♦♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s ❛♥❞ y ♦♥ t❤❡ ✈❡rt✐❝❛❧✳ ▲❡t✬s ❝♦♥❝❡♥tr❛t❡ ♦♥ ❥✉st ♦♥❡ ❝❡❧❧✿

[x, x + ∆x] × [y, y + ∆y] .

✹✳✷✳

●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✶

❲❡ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ ❛♥ ❡❞❣❡ t♦ ❜❡ t❤❡ ♣r✐♠❛r② ♥♦❞❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❡❞❣❡✿

• •

(x, y) ✐♥ [x, x + ∆x] × {y}✱ ❡t❝✳ (x, y) ✐♥ {x} × [y, y + ∆y]✱ ❡t❝✳

❤♦r✐③♦♥t❛❧✿ ✈❡rt✐❝❛❧✿

❚❤❡♥ ♦✉r ❡q✉❛t✐♦♥ ❞❡✈❡❧♦♣s ❛s ❢♦❧❧♦✇s✿

  ∆ f =x x =⇒  ∆x f = y

  f (x + ∆x, y) − f (x, y) = x =⇒  f (x, y + ∆y) − f (x, y) = y

❚❤❡s❡ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥s ❛❧❧♦✇ ✉s t♦ ❝♦♥str✉❝t

f

  f (x + ∆x, y) = f (x, y) + x =⇒  f (x, y + ∆y) = f (x, y) + y

♦♥❡ ♥♦❞❡ ❛t ❛ t✐♠❡✳ ❲❡ ✉s❡ t❤❡ ✜rst ♦♥❡ t♦ ♣r♦❣r❡ss

❤♦r✐③♦♥t❛❧❧② ❛♥❞ t❤❡ s❡❝♦♥❞ t♦ ♣r♦❣r❡ss ✈❡rt✐❝❛❧❧②✿

↑

↑

↑

↑

(x, y + ∆y) → (x + ∆x, y + ∆y) → →

(x, y) ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦

❍♦✇❡✈❡r✱ t❤❡r❡ ♠❛② ❜❡ ❛

❝♦♥s❡❝✉t✐✈❡❧② ❜✉t ✐♥ ❛ ❞✐✛❡r❡♥t ♦r❞❡r❄

→

(x + ∆x, y)

❝♦♥✢✐❝t ✿

✇❤❛t ✐❢ ✇❡ ❛♣♣❧② t❤❡s❡ t✇♦ ❢♦r♠✉❧❛s

❋♦rt✉♥❛t❡❧②✱ ❣♦✐♥❣ ❤♦r✐③♦♥t❛❧❧② t❤❡♥ ✈❡rt✐❝❛❧❧② ♣r♦❞✉❝❡s t❤❡

s❛♠❡ ♦✉t❝♦♠❡ ❛s ❣♦✐♥❣ ✈❡rt✐❝❛❧❧② t❤❡♥ ❤♦r✐③♦♥t❛❧❧②✿

f (x + ∆x, y + ∆y) = f (x, y) + x + y . ❚❤✐s ✐s t❤❡ ✜rst ✐♥st❛♥❝❡ ♦❢

♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡✳

◆♦✇ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳ ❙✉♣♣♦s❡

z = f (x, y)✳

V

✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ s♦♠❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❚❤❡ r❡s✉❧t ♦❢ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ❛ ✈❡❝t♦r ❡q✉❛t✐♦♥ t❤❛t ❜r❡❛❦s ✐♥t♦ t✇♦✿

  x = f (x, y), x V = ∇f =⇒ V (x, y) =< x, y >=< fx (x, y) , fy (x, y) > =⇒  y = fy (x, y).

❲❡ ♥♦✇ ✐♥t❡❣r❛t❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✿

Z

x2 +C 2 Z y2 +K y = fy (x, y) =⇒ f (x, y) = y dy = 2

x = fx (x, y) =⇒ f (x, y) =

x dx =

x2 + C(y) 2 y2 = + K(x) 2

=

◆♦t❡ t❤❛t ✐♥ ❡✐t❤❡r ❝❛s❡ ✇❡ ❛❞❞ t❤❡ ❢❛♠✐❧✐❛r ❝♦♥st❛♥ts ♦❢ ✐♥t❡❣r❛t✐♦♥ ✏ +C ✑ ❛♥❞ ✏ +K ✑ ✭❞✐✛❡r❡♥t ❢♦r t❤❡ t✇♦ ❞✐✛❡r❡♥t ✐♥t❡❣r❛t✐♦♥s✮✳✳✳ ❤♦✇❡✈❡r✱ t❤❡s❡ ❝♦♥st❛♥ts ❛r❡ ♦♥❧② ❝♦♥st❛♥t r❡❧❛t✐✈❡ t♦ ❚❤❛t ♠❛❦❡s t❤❡♠

❢✉♥❝t✐♦♥s

♦❢

y

❛♥❞

x

x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✳

r❡s♣❡❝t✐✈❡❧②✦ P✉tt✐♥❣ t❤❡ t✇♦ t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣

r❡str✐❝t✐♦♥ ♦♥ t❤❡ t✇♦ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s✿

f (x, y) = ❈❛♥ ✇❡ ✜♥❞ s✉❝❤ ❢✉♥❝t✐♦♥s

C

❛♥❞

K❄

x2 y2 + C(y) = + K(x) . 2 2

■❢ ✇❡

❣r♦✉♣

t❤❡ t❡r♠s✱ t❤❡ ❝❤♦✐❝❡ ❜❡❝♦♠❡s ♦❜✈✐♦✉s✿

x2 K(x) 2 = y2 + +C(y) 2 ■❢ ✭♦r ✇❤❡♥✮ ✐t ❞♦❡s ♥♦t✱ ✇❡ ❝♦✉❧❞ ❥✉st ♣❧✉❣ s♦♠❡ ✈❛❧✉❡s ✐♥t♦ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ ❡①❛♠✐♥❡ t❤❡ r❡s✉❧ts✿

y2 y2 + K(0) =⇒ C(y) = + 22 22 x x y = 0 =⇒ C(0) + = K(x) =⇒ K(x) = + 2 2 x = 0 =⇒ C(y) =

❝♦♥st❛♥t ❝♦♥st❛♥t

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✷

❙♦✱ ❡✐t❤❡r ♦❢ t❤❡ ❢✉♥❝t✐♦♥s C(y) ❛♥❞ K(x) ❞✐✛❡rs ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①♣r❡ss✐♦♥ ❜② ❛ ❝♦♥st❛♥t✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ f (x, y) =

x2 y 2 + + L, 2 2

❢♦r s♦♠❡ ❝♦♥st❛♥t L✳ ❚❤❡ s✉r❢❛❝❡ ✐s ✐♥❞❡❡❞ ❛ ♣❛r❛❜♦❧♦✐❞ ♦❢ r❡✈♦❧✉t✐♦♥✳ ❲❤❡♥ ❛ ✈❡❝t♦r ✜❡❧❞ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ♦r s✐♠♣❧② ❛ ♣♦t❡♥t✐❛❧✱ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ◆♦t❡ t❤❛t ✜♥❞✐♥❣ ❢♦r ❛ ❣✐✈❡♥ ✈❡❝t♦r ✜❡❧❞ V ❛ ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t ∇f = V ❛♠♦✉♥ts t♦ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❛♥❛❧♦❣ ♦❢ s❡✈❡r❛❧ ❢❛♠✐❧✐❛r r❡s✉❧ts✿ ❛♥② t✇♦ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ s❛♠❡ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ❞✐s❦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ❙♦✱ ②♦✉✬✈❡ ❢♦✉♥❞ ♦♥❡ ✕ ②♦✉✬✈❡ ❢♦✉♥❞ ❛❧❧✱ ❥✉st ❧✐❦❡ ✐♥ ❱♦❧✉♠❡ ✷✳ ❚❤❡ ♣r♦♦❢ ✐s ❡①❛❝t❧② t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡ ❜✉t ✐t r❡❧✐❡s✱ ❥✉st ❛s ❜❡❢♦r❡✱ ♦♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ✭✐✳❡✳✱ ❣r❛❞✐❡♥ts✮ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s t❤❡♥ ❞✐✛❡r ❜② ❛ ✈❡rt✐❝❛❧ s❤✐❢t✳

■t ✐s ❛s ✐❢ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣ ✐♥ ❛ ❝❛✈❡ ❤❛✈❡ t❤❡ ❡①❛❝t s❛♠❡ s❧♦♣❡ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥❀ t❤❡♥ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❝❡✐❧✐♥❣ ✐s t❤❡ s❛♠❡ t❤r♦✉❣❤♦✉t t❤❡ ❝❛✈❡✳

❊①❛♠♣❧❡ ✹✳✷✳✷✿ r♦t❛t✐♦♥❛❧ ✢♦✇ ❲❡ ❝♦♥s✐❞❡r t❤✐s ✈❡❝t♦r ✜❡❧❞ ❛❣❛✐♥✿ ❖✉r t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡✿

V (x, y) =< y , −x > .

V (x, y) =< p(x, y) , q(x, y) > ✇✐t❤ p(x, y) = y, q(x, y) = −x .

❲❡ ♣✐❝❦ ❛ ❢❡✇ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✱ ❝♦♠♣✉t❡ t❤❡ ♦✉t♣✉t ❢♦r ❡❛❝❤✱ ❛♥❞ ❛ss✐❣♥ ✐t t♦ t❤❛t ♣♦✐♥t✿ (−1, 1)

(0, 1)

(1, 1)

(−1, 0)

(0, 0)

(1, 0)

(−1, −1) (0, −1) (1, −1)

❧❡❛❞✐♥❣ t♦

< 1, −1 > < 0, 1 >

< 1, 0 > < 0, 0 >

< 1, −1 >

< 0, −1 >

< −1, 1 > < −1, 0 > < −1, 1 >

❊❛❝❤ ✈❡❝t♦r ♦♥ r✐❣❤t st❛rts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t ♦♥ ❧❡❢t✿

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✸

◆♦✇✱ ✐❢ t❤❡ ✈❡❝t♦rs r❡♣r❡s❡♥t ✈❡❧♦❝✐t✐❡s ♦❢ ♣❛rt✐❝❧❡s✱ ✇❤❛t ❦✐♥❞ ♦❢ ✢♦✇ ✐s t❤✐s❄ ■t ❧♦♦❦s ❧✐❦❡ t❤❡ ✇❛t❡r ✐s ✢♦✇✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✳ ■s ✐t ❛ ✇❤✐r❧ ❄ ▲❡t✬s ♣❧♦t s♦♠❡ ♠♦r❡✿

❚❤❡s❡ ❧✐❡ ♦♥ t❤❡ ❛①❡s ❛♥❞ t❤❡② ❛r❡ ❛❧❧ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤♦s❡ ❛①❡s✳ ❲❡ r❡❛❧✐③❡ t❤❛t t❤❡r❡ ✐s ❛ ♣❛tt❡r♥✿ V (x, y) ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ < x, y >✳ ■♥❞❡❡❞✱ < y, −x > · < x, y >= yx − xy = 0 .

❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ t♦ ❢♦❧❧♦✇ t❤❡s❡ ❛rr♦✇s ❛ ❝✉r✈❡ ✇♦✉❧❞ ❜❡ r♦✉♥❞✐♥❣ t❤❡ ♦r✐❣✐♥ ♥❡✈❡r ❣❡tt✐♥❣ ❝❧♦s❡r t♦ ♦r ❢❛rt❤❡r ❛✇❛② ❢r♦♠ ✐t❀ t❤✐s ♠✉st ❜❡ ❛ r♦t❛t✐♦♥✳ ◆♦✇✱ ✐s t❤✐s ❛ ✢♦✇ ♦♥ ❛ s✉r❢❛❝❡ ♣r♦❞✉❝❡❞ ❜② ❣r❛✈✐t② ❧✐❦❡ ❧❛st t✐♠❡❄ ■❢ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❛s ❛ s②st❡♠ ♦❢ ♣✐♣❡s✱ t❤❡ q✉❡st✐♦♥ ❛❜♦✈❡ ❜❡❝♦♠❡s✿ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦❞✉❝❡ t❤✐s ♣❛tt❡r♥ ♦❢ ✢♦✇ ✐♥ t❤❡ ♣✐♣❡s ❜② ❝♦♥tr♦❧❧✐♥❣ t❤❡ ♣r❡ss✉r❡ ❛t t❤❡ ❥♦✐♥ts❄

▲❡t✬s ✜♥❞ ♦✉t✳ ❲❡ ✇✐❧❧ tr② t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢♦r z = f (x, y)✿ ∆f =< y , −x > .

❏✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ t♦ ❜❡ t❤❡ ♣r✐♠❛r② ♥♦❞❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❡❞❣❡✳ ❲❡ ❤❛✈❡✿   ∆ f =y x =⇒  ∆y f = −x

  f (x + ∆x, y) − f (x, y) = y =⇒  f (x, y + ∆y) − f (x, y) = −x

  f (x + ∆x, y) = f (x, y) + y =⇒  f (x, y + ∆y) = f (x, y) − x

✹✳✷✳

●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✹

❈❛♥ ✇❡ ✉s❡ t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s t♦ ❝♦♥str✉❝t ❣❡t t♦

(x + ∆x, y + ∆y)

f❄

■s t❤❡r❡ ❛

❝♦♥✢✐❝t ✿

✐❢ ✇❡ st❛rt ❛t

(x, y)

❛♥❞ t❤❡♥

✐♥ t❤❡ t✇♦ ❞✐✛❡r❡♥t ✇❛②s✱ ✇✐❧❧ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ♦✉t❝♦♠❡❄

↑

↑

↑

↑

(x, y + ∆y) → (x + ∆x, y + ∆y) → →

(x, y) ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♦✉t❝♦♠❡ ✐s

♥♦t

→

(x + ∆x, y)

t❤❡ s❛♠❡✿

f (x + ∆x, y + ∆y) = f (x, y) + y − (x + ∆x) = f (x, y) − x + y − ∆x

6= f (x + ∆x, y + ∆y) = f (x, y) − x + (y + ∆y) = f (x, y) − x + y + ∆y ❚❤✐s ✐s

♣❛t❤✲❞❡♣❡♥❞❡♥❝❡ ✦ V

❙✉♣♣♦s❡

✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

f✿

  y = fx (x, y), V = ∇f =⇒ V (x, y) =< y , −x >=< fx (x, y) , fy (x, y) > =⇒  −x = fy (x, y).

❲❤❛t ❞♦ ✇❡ ❞♦ ✇✐t❤ t❤♦s❡❄

❚❤❡② ❛r❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s s♦ ❧❡t✬s s♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s ❜② ♣❛rt✐❛❧

✐♥t❡❣r❛t✐♦♥✱ ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✿

y

= fx (x, y) =⇒ f (x, y) =

−x = fy (x, y) =⇒ f (x, y) =

Z

Z

y dx

= xy + C

−x dy = −xy + K

P✉tt✐♥❣ t❤❡ t✇♦ t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡✿

f (x, y) = xy + C(y) = −xy + K(x) . ❈❛♥ ✇❡ ✜♥❞ s✉❝❤ ❢✉♥❝t✐♦♥s

C

❛♥❞

K❄

■❢ ✇❡ tr② t♦

xy +C(y)

❣r♦✉♣

=

t❤❡ t❡r♠s✱ t❤❡② ❞♦♥✬t ❣r♦✉♣ ✇❡❧❧✿

−xy

+K(x)

❚♦ ❝♦♥✜r♠ t❤❛t t❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠✱ ❧❡t✬s ♣❧✉❣ s♦♠❡ ✈❛❧✉❡s ✐♥t♦ t❤✐s ❡q✉❛t✐♦♥✿

❙♦✱ ❜♦t❤

C

❛♥❞

K

x = 0 =⇒ C(y) = K(0) =⇒ C(y) =

❝♦♥st❛♥t

y = 0 =⇒ C(0) = K(x) =⇒ K(x) =

❝♦♥st❛♥t

❛r❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡✦ ■♥❞❡❡❞✱ ♦♥ ❧❡❢t ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢

t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ ❛ ❝♦♥st❛♥t ♦♥ r✐❣❤t✿

2xy = −C + K = ❚❤✐s ❝♦♥tr❛❞✐❝t✐♦♥ ♣r♦✈❡s t❤❛t ♦✉r ❛ss✉♠♣t✐♦♥ t❤❛t s✉❝❤

f✳

V

❝♦♥st❛♥t . ❤❛s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✇❛s ✇r♦♥❣❀ t❤❡r❡ ✐s ♥♦

❲❡ ♠❛② ❡✈❡♥ s❛② t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s♥✬t ✏✐♥t❡❣r❛❜❧❡✑✦ ●❡♦♠❡tr✐❝❛❧❧②✱ t❤❡r❡ ✐s ♥♦ s✉r❢❛❝❡ ❛

✢♦✇ ♦❢ ✇❛t❡r ♦♥ ✇❤✐❝❤ ✇♦✉❧❞ ♣r♦❞✉❝❡ t❤✐s ♣❛tt❡r♥✳

✹✳✷✳ ●r❛❞✐❡♥ts ✈s✳ ✈❡❝t♦r ✜❡❧❞s

✸✵✺

❊①❛♠♣❧❡ ✹✳✷✳✸✿ ❝❧✐♠❜✐♥❣

❆♥ ✐♥s✐❣❤t❢✉❧ ✐❢ ✐♥❢♦r♠❛❧ ❛r❣✉♠❡♥t t♦ t❤❡ s❛♠❡ ❡✛❡❝t ✐s ❛s ❢♦❧❧♦✇s✳ ❙✉♣♣♦s❡ ✇❡ tr❛✈❡❧ ❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞✳ ❙✉♣♣♦s❡ t❤❛t ❡✈❡♥t✉❛❧❧② ✇❡ ❛rr✐✈❡ t♦ ♦✉r ♦r✐❣✐♥❛❧ ❧♦❝❛t✐♦♥✳ ■s ✐t ♣♦ss✐❜❧❡ t❤❛t t❤✐s ✈❡❝t♦r ✜❡❧❞ ❤❛s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥❄ ■s ✐t t❤❡ ❣r❛❞✐❡♥t ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ■❢ ✐t ✐s✱ ✇❡ ❤❛✈❡ ❢♦❧❧♦✇❡❞ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✭❢❛st❡st✮ ✐♥❝r❡❛s❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳✳✳ ❜✉t ♦♥❝❡ ✇❡ ❤❛✈❡ ❝♦♠❡ ❜❛❝❦✱ ✇❤❛t ✐s t❤❡ ❡❧❡✈❛t✐♦♥❄ ❆❢t❡r ❛❧❧ t❤✐s ❝❧✐♠❜✐♥❣✱ ✐t ❝❛♥✬t ❜❡ t❤❡ s❛♠❡✿

❚❤✐s ❢✉♥❝t✐♦♥ t❤❡r❡❢♦r❡ ❝❛♥♥♦t ❜❡ ❝♦♥t✐♥✉♦✉s ❛t t❤✐s ❧♦❝❛t✐♦♥✳ ❚❤❡♥ ✐t ❛❧s♦ ❝❛♥♥♦t ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✦ ❖✉r ❝♦♥❝❧✉s✐♦♥ t❤❛t s♦♠❡ ❝♦♥t✐♥✉♦✉s ✈❡❝t♦r ✜❡❧❞s ♦♥ t❤❡ ♣❧❛♥❡ ❛r❡♥✬t ❞❡r✐✈❛t✐✈❡s ❤❛s ♥♦ ❛♥❛❧♦❣ ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧✱ ♥✉♠❡r✐❝❛❧✱ ❝❛s❡ ❞✐s❝✉ss❡❞ ✐♥ ❱♦❧✉♠❡s ✷ ❛♥❞ ✸✳ ❊①❛♠♣❧❡ ✹✳✷✳✹✿ s♦✉r❝❡

❚❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞s ❛r❡ ♠♦r❡ ❝♦♠♣❧❡①✳ ❚❤❡ ♦♥❡ ❜❡❧♦✇ ✐s s✐♠✐❧❛r t♦ t❤❡ ✜rst ❡①❛♠♣❧❡ ❛❜♦✈❡✿ V (x, y, z) =< x, y, z > .

❚❤❡ ✈❡❝t♦rs ♣♦✐♥t ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ t♦ t❤❡ ♦r✐❣✐♥✿

❏✉st ❛s ✐ts t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✱ t❤❡r❡ ✐s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿ f (x, y, z) =

x2 y 2 z 2 + + . 2 2 2

❚❤❡ ❧❡✈❡❧ s✉r❢❛❝❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛r❡ ❝♦♥❝❡♥tr✐❝ s♣❤❡r❡s✿

✹✳✸✳

❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✸✵✻

✹✳✸✳ ❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈✲ ❡r❛❧ ✈❛r✐❛❜❧❡s

❲❡ ❝♦♥s✐❞❡r ❢✉♥❝t✐♦♥s ♦❢

n

✈❛r✐❛❜❧❡s ❛❣❛✐♥✳

❧✐♥❡❛r ❢✉♥❝t✐♦♥s ✿

❋✐rst ❧❡t✬s ❧♦♦❦ ❛t t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢

l(x1 , ..., xn ) = p + m1 (x1 − a1 ) + ... + mn (xn − an ) , p ✐s t❤❡ z ✲✐♥t❡r❝❡♣t✱ m1 , ..., mn ❛r❡ t❤❡ ❝❤♦s❡♥ s❧♦♣❡s ♦❢ t❤❡ ♣❧❛♥❡ ❛❧♦♥❣ t❤❡ ❛①❡s✱ ❛♥❞ a1 , ..., an ❛r❡ t❤❡ n ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝❤♦s❡♥ ♣♦✐♥t ✐♥ R ✳ ▲❡t✬s r❡❝❛st t❤✐s ❡①♣r❡ss✐♦♥✱ ❛s ❜❡❢♦r❡✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t ✇❤❡r❡

✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✿

l(X) = p + M · (X − A) ❍❡r❡ ✇❡ ❤❛✈❡✿

• M =< m1 , ..., mn > • A = (a1 , ..., an ) • X = (x1 , ..., xn ) • X −A

✐s t❤❡ ✈❡❝t♦r ♦❢ s❧♦♣❡s✳

✐s t❤❡ ♣♦✐♥t ✐♥

Rn ✳

✐s ♦✉r ✈❛r✐❛❜❧❡ ♣♦✐♥t ✐♥

Rn ✳

✐s ❤♦✇ ❢❛r ✇❡ st❡♣ ❛✇❛② ❢r♦♠ ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st

A✳

❚❤❡♥ ✇❡ ❝❛♥ s❛② t❤❛t t❤❡ ✈❡❝t♦r N =< m1 , ..., mn , 1 > ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✭❛ n+1 ✏♣❧❛♥❡✑ ✐♥ R ✮✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❤♦❧❞s ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❛♥② ❝❤♦✐❝❡ ♦❢ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠✦ ❙✉♣♣♦s❡ ♥♦✇ ✇❡ ❤❛✈❡ ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ (x1 , ..., xn ) ✐♥ Rn ✳

z = f (X)

♦❢

n

✈❛r✐❛❜❧❡s✱ ✐✳❡✳✱

z

✐s ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞

X =

❲❡ st❛rt ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✳ ❙✉♣♣♦s❡

Rn

✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ s✐❞❡s

∆xk

❛❧♦♥❣ t❤❡ ❛①✐s ♦❢ ❡❛❝❤ ✈❛r✐❛❜❧❡

xk ✳

■t s❡r✈❡s

❛s ❛ ♣❛rt✐t✐♦♥ ✇✐t❤ s❡❝♦♥❞❛r② ♥♦❞❡s ♣r♦✈✐❞❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❣r✐❞✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❡①❛❝t❧② t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✿

❉❡✜♥✐t✐♦♥ ✹✳✸✳✶✿ ❞✐✛❡r❡♥❝❡ ❚❤❡

❞✐✛❡r❡♥❝❡

♦❢

z = f (X) = f (x1 , ..., xn )

❛t

C

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❝❤❛♥❣❡ ♦❢

z

✹✳✸✳

❚❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✸✵✼

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❝r❡♠❡♥t ∆X ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆f (C) = f (X + ∆X) − f (X)

✇❤❡r❡ C ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ❡❞❣❡ ❜❡t✇❡❡♥ X ❛♥❞ X + ∆X ✳ ❲❤❛t ✉s❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♥♦❡s ❤❛s ❜❡❝♦♠❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s✿

❊❛❝❤ ❞✐✛❡r❡♥❝❡ ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t✐♣ t♦ t♦❡ ♦❢ t❤❡ ♦r✐❡♥t❡❞ ❡❞❣❡✳ ◆♦✇✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✱ t❤❡ ❝❤❛♥❣❡ ♦❢ z r❡❧❛t✐✈❡ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ X ✳ ■t ❤❛s ❜❡❡♥ t❤❡ r❛t✐♦ ♦❢ t❤❡ ❢♦r♠❡r ♦✈❡r t❤❡ ❧❛tt❡r✳ ❍♦✇❡✈❡r✱ ∆X ✐s ❛ ✈❡❝t♦r t❤✐s t✐♠❡✳ ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦ t❤✐s ❢r❛❝t✐♦♥❄ ❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ X ♦❢ ❛ s♣❡❝✐✜❝ ❦✐♥❞✳ ❲❡ ♦♥❧② ❝♦♥s✐❞❡r t❤❡ ✐♥❝r❡♠❡♥t ♦❢ X ✐♥ ♦♥❡ ♦❢ t❤♦s❡ ❞✐r❡❝t✐♦♥s✿ ∆Xk = < 0, ..., 1

0,

... k − 1

∆xk , k

0,

..., 0 >

k + 1 ... n

❚❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ♦❢ z = f (X) = f (x1 , ..., xn ) ✇✐t❤ r❡s♣❡❝t xk ❛t C ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ xk ✿ f (X + ∆Xk ) − f (X) .

❲❡ ❝♦❧❧❡❝t t❤❡s❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ✐♥t♦ ♦♥❡ ❢✉♥❝t✐♦♥✱ ∆f ✳ ◆♦✇ ✇❡ ❥✉st ❞✐✈✐❞❡ ❜② t❤❡ r❡s♣❡❝t✐✈❡ ✐♥❝r❡♠❡♥t ♦❢ X ✿

✹✳✹✳ ❚❤❡ ❣r❛❞✐❡♥t

✸✵✽

❉❡✜♥✐t✐♦♥ ✹✳✸✳✷✿ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ z = f (X) = f (x1 , ..., xn ) ✇✐t❤ r❡s♣❡❝t xk ❛t X ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ z ✇✐t❤ r❡s♣❡❝t t♦ xk ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ f (X + ∆Xk ) − f (X) ∆f (C) = ∆xk ∆xk

✇❤❡r❡ C ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ❡❞❣❡ ❜❡t✇❡❡♥ X ❛♥❞ X + ∆Xk ✳ ❲❡ ❝♦❧❧❡❝t t❤❡s❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ✐♥t♦ ♦♥❡ ❢✉♥❝t✐♦♥✦

❉❡✜♥✐t✐♦♥ ✹✳✸✳✸✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ z = f (X) = f (x1 , ..., xn ) ❛t ❡❞❣❡ C ✐s ❡q✉❛❧ t♦ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ z = f (X) ✇✐t❤ r❡s♣❡❝t xk ❛t C ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆f ∆f (C) = (C) ∆X ∆xk

✇❤❡r❡ C ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ❡❞❣❡ ❜❡t✇❡❡♥ X ❛♥❞ X + ∆Xk ✳ ❚❤✐s ✐s ❛ 1✲❢♦r♠ ✐♥ Rn ✳ ❈❛♥ ✇❡ s❡❡ t❤✐s q✉❛♥t✐t② ❛s ❛ ✈❡❝t♦r❄ ❋♦r ❡✈❡r② ❡❞❣❡ C ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ kt❤ ❛①✐s✱ ✇❡ ❝♦♥s✐❞❡r✿   ∆f (C), 0, ..., 0 0, ..., 0, ∆xk

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡r❡ ✐s ♥♦ ✢♦✇ ✐♥ ❛♥ ❞✐r❡❝t✐♦♥ ♦t❤❡r t❤❛♥ xk ✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ♠♦✈❡ ❢r♦♠ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❛❧s♦ ♠♦✈❡ ❢r♦♠ t❤❡ ❤②❞r❛✉❧✐❝ ❛♥❛❧♦❣② t♦ ✢♦✇ ♦♥ s✉r❢❛❝❡ ✿

✹✳✹✳ ❚❤❡ ❣r❛❞✐❡♥t

❋♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ ✇❡ r❡♣❡❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✭❜❡❧♦✇✮ ❛♥❞ t❤❡♥ ❥✉st t❛❦❡ t❤❡ ❧✐♠✐t✿

✹✳✹✳

❚❤❡ ❣r❛❞✐❡♥t

✸✵✾

❲❡ ❢♦❝✉s ♦♥ ♦♥❡ ♣♦✐♥t A = (a1 , ..., an ) ✐♥ Rn ✳

❉❡✜♥✐t✐♦♥ ✹✳✹✳✶✿ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ❚❤❡

♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ z = f (X) = f (x1, ..., xn ) ✇✐t❤ r❡s♣❡❝t xk ❛t

X =A=

(a1 , ..., an ) ❛r❡ ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ xk ❛t xk = ak ✱ ✐❢ ✐t ❡①✐sts✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆f ∂f (A) = lim (A) ∆xk →0 ∆xk ∂xk

♦r fk′ (A)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❞❡✜♥✐t✐♦♥✿

❚❤❡♦r❡♠ ✹✳✹✳✷✿ P❛rt✐❛❧ ❉❡r✐✈❛t✐✈❡s ❚❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢

z = f (X)

✇✐t❤ r❡s♣❡❝t t♦

xk

❛t

X = A = (a1 , ..., an )

✐s

❢♦✉♥❞ ❛s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

g(x) = f (a1 , ..., ak−1 , x, ak+1 , ..., an ) , ❡✈❛❧✉❛t❡❞ ❛t

x = ak ❀

✐✳❡✳✱

 ∂f d (A) = f a1 , ..., ak−1 , x, ak+1 , ..., an ∂xk dx x=ak

❙♦✱ t❤✐s ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ z = f (x1 , ..., xn ) ✇✐t❤ r❡s♣❡❝t t♦ xk ✇✐t❤ t❤❡ r❡st ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✜①❡❞✳ ❚❤❡s❡ ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ❥✉st t❤❡ s❧♦♣❡s t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ ❣r❛♣❤✳ ❚❤❡r❡ ✐s ❛♥♦t❤❡r ❝♦♥❝❡♣t t❤❛t r❡❛♣♣❡❛rs ✐♥ ❞✐♠❡♥s✐♦♥ n✿

✹✳✹✳

❚❤❡ ❣r❛❞✐❡♥t

✸✶✵

❉❡✜♥✐t✐♦♥ ✹✳✹✳✸✿ ❣r❛❞✐❡♥t ❚❤❡

❣r❛❞✐❡♥t✱ ♦r t❤❡ ❞❡r✐✈❛t✐✈❡✱ ♦❢ f

❛t

X=A

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛s ✇❡❧❧ ❛s t❤❡ ✈❡❝t♦r ♦❢ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

df ∇f (A) = (A) = dX



∂f ∂f (A), ..., (A) ∂x1 ∂xn



❲❛r♥✐♥❣✦ ❚❤❡ ❣r❛❞✐❡♥t ✐s ♥♦t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦✲ t✐❡♥t✳

❲❛r♥✐♥❣✦ ❚❤❡ ❣r❛❞✐❡♥t ♥♦t❛t✐♦♥ ✐s t♦ ❜❡ r❡❛❞ ❛s ❢♦❧❧♦✇s✿  ∇f (A),

 grad f (A ,

✐✳❡✳✱ t❤❡ ❣r❛❞✐❡♥t ✐s ❝♦♠♣✉t❡❞ ❛♥❞ t❤❡♥ ❡✈❛❧✉❛t❡❞ ❛t X = A✳

❉❡✜♥✐t✐♦♥ ✹✳✹✳✹✿ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❙✉♣♣♦s❡

z = f (X)

✐s ❞❡✜♥❡❞ ❛t

X=A

❛♥❞

l(X) = f (A) + M · (X − A) ✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛t t❤❛t ♣♦✐♥t✳ ❚❤❡♥✱

❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

♦❢

f

❛t

lim

X→A ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❢✉♥❝t✐♦♥

f

X=A

z = l(X)

✐s ❝❛❧❧❡❞ t❤❡

✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s❛t✐s✜❡❞✿

f (X) − l(X) = 0. ||X − A||

✐s ❝❛❧❧❡❞

❞✐✛❡r❡♥t✐❛❜❧❡

❛t

X = A✳

❚❤❡ ♥✉♠❡r❛t♦r ✐♥ t❤❡ ❢♦r♠✉❧❛ ✐s t❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✏r✉♥✑✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐♥ ❞✐♠❡♥s✐♦♥

1

❛♥❞ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ ✐♥ ❞✐♠❡♥s✐♦♥

2✿

✹✳✹✳

❚❤❡ ❣r❛❞✐❡♥t

✸✶✶

❊①❛♠♣❧❡ ✹✳✹✳✺✿ ❞✐✛❡r❡♥t✐❛❜❧❡ ❙♦✱ ✇❡ st✐❝❦ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥s ♦❢ n ✈❛r✐❛❜❧❡s t❤❡ ❣r❛♣❤s ♦❢ ✇❤✐❝❤ ✕ ♦♥ ❛ s♠❛❧❧ s❝❛❧❡ ✕ ❧♦♦❦ ❧✐❦❡ ❧✐♥❡s ✐♥ ❞✐♠❡♥s✐♦♥ n = 1✱ ❧✐❦❡ ♣❧❛♥❡s ✐♥ ❞✐♠❡♥s✐♦♥ n = 2✱ ❛♥❞ ❣❡♥❡r❛❧❧② ❧✐❦❡ Rn ✦ ❇❡❧♦✇ ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ ✐ts ❧❡✈❡❧ ❝✉r✈❡s✿

◆♦t ♦♥❧② t❤❡s❡ ❝✉r✈❡s ❧♦♦❦ ❧✐❦❡ str❛✐❣❤t ❧✐♥❡s ✇❤❡♥ ✇❡ ③♦♦♠ ✐♥✱ t❤❡② ❛❧s♦ ♣r♦❣r❡ss ❛t ❛ ✉♥✐❢♦r♠ r❛t❡✳ ❏✉st ❛s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❤❛✈❡ ❛ s✐♥❣❧❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥ ✇✐t❤ t❤❡ ❣r❛♣❤✱ s❛♠❡ ❛♣♣❧✐❡s t♦ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡✿

❊①❛♠♣❧❡ ✹✳✹✳✻✿ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜❧❡ ❚❤❡ s✐♠♣❧❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ✐ts ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❜♦t❤ ③❡r♦ ❛t t❤❡ ♦r✐❣✐♥✿

✹✳✹✳

❚❤❡ ❣r❛❞✐❡♥t

✸✶✷

❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❝❧✐✛✱ t❤❡r❡ ✐s ♥♦ ♣❧❛♥❡ t❤❛t ❛♣♣r♦①✐♠❛t❡s t❤❡ s✉r❢❛❝❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✿

❚❤❡♦r❡♠ ✹✳✹✳✼✿ ❯♥✐q✉❡♥❡ss ♦❢ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥

❋♦r ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛s X = A✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛✲ t✐♦♥ ❛t A✳ Pr♦♦❢✳

❇② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ ❢✉♥❝t✐♦♥s l(X) = f (A) + M · (X − A) ✇✐t❤

f (X) − l(X) = 0, X→A ||X − A||

p(X) = f (A) + Q · (X − A) ✇✐t❤

f (X) − q(X) = 0. X→A ||X − A||

❛♥❞ ❚❤❡♥ ✇❡ ❤❛✈❡✿ lim

lim



f (X) − f (A) Q · (X − A) + ||X − A|| ||X − A||



= 0.



M · (X − A) Q · (X − A) − ||X − A|| ||X − A||



= 0,

❛♥❞ X→A

f (X) − f (A) M · (X − A) + ||X − A|| ||X − A||

❘✉❧❡ ✇❡ ❤❛✈❡✿ lim

X→A

♦r

lim



X→A

❚❤❡r❡❢♦r❡✱ ❜② t❤❡ ❙✉♠

lim



= 0,

M · (X − A) − Q · (X − A) = 0, X→A ||X − A|| lim

✹✳✹✳ ❚❤❡ ❣r❛❞✐❡♥t

✸✶✸

♦r lim

X→A

♦r

(M − Q) · (X − A) = 0, ||X − A||

lim (M − Q) ·

X→A

X −A = 0. ||X − A||

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❢r❛❝t✐♦♥✱ ❤♦✇❡✈❡r✱ ❞♦❡s ♥♦t ❡①✐st✳ ❚❤❡r❡❢♦r❡✱ M − Q = 0✳ ❊①❛♠♣❧❡ ✹✳✹✳✽✿

3

✈❛r✐❛❜❧❡s

❇❡❧♦✇ ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s ❣✐✈❡♥ ❜② ✐ts ❧❡✈❡❧ s✉r❢❛❝❡s✿

◆♦t ♦♥❧② t❤❡s❡ s✉r❢❛❝❡s ❧♦♦❦ ❧✐❦❡ ♣❧❛♥❡s ✇❤❡♥ ✇❡ ③♦♦♠ ✐♥✱ t❤❡② ❛❧s♦ ♣r♦❣r❡ss ❛t ❛ ✉♥✐❢♦r♠ r❛t❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✿

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❤❛s ❜❡❡♥ s♦❧✈❡❞✿ ❚❤❡♦r❡♠ ✹✳✹✳✾✿ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ■❢

l(X) = f (A) + M · (X − A) ✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

z = f (X)

❛t

X = A✱

t❤❡♥

M = grad f (A)

❊①❡r❝✐s❡ ✹✳✹✳✶✵

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ◆♦✇✱ s✉♣♣♦s❡ ✇❡ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥ t❤r♦✉❣❤♦✉t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❡ ✇✐❧❧ ❤❛✈❡ ❛ ✈❡❝t♦r ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✳ ❚❤✐s ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ✦

✹✳✺✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

❚❤❡ ❣r❛❞✐❡♥t s❡r✈❡s ❛s

t❤❡

✸✶✹

❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✳

❲❛r♥✐♥❣✦

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❝♦♠❜✐♥✐♥❣ ✐ts ✈❛r✐❛❜❧❡s✱ s❛② x ❛♥❞ y ✱ ✐♥t♦ ♦♥❡✱ X = (x, y)✱ ♠❛② ❜❡ ✐❧❧✲❛❞✈✐s❡❞ ❡✈❡♥ ✇❤❡♥ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♠❛❦❡ s❡♥s❡✳

✹✳✺✳ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ❏✉st ❛s ✐♥ ❞✐♠❡♥s✐♦♥

1✱

❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ t♦♦✱ ❛

f →

d dX

❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✿

→ G = ∇f

❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛r❡ ❞✐✛❡r❡♥t✳ ❲❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤✐s ❢✉♥❝t✐♦♥ ♦♣❡r❛t❡s✳

❲❛r♥✐♥❣✦

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ Rn ❛t ❛ ♣♦✐♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ❛t ❛ ♣♦✐♥t ❛r❡ ❜♦t❤ ✈❡❝t♦rs ✐♥ Rn ✱ t❤✐s ❞♦❡s♥✬t ♠❛❦❡ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s s✐♠✐❧❛r✳ ❲❡ st❛rt ✇✐t❤

❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳

❆❢t❡r ❛❧❧✱ t❤❡② s❡r✈❡ ❛s ❣♦♦❞✲❡♥♦✉❣❤ s✉❜st✐t✉t❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❛r♦✉♥❞ ❛

✜①❡❞ ♣♦✐♥t✳ ❚❤✐s ✐s t❤❡ ✏▲✐♥❡❛r ❙✉♠ ❘✉❧❡✑✿

f (X) + g(X)

❧✐♥❡❛r ❢✉♥❝t✐♦♥

✐ts ❣r❛❞✐❡♥t

= p + M (X − A)

M

= q + N (X − A)

N

f (X) + g(X) = p + (M + N )(X − A) M + N

✹✳✺✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

❲❡ ✉s❡❞ t❤❡

▲✐♥❡❛r✐t② ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t✳ f (X) ·k

✸✶✺

❚❤❡ ✏▲✐♥❡❛r ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✑ r❡❧✐❡s ♦♥ t❤❡ s❛♠❡ ♣r♦♣❡rt②✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥

✐ts ❣r❛❞✐❡♥t

= p + M (X − A)

M

k · f (X) = kp + (kM )(X − A) kM ❚❤❡ ✜rst ♦♣❡r❛t✐♦♥ ♦♥ ♦♥ ❢✉♥❝t✐♦♥s ✐s ❛❞❞✐t✐♦♥✳ ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

f, g

❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s

t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡

C

X

❛♥❞

X + ∆X

♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡

s❛t✐s❢②✿

∆(f + g) (C) = ∆f (C) + ∆g (C) ❚❤❡ ✇♦r❞✐♥❣s ♦❢ t❤❡ ❛❝t✉❛❧ t❤❡♦r❡♠s ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤♦s❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ ❱♦❧✉♠❡ ✷✿

❚❤❡♦r❡♠ ✹✳✺✳✶✿ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ✱ t❤❡✐r ❞✐✛❡r❡♥❝❡s s❛t✐s❢②✿ ∆(f + g) = ∆f + ∆g

Pr♦♦❢✳ ❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥

f + g✱

✇❡ ❤❛✈❡✿

∆(f + g)(C) = (f + g)(X + ∆X) − (f + g)(X)

= f (X + ∆X) + g(X + ∆X) − f (X) − g(X)   = f (X + ∆X) − f (X) + g(X + ∆X) − g(X) = ∆f (C) + ∆g(C) .

❚❤❡ ❢♦r♠✉❧❛ ❤♦❧❞s ❢♦r ❡✈❡r②

∆xk ✿

∆X ✳

❲❡ ❝❤♦♦s❡ ✐t t♦ ❢♦❧❧♦✇ t❤❡

k t❤

❛①✐s ❛♥❞ t❤❡♥ ❞✐✈✐❞❡ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❜②

∆f ∆g ∆(f + g) (C) = (C) + (C) . ∆xk ∆xk ∆xk

n ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❲❡ ❝♦♠❜✐♥❡ t❤❡♠ ✐♥t♦ ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛✳ ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s f, g ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ❚❤❡s❡ ❛r❡ t❤❡

∆(f + g) ∆f ∆g (C) = (C) + (C) ∆X ∆X ∆X ❲❡ ❤❛✈❡ t❤❡ ❛♥❛❧♦❣ ❢♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

❚❤❡♦r❡♠ ✹✳✺✳✷✿ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳

✹✳✺✳ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

✸✶✻

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡✮ s❛t✐s❢②✿ ∆f ∆g ∆(f + g) = + ∆X ∆X ∆X

■♥ ❛ ✇❛②✱ ✇❡ ❥✉st t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❛s ||∆X|| → 0 .

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A✱ ✇❡ ❤❛✈❡✿ df dg d(f + g) (A) = (A) + (A) dX dX dX

❙♦✱ ✇❡ ❤❛✈❡ t❤❡ ❛♥❛❧♦❣ ❢♦r ❞❡r✐✈❛t✐✈❡s✿ ❚❤❡♦r❡♠ ✹✳✺✳✸✿ ❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❚❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X ✱ ✇❡ ❤❛✈❡ ❛t X ✿ d(f + g) df dg = + dX dX dX Pr♦♦❢✳

❙✉♣♣♦s❡

l(X) = f (A) + M · (X − A) ❛♥❞ k(X) = g(A) + N · (X − A)

❛r❡ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛t A ♦❢ f ❛♥❞ g r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s❛t✐s✜❡❞✿ lim

X→A

M · (X − A) = 0 ❛♥❞ ||X − A||

lim

X→A

N · (X − A) = 0. ||X − A||

❲❡ ❝❛♥ ❛❞❞ t❤❡ t✇♦ ❧✐♠✐t t♦❣❡t❤❡r✱ ❛s ❛❧❧♦✇❡❞ ❜② t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ▲✐♠✐ts✱ ❛♥❞ t❤❡♥ ♠❛♥✐♣✉❧❛t❡ t❤❡ ❡①♣r❡ss✐♦♥✿ M · (X − A) N · (X − A) + lim X→A ||X − A|| X→A ||X − A|| . (M + N ) · (X − A) = lim X→A ||X − A||

0 = lim

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱

l(X) + k(X) = f (A) + g(A) + (M + N ) · (X − A)

✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f + g ✳ ❊①❡r❝✐s❡ ✹✳✺✳✹

❉❡r✐✈❡ t❤❡ ❧❛st t❤❡♦r❡♠ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s✳ ❙♦✱ ∆✬s ❜❡❝♦♠❡ d✬s✦

✹✳✺✳ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

✸✶✼

■♥ t❤❡ ❛❧t❡r♥❛t✐✈❡ ♥♦t❛t✐♦♥✿ ∇(f + g) = ∇f + ∇g

❊✐t❤❡r ♦❢ t❤❡ t✇♦ ❧❛st t❤❡♦r❡♠s ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡ s❡❝♦♥❞ ♦♣❡r❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥s ✐s s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❋♦r ❛♥② r❡❛❧ k ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ∆(k · f ) (C) = k · ∆f (C)

❚❤❡ r❡s✉❧t ✐s ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t❤❡ ♦♥❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿

❚❤❡♦r❡♠ ✹✳✺✳✺✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ✱ t❤❡ ✐ts ❞✐✛❡r❡♥❝❡ s❛t✐s✜❡s✿ ∆(kf ) = k∆f

Pr♦♦❢✳ ❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ k f ✱ ✇❡ ❤❛✈❡✿ ∆(k · f )(C) = (k · f )(X + ∆X) − (k · f )(X) = k · f (X + ∆X) − k · f (X)  = k · f (X + ∆X) − f (X)

= k · ∆f (C) .

❚❤❡ ❢♦r♠✉❧❛ ❤♦❧❞s ❢♦r ❡✈❡r② ∆X ✳ ❲❡ ❝❤♦♦s❡ ✐t t♦ ❢♦❧❧♦✇ t❤❡ kt❤ ❛①✐s ❛♥❞ t❤❡♥ ❞✐✈✐❞❡ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❜②

∆xk ✿

∆(k · f ) ∆f (C) = k · (C) . ∆xk ∆xk

❚❤❡s❡ ❛r❡ t❤❡ n ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❲❡ ❝♦♠❜✐♥❡ t❤❡♠ ✐♥t♦ ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛✳ ❋♦r ❛♥② r❡❛❧ k ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ∆(k · f ) ∆f (C) = k · (C) ∆X ∆X

✹✳✺✳ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

✸✶✽

❲❡ ❤❛✈❡ t❤❡ ❛♥❛❧♦❣ ❢♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

❚❤❡♦r❡♠ ✹✳✺✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X+∆X ♦❢ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛♥② r❡❛❧ k✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡✮ s❛t✐s❢②✿ ∆f ∆(kf ) =k ∆X ∆X

❆❣❛✐♥✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❛s ||∆X|| → 0✳ ❋♦r ❛♥② r❡❛❧ k ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ f ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A✱ ✇❡ ❤❛✈❡✿ df d(k · f ) (A) = k · (A) dX dX

❲❡ ❤❛✈❡ t❤❡ ❛♥❛❧♦❣ ❢♦r ❞❡r✐✈❛t✐✈❡s✿

❚❤❡♦r❡♠ ✹✳✺✳✼✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s ❆ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t✱ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❞❡r✐✈❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x ❛♥❞ ❛♥② r❡❛❧ k✱ ✇❡ ❤❛✈❡ ❛t X✿ df d(kf ) =k dX dX

❊①❡r❝✐s❡ ✹✳✺✳✽ ❉❡r✐✈❡ t❤❡ ❧❛st t❤❡♦r❡♠ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s✳ ❆♥❞ ∆✬s ❜❡❝♦♠❡ d✬s ❛❣❛✐♥✳ ■♥ t❤❡ ❛❧t❡r♥❛t✐✈❡ ♥♦t❛t✐♦♥✿ ∇(kf ) = k∇f

❊✐t❤❡r ♦❢ t❤❡ t✇♦ ❧❛st t❤❡♦r❡♠s ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❆s ✇❡ s❡❡ ✭❛♥❞ ✇✐❧❧ s❡❡ ❛❣❛✐♥ ✐♥ t❤✐s ❝❤❛♣t❡r✮✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢♦❧❧♦✇s t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ❡✈❡r② t✐♠❡✿

✹✳✺✳

❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts

❚❤❡s❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♦♥❡ ♣r♦❞✉❝✐♥❣

✸✶✾

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ✿

αx + βy , ✇❤❡r❡

α, β

❛r❡ t✇♦ ❝♦♥st❛♥t ♥✉♠❜❡rs✳ ❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❢✉♥❝t✐♦♥s❀ ❢♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t✇♦

❢✉♥❝t✐♦♥s ✭❧❡❢t✮✿

❲❡ ❛❧s♦ ♥♦t✐❝❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ✭r✐❣❤t✮✿

◮

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳

❚❤❡ q✉❡st✐♦♥ ❜❡❝♦♠❡s✿ ❲❤❛t ❤❛♣♣❡♥s t♦ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✉♥❞❡r ❞✐✛❡r❡♥t✐❛t✐♦♥❄ ❘❡❝❛❧❧ t❤❛t ❛ ❢✉♥❝t✐♦♥

F

✐s ❧✐♥❡❛r ✐❢ ✐t ✏♣r❡s❡r✈❡s✑ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✿

αx + βy →

F

→ αF (x) + βF (y)

❲✐t❤ t❤✐s ✐❞❡❛✱ t❤❡s❡ t✇♦ ❢♦r♠✉❧❛s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♦♥❡✿ ❚❤❡ ❞✐✛❡r❡♥❝❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✳ ❆ ♣r❡❝✐s❡ ✈❡rs✐♦♥ ✐s ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✹✳✺✳✾✿ ▲✐♥❡❛r✐t② ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ❞✐✛❡r❡♥❝❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❡r❛✐❜❧❡s ✐s t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡s✱ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ❛♥❞ ❞❡r✐✈❛t✐✈❡s r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡♥❡✈❡r t❤❡② ❡①✐st✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆(αf + βg)

= α∆f

∆f ∆(αf + βg) =α ∆X ∆X df d(αf + βg) =α dX dX

+β∆g ∆g ∆X dg +β dX

+β

❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠✿

αf + βg →

d dX

→ α∇f + β∇g

✹✳✻✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✷✵

✹✳✻✳ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

❍♦✇ ❞♦❡s ♦♥❡ ❧❡❛r♥ t❤❡ t❡rr❛✐♥ ❛r♦✉♥❞ ❤✐♠ ✇✐t❤♦✉t t❤❡ ❛❜✐❧✐t② t♦ ✢②❄ ❇② t❛❦✐♥❣ ❤✐❦❡s ❛r♦✉♥❞ t❤❡ ❛r❡❛✦ ▼❛t❤❡♠❛t✐❝❛❧❧②✱ t❤❡ ❢♦r♠❡r ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❧❛tt❡r ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✈✐❛ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡s❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ✇❤✐❝❤ t♦ ❝♦♠♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✿ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❜❡❢♦r❡ ♦r ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛❢t❡r✳ ❚❤✐s ✐s t❤❡ ❢♦r♠❡r✿ tr✐♣ ♠❛♣ −→

t

(x, y)



−→

z

t❡rr❛✐♥ ♠❛♣

❘❡❝❛❧❧ ❤♦✇ ✇❡ ✐♥t❡r♣r❡t t❤✐s ❝♦♠♣♦s✐t✐♦♥✳ ❲❡ ✐♠❛❣✐♥❡ ❝r❡❛t✐♥❣ ❛ tr✐♣ ♣❧❛♥ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t)✿ t❤❡ t✐♠❡s ❛♥❞ t❤❡ ♣❧❛❝❡s ♣✉t ♦♥ ❛ s✐♠♣❧❡ ❛✉t♦♠♦t✐✈❡ ♠❛♣✱ ❛♥❞ t❤❡♥ ❜r✐♥❣ t❤❡ t❡rr❛✐♥ ♠❛♣ ♦❢ t❤❡ ❛r❡❛ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (x, y)✿

❚❤❡ ❢♦r♠❡r ❣✐✈❡ ✉s t❤❡ ❧♦❝❛t✐♦♥ ❢♦r ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ ❡❧❡✈❛t✐♦♥ ❢♦r ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡s ✉s t❤❡ ❡❧❡✈❛t✐♦♥ ❢♦r ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♠❜✐♥❡❞✱ ✇❡ st❛rt ✇✐t❤ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ✐❢ ✇❡ tr❛✈❡❧ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ♦♥ ❛ ✢❛t✱ ♥♦t ♥❡❝❡ss❛r✐❧② ❤♦r✐③♦♥t❛❧✱ s✉r❢❛❝❡ ✭♠❛②❜❡ ❛ r♦♦❢✮❄ ❆❢t❡r t❤✐s s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❢♦✉♥❞ ❜② ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ ❚❤✉s✱ t❤❡

✐ts ❞❡r✐✈❛t✐✈❡

X = F (t) = A + D(t − a)

D

✐♥ Rn

z = f (X) = p + M · (X − A)

M

✐♥ Rn

M ·D

✐♥ R

◦

f (F (t))

= p + M · (A + D(t − a) − A)

= p + (M · D)(t − a)

❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳

✹✳✻✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✷✶

❲❡ ✉s❡ t❤✐s r❡s✉❧t ❢♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ ❛r❜✐tr❛r② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✈✐❛ t❤❡✐r ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡ r❡s✉❧t ✐s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ • ■❢ ✇❡ ❞♦✉❜❧❡ ♦✉r ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ✭✇✐t❤ t❤❡ s❛♠❡ t❡rr❛✐♥✮✱ t❤❡ ❝❧✐♠❜ ✇✐❧❧ ❜❡ t✇✐❝❡ ❛s ❢❛st✳

• ■❢ ✇❡ ❞♦✉❜❧❡ st❡❡♣♥❡ss ♦❢ t❤❡ t❡rr❛✐♥ ✭✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ s♣❡❡❞✮✱ t❤❡ ❝❧✐♠❜ ✇✐❧❧ ❜❡ t✇✐❝❡ ❛s ❢❛st✳

■t ❢♦❧❧♦✇s t❤❛t t❤❡ s♣❡❡❞ ♦❢ t❤❡ ❝❧✐♠❜ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ❜♦t❤ ♦✉r ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ❛♥❞ t❤❡ st❡❡♣♥❡ss ♦❢ t❤❡ t❡rr❛✐♥✳ ❚❤✐s ♥✉♠❜❡r ✐s ❝♦♠♣✉t❡❞ ❛s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢✿ • ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ F ♦❢ t❤❡ tr✐♣✱ ✐✳❡✳✱ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② E D ∂z ∂z , ∂y ✳ • ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❡rr❛✐♥ ❢✉♥❝t✐♦♥ f ✱ ✐✳❡✳✱ ∂x

dx

dy , ✱ ❛♥❞ dt dt

❋♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ ✇❡ ♥❡❡❞ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) t♦ ♠❛♣ t❤❡ ♣❛rt✐t✐♦♥ ❢♦r t t♦ t❤❡ ♣❛rt✐t✐♦♥ ❢♦r X ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ❤❛s t♦ ❢♦❧❧♦✇ t❤❡ ❣r✐❞✳ ❉✐♠❡♥s✐♦♥ 1✿

❉✐♠❡♥s✐♦♥ 2✿

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✐s ❛s ❢♦❧❧♦✇s✿ g

F

❇♦t❤ t ❛♥❞ X ❤❛✈❡ ♣❛rt✐t✐♦♥s✳

t −−−−→ X −−−−→ z

❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡ ♥♦t✐❝❡❞ t❤✐s ♣❛tt❡r♥ ♦❢ ❝❛♥❝❡❧❧❛t✐♦♥✿ ∆X ∆t

·

∆z ∆X

=

∆z ∆t

✹✳✻✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✷✷

❚❤❡♦r❡♠ ✹✳✻✳✶✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ X = F (t) ❞❡✜♥❡❞ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ z = g(X) ❞❡✜♥❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s X = F (t) ❛♥❞ X + ∆X = F (t + ∆t) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c ❛♥❞ Q = F (c) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢

t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿

∆(g ◦ F )(C) = ∆g(Q)

❲❡ ❤❛✈❡ t❤❡ ❛♥❛❧♦❣ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

❚❤❡♦r❡♠ ✹✳✻✳✷✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ X = F (t) ❞❡✜♥❡❞ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ z = g(X) ❞❡✜♥❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s X = F (t) ❛♥❞ X + ∆X = F (t + ∆t) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c ❛♥❞ Q = F (c) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✱ ♣r♦✈✐❞❡❞ ∆X 6= 0✿ ∆g ∆F ∆(g ◦ F ) (c) = (Q) · (c) ∆t ∆X ∆t

Pr♦♦❢✳ ❙✉♣♣♦s❡ F ♠♦✈❡s ❛❧♦♥❣ t❤❡ ❣r✐❞ ❛t x = c✳ ❚❤❡♥ ∆F (c) = ∆Xk =< 0, ..., 0, ∆xk , 0, ..., 0 > ,

❢♦r s♦♠❡ k ❛♥❞ ∆xk 6= 0✳ ❚❤❡♥ t♦ ❣❡t t♦ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✇❡ ❥✉st ♠✉❧t✐♣❧② t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡s ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t ❜② t❤❡s❡ r❡s♣❡❝t✐✈❡❧②✿ 1 ∆xk 1 = · . ∆t ∆xk ∆t

❆s ❛ r❡s✉❧t✱ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ g ❡♠❡r❣❡s✿ ∆(g ◦ F ) ∆g(Q) ∆xk ∆g ∆F (c) = · = (Q) · (c) . ∆t ∆xk ∆t ∆X ∆t

❆❧t❡r♥❛t✐✈❡❧②✱ ❜❡❝❛✉s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ X ✐s ✈❡r② s✐♠♣❧❡✱ s♦ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ F ✿ 1 ∆F (c) = ∆Xk = ∆t ∆t



 ∆xk 0, ..., 0, , 0, ..., 0 . ∆t

❚❤❡ ❞♦t ♣r♦❞✉❝t ❡♠❡r❣❡s✿ 

   ∆g(Q) ∆xk ∆g(Q) ∆xk ..., , ... · 0, ..., 0, , 0, ..., 0 = . ∆xk ∆t ∆xk ∆t

✹✳✻✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✷✸

❚❤❡ ❧✐♠✐t ❣✐✈❡s ✉s t❤❡ ❞❡r✐✈❛t✐✈❡✿

❚❤❡♦r❡♠ ✹✳✻✳✸✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❛t t❤❡ ✈❛❧✉❡ ♦❢ t❤❛t ♣♦✐♥t ✉♥❞❡r t❤❡ ✜rst ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t✱ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ X = F (t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = c ❛♥❞ z = g(X) ✐s ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❛t X = Q = F (c)✱ t❤❡♥ ✇❡ ❤❛✈❡✿ dg dF d(g ◦ F ) (c) = (Q) · (c) dt dX dt

❲❛r♥✐♥❣✦ ❲❡ ❤❛✈❡ ❛ ❞♦t ♣r♦❞✉❝t ♦♥ t❤❡ r✐❣❤t✳

❊①❡r❝✐s❡ ✹✳✻✳✹ ❉❡r✐✈❡ t❤❡ ❧❛st t❤❡♦r❡♠ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s✳

❊①❡r❝✐s❡ ✹✳✻✳✺ ❋✐♥❞ ❛♥♦t❤❡r✱ ♥♦♥✲❝♦♥st❛♥t✱ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✈❛❧✉❡s ♦❢

X = F (t) s✉❝❤ t❤❛t ∆F

♠❛② ❜❡ ③❡r♦ ❡✈❡♥ ❢♦r s♠❛❧❧

∆t✳ ❲❛r♥✐♥❣✦ ❚❤❡r❡ ❛r❡ t✇♦ ✐♥♣✉t ✈❛r✐❛❜❧❡s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❡❛❝❤ ❢♦r♠✉❧❛❀ t❤❡② ❛r❡ ❧✐♥❦❡❞ ❜② s✉❜st✐t✉t✐♦♥✳

❲❡ s❡❡ t❤❡ s❛♠❡ ♣❛tt❡r♥ ♦❢ ❝❛♥❝❡❧❧❛t✐♦♥✿

dX dt

·

dz dX

=

dz dt

❚❤❡s❡ ❛r❡♥✬t ❢r❛❝t✐♦♥s t❤♦✉❣❤✳ ❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

(g ◦ F )′ (t) = ∇g(F (t)) · F ′ (t) ❲✐t❤♦✉t t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✿

(g ◦ F )′ = (∇g ◦ F ) · F ′ ❆ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ♠❛② ❛♣♣❡❛r ✐♥ ❛♥♦t❤❡r ❝♦♥t❡①t✳✳✳ ❚❤✐s ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✇❤❡♥ ♦✉r ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ t❡rr❛✐♥ ♠❛♣

(x, y)

−→

z



−→ ♣r❡ss✉r❡

u

✹✳✻✳

❈♦♠♣♦s✐t✐♦♥s ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✷✹

t❡rr❛✐♥ ♠❛♣ ♦❢ t❤❡ ❛r❡❛ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ❛t♠♦s♣❤❡r✐❝ ♣r❡ss✉r❡ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❡❧❡✈❛t✐♦♥ ✭❛❜♦✈❡ t❤❡ s❡❛ ❧❡✈❡❧✮ ❛s

❘❡❝❛❧❧ ❤♦✇ ✇❡ ✐♥t❡r♣r❡t t❤✐s ❝♦♠♣♦s✐t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✈❛r✐❛❜❧❡s

z = f (x, y)✱

✇❡ ❤❛✈❡ t❤❡

❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿

❚❤❡ ❢♦r♠❡r ❣✐✈❡ ✉s t❤❡ ❡❧❡✈❛t✐♦♥ ❢♦r ❡✈❡r② ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ ♣r❡ss✉r❡ ❢♦r ❡✈❡r② ❡❧❡✈❛t✐♦♥✳ ❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡s ✉s t❤❡ ♣r❡ss✉r❡ ❢♦r ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♠❜✐♥❡❞✱ ✇❡ st❛rt ✇✐t❤ t❤✐s s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❢♦✉♥❞ ❜② ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✿

♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿

u = g(z)

❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✿

g(f (X))

❚❤✉s✱

✐ts ❞❡r✐✈❛t✐✈❡

M

✐♥

Rn

= q + m(z − p)

m

✐♥

R

= q + (mM ) · (X − A))

mM

✐♥

Rn

z = f (X) = p + M · (X − A) ◦

❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ❆❢t❡r

= q + m(p + M · (X − A) − p)

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳

❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

z = f (X)

❤❛s t♦ ♠❛♣ t❤❡ ♣❛rt✐t✐♦♥ ❢♦r

X

t♦ t❤❡ ♣❛rt✐t✐♦♥ ❢♦r

z✳

❚❤❡♦r❡♠ ✹✳✻✳✻✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ■■ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❧❛tt❡r❀ ✐✳❡✳✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s z = f (X) ❞❡✜♥❡❞ ❛t ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X +∆X ♦❢ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛♥② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ u = g(z) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s z = f (X) ❛♥❞ z + ∆z = f (X + ∆X) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s A ❛♥❞ a = f (A) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿ ∆(g ◦ f )(A) = ∆g (a)

❚❤❡♦r❡♠ ✹✳✻✳✼✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ■■ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts❀ ✐✳❡✳✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐✲ ❛❜❧❡s z = f (X) ❞❡✜♥❡❞ ❛t ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ❛♥② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ u = g(z) ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s z = f (X) ❛♥❞ z + ∆z = f (X + ∆X) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s A ❛♥❞ a = f (A) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s

✹✳✼✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✺

r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿ ∆g ∆f ∆(g ◦ f ) (A) = (a) · (A) ∆X ∆z ∆X

◆♦t❡✿ ❲❤✐❧❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❛❜♦✈❡ ✐♥✈♦❧✈❡s ❛ ❞♦t ♣r♦❞✉❝t✱ t❤❡ ♦♥❡ ❜❡❧♦✇ ✐s ❛ s❝❛❧❛r ♣r♦❞✉❝t✳

❚❤❡♦r❡♠ ✹✳✻✳✽✿ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ■■

❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❛t ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s z = f (X) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A ❛♥❞ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ u = g(z) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t a = f (A)✱ t❤❡♥ ✇❡ ❤❛✈❡✿ dg df d(g ◦ f ) (A) = (a) · (A) dX dz dX

◆♦t✐❝❡ ❤♦✇ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡ ✐s ✏❝❛♥❝❡❧❧❡❞✑ ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥ ✐♥ ❜♦t❤ ♦❢ t❤❡ t✇♦ ❢♦r♠s ♦❢ t❤❡ ❈❤❛✐♥ ❘✉❧❡❀ ✜rst✿

dz dz 6 dX · = ; 6 dX dt dt

❛♥❞ s❡❝♦♥❞✿

du du 6 dz · = . 6 dz dX dX

❚❤✉s✱ ✐♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ t✇♦ ❝♦♠♣♦s✐t✐♦♥s ❛r❡ ✈❡r② ❞✐✛❡r❡♥t✱ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❤❛s ❛ s♦♠❡✇❤❛t ✐♥❢♦r♠❛❧ ✕ ❜✉t s✐♥❣❧❡ ✕ ✈❡r❜❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✿

♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ♣r♦❞✉❝t

❚❤❡ ✇♦r❞ ✏♣r♦❞✉❝t✑✱ ❛s ✇❡ ❥✉st s❛✇✱ ✐s ❛❧s♦ ❛♠❜✐❣✉♦✉s✳ ❲❡ s❛✇ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢

t✇♦ ♥✉♠❜❡rs ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❜♦♦❦✱ t❤❡♥ t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ✈❡❝t♦rs✱ ❛♥❞ ✜♥❛❧❧② ❛ ✈❡❝t♦r ❛♥❞ ❛ ♥✉♠❜❡r✿

(f ◦ g)′ (x)

(g ◦ F )′ (x)

= f ′ (g(x))

· g ′ (x)

= ∇g(F (t)) · F ′ (t)

∇(g ◦ f )(X) = g ′ (f (X))

· ∇f (X)

❚❤❡ ❝♦♥t❡①t ❞❡t❡r♠✐♥❡s t❤❡ ♠❡❛♥✐♥❣ ❛♥❞ t❤✐s ❛♠❜✐❣✉✐t② s❡r✈❡s ❛ ♣✉r♣♦s❡✿ ✇❡ ✇✐❧❧ s❡❡ ❧❛t❡r ❤♦✇ t❤✐s ✇♦r❞✐♥❣ ✐s✱ ✐♥ ❛ r✐❣♦r♦✉s ✇❛②✱ ❛♣♣❧✐❝❛❜❧❡ t♦ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢

❛♥②

t✇♦ ❢✉♥❝t✐♦♥s✳

✹✳✼✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❯♥❢♦rt✉♥❛t❡❧②✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ♦❢ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❞♦ ♥♦t ♣r♦❞✉❝❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳✳✳

❲❛r♥✐♥❣✦ ❏✉st ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ ✇❡ ❢❛❝❡ ❛♥❞ r❡❥❡❝t t❤❡ ✏♥❛✐✈❡✑ ♣r♦❞✉❝t r✉❧❡✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♣r♦❞✉❝t ✐s ♥♦t t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❞❡r✐✈❛✲ t✐✈❡s✦

◆♦t ♦♥❧② t❤❡ ✉♥✐ts ❞♦♥✬t ♠❛t❝❤✱ ✐t✬s ✇♦rs❡

t❤✐s t✐♠❡✿ ❛❧❧ t❤r❡❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ✈❡❝t♦rs

✹✳✼✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✻ ❛♥❞ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❝❛♥✬t ❣✐✈❡ ✉s t❤❡ t❤✐r❞✳✳✳

❘❡❝❛❧❧ t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✐s ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥s ✭t♦♣✮✿

❆s t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❞❡♣t❤ ❛r❡ ✐♥❝r❡❛s✐♥❣✱ s♦ ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳ ❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❛r❡❛ ✭❜♦tt♦♠ ❧❡❢t✮ ❝❛♥♥♦t ❜❡ ❡①♣r❡ss❡❞ ❡♥t✐r❡❧② ✐♥ t❡r♠s ♦❢ t❤❡ ✐♥❝r❡❛s❡s ♦❢ t❤❡ ✇✐❞t❤ ❛♥❞ ❞❡♣t❤✦ ❚❤✐s ✐♥❝r❡❛s❡ ✐s s♣❧✐t ✐♥t♦ t✇♦ r❡❝t❛♥❣❧❡s ✭❜♦tt♦♠ r✐❣❤t✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ t✇♦ t❡r♠s ✐♥ ♦✉r ❢♦r♠✉❧❛✳ ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ∆(f · g) (C) = f (X + ∆X) · ∆g (C) + ∆f (C) · g(X)

❚❤❡♦r❡♠ ✹✳✼✳✶✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆(f · g)(C) = f (X + ∆X) · ∆g(C) + ∆f (C) · g(X)

✹✳✼✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✼

Pr♦♦❢✳ ❚❤❡ tr✐❝❦ ✐s t♦ ✐♥s❡rt ❡①tr❛ t❡r♠s✿

∆(f · g)(C) = (f · g)(X + ∆X) − (f · g)(X)

= f (X + ∆X) · g(X + ∆X) − f (X) · g(X)

= f (X + ∆X) · g(X + ∆X)−f (X + ∆X) · g(X) + f (X + ∆X) · g(X) − f (X) · g(X) = f (X + ∆X) · (g(X + ∆X) − g(X)) + (f (X + ∆X) − f (X)) · g(X)

= f (X + ∆X) · ∆g(C) + ∆f (X) · g(X) .

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿

∆g ∆f ∆(f · g) (C) = f (X + ∆X) · (C) + (C) · g(X) ∆X ∆X ∆X ❙♦✱ ✇❡ ❥✉st ❞✐✈✐❞❡ ❜② ∆X ✿

❚❤❡♦r❡♠ ✹✳✼✳✷✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C ✮ s❛t✐s❢②✿ ∆(f · g) ∆g ∆f (C) + (C) · g(X) (C) = f (X + ∆X) · ∆X ∆x ∆x ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A✱ ✇❡ ❤❛✈❡✿

d(f · g) dg df (A) = f (A) · (A) + (A) · g(A) dX dX dX ❙♦✱ ∆X ❣♦❡s t♦ 0✱ ❛♥❞ ❛❧❧ ∆s ✕ ❡①❝❡♣t ❢♦r ♦♥❡ ✕ ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ d✬s✿

❚❤❡♦r❡♠ ✹✳✼✳✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s ❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞❡r✐✈❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X ✱ ✇❡ ❤❛✈❡✿ d(f · g) dg df + ·g =f· dX dX dX

❊①❡r❝✐s❡ ✹✳✼✳✹ ❉❡r✐✈❡ t❤❡ ❧❛st t❤❡♦r❡♠ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s✳ ❆s ❛♥ ✐♥❢♦r♠❛❧ ❛❜❜r❡✈✐❛t✐♦♥✿

✹✳✼✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ◮ ❲❤❡♥ ✇❡

✸✷✽

♠✉❧t✐♣❧② ❢✉♥❝t✐♦♥s✱ ✇❡ ❝r♦ss✲♠✉❧t✐♣❧② t❤❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳

■♥ s✉♠♠❛r②✱ t❤✐s ✐s ❤♦✇ t❤❡ ❝r♦ss✲♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇♦r❦s✿ ❢✉♥❝t✐♦♥s ❞❡r✐✈❛t✐✈❡s ✜rst s❡❝♦♥❞

−→ ∇(f g) = f ∇g + ∇f g

∇f

f

∇g

g

❚❤❡ ❢♦r♠✉❧❛ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❜✉t ✇❡ ❤❛✈❡ t♦ ❡①❛♠✐♥❡ ✐t ❝❛r❡❢✉❧❧②❀ s❛♠❡ t❤✐♥❣s ❤❛✈❡ ❝❤❛♥❣❡❞✦ ■♥❞❡❡❞✱ ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❡✐t❤❡r t❡r♠ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✭❛ ♥✉♠❜❡r✮ ❛♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♦t❤❡r ✭❛ ✈❡❝t♦r✮✳ ❋✉rt❤❡r♠♦r❡ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥✿ s❝❛❧❛r ✈❡❝t♦r ✈❡❝t♦r s❝❛❧❛r f (A)

·

∇g (A)

✈❡❝t♦r

+

∇f (A)

·

g(A)

✈❡❝t♦r

✈❡❝t♦r ■t ♠❛t❝❤❡s t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✳ ▼♦r❡♦✈❡r✱ ✇❤❡♥ A ✈❛r✐❡s✱ t❤❡ ❢♦r♠✉❧❛s t❛❦❡ t❤❡ ❢♦r♠ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ ∇(f · g) = f · ∇g + ∇f · g .

❍❡r❡✱ ❡✐t❤❡r t❡r♠ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✱ ❛ s❝❛❧❛r ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♦t❤❡r✱ ❛ ✈❡❝t♦r ✜❡❧❞✳ ❙✉❝❤ ❛ ♣r♦❞✉❝t ✐s ❛❣❛✐♥ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ s♦ ✐s t❤❡✐r s✉♠✳ ■t ♠❛t❝❤❡s t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✳ ◆♦✇ ❞✐✈✐s✐♦♥✳ ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ∆(f /g) (C) =

f (X + ∆X) · ∆g (C) − ∆f (C) · g(X) g(X)g(X + ∆X)

❚❤❡♦r❡♠ ✹✳✼✳✺✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C ✮ s❛t✐s❢②✿ ∆(f /g)(C) =

f (X + ∆X) · ∆g(C) − ∆f (C) · g(X) g(X)g(X + ∆X)

Pr♦♦❢✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❝❛s❡ f = 1✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ 1 1 − g(X + ∆X) g(X) g(X) − g(X + ∆X) . = g(X + ∆X)g(X)

∆(1/g)(X) =

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳

✹✳✼✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✾

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s X ❛♥❞ X + ∆X ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ C s❛t✐s❢②✿ ∆f ∆g (C) − ∆X (C) · g(X) f (X + ∆X) · ∆X ∆(f /g) (C) = ∆X g(X)g(X + ∆X)

♣r♦✈✐❞❡❞ g(X), g(X + ∆X) 6= 0✳

❚❤❡♦r❡♠ ✹✳✼✳✻✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆g ∆f f (X + ∆X) · ∆X (C) − ∆X (C) · g(X) ∆(f /g) (C) = ∆x g(X) · g(X + ∆X)

♣r♦✈✐❞❡❞ g(X), g(X + ∆X) 6= 0✳

Pr♦♦❢✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❝❛s❡ f = 1✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ g(X) − g(X + ∆X) ∆(1/g)(X) = ∆X ∆Xg(X + ∆X)g(X) 1 g(X + ∆X) − g(X) =− · ∆X g(X + ∆X) · g(X) ∆g 1 =− (C) · ∆X g(X + ∆X) · g(X)

✇✐t❤ C = X .

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳ ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A✱ ✇❡ ❤❛✈❡✿ d(f /g) (A) = dX

df (A) dX

· g(A) − f (A) · g(A)2

dg (A) dX

♣r♦✈✐❞❡❞ g(A) 6= 0✳ ❙♦✱ ∆X ❣♦❡s t♦ 0 ❛♥❞ ❛❧❧ ∆s ❡①❝❡♣t ❢♦r ♦♥❡ ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ d✬s✿

❚❤❡♦r❡♠ ✹✳✼✳✼✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X ✱ ✇❡ ❤❛✈❡✿ dg f (x) · dX − d(f /g) = dX g2

♣r♦✈✐❞❡❞ g(X) 6= 0✳

df dX

·g

✹✳✽✳

✸✸✵

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

Pr♦♦❢✳ ❲❡ r❡♣r❡s❡♥t t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ g ❛s ❛ ❝♦♠♣♦s✐t✐♦♥✿ z=

1 1 dz 1 dy dz 1 ∇g(X) , =⇒ z = , y = g(X) =⇒ = − 2, = ∇g(X) =⇒ =− g(X) y dy y dX dX g(X)2

❜② t❤❡ ❈❤❛✐♥

❘✉❧❡✳

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ Pr♦❞✉❝t

❚❤❡ ❢♦r♠✉❧❛ ✐s s✐♠✐❧❛r t♦ t❤❡ Pr♦❞✉❝t

❘✉❧❡

✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❛❧s♦ ✐♥✈♦❧✈❡s ❝r♦ss✲♠✉❧t✐♣❧✐❝❛t✐♦♥ ✿

❢✉♥❝t✐♦♥s ❞❡r✐✈❛t✐✈❡s ✜rst

f

s❡❝♦♥❞

g

❘✉❧❡✳

∇f

∇g

  f f ∇g − ∇f g −→ ∇ = g g2

❙✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ❡✐t❤❡r t❡r♠ ✐♥ t❤❡ ♥✉♠❡r❛t♦r ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ❛ s❝❛❧❛r ❢✉♥❝t✐♦♥ ❛♥❞ ❛ ✈❡❝t♦r ✜❡❧❞✳ ❚❤❡✐r s✉♠ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ✐t✬s st✐❧❧ ❛ ✈❡❝t♦r ✜❡❧❞ ✇❤❡♥ ✇❡ ❞✐✈✐❞❡ ❜② ❛ s❝❛❧❛r ❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❢♦✉r ♣r♦♣❡rt✐❡s r❡✲st❛t❡❞ ✐♥ t❤❡ ❣r❛❞✐❡♥t ♥♦t❛t✐♦♥✿ ❙❘✿

∇(f + g) = ∇f + ∇g ❈▼❘✿

∇(kf ) = k ∇f

❢♦r ❛♥② r❡❛❧ k

P❘✿

∇(f g) = ∇f g + f ∇g ◗❘✿

∇(f /g) =

✇❤❡r❡✈❡r g 6= 0

∇f g − f ∇g g2

✹✳✽✳ ❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

❚❤❡ r❡s✉❧t ✇❡ ❤❛✈❡ ❜❡❡♥ ❛❧❧✉❞✐♥❣ t♦ ✐s t❤❛t t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ■t ✐s ♣r♦✈❡♥ ❧❛t❡r ❜✉t ❤❡r❡ ✇❡ ❥✉st ❝♦♥s✐❞❡r t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❣r❛❞✐❡♥t ❛♥❞ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s✱ ✐✳❡✳✱ t❤❡ ❝✉r✈❡s ♦❢ ❝♦♥st❛♥t ✈❛❧✉❡✱ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❡ st❛rt ✇✐t❤ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥s✿

❚❤❡♦r❡♠ ✹✳✽✳✶✿ ▲❡✈❡❧ ❙❡ts ❛♥❞ ❉✐✛❡r❡♥❝❡s ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ z = f (X) ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X 6= X ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ✐❢ t❤❡s❡ t✇♦ ♥♦❞❡s ❧✐❡ ✇✐t❤✐♥ ❛ ❧❡✈❡❧ s❡t ♦❢ z = f (X)✱ ✐✳❡✳✱ f (X) = f (X + ∆X)✱ t❤❡♥ ∆f (A) = 0 ,

✇❤❡r❡ A ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤✐s ❡❞❣❡✳

❚❤❡♦r❡♠ ✹✳✽✳✷✿ ▲❡✈❡❧ ❙❡ts ❛♥❞ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ z = f (X) ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s X ❛♥❞ X + ∆X 6= X ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥✱ ✐❢ t❤❡s❡ t✇♦ ♥♦❞❡s ❧✐❡ ✇✐t❤✐♥ ❛ ❧❡✈❡❧ s❡t ♦❢ z = f (X)✱ ✐✳❡✳✱ f (X) = f (X + ∆X)✱ t❤❡♥ ∆f (A) = 0 , ∆X

✇❤❡r❡ A ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤✐s ❡❞❣❡✳

✹✳✽✳

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

✸✸✶

❊①❛♠♣❧❡ ✹✳✽✳✸✿ s♣r❡❛❞s❤❡❡t

❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦♥ t❤❡ ♣❧❛♥❡ ✐s s❤♦✇♥ ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ✇✐t❤ ✐ts ❧❡✈❡❧ ❝✉r✈❡ ✈✐s✉❛❧✐③❡❞✿

■♥ t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝♦♠♣✉t❡❞ ❛♥❞ t❤❡♥ ❜❡❧♦✇ ✐t ✐s ✈✐s✉❛❧✐③❡❞✳ ❚❤✐s ❝✉r✈❡ ❛♥❞ t❤✐s ✈❡❝t♦r ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳ ❊①❡r❝✐s❡ ✹✳✽✳✹

❈♦♥s✐❞❡r ♦t❤❡r ♣♦ss✐❜❧❡ ❛rr❛♥❣❡♠❡♥ts ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❝♦♥✜r♠ t❤❡ ❝♦♥❥❡❝t✉r❡✳ ❊①❛♠♣❧❡ ✹✳✽✳✺✿ ♣❛r❛❧❧❡❧

■♥ t❤❡ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡ ♦❢ ❛ ♣❧❛♥❡✿ f (x, y) = 2x + 3y ,

t❤❡ ❣r❛❞✐❡♥t ✐s ❛ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞✿ ∇f (x, y) =< 2, 3 > .

▼❡❛♥✇❤✐❧❡✱ ✐ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ♣❛r❛❧❧❡❧ str❛✐❣❤t ❧✐♥❡s✿ 2x + 3y = c .

❚❤❡ s❧♦♣❡ ✐s −2/3 ✇❤✐❝❤ ♠❛❦❡s t❤❡♠ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❣r❛❞✐❡♥t ✈❡❝t♦r < 2, 3 >✦ ❲❡ t❤❡♥ ❝♦♥❥❡❝t✉r❡ t❤❛t t❤❡

❣r❛❞✐❡♥t ❛♥❞ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ❣❡♥❡r❛❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ z = f (x, y) = c + m(x − a) + n(y − b) ,

❛♥❞ M = ∇f =< m, n > t❤❡ ❣r❛❞✐❡♥t ♦❢ f ✳ ▲❡t✬s ♣✐❝❦ ❛ s✐♠♣❧❡ ✈❡❝t♦r D =< −n, m > ♣❡r♣❡♥❞✐❝✉❧❛r t♦ M =< m, n >✳ ❈♦♥s✐❞❡r t❤✐s str❛✐❣❤t ❧✐♥❡ ✇✐t❤ D ❛s ❛ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✿ F (t) = (a, b)+ < −n, m > t .

✹✳✽✳

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

✸✸✷

❲❡ s✉❜st✐t✉t❡ ✐t ✐♥t♦ f ✿

f (F (t)) = f (a − nt, b + mt) = c + m(−nt) + n(mt) = c . ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s ❝♦♥st❛♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❧✐♥❡ st❛②s ✇✐t❤✐♥ ❛ ❧❡✈❡❧ ❝✉r✈❡ ♦❢ f ✳ ❚❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ❝♦♥✜r♠❡❞✳ ❊①❛♠♣❧❡ ✹✳✽✳✻✿ ♣❛r❛❜♦❧♦✐❞

■♥ t❤❡ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡ ♦❢ ❛ ❝✐r❝✉❧❛r ♣❛r❛❜♦❧♦✐❞✿

f (x, y) = x2 + y 2 , t❤❡ ❣r❛❞✐❡♥t ❝♦♥s✐sts ♦❢ t❤❡ r❛❞✐❛❧ ✈❡❝t♦rs✿

∇f (x, y) =< 2x, 2y > . ▼❡❛♥✇❤✐❧❡✱ ✐ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ❝✐r❝❧❡s✿

x2 + y 2 = c ❢♦r c > 0 .

❚❤❡ r❛❞✐✐ ♦❢ ❛ ❝✐r❝❧❡ ❛r❡ ❦♥♦✇♥ ✭❛♥❞ ❛r❡ s❡❡♥ ❛❜♦✈❡✮ t♦ ❜❡

♣❡r♣❡♥❞✐❝✉❧❛r

t♦ t❤❡ ❝✐r❝❧❡✦

❲❡ ♥❡❡❞ t♦ ♠❛❦❡ ♦✉r ❝♦♥❥❡❝t✉r❡ ♣r❡❝✐s❡ ❜❡❢♦r❡ ♣r♦✈✐♥❣ ✐t✳ ❋✐rst✱ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡♥✬t ♥❡❝❡ss❛r✐❧② ❝✉r✈❡s✳ ❚❤❡② ❛r❡ ❥✉st s❡ts ✐♥ t❤❡ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❡♥ f ✐s ❝♦♥st❛♥t✱ ❛❧❧ t❤❡ ❧❡✈❡❧ s❡ts ❛r❡ ❡♠♣t② ❜✉t ♦♥❡ ✇❤✐❝❤ ✐s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿  f (x, y) = c =⇒ (x, y) : f (x, y) = b} = ∅ ✇❤❡♥ b 6= c ❛♥❞ {(x, y) : f (x, y) = c = R2 .

❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ ✇❤❡♥ ❛ ❧❡✈❡❧ ❝✉r✈❡ ✐s ❛ ❝✉r✈❡✱ ✐t✬s ❛♥ ✐♠♣❧✐❝✐t ❝✉r✈❡ ❛♥❞ ✐s♥✬t r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❲❛r♥✐♥❣✦ ❚❤✐s ❞♦❡s♥✬t ♠❡❛♥ t❤❛t t❤❡ ♣❛t❤ ♦❢

F

✐s ❛ ❧❡✈❡❧ s❡t

❜✉t s✐♠♣❧② ✐ts s✉❜s❡t✳ ❚❤❡ q✉❡st✐♦♥ ♦❢ ✇❤❡♥ ❡①❛❝t❧② ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ❝✉r✈❡s ✐s ❛❞❞r❡ss❡❞ ❡❧s❡✇❤❡r❡✳

❍♦✇ ❞♦ ✇❡ s♦rt t❤✐s ♦✉t❄ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ st✉❞② t❤❡ t❡rr❛✐♥ ❜② t❛❦✐♥❣ t❤❡s❡ ❤✐❦❡s ✕ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✕ ❛♥❞ t❤✐s t✐♠❡ ✇❡ ❝❤♦♦s❡ ❛♥ ❡❛s② ♦♥❡✿ ♥♦ ❝❧✐♠❜✐♥❣✳ ❲❡ st❛② ❛t t❤❡ s❛♠❡ ❡❧❡✈❛t✐♦♥ ✿

✹✳✽✳

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

✸✸✸

■♥ ♦t❤❡r ✇♦r❞s✱ ♦✉r ❢✉♥❝t✐♦♥ z = f (X) ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❛❧♦♥❣ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t)✱ ✐✳❡✳✱ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s ❝♦♥st❛♥t✿ f (F (t)) = ❝♦♥st❛♥t . ❲❡ ❝❛♥ ❣♦ s❧♦✇ ♦r ❢❛st ❛♥❞ ✇❡ ❝❛♥ ❣♦ ✐♥ ❡✐t❤❡r ❞✐r❡❝t✐♦♥✳ ❙❡❝♦♥❞✱ ✇❤❛t ❞♦ ✇❡ ♠❡❛♥ ✇❤❡♥ ✇❡ s❛② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ❛ ✈❡❝t♦r ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r❄ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ❛t ❛ ♣♦✐♥t ✐s ✐ts t❛♥❣❡♥t ✈❡❝t♦r ❛t t❤❛t ♣♦✐♥t✱ ❜② ❞❡✜♥✐t✐♦♥✦

❲❡ ❛r❡ t❤❡♥ ❝♦♥❝❡r♥❡❞ ✇✐t❤✿ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ∇f (A) ❛♥❞ F ′ (a), ✇❤❡r❡ A = F (a) ,

❛♥❞✱ t❤❡r❡❢♦r❡✱ ✇✐t❤ t❤❡✐r ❞♦t

♣r♦❞✉❝t ✿

∇f (F (a)) · F ′ (a) .

■s ✐t ③❡r♦❄ ❇✉t ✇❡ ❥✉st s❛✇ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✦ ■t✬s t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❈❤❛✐♥

❘✉❧❡ ✿

(f ◦ F )′ (a) = ∇f (F (a)) · F ′ (a) .

❲❤② ✐s t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ③❡r♦❄ ❇❡❝❛✉s❡ ✐t✬s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✦ ■♥❞❡❡❞✱ t❤❡ ♣❛t❤ ♦❢ F ❧✐❡ ✇✐t❤✐♥ ❛ ❧❡✈❡❧ ❝✉r✈❡ ♦❢ f ✳ ❙♦✱ ✇❡ ❤❛✈❡✿ d 0 = f (F (t)) = (f ◦ F )′ (a) = ∇f (F (a)) · F ′ (a) . dt t=a

❙♦✱ ✇❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ❧❡✈❡❧ ❝✉r✈❡s ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ✈❡❝t♦rs ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r✿ ∇f (A) ⊥ F ′ (a) .

❲❤❛t r❡♠❛✐♥s ✐s ❥✉st s♦♠❡ ❝❛✈❡❛ts✳ ❋✐rst✱ t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s t♦ ♠❛❦❡ s❡♥s❡✳ ❙❡❝♦♥❞✱ ♥❡✐t❤❡r ♦❢ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s s❤♦✉❧❞ ❜❡ ③❡r♦ ♦r t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠ ✇✐❧❧ ❜❡ ✉♥❞❡✜♥❡❞ ✭∇f (A) 6= 0 ❛♥❞ F ′ (a) 6= 0✮✳ ❚❤❡♦r❡♠ ✹✳✽✳✼✿ ▲❡✈❡❧ ❈✉r✈❡s ❆r❡ P❡r♣❡♥❞✐❝✉❧❛r t♦ ●r❛❞✐❡♥t

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s z = f (X) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t X = A ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I t❤❛t ❝♦♥t❛✐♥s a ✇✐t❤ F (a) = A✳ ❚❤❡♥✱ ✐❢ t❤❡ ♣❛t❤ ♦❢ X = F (t) ❧✐❡s ✇✐t❤✐♥ ❛ ❧❡✈❡❧ s❡t

✹✳✽✳

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

✸✸✹

♦❢ z = f (X)✱ t❤❡♥ df dF (A) ⊥ (a) dX dt

♣r♦✈✐❞❡❞ ❜♦t❤ ❛r❡ ♥♦♥✲③❡r♦✳

❊①❡r❝✐s❡ ✹✳✽✳✽

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❲❡ ♥♦✇ ❞❡♠♦♥str❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t t❤❛t ♠✐①❡s t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s✿

❈♦r♦❧❧❛r② ✹✳✽✳✾✿ ▲❡✈❡❧ ❈✉r✈❡s ❆r❡ P❡r♣❡♥❞✐❝✉❧❛r t♦ ●r❛❞✐❡♥t

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s z = f (X) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ s❡t U ✐♥ Rn ✳ ❙✉♣♣♦s❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = F (t) ✐s ❞❡✜♥❡❞ ❛t ❛❞❥❛❝❡♥t ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❡ ♣♦✐♥ts P = F (t) ❛♥❞ Q = F (t + ∆t) ❛r❡ ❞✐st✐♥❝t ❛♥❞ ❧✐❡ ✇✐t❤✐♥ ❛ ❧❡✈❡❧ s❡t ♦❢ z = f (X)✱ ✐✳❡✳✱ f (P ) = f (Q)✱ ❛♥❞ t❤❡ s❡❣♠❡♥t P Q ❜❡t✇❡❡♥ t❤❡♠ ❧✐❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ U ✳ ❚❤❡♥✱ ❢♦r s♦♠❡ ♣♦✐♥t A ♦♥ P Q ❛♥❞ ❛ s❡❝♦♥❞❛r② ♥♦❞❡ a ♦❢ [t, t + ∆t]✱ ✇❡ ❤❛✈❡✿ ∆F df (A) ⊥ (a) , dX ∆t

♣r♦✈✐❞❡❞ t❤❡ ❣r❛❞✐❡♥t ✐s ♥♦♥✲③❡r♦ ✐♥ U ✳ Pr♦♦❢✳

▲❡t z = L(t) ❜❡ t❤❡ ❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✇✐t❤ L(t) = P ❛♥❞ L(t + ∆t) = Q✳ ❚❤❡♥✱ dL ∆F (a) = (a) , dt ∆t

❢♦r ❛♥② ❝❤♦✐❝❡ ♦❢ ❛ s❡❝♦♥❞❛r② ♥♦❞❡ a ♦❢ t❤❡ ✐♥t❡r✈❛❧ [t, t + ∆t]✳ ❲❡ ❞❡✜♥❡ ❛ ♥❡✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

✹✳✽✳

❚❤❡ ❣r❛❞✐❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

✸✸✺

❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s t ❛♥❞ t + ∆t✿ ❚❤❡♥ ❜② t❤❡

h=f ◦F .

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ t❤❡r❡ ✐s s✉❝❤ ❛ s❡❝♦♥❞❛r② ♥♦❞❡ a t❤❛t✿ dh ∆h (a) = (a) . ∆t dt

❙✐♥❝❡ t❤❡ ❢♦r♠❡r ✐s ③❡r♦ ❜② t❤❡ ❛ss✉♠♣t✐♦♥✱ ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t t❤❡ ❧❛tt❡r✿ dh df dL df ∆F 0= (a) = (A) · (a) = (A) · (a) , dt dX dt dX ∆t ✇❤❡r❡ A = L(a)✳ ❚❤❡ t❤❡♦r❡♠ r❡♠❛✐♥s ✈❛❧✐❞ ♥♦ ♠❛tt❡r ❤♦✇ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ tr❛❝❡s t❤❡ ❧❡✈❡❧ ❝✉r✈❡ ❛s ❧♦♥❣ ❛s ✐t ❞♦❡s♥✬t st♦♣✳ ❚❤❡r❡ ❛r❡ t❤❡♥ ♦♥❧② t✇♦ ♠❛✐♥ ✇❛②s ✕ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ✕ t❤❛t ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❝❛♥ ❢♦❧❧♦✇ t❤❡ ❧❡✈❡❧ ❝✉r✈❡✳

❇✉t ✇❛✐t ❛ ♠✐♥✉t❡✱ t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t s♣❡❛❦ ❡①❝❧✉s✐✈❡❧② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ■t s❡❡♠s t♦ ❛♣♣❧② t♦ ❧❡✈❡❧ s✉r❢❛❝❡s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✳ ■♥❞❡❡❞✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ✐s t❤❡ s❛♠❡✿ ❆ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❣r❛❞✐❡♥t✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ❢♦r t❤❡ ❝✉r✈❡ t♦ ❣♦ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳

❲✐t❤ ❛❧❧ t❤❡ ✈❛r✐❡t② ♦❢ ❛♥❣❧❡s ❜❡t✇❡❡♥ t❤❡✐r t❛♥❣❡♥ts✱ t❤❡② ❛❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❛♥❣❧❡ ✇✐t❤ t❤❡ ❣r❛❞✐❡♥t✳ ■♥ t❤✐s ❡①❛❝t s❡♥s❡ ✇❡ s♣❡❛❦ ♦❢ t❤❡ ❣r❛❞✐❡♥t ❜❡✐♥❣ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧❡✈❡❧ s✉r❢❛❝❡✳

❚❤✐s r❡s✉❧t ✐s ❛ ❢r❡❡ ❣✐❢t ❝♦✉rt❡s② ♦❢ ❛❜str❛❝t t❤✐♥❦✐♥❣ ❛♥❞ t❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥✦

✹✳✾✳

▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✸✸✻

❊①❛♠♣❧❡ ✹✳✽✳✶✵✿ r❛❞✐❛❧ ✈❡❝t♦r ✜❡❧❞ ❚❤❡ ❧❡✈❡❧ s✉r❢❛❝❡s ♦❢ t❤❡ r❛❞✐❛❧ ✈❡❝t♦r ✜❡❧❞✱

V (x, y, z) =< x, y, z > , ❛s ✇❡❧❧ ❛s t❤❡ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥✱

W (x, y, z) = −

c < x, y, z > , || < x, y, z > ||3

❛r❡ ❝♦♥❝❡♥tr✐❝ s♣❤❡r❡s✳ ❚❤❡ ❣r❛❞✐❡♥t ✈❡❝t♦rs ♣♦✐♥t ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞ t♦✇❛r❞s ✐t ✐♥ t❤❡ ❧❛tt❡r✳ ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ❤♦✇✱ ♥♦ ♠❛tt❡r ✇❤❛t ♣❛t❤ ②♦✉ t❛❦❡ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✱ ②♦✉r ❜♦❞② ✇✐❧❧ ♣♦✐♥t ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✳

❲✐t❤ t❤✐s t❤❡♦r❡♠ ✇❡ ❝❛♥ ✐♥t❡r♣r❡t t❤❡ ✐❞❡❛ t❤❛t

♦❢ t❤❡ ❢✉♥❝t✐♦♥

t❤❡ ❣r❛❞✐❡♥t ♣♦✐♥ts ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤

✳ ■♥❞❡❡❞✱ ✐t s✉❣❣❡sts t❤❡ s❤♦rt❡st ♣❛t❤ t♦✇❛r❞ t❤❡ ✏♥❡①t✑ ❧❡✈❡❧ ❝✉r✈❡✿

❚❤✐s ✐♥❢♦r♠❛❧ ❡①♣❧❛♥❛t✐♦♥ ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✳ ❲❡ ✇✐❧❧ ♠❛❦❡ t❤❡ t❡r♠s ✐♥ t❤✐s st❛t❡♠❡♥t ❢✉❧❧② ♣r❡❝✐s❡ ♥❡①t✳

✹✳✾✳ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❙✉♣♣♦s❡

z = f (X)

✐s ❛ ❢✉♥❝t✐♦♥ ♦❢

✈❡❝t♦r✳ ❆s s✉❝❤ ✐t ❤❛s ❛

V /||V ||✳

❞✐r❡❝t✐♦♥

n

✈❛r✐❛❜❧❡s✳ ❙✉♣♣♦s❡

A

✐s ❛ ♣♦✐♥t ✐♥

✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢

V

Rn ✳

❚❤❡♥

V = ∇f (A) 6= 0

✐s ❛

✐s ✐ts ♥♦r♠❛❧✐③❛t✐♦♥✱ t❤❡ ✉♥✐t ✈❡❝t♦r

❚❤✉s✱ t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ st❛t❡♠❡♥t ✏t❤❡ ❣r❛❞✐❡♥t ♣♦✐♥ts ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢

t❤❡ ❢✉♥❝t✐♦♥✑ ✐s ✇❡❧❧ ✉♥❞❡rst♦♦❞✳ ❇✉t ✇❤❛t ❞♦❡s ✏t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✑ ♠❡❛♥❄ ❋✐rst✱ t❤❡ ❣r❛❞✐❡♥t ✇✐❧❧ ❜❡ ❝❤♦s❡♥ ❢r♦♠

❛❧❧ ♣♦ss✐❜❧❡ ❞✐r❡❝t✐♦♥s

✱ ✐✳❡✳✱ ❛❧❧ ✉♥✐t ✈❡❝t♦rs✿

✹✳✾✳

▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✸✸✼

❚❤❡♥✱ ✇❤❛t ❞♦❡s t❤❡ ✏❣r♦✇t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ✉♥✐t ✈❡❝t♦r✑ ♠❡❛♥❄

t✇♦

▲❡t✬s ✜rst t❛❦❡ ❛ ❧♦♦❦ ❛t ❞✐♠❡♥s✐♦♥ n = 1✳ ❚❤❡r❡ ❛r❡ ♦♥❧② ✉♥✐t ✈❡❝t♦rs✱ i ❛♥❞ −i✱ ❛❧♦♥❣ t❤❡ x✲❛①✐s✳ ′ ′ ❚❤❡r❡❢♦r❡✱ ✐❢ f (A) > 0✱ t❤❡♥ i ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤❀ ♠❡❛♥✇❤✐❧❡✱ ✐❢ f (A) < 0✱ ✐t✬s −i✳

❋♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✱ ✇❡ ❝❡rt❛✐♥❧② ❦♥♦✇ ✇❤❛t t❤✐s st❛t❡♠❡♥t ♠❡❛♥s ✇❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦♥❡ ♦❢ t❤❡ ❛①❡s✿ ✐t✬s t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ✭✈❡❝t♦rs

i, −i✱ j, −j

❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❛r❡

❡①♣❧♦r✐♥❣ t❤❡ t❡rr❛✐♥ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ ❣♦✐♥❣ ♦♥❧② ♥♦rt❤✲s♦✉t❤ ♦r ❡❛st✲✇❡st ✐s ♥♦t ❡♥♦✉❣❤✳

❚❤❡ ✐❞❡❛ ❝♦♠❡s ❢r♦♠ t❤❡ ❡❛r❧✐❡r ♣❛rt ♦❢ t❤✐s s❡❝t✐♦♥✿ ✇❡✱ ❛❣❛✐♥✱ t❛❦❡ ✈❛r✐♦✉s tr✐♣s ❛r♦✉♥❞ t❤✐s

t❡rr❛✐♥✳ ❚❤✐s t✐♠❡ ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ❣♦ ❢❛r ♦r ❢♦❧❧♦✇ ❛♥② ❝♦♠♣❧❡① r♦✉ts✿ ✇❡✬❧❧ ❣♦ ❛❧♦♥❣ str❛✐❣❤t ❧✐♥❡s✳ ❆❧s♦✱ ✐♥ ♦r❞❡r t♦ ❝♦♠♣❛r❡ t❤❡ r❡s✉❧ts✱ ✇❡ ✇✐❧❧ tr❛✈❡❧ ❛t t❤❡ s❛♠❡ s♣❡❡❞✱ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ❛❧❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

•

st❛rt ❛t

•

❛r❡ ❧✐♥❡❛r✱ ✐✳❡✳✱

•

❤❛✈❡ ✉♥✐t ❞✐r❡❝t✐♦♥ ✈❡❝t♦r

X = A✱

✐✳❡✳✱

X = FU (t)

1✱

❞✉r✐♥❣ ❛❧❧ tr✐♣s✳

t❤❛t

FU (0) = A✱

FU (t) = A + tU ✱

❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱

||U || = 1✳ ❲❛r♥✐♥❣✦

■t ✐s s❛❢❡ t♦ ❞✐sr❡❣❛r❞ ♥♦♥✲❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥❧② ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✳ ◆♦✇ ✇❡ ❝♦♠♣❛r❡ t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ ✇✐t❤

f

❛❧♦♥❣ t❤❡s❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥

f✿ hU (t) = f (FU (t)) .

✹✳✾✳

▼♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✸✸✽

❙♦✱ t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ✇❡ ❛r❡ ❛❢t❡r ✐s t❤✐s✿ h′U (0)

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❤❛✐♥

d ′ = f (FU (t)) = ∇f (FU (t)) · FU (t) = ∇f (A) · U , dt t=0 t=0

❘✉❧❡✳

❚❤❡r❡ ✐s ❛ ❝♦♥✈❡♥✐❡♥t t❡r♠ ❢♦r t❤✐s q✉❛♥t✐t②✳

❉❡✜♥✐t✐♦♥ ✹✳✾✳✶✿ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❚❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (X) ❛t ♣♦✐♥t X = A ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ✉♥✐t ✈❡❝t♦r U ✐s ❞❡✜♥❡❞ t♦ ❜❡ DU (f, A) = ∇f (A) · U

❲❡ ❝♦♥t✐♥✉❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥✿ DU (f, A) = ||∇f (A)|| · ||U || cos α = ||∇f (A)|| cos α ,

✇❤❡r❡ α ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ∇f (A) ❛♥❞ U ✳ ❆s t❤❡ ❣r❛❞✐❡♥t ✐s ❦♥♦✇♥ ❛♥❞ ✜①❡❞✱ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❞✐r❡❝t✐♦♥ ❞❡♣❡♥❞s ♦♥ ✐ts ❛♥❣❧❡ ✇✐t❤ t❤❡ ❣r❛❞✐❡♥t✱ ❛s ❡①♣❡❝t❡❞✿

◆♦✇✱ t❤✐s ❡①♣r❡ss✐♦♥ ✐s ❡❛s② t♦ ♠❛①✐♠✐③❡ ♦✈❡r t❤❡ ✈❡❝t♦rs U ✳ ❲❤❛t ❞✐r❡❝t✐♦♥✱ ✐✳❡✳✱ ❛ ✉♥✐t ✈❡❝t♦r U ✱ ♣r♦✈✐❞❡ t❤❡ ❤✐❣❤❡st ✈❛❧✉❡ ♦❢ DU (f, A)❄ ❖♥❧② cos α ♠❛tt❡rs✦ ❆♥❞ ✐t r❡❛❝❤❡s ✐s ♠❛①✐♠✉♠ ✈❛❧✉❡s✱ ✇❤✐❝❤ ✐s 1✱ ❛t α = 0✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♠❛①✐♠✉♠ ✐s r❡❛❝❤❡❞ ✇❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣r❛❞✐❡♥t✦

❚❤❡♦r❡♠ ✹✳✾✳✷✿ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ z = f (X) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t A ✐♥ Rn ✳ ❚❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ DU (f, A) r❡❛❝❤❡s ✐t ♠❛①✐♠✉♠ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ U ♦❢ t❤❡ ❣r❛❞✐❡♥t ∇f (A) ♦❢ f ❛t A❀ t❤✐s ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s ||∇f (A)||✳

❊①❡r❝✐s❡ ✹✳✾✳✸ ❙❤♦✇ t❤❛t t❤❡ t❤❡♦r❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 1 r❡✈❡❛❧s ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ❛♥❛❧②s✐s✿

✹✳✶✵✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✾

❊①❡r❝✐s❡ ✹✳✾✳✹

❊①♣❧❛✐♥ t❤❡ ❞✐❛❣r❛♠✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ❞❡✜♥✐t✐♦♥ ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ ❚❤❡♦r❡♠ ✹✳✾✳✺✿ ❉✐r❡❝t✐♦♥❛❧ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ✉♥✐t ✈❡❝t♦r

U

z = f (X) ❛t ♣♦✐♥t X = A ✐♥ t❤❡ ❞✐r❡❝t✐♦♥

✐s ❛❧s♦ ❢♦✉♥❞ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t✿

f (A + hU ) − f (A) h→0 h

DU (f, A) = lim

❊①❡r❝✐s❡ ✹✳✾✳✻

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✹✳✾✳✼

❘❡♣r❡s❡♥t ❡❛❝❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡✳ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✇✐t❤ ❢✉♥❝t✐♦♥s ♦❢ 3 ✈❛r✐❛❜❧❡s✿

❆❧❧ ✈❡❝t♦rs ♦♥ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ❧❡✈❡❧ s✉r❢❛❝❡ ❛r❡ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ✐♥❝r❡❛s✐♥❣ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❛❧❧ ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ❛r❡ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ❞❡❝r❡❛s✐♥❣ ✈❛❧✉❡s✳

✹✳✶✵✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ❛r❡ t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✦

✹✳✶✵✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥

✸✹✵

❚❤❡♦r❡♠ ✹✳✶✵✳✶✿ ❆❧❣❡❜r❛ ♦❢ ●r❛❞✐❡♥ts ❋♦r ❛♥② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s✱ ✇❡ ❤❛✈❡ ✐♥ t❤❡ ❣r❛❞✐❡♥t ♥♦t❛t✐♦♥✿ ❙❘✿

∇(f + g) = ∇f + ∇g ❈▼❘✿ ∇(cf ) = c∇f r❡❛❧ c ∇f g − f ∇g g 6= 0 P❘✿ ∇(f g) = ∇f g + f ∇g ◗❘✿ ∇(f /g) = g2 ❈❘✷✿ (g ◦ f )′ = g ′ ∇f ❈❘✶✿ (f ◦ F )′ = ∇f · F ′ ❚❤❡

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✭❈❤❛♣t❡r ✷❉❈✲✺✮ ✇✐❧❧ ❤❡❧♣ ✉s t♦ ❞❡r✐✈❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ ❢❛❝ts

❛❜♦✉t ✐ts ❣r❛❞✐❡♥t✳ ❋♦r ❡①❛♠♣❧❡✿ ✐♥❢♦ ❛❜♦✉t

f

f

✐♥❢♦ ❛❜♦✉t

✐s ❝♦♥st❛♥t

∇f

=⇒ ∇f

✐s ③❡r♦

=⇒ ∇f

✐s ❝♦♥st❛♥t

?

⇐=

f

✐s ❧✐♥❡❛r

?

⇐=

❆r❡ t❤❡s❡ ❛rr♦✇s r❡✈❡rs✐❜❧❡❄

■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s

❝♦♥st❛♥t❄ ❆t t❤✐s t✐♠❡✱ ✇❡ ❤❛✈❡ ❛ t♦♦❧ t♦ ♣r♦✈❡ t❤✐s ❢❛❝t✳ ❈♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ st❛t❡♠❡♥t ❛❜♦✉t t❡rr❛✐♥s✿

◮

✏■❢ t❤❡r❡ ✐s ♥♦ s❧♦♣✐♥❣ ❛♥②✇❤❡r❡ ✐♥ t❤❡ t❡rr❛✐♥✱ ✐t✬s ✢❛t✑✳

■❢ ❛ ❢✉♥❝t✐♦♥

y = f (x)

r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✿

❚❤❡♦r❡♠ ✹✳✶✵✳✷✿ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ■❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧ ✐♥ Rn ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ t❤r♦✉❣❤♦✉t t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡♥ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s❀ ✐✳❡✳✱ ∆f (C) = 0 =⇒ f = ❝♦♥st❛♥t

Pr♦♦❢✳ ■❢

X

❛♥❞

Y

❛r❡ t✇♦ ♥♦❞❡s ❝♦♥♥❡❝t❡❞ ❜② ❛♥ ❡❞❣❡ ✇✐t❤ ❛ s❡❝♦♥❞❛r② ♥♦❞❡

C✱

t❤❡♥ ✇❡ ❤❛✈❡✿

∆f (C) = 0 =⇒ f (X) − f (Y ) = 0 =⇒ f (X) = f (Y ) . ■♥ ❛ ❝❡❧❧✱ ❛♥② t✇♦ ♥♦❞❡s ❝❛♥ ❜❡ ❝♦♥♥❡❝t❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ❛❞❥❛❝❡♥t ♥♦❞❡s✱ ✇✐t❤ ♥♦ ❝❤❛♥❣❡ ✐♥ t❤❡ ✈❛❧✉❡ ♦❢

f✳

❚❤❡♦r❡♠ ✹✳✶✵✳✸✿ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ■❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧ ✐♥ Rn ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t t❤r♦✉❣❤♦✉t t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡♥ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s❀ ✐✳❡✳✱ ∆f (C) = 0 =⇒ f = ❝♦♥st❛♥t ∆X

✹✳✶✵✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥

✸✹✶

❚❤❡♦r❡♠ ✹✳✶✵✳✹✿ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s

■❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ♣❛t❤✲❝♦♥♥❡❝t❡❞ s❡t I ✐♥ Rn ❤❛s ❛ ③❡r♦ ❣r❛❞✐❡♥t ❢♦r ❛❧❧ X ✐♥ I ✱ t❤❡♥ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦♥ I ❀ ✐✳❡✳✱ df = 0 =⇒ f = ❝♦♥st❛♥t dX Pr♦♦❢✳

❙✉♣♣♦s❡ t✇♦ ♣♦✐♥ts

A, B

✐♥s✐❞❡

✇✐t❤ ✐ts ♣❛t❤ t❤❛t ❣♦❡s ❢r♦♠

A

I

❛r❡ ❣✐✈❡♥✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

t♦

B

❛♥❞ ❧✐❡s ❡♥t✐r❡❧② ✐♥

I✿

P (a) = A, P (b) = B, P (t) ❉❡✜♥❡ ❛ ♥❡✇

X = P (X)

✐♥

I.

♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✿ h(t) = f (P (t)) .

❚❤❡♥✱ ❜② t❤❡

❈❤❛✐♥ ❘✉❧❡

✇❡ ❤❛✈❡✿

 dh d f (P (t)) = ∇f (P (t)) · F ′ (t) = 0 · F ′ (t) = 0 . (t) = dt dt

❚❤❡♥✱ ❜② t❤❡ ❝♦r♦❧❧❛r② t♦ t❤❡

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✐♥ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✺✮✱

h

✐s ❛ ❝♦♥st❛♥t

❢✉♥❝t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡

f (A) = f (B) . ❲❡ ✇✐❧❧ s❡❡ ❧❛t❡r t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② r❡q✉✐r❡♠❡♥t ✐s ✉♥♥❡❝❡ss❛r②✳

❊①❡r❝✐s❡ ✹✳✶✵✳✺

❲❤❛t ✐❢

∇f = 0

♦♥ ❛ s❡t t❤❛t ✐s♥✬t ♣❛t❤✲❝♦♥♥❡❝t❡❞❄ ■s ✐t st✐❧❧ tr✉❡ t❤❛t

∇f = 0 =⇒ f = ❏✉st ❛s ✐♥ ❞✐♠❡♥s✐♦♥

1✱

❝♦♥st❛♥t

?

t❤❡ ♦♣❡♥♥❡ss ♦❢ t❤❡ ❞♦♠❛✐♥ ✐s ❝r✉❝✐❛❧✳

❚❤❡ ♣r♦❜❧❡♠ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ r❡❝♦✈❡r✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ♣r♦❝❡ss ✇❡ ❤❛✈❡ ❝❛❧❧❡❞

❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

f

❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡ ✭✐✳❡✳✱ ❣r❛❞✐❡♥t✮

∇f ✱

t❤❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ r❡❝♦♥str✉❝t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ✏✜❡❧❞ ♦❢

t❛♥❣❡♥t ❧✐♥❡s ♦r ♣❧❛♥❡s✑✿

◆♦✇✱ ❡✈❡♥ ✐❢ ✇❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ s✉❝❤ ❛s

g =f +C

f

❢r♦♠ ✐t ❞❡r✐✈❛t✐✈❡

❢♦r ❛♥② ❝♦♥st❛♥t ✈❡❝t♦r

C✳

∇f ✱ t❤❡r❡ ♠❛♥② ♦t❤❡rs ✇✐t❤ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡✱

❆r❡ t❤❡r❡ ♦t❤❡rs❄ ◆♦✳

✹✳✶✶✳ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄

✸✹✷

❚❤❡♦r❡♠ ✹✳✶✵✳✻✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉✐✛❡r❡♥❝❡s ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧ ✐♥ Rn ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡✱ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t❀ ✐✳❡✳✱ ∆f (C) = ∆g (C) =⇒ f (X) − g(X) = ❝♦♥st❛♥t

❚❤❡♦r❡♠ ✹✳✶✵✳✼✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧ ✐♥ Rn ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t❀ ✐✳❡✳✱ ∆f ∆g (C) = (C) =⇒ f (X) − g(X) = ❝♦♥st❛♥t ∆X ∆X

❚❤❡♦r❡♠ ✹✳✶✵✳✽✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉❡r✐✈❛t✐✈❡s ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ♣❛t❤✲❝♦♥♥❡❝t❡❞ s❡t I ✐♥ Rn ❤❛✈❡ t❤❡ s❛♠❡ ❣r❛❞✐❡♥t✱ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t❀ ✐✳❡✳✱ dg df = =⇒ f − g = ❝♦♥st❛♥t dX dX

Pr♦♦❢✳ ❉❡✜♥❡ h(X) = f (X) − g(X) .

❚❤❡♥✱ ❜② ❙❘✱ ✇❡ ❤❛✈❡✿

∇h (X) = ∇ (f (X) − g(X)) = ∇f (X) − ∇g (X) = 0 ,

❢♦r ❛❧❧ X ✳ ❚❤❡♥ h ✐s ❝♦♥st❛♥t✱ ❜② t❤❡ ❈♦♥st❛♥t ❚❤❡♦r❡♠✳ ●❡♦♠❡tr✐❝❛❧❧②✱

∇f = ∇g =⇒ f − g = ❝♦♥st❛♥t ,

♠❡❛♥s t❤❛t t❤❡ ❣r❛♣❤ ♦❢ f s❤✐❢t❡❞ ✈❡rt✐❝❛❧❧② ❣✐✈❡s ✉s t❤❡ ❣r❛♣❤ ♦❢ g ✳

❲❡ ❝❛♥ ❝✉t t❤❡ ❧✐st ♦❢ ❛❧❣❡❜r❛✐❝ r✉❧❡s ❞♦✇♥ t♦ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♦♥❡s✿ ▲✐♥❡❛r✐t② ❘✉❧❡✿ ❈❤❛✐♥ ❘✉❧❡✿

∇(λf + µg) = λ∇f + µ∇g ❢♦r ❛❧❧ r❡❛❧ λ, µ

(f ◦ F )′ = ∇f · F ′

✹✳✶✶✳ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄ ❇❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✳ ❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❞✐♠❡♥s✐♦♥ 2 ❤❡r❡✳

✹✳✶✶✳ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄

✸✹✸

❘❡❝❛❧❧ t❤❡ ❞✐❛❣r❛♠ ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ f ∆f =< ∆x f

ւx

y

,

ց

∆y f >

■t ♣r♦❞✉❝❡s t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❜♦t❤ ♦❢ ✇❤✐❝❤ ❛r❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ■♥ ❝♦♥tr❛st t♦ ❞✐♠❡♥s✐♦♥ 1✱ ✐t ✐s ❡❛s② t♦ t❤✐♥❦ ♦❢ ❛ 1✲❢♦r♠ t❤❛t ✐s♥✬t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✿

❊①❡r❝✐s❡ ✹✳✶✶✳✶ Pr♦✈❡ t❤❛t t❤❡r❡ ✐s ♥♦ s✉❝❤ ❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✷✿ ❡①❛❝t ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ❝❛❧❧❡❞ ❡①❛❝t ✐❢ ∆f = G✱ ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❤❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡♥✬t s♣❡❝✐✜❡❞✱ ✇❡ s♣❡❛❦ ♦❢ ❛♥ ❡①❛❝t 1✲❢♦r♠✳ ■♥ ❢❛❝t✱ ❝❤♦♦s✐♥❣ ♥✉♠❜❡rs ❛t r❛♥❞♦♠ ✐s ❧✐❦❡❧② t♦ ♣r♦❞✉❝❡ ♥♦♥✲❡①❛❝t ❢♦r♠✳

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✸✿ ❣r❛❞✐❡♥t ❆ ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ❝❛❧❧❡❞ ❣r❛❞✐❡♥t ✐❢ F (N ) · N = G(N ) ❢♦r s♦♠❡ ❡①❛❝t ❢✉♥❝t✐♦♥ G ❛♥❞ ❛♥② s❡❝♦♥❞❛r② ♥♦❞❡ N ✳ ◆♦t ❛❧❧ ✈❡❝t♦r ✜❡❧❞s ❛r❡ ❣r❛❞✐❡♥t❀ t❤❡ ❡①❛♠♣❧❡ ✇❡ s❛✇ ✇❛s F (x, y) =< y, −x >✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s r❡✈❡rs✐♥❣ t❤❡ ❛rr♦✇s ✐♥ t❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠✿ f =? < p

րx

y

,

տ

q >

❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐s ❡①❛❝t❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐s ✐t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥❄ ❲❡ ❤❛✈❡ ♣r❡✈✐♦✉s❧② s♦❧✈❡❞ t❤✐s ♣r♦❜❧❡♠ ❜② ✜♥❞✐♥❣ s✉❝❤ ❛ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❡①❛♠♣❧❡s r❡q✉✐r❡❞ ♣r♦❞✉❝✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r x ❛♥❞ y ❛♥❞ t❤❡♥ ♠❛t❝❤✐♥❣ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ✐♥ r❡✈❡rs❡ ♦r❞❡r✳ ❚❤❡ ♠❡t❤♦❞s ♦♥❧② ✇♦r❦ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ s✐♠♣❧❡ ❡♥♦✉❣❤✳ ❈❛♥ ✇❡ ❦♥♦✇ ✐♥ ❛❞✈❛♥❝❡❄ ❚❤❡ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ❜❡❧♦✇ ❣✐✈❡s ✉s ❛ ❜❡tt❡r t♦♦❧✳ ❙✉r♣r✐s✐♥❣❧②✱ t❤✐s t♦♦❧ ✐s ❢✉rt❤❡r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

✹✳✶✶✳

❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄

✸✹✹

❲❡ ❝♦♥t✐♥✉❡ t❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠✿

f ∆f =< ∆x f ∆2xx f

ւx

ւx y

ց

y

, ∆2yx f = ∆2xy f

ց

∆y f

ւx

> y

ց

∆2yy f

❘❡❝❛❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢r♦♠ ❈❤❛♣t❡r ✸✿ ❚❤❡♦r❡♠ ✹✳✶✶✳✹✿ ❉✐s❝r❡t❡ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠

❖✈❡r ❛ ♣❛rt✐t✐♦♥ ✐♥ Rn ✱ ✜rst✱ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ ∆2yx f = ∆2xy f

❛♥❞✱ s❡❝♦♥❞✱ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ ∆2 f ∆2 f = ∆y∆x ∆x∆y ❚❤❛♥❦s t♦ t❤✐s t❤❡♦r❡♠ ✇❡ ❝❛♥ ❞r❛✇ ❝♦♥❝❧✉s✐♦♥s ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✇❡ ❢❛❝❡ ❛ ❞✐✛❡r❡♥❝❡✳ ❙♦✱ t❤❡ ♣❧❛♥ ✐s✱ ✐♥st❡❛❞ ♦❢ tr②✐♥❣ t♦ r❡✈❡rs❡ t❤❡ ❛rr♦✇s ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ ❝♦♥t✐♥✉❡ ❞♦✇♥ ❛♥❞ s❡❡ ✇❤❡t❤❡r ✇❡ ❤❛✈❡ ❛ ♠❛t❝❤✿ ∆ y p = ∆x q . ❆s ❛ s✉♠♠❛r②✱ ❝♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ♦♥ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ■t ❤❛s ❛r❜✐tr❛r② ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱ ✇✐t❤ ♥♦ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡♠ ✇❤❛ts♦❡✈❡r✦ ❊✈❡r②t❤✐♥❣ ❝❤❛♥❣❡s ♦♥❝❡ ✇❡ ♠❛❦❡ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✐t ✐s ❡①❛❝t✳ ❚❤❡ t❤❡♦r❡♠ ❡♥s✉r❡s t❤❛t t❤❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ♦❢ s♦♠❡ V =< p, q >✱ ♦♥ ❧❡❢t✱ t✉r♥s ✕ ✉♥❞❡r t❤✐s ❛ss✉♠♣t✐♦♥ ✕ ✐♥t♦ s♦♠❡t❤✐♥❣ r✐❣✐❞✱ ♦♥ r✐❣❤t✿

..

f

.. V =< p

.. q > → p = ∆x f

, y

ց

∆y p ... ∆x q

ւx

ւx y

ց

y

∆y p = ∆2yx f = ∆2xy f = ∆x q

ց

q = ∆y f

ւx

❚❤✐s r✐❣✐❞✐t② ♦❢ t❤❡ ❞✐❛❣r❛♠ ♠❡❛♥s t❤❛t t❤❡ t✇♦ tr✐♣s ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳ ❲❡ ❤❛✈❡ ❞❡s❝r✐❜❡❞ t❤✐s ♣r♦♣❡rt② ❛s ❝♦♠♠✉t❛t✐✈✐t②✳ ■♥❞❡❡❞✱ ✐t✬s ❛❜♦✉t ✐♥t❡r❝❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ∆ x ∆ y = ∆y ∆ x . ❚❤❡♦r❡♠ ✹✳✶✶✳✺✿ ❊①❛❝t♥❡ss ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

2

■❢ G ✐s ❡①❛❝t ♦♥ ❛ r❡❝t❛♥❣❧❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✇✐t❤ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p ❛♥❞ q ✱ t❤❡♥ ∆ y p = ∆x q

✹✳✶✶✳ ❲❤❡♥ ✐s ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣♦ss✐❜❧❡❄

✸✹✺

❈♦r♦❧❧❛r② ✹✳✶✶✳✻✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

2

❙✉♣♣♦s❡ ❛ ✈❡❝t♦r ✜❡❧❞ V ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐♥ t❤❡ xy ✲♣❧❛♥❡ ✇✐t❤ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p ❛♥❞ q ✳ ■❢ V ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ ∆p ∆q = ∆y ∆x

❚❤❡ q✉❛♥t✐t② t❤❛t ✈❛♥✐s❤❡s ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣r❛❞✐❡♥t ✐s ❝❛❧❧❡❞ ✐ts r♦t♦r✳ ■t ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ❢❛❝❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ∆y p − ∆x q .

❋♦r t❤r❡❡ ✈❛r✐❛❜❧❡s✱ ✇❡ ❥✉st ❝♦♥s✐❞❡r t✇♦ ❛t ❛ t✐♠❡ ✇✐t❤ t❤❡ t❤✐r❞ ❦❡♣t ✜①❡❞✳ ❇❡❧♦✇ ✐s t❤❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡s ❢♦r t❤r❡❡ ✈❛r✐❛❜❧❡s ✇✐t❤ ♦♥❧② t❤❡ ♠✐①❡❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡s s❤♦✇♥✿

∆x f ↓y

∆2yx f

ւx z

ց

←−−− ←−−− ←−−−

−−−→ −−−→ −−−→

f ↓y

z

∆y f ∆2zx f

∆2xy f

ւx

z

ց

∆2zy f

ց

ւx

∆2xz f

∆z f ↓y

∆2yz f

❚❤❡ s✐① t❤❛t ❛r❡ ❧❡❢t ❛r❡ ♣❛✐r❡❞ ✉♣ ❛❝❝♦r❞✐♥❣ t♦ ❉✐s❝r❡t❡ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠ ❛❜♦✈❡✳ ▲❡t✬s ❛ss✉♠❡✿ V

=< p, q, r > =< ∆x f, ∆y f, ∆z f >= ∆f

❚❤❡♥ ✇❡ ❝❛♥ tr❛❝❡ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❞✐❛❣r❛♠✿ p ↓y

∆y p

q z

ց

∆z p ∆x q

ւx

r z

ց

∆z q ∆x r

ւx

↓y

∆y r

❲❡ ✇r✐t❡ ❞♦✇♥ t❤❡ r❡s✉❧ts ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✹✳✶✶✳✼✿ ❊①❛❝t♥❡ss ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

3

■❢ G ✐s ❡①❛❝t ♦♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❜♦① ✐♥ t❤❡ xyz ✲s♣❛❝❡ ✇✐t❤ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p✱ q ✱ ❛♥❞ r✱ t❤❡♥ ∆y p = ∆x q, ∆z q = ∆y r, ∆x r = ∆z p

❈♦r♦❧❧❛r② ✹✳✶✶✳✽✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

3

❙✉♣♣♦s❡ ❛ ✈❡❝t♦r ✜❡❧❞ V ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❜♦① ✐♥ t❤❡ xyz ✲s♣❛❝❡ ✇✐t❤ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p✱ q ✱ ❛♥❞ r✳ ■❢ V ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ ∆p ∆q ∆q ∆r ∆r ∆p = , = , = ∆y ∆x ∆z ∆y ∆x ∆z

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✹✻

❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ ♣❛tt❡r♥ ✐♥ t❤❡s❡ ❢♦r♠✉❧❛s✳ ❋✐rst t❤❡ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦♠♣♦♥❡♥ts ❛r❡ ❛rr❛♥❣❡❞ ❛r♦✉♥❞ ❛ tr✐❛♥❣❧❡✿ p

x z

ր

←

ց

r

y

ր

←

ց

q

❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✐s ♦♠✐tt❡❞ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦✈❡r t❤❡ ♦t❤❡r t✇♦ ✐s s❡t t♦ 0✿ •

· •

∆z q = ∆y r

•

∆y p = ∆x q

·

∆ x r = ∆z p •

•

• ·

❚❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡s❡ t✇♦ t❤❡♦r❡♠s✱ ❛♥❞ t❤❡✐r ❛♥❛❧♦❣s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✱ ♣✉t s❡✈❡r❡ ❧✐♠✐t❛t✐♦♥s ♦♥ ✇❤❛t ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❡①❛❝t✳

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄ ❲❡ ♣r♦❝❡❡❞ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳ ❘❡❝❛❧❧ t❤❡ ❞✐❛❣r❛♠ ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s t❤❛t ♣r♦❞✉❝❡s t❤❡ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ❣r❛❞✐❡♥t✿ f ∇f =< fx

ւx

y

,

ց

fy >

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❛❧♦❣ ♦❢ ❛♥ ❡①❛❝t 1✲❢♦r♠✿

❉❡✜♥✐t✐♦♥ ✹✳✶✷✳✶✿ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞ ❆ ✈❡❝t♦r ✜❡❧❞ t❤❛t ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s ❝❛❧❧❡❞ ❛ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞✳ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡♥ ❝❛❧❧❡❞ ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ◆♦t❡ t❤❛t ✜♥❞✐♥❣ ❢♦r ❛ ❣✐✈❡♥ ✈❡❝t♦r ✜❡❧❞ V ❛ ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t ∇f = V ❛♠♦✉♥ts t♦ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ✇❡ tr② t♦ r❡✈❡rs❡ t❤❡ ❛rr♦✇s ✐♥ t❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠✳ ❚❤❡ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠ ✐s ❛♥ ❛♥❛❧♦❣ ♦❢ s❡✈❡r❛❧ ❢❛♠✐❧✐❛r r❡s✉❧t ❢r♦♠ ❱♦❧✉♠❡ ✷✿ ∇f = ∇g =⇒ f − g = ❝♦♥st❛♥t .

❈♦r♦❧❧❛r② ✹✳✶✷✳✷✿ ❚✇♦ P♦t❡♥t✐❛❧ ❋✉♥❝t✐♦♥s ❆♥② t✇♦ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ s❛♠❡ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ♣❛t❤✲ ❝♦♥♥❡❝t❡❞ s❡t ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ✇✐t❤✐♥ t❤✐s s❡t✳ ◆♦t ❛❧❧ ✈❡❝t♦r ✜❡❧❞s ❛r❡ ❣r❛❞✐❡♥t✳ ❚❤❡ ❡①❛♠♣❧❡ ✇❡ s❛✇ ✇❛s V (x, y) =< y, −x >✳ ❚❤❡r❡ ❛r❡ ♠❛♥② ♠♦r❡✳✳✳

✹✳✶✷✳

❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✹✼

❊①❛♠♣❧❡ ✹✳✶✷✳✸✿ s♣✐r❛❧

❈♦♥s✐❞❡r t❤❡ s♣✐r❛❧ ❜❡❧♦✇✳ ❈❛♥ ✐t ❜❡ ❛ ❧❡✈❡❧ ❝✉r✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s r❡✈❡rs✐♥❣ t❤❡ ❛rr♦✇s ✐♥ t❤❡ ❛❜♦✈❡ ❞✐❛❣r❛♠✿

f =? րx

< p

y

,

տ

q >

❙♦✱ ❤♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ❛ ❣✐✈❡♥ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t❄ ❲❡ ❤❛✈❡ ♣r❡✈✐♦✉s❧② s♦❧✈❡❞ t❤✐s ♣r♦❜❧❡♠ ❜② ✜♥❞✐♥❣✱ ❛♥❞ tr②✐♥❣ ❛♥❞ ❢❛✐❧✐♥❣ t♦ ✜♥❞✱ ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐♥ ❞✐♠❡♥s✐♦♥ 2✳ ❚❤❡ ❡①❛♠♣❧❡s r❡q✉✐r❡❞ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❜♦t❤ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡♥ ♠❛t❝❤✐♥❣ t❤❡ r❡s✉❧ts✳ ❚❤❡ ♠❡t❤♦❞s ♦♥❧② ✇♦r❦ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ s✐♠♣❧❡ ❡♥♦✉❣❤✳ ❈❛♥ ✇❡ ❦♥♦✇ ✐♥ ❛❞✈❛♥❝❡❄ ❚❤❡ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ❜❡❧♦✇ ❣✐✈❡s ✉s s✉❝❤ ❛ t♦♦❧✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ t❤❡ ❛♥s✇❡r ✐s ❢✉rt❤❡r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ❝♦♥t✐♥✉❡ t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❞✐❛❣r❛♠✿

f ∇f =< fx fxx

ւx

ւx y

ց

y

, fyx = fxy

ց

fy

ւx

> y

ց

fyy

❲❛r♥✐♥❣✦

❚❤❡ ❧❛st r♦✇ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✭♦❢ t❤❡ ❣r❛❞✐❡♥t✮ s♦♠❡❤♦✇✳✳✳ ❘❡❝❛❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢r♦♠ ❈❤❛♣t❡r ✸ t❤❛t ❣✐✈❡s ✉s t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿ ❚❤❡♦r❡♠ ✹✳✶✷✳✹✿ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠ ❚❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛t ❛ ♣♦✐♥t

(a, b)

f

♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐t❤ ❝♦♥t✐♥✉♦✉s

❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r❀ ✐✳❡✳✱

fxy (a, b) = fyx (a, b)

❚❤❛♥❦s t♦ t❤✐s t❤❡♦r❡♠ ✇❡ ❝❛♥ ❞r❛✇ ❝♦♥❝❧✉s✐♦♥s ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛ ❣✐✈❡♥ ✈❡❝t♦r ✜❡❧❞ V =< p, q > ✐s ❣r❛❞✐❡♥t ✕ ❛s ❧♦♥❣ ❛s ✐ts ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p ❛♥❞ q ❛r❡ t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❙♦✱ t❤❡ ♣❧❛♥ ✐s✱ ✐♥st❡❛❞ ♦❢ tr②✐♥❣ t♦ r❡✈❡rs❡ t❤❡ ❛rr♦✇s ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ ❞✐❛❣r❛♠ ❛♥❞ ✜♥❞ f ✇✐t❤ ∇f = V ✱ ✇❡ ❝♦♥t✐♥✉❡ ❞♦✇♥ ❛♥❞ s❡❡ ✇❤❡t❤❡r ✇❡ ❤❛✈❡ ❛ ♠❛t❝❤✿ p y = qx .

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✹✽

❊①❛♠♣❧❡ ✹✳✶✷✳✺✿ t❡st✐♥❣

■t✬s ❡❛s②✳ ❋♦r V =< x, y >✱ ✇❡ ❤❛✈❡✿

p = x =⇒ py = 0 q = y =⇒ qx = 0

=⇒ ♠❛t❝❤✦

❚❤❡ t❡st ✐s ♣❛ss❡❞✦ ❙♦ ✇❤❛t❄ ❲❤❛t ❞♦ ✇❡ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤❛t❄ ◆♦t❤✐♥❣✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ V =< y, −x >✱ ✇❡ ❤❛✈❡✿

p = y =⇒ py = 1 q = x =⇒ qx = −1

=⇒ ♥♦ ♠❛t❝❤✦

❚❤❡ t❡st ✐s ❢❛✐❧❡❞✳ ■t✬s ♥♦t ❣r❛❞✐❡♥t✦ ❲❡ ❞r❛✇ ♥♦ ❝♦♥❝❧✉s✐♦♥ ✇❤❡♥ t❤❡ t❡st ✐s ♣❛ss❡❞ ❛♥❞ ✇❤❡♥ ✐t ✐s♥✬t✱ ✇❡ ✇♦✉❧❞ st✐❧❧ ❤❛✈❡ t♦ ✐♥t❡❣r❛t❡ t♦ ✜♥❞ ♦✉t ✐❢ ✐t ✐s ❣r❛❞✐❡♥t ❛♥❞✱ ❛t t❤❡ s❛♠❡ t✐♠❡✱ tr② t♦ ✜♥❞ t❤❡ ❣r❛❞✐❡♥t✳ ▼❡❛♥✇❤✐❧❡ t❤❡ ❢❛✐❧✉r❡ t♦ s❛t✐s❢② t❤❡ t❡st ♣r♦✈❡s t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ♥♦t ❣r❛❞✐❡♥t✳ ❆s ❛ s✉♠♠❛r②✱ ❝♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r ✜❡❧❞✳ ■t ❤❛s ❛r❜✐tr❛r② ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱ ✇✐t❤ ♥♦ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡♠ ✇❤❛ts♦❡✈❡r✦ ❊✈❡r②t❤✐♥❣ ❝❤❛♥❣❡s ♦♥❝❡ ✇❡ ♠❛❦❡ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤✐s ✐s ❛ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞✳ ❚❤❡ t❤❡♦r❡♠ ❡♥s✉r❡s t❤❛t t❤❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ♦❢ ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r ✜❡❧❞ V =< p, q > ♦♥ ❧❡❢t t✉r♥s ✕ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✐t✬s ❣r❛❞✐❡♥t ✕ ✐♥t♦ s♦♠❡t❤✐♥❣ r✐❣✐❞ ♦♥ r✐❣❤t✿

f

.. .. V =< p

.. ,

y

ց

q >

py ... qx

ւx

❧❡❛❞✐♥❣ t♦

p = fx

ւx y

ց

y

py = fyx = fxy = qx

ց

q = fy

ւx

❚❤✐s r✐❣✐❞✐t② ♦❢ t❤❡ ❞✐❛❣r❛♠ ♠❡❛♥s t❤❛t t❤❡ t✇♦ tr✐♣s ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳ ❲❡ ❤❛✈❡ ❞❡s❝r✐❜❡❞ t❤✐s ♣r♦♣❡rt② ❛s ❝♦♠♠✉t❛t✐✈✐t②✳ ■♥❞❡❡❞✱ ✐t✬s ❛❜♦✉t ✐♥t❡r❝❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ ∂ ∂ ∂ ∂ = . ∂x ∂y ∂y ∂x ❚❤❡♦r❡♠ ✹✳✶✷✳✻✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

2

❙✉♣♣♦s❡ V =< p, q > ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ❞✐s❦ ✐♥ R2 ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p ❛♥❞ q ✳ ■❢ V ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ p y = qx ❚❤❡ q✉❛♥t✐t② t❤❛t ✈❛♥✐s❤❡s ✇❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t ✐s ❝❛❧❧❡❞ t❤❡ r♦t♦r ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ■t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ p y − qx . ❲❡ ✇✐❧❧ s❡❡ ❧❛t❡r ❤♦✇ t❤❡ r♦t♦r ✐s ✉s❡❞ t♦ ♠❡❛s✉r❡ ❤♦✇ ❝❧♦s❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s t♦ ❜❡✐♥❣ ❣r❛❞✐❡♥t✳ ❋♦r t❤r❡❡ ✈❛r✐❛❜❧❡s✱ ✇❡ ❥✉st ❝♦♥s✐❞❡r t✇♦ ❛t ❛ t✐♠❡ ✇✐t❤ t❤❡ t❤✐r❞ ❦❡♣t ✜①❡❞✳

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✹✾

❇❡❧♦✇ ✐s t❤❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❢♦r t❤r❡❡ ✈❛r✐❛❜❧❡s ✇✐t❤ ♦♥❧② t❤❡ ♠✐①❡❞ ❞❡r✐✈❛t✐✈❡s s❤♦✇♥✿

ւx

fx ↓y

z

fyx

←−−− ←−−− ←−−− f

↓y

−−−→ −−−→ −−−→ z

fy

ց

fzx

ւx

fxy

z

ց

fzy

ց

ւx

fxz

fz ↓y

fyz

❚❤❡ s✐① t❤❛t ❛r❡ ❧❡❢t ❛r❡ ♣❛✐r❡❞ ✉♣ ❛❝❝♦r❞✐♥❣ t♦ ❈❧❛✐r❛✉t✬s t❤❡♦r❡♠✳ ■❢ ✇❡ ❝❤♦♦s❡ V =< p, q, r >=< fx , fy , fz >= ∇f ,

✇❡ ❝❛♥ tr❛❝❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❞✐❛❣r❛♠✿ p

q

↓y

z

py

ց

p z qx

ւx

r z

ց

qz r x

ւx ↓y ry

❲❡ ✇r✐t❡ ❞♦✇♥ t❤❡ r❡s✉❧ts ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✹✳✶✷✳✼✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ 3 ❙✉♣♣♦s❡ V =< p, q, r > ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ❜❛❧❧ ✐♥ R3 ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s p✱ q ✱ ❛♥❞ r✳ ■❢ V ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ p y = qx , qz = r y , r x = p z

❲❡ ❤❛✈❡ t❤❡ s❛♠❡ s✐♠♣❧❡ ♣❛tt❡r♥✳ ❚❤❡ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦♠♣♦♥❡♥ts ❛r❡ ❛rr❛♥❣❡❞ ❛r♦✉♥❞ ❛ tr✐❛♥❣❧❡✿ x z

ր

←

p ց

y

r

ր

←

ց

q

❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✐s ♦♠✐tt❡❞ ❛♥❞ t❤❡ r♦t♦r ♦✈❡r t❤❡ ♦t❤❡r t✇♦ ✐s s❡t t♦ 0✿ · •

qz = ry

• •

·

p y = qx

rx = p z •

•

• ·

❆❧❧ t❤r❡❡ ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s✿ py − qx , qz − ry , rx − pz ✱ ✈❛♥✐s❤ ✇❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t✳ ■♥ ♦r❞❡r t♦ ❤❛✈❡ ♦♥❧② ♦♥❡✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡♠ ❛s ❝♦♠♣♦♥❡♥ts t♦ ❢♦r♠ ❛ ♥❡✇ ✈❡❝t♦r ✜❡❧❞✱ ❝❛❧❧❡❞ t❤❡ ❝✉r❧ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡s❡ t✇♦ t❤❡♦r❡♠s✱ ❛♥❞ t❤❡✐r ❛♥❛❧♦❣s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✱ ♣✉t s❡✈❡r❡ ❧✐♠✐t❛t✐♦♥s ♦♥ ✇❤❛t ✈❡❝t♦r ✜❡❧❞s ❝❛♥ ❜❡ ❣r❛❞✐❡♥t✳ ❚❤❡ s♦✉r❝❡ ♦❢ t❤❡s❡ ❧✐♠✐t❛t✐♦♥s ✐s t❤❡ t♦♣♦❧♦❣② ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ♦❢ ❞✐♠❡♥s✐♦♥ 2 ❛♥❞ ❤✐❣❤❡r✳ ❚❤❡② ❛r❡ t♦ ❜❡ ❞✐s❝✉ss❡❞ ❧❛t❡r✳ ❇❛❝❦ t♦ ❞✐♠❡♥s✐♦♥ 2✳

❊①❛♠♣❧❡ ✹✳✶✷✳✽✿ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ❚❤❡ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞✱ V =< y, −x > ,

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✺✵

✐s ♥♦t ❣r❛❞✐❡♥t ❛s ✐t ❢❛✐❧s t❤❡ ●r❛❞✐❡♥t ❚❡st✳

▲❡t✬s ❝♦♥s✐❞❡r ✐ts ♥♦r♠❛❧✐③❛t✐♦♥✿ V 1 U= =p < y , −x >= ||V || x2 + y 2

*

y p

x

x2 + y 2

, −p x2 + y 2

+

=< p, q > .

❆❧❧ ✈❡❝t♦rs ❛r❡ ✉♥✐t ✈❡❝t♦rs ✇✐t❤ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥s ❛s t❤❡ ❧❛st ✈❡❝t♦r ✜❡❧❞✿

▲❡t✬s t❡st t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ●r❛❞✐❡♥t ❚❡st ✿ y ∂ p = py = ∂y x2 + y 2

qx =

1·

∂ −x p =− ∂x x2 + y 2

p x2 + y 2 − y √

1·

p

y x2 +y 2

x2 + y 2 x2 + y 2 − x √

x x2 +y 2

✳✳✳♥♦ ♠❛t❝❤✦

x2 + y 2

❚❤✐s ✈❡❝t♦r ✜❡❧❞ ❛❧s♦ ❢❛✐❧s t❤❡ t❡st ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s♥✬t ❣r❛❞✐❡♥t✳ ▲❡t✬s t❛❦❡ t❤✐s ♦♥❡ st❡♣ ❢✉rt❤❡r✿ 1 V = < y, −x >= W = ||V ||2 x2 + y 2



x y , − x2 + y 2 x2 + y 2



=< p, q > .

❚❤❡ ♥❡✇ ✈❡❝t♦r ✜❡❧❞ ❤❛s t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥s ❜✉t t❤❡ ♠❛❣♥✐t✉❞❡ ✈❛r✐❡s❀ ✐t ❛♣♣r♦❛❝❤❡s 0 ❛s ✇❡ ♠♦✈❡ ❢❛rt❤❡r ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ❛♥❞ ✐♥✜♥✐t❡ ❛s ✇❡ ❛♣♣r♦❛❝❤ t❤❡ ♦r✐❣✐♥❀ ✐✳❡✳✱ ✇❡ ❤❛✈❡✿ W (X) → 0 ❛s ||X|| → ∞ ❛♥❞ ||W (X)|| → ∞ ❛s X → 0 .

▲❡t✬s t❡st t❤❡ ❝♦♥❞✐t✐♦♥✿ y 1 · (x2 + y 2 ) − y · 2y x2 − y 2 ∂ = = ∂y x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )2 2 2 ∂ −x 1 · (x + y ) − x · 2x y 2 − x2 qx = = − = − ∂x x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )2

py =

✳✳✳♠❛t❝❤✦

✹✳✶✷✳ ❲❤❡♥ ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ❛ ❣r❛❞✐❡♥t❄

✸✺✶

❚❤❡ ✈❡❝t♦r ✜❡❧❞ ♣❛ss❡s t❤❡ t❡st✦ ❉♦❡s ✐t ♠❡❛♥ t❤❛t ✐t ✐s ❣r❛❞✐❡♥t t❤❡♥❄ ◆♦✱ ✐t ❞♦❡s♥✬t ❛♥❞ ✇❡ ✇✐❧❧ ❞❡♠♦♥str❛t❡ t❤❛t ✐t ✐s ♥♦t✦ ❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡ t❤❛t ✇❡ st❛rt❡❞ ✇✐t❤ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✿ ❆ r♦✉♥❞ tr✐♣ ❛❧♦♥❣ t❤❡ ❣r❛❞✐❡♥ts ✐s ✐♠♣♦ss✐❜❧❡ ❛s ✐t ❧❡❛❞s t♦ ❛ ♥❡t ✐♥❝r❡❛s❡ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✳ ■t ✐s ❝r✉❝✐❛❧ t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ✉♥❞❡✜♥❡❞ ❛t t❤❡ ♦r✐❣✐♥✳ ❲❡ ✇✐❧❧ s❤♦✇ ❧❛t❡r t❤❛t t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ●r❛❞✐❡♥t ❚❤❡♦r❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 2 ✐s♥✬t tr✉❡✿ py = qx 6=⇒ < p, q >= ∇f ,

✉♥❧❡ss ❛ ❝❡rt❛✐♥ ❢✉rt❤❡r r❡str✐❝t✐♦♥ ✐s ♣❧❛❝❡❞✳ ❚❤✐s r❡str✐❝t✐♦♥ ✐s t♦♣♦❧♦❣✐❝❛❧ ✿ t❤❡r❡ ❝❛♥ ❜❡ ♥♦ ❤♦❧❡s ✐♥ t❤❡ ❞♦♠❛✐♥✳ ❋✉rt❤❡r♠♦r❡✱ ✐♥t❡❣r❛t✐♥❣ py − qx ✇✐❧❧ ❜❡ ✉s❡❞ t♦ ♠❡❛s✉r❡ ❤♦✇ ❝❧♦s❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s t♦ ❜❡✐♥❣ ❣r❛❞✐❡♥t✳

❈❤❛♣t❡r ✺✿ ❚❤❡ ✐♥t❡❣r❛❧

❈♦♥t❡♥ts ✺✳✶ ❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✷ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✸ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦✈❡r r❡❝t❛♥❣❧❡s ✺✳✹ ❚❤❡ ✇❡✐❣❤t ❛s t❤❡ ✸❞ ❘✐❡♠❛♥♥ s✉♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✺ ❚❤❡ ✇❡✐❣❤t ❛s t❤❡ ✸❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✻ ▲❡♥❣t❤s✱ ❛r❡❛s✱ ✈♦❧✉♠❡s✱ ❛♥❞ ❜❡②♦♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼ ❖✉ts✐❞❡ t❤❡ s❛♥❞❜♦① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✽ ❚r✐♣❧❡ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✾ ❚❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡

✺✳✶✵ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺✷ ✸✻✵ ✸✻✹ ✸✼✷ ✸✼✻ ✸✽✶ ✸✽✺ ✸✾✵ ✸✾✸ ✸✾✺

✺✳✶✳ ❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

❆❧❧ ❢✉♥❝t✐♦♥s ✐♥ t❤✐s ❝❤❛♣t❡r ❛r❡ r❡❛❧✲✈❛❧✉❡❞✳ ❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ✈♦❧✉♠❡s ✐s ❧✐♠✐t❡❞ t♦ t❤❛t ♦❢ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✸✮✳ ■❢ ❛♥❞ ✐t ✐s ❧✐❢t❡❞ ♦✛ t❤❡ ♣❧❛♥❡ t♦ t❤❡ ❤❡✐❣❤t

h

D

✐s ❛ r❡❣✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡

t❤❡♥ t❤❡ ❝②❧✐♥❞❡r✲❧✐❦❡ s♦❧✐❞ ✭❛ ✏s❤❡❧❧✑✮ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣❧❛♥❡

r❡❣✐♦♥s ✐s ❛ss✉♠❡❞ t♦ ❤❛✈❡ t❤❡ ✈♦❧✉♠❡ ♦❢✿

V = A · h.

❲❡ ❢✉rt❤❡r♠♦r❡ r❡♣r❡s❡♥t❡❞ ♠♦r❡ ❝♦♠♣❧❡① s♦❧✐❞ ✐♥ t❡r♠s ♦❢ t❤❡s❡ s❤❡❧❧s✳ ❲❡ ✇✐❧❧ ❤❛✈❡ t♦ st❛rt ♦✈❡r t❤♦✉❣❤✳ ❊①❛♠♣❧❡ ✺✳✶✳✶✿ ❝✐r❝❧❡

▲❡t✬s r❡✈✐❡✇ t❤❡ ❆r❡❛ Pr♦❜❧❡♠✳ ❲❡ ❝♦♥✜r♠ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢

y = f (x) = ✇✐t❤

21 ♣♦✐♥ts ✭20 ✐♥t❡r✈❛❧s✮✳

√

1

✐s

A = π✳

❋✐rst ✇❡

1 − x2 , −1 ≤ x ≤ 1 ,

❲❡ ❧❡t t❤❡ ✈❛❧✉❡s ♦❢

x r✉♥ ❢r♦♠ −1 t♦ 1 ❡✈❡r② 0.1 ❛♥❞ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛✿

❂❙◗❘❚✭✶✲❘❈✸✂✷✮

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✸

❲❡ ❣❡t t❤❡ ✈❛❧✉❡s ♦❢ y ✿

❲❡ ♥❡①t ❝♦✈❡r t❤✐s ❤❛❧❢✲❝✐r❝❧❡ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs ❜❛s❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [−1, 1]✿ t❤❡ ❜❛s❡s ♦❢ t❤❡ ❜❛rs ❛r❡ ♦✉r ✐♥t❡r✈❛❧s ✐♥ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ ❤❡✐❣❤ts ❛r❡ ✈❛❧✉❡s ♦❢ y = f (x)✳ ❚♦ s❡❡ t❤❡ ❜❛rs✱ ✇❡ s✐♠♣❧② ❝❤❛♥❣❡ t❤❡ t②♣❡ ♦❢ t❤❡ ❝❤❛rt ♣❧♦tt❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t✿

❚❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❜❛rs✿ ✇❡ ♠✉❧t✐♣❧② t❤❡ ✇✐❞t❤s ♦❢ t❤❡ ❜❛rs ❜② t❤❡ ❤❡✐❣❤ts✱ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥✱ ❛♥❞ ✜♥❛❧❧② ❛❞❞ ❛❧❧ ❡♥tr✐❡s ✐♥ t❤✐s ❝♦❧✉♠♥✳ ❚❤❡ r❡s✉❧t 1.552 ✐s ❝❧♦s❡ t♦ π/2 ≈ 1.571✳ ❲❡ ♣r♦❝❡❡❞ t♦ t❤❡ ❱♦❧✉♠❡ Pr♦❜❧❡♠✳ ❲❡ ✇✐❧❧ ❝♦♥✜r♠ t❤❛t t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♣❤❡r❡ ♦❢ r❛❞✐✉s 1 ✐s V = 43 π ✳ ❋✐rst ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ z = f (x, y) =

p

1 − x2 − y 2 ,

−1 ≤ x ≤ 1, −1 ≤ y ≤ 1 .

❲❡ r❡❝②❝❧❡ ♦✉r s♣r❡❛❞s❤❡❡t ❢♦r t❤❡ s♣❤❡r❡✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ 20 ✐♥t❡r✈❛❧s ❢♦r x ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥✳ ◆♦✇✱ ❥✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝♦♥str✉❝t 20 ✐♥t❡r✈❛❧s ❢♦r y ✐♥ t❤❡ ✜rst r♦✇✳ ❲❡ ❧❡t t❤❡ ✈❛❧✉❡s ♦❢ x ❛♥❞ y r✉♥ ❢r♦♠ −1 t♦ 1 ❡✈❡r② 0.1 ❛♥❞ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛✿ ❂❙◗❘❚✭✶✲❘❈✸✂✷✲❘✹❈✂✷✮

t♦ ❣❡t t❤❡ ✈❛❧✉❡s ♦❢ z ✿

❲❡ ♥❡①t ✜❧❧ t❤✐s ❤❛❧❢✲s♣❤❡r❡ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs ❜❛s❡❞ ♦♥ t❤❡ sq✉❛r❡ [−1, 1] × [−1, 1]✿ t❤❡ ❜❛s❡s ♦❢ t❤❡

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✹

❜❛rs ✭♣✐❧❧❛rs✮ ❛r❡ ♦✉r ❧✐tt❧❡ sq✉❛r❡s ✐♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ t❤❡ ❤❡✐❣❤ts ❛r❡ ✈❛❧✉❡s ♦❢ z ✳ ❚♦ s❡❡ t❤❡ ❜❛rs✱ ✇❡ s✐♠♣❧② ❝❤❛♥❣❡ t❤❡ t②♣❡ ♦❢ t❤❡ ❝❤❛rt ♣❧♦tt❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t✿

❲❡ ❝❛♥ s❡❡ ❡❛❝❤ r♦✇ ♦❢ ❜❛rs ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❛ s❧✐❝❡ ♦❢ t❤❡ s♣❤❡r❡✱ ✇❤✐❝❤ ✐s ❛♥♦t❤❡r ❝✐r❝❧❡✳✳✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♣❤❡r❡ ✐s ♥♦✇ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡ ❜❛rs✳ ❊❛❝❤ ♦❢ t❤❡s❡ ✈♦❧✉♠❡s ✐s t❤❡ ♣r♦❞✉❝t ♦❢ • ❚❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❜❛r ✐♥ t❤✐s r❡❝t❛♥❣❧❡ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✇✐t❤ t❤❡ ♦♥❡s ♦✉ts✐❞❡ t❤❡ ❞♦♠❛✐♥ r❡♣❧❛❝❡❞ ✇✐t❤ 0s✳ • ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❜❛s❡ ✐s ❡q✉❛❧ t♦ 0.01✳ ❚❤❡✐r s✉♠ ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡s❡ ❤❡✐❣❤ts ♠✉❧t✐♣❧✐❡❞ ❜② 0.01✳ ❚❤❡ r❡s✉❧t ♣r♦❞✉❝❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆♣♣r♦①✐♠❛t❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❤❡♠✐s♣❤❡r❡ = 2.081 . ■t ✐s ❝❧♦s❡ t♦ t❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧t✿ ❊①❛❝t ✈♦❧✉♠❡ ♦❢ t❤❡ ❤❡♠✐s♣❤❡r❡ = 2π/3 ≈ 2.094 . ❊①❡r❝✐s❡ ✺✳✶✳✷

❆♣♣r♦①✐♠❛t❡ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♣❤❡r❡ r❛❞✐✉s 1 ✇✐t❤✐♥ 0.0001✳ ❲❡ ❤❛✈❡ s❤♦✇❡❞ t❤❛t ✐♥❞❡❡❞ t❤❡ ❛r❡❛ ♦❢ ❛ s♣❤❡r❡ ♦❢ r❛❞✐✉s 1 ✐s ❝❧♦s❡ t♦ A = 43 π ✳ ❇✉t t❤❡ r❡❛❧ q✉❡st✐♦♥ ✐s✿ ❲❤❛t ✐s t❤❡ ✈♦❧✉♠❡❄ ❖♥❡ t❤✐♥❣ ✇❡ ❞♦ ❦♥♦✇✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❜♦① a × b × c ✐s abc✳ ❲✐t❤ t❤❛t ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ✈♦❧✉♠❡ ♦❢ ✈❛r✐♦✉s ❣❡♦♠❡tr✐❝ ✜❣✉r❡s ✇✐t❤ str❛✐❣❤t ❡❞❣❡s ❜✉t ✇❤❛t ❛r❡ t❤❡ ✈♦❧✉♠❡s ♦❢ ❝✉r✈❡❞ ♦❜❥❡❝ts ❄ ❚❤❡ ✐❞❡❛✱ ♦♥❝❡ ❛❣❛✐♥✱ ❝♦♠❡s ❢r♦♠ t❤❡ ❛♥❝✐❡♥t ●r❡❡❦✬s ❛♣♣r♦❛❝❤ t♦ ✉♥❞❡rst❛♥❞✐♥❣ ❛♥❞ ❝♦♠♣✉t✐♥❣ t❤❡ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s✳ ❚❤❡② ❛♣♣r♦①✐♠❛t❡❞ t❤❡ ❝✐r❝❧❡ ✇✐t❤ r❡❣✉❧❛r ♣♦❧②❣♦♥s ❛♥❞ t❤❡ s♣❤❡r❡ ✇✐t❤ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✿

❚❤❡ s❡t✉♣ ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❢♦r ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❜✉t ❜② ❢❛r ♠♦r❡ ❝✉♠❜❡rs♦♠❡✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d], a < b, c < d✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ✇❡ ❤❛✈❡ t✇♦ ✐♥t❡❣❡rs n, m ≥ 1✳

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✺

❋✐rst✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✐♥t♦ n ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] ,

✇✐t❤ x0 = a, xn = b✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ x ❛r❡✿ ∆xi = xi − xi−1 , i = 1, 2, ..., n .

❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [c, d] ✐♥t♦ m ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [y0 , y1 ], [y1 , y2 ], ..., [ym−1 , ym ] ,

✇✐t❤ y0 = c, ym = d✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ y ❛r❡✿ ∆yj = yj − yj−1 , j = 1, 2, ..., m .

❆❧t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ P ♦❢ t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d] ✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s ❚❤❡s❡ ❛r❡ 2✲❝❡❧❧s✦ ❚❤❡ ♣♦✐♥ts

Rij = [xi−1 , xi ] × [yj−1 , yj ] . (xi , yj ), i = 1, 2, ..., n, j = 1, 2, ..., m ,

✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ✭♣r✐♠❛r②✮ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

❲❡ ✇♦♥✬t ♥❡❡❞ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❲❡ ❛r❡ ❛❧s♦ ❣✐✈❡♥ t❤❡ t❡rt✐❛r②

♥♦❞❡s ♦❢ P ❢♦r ❡❛❝❤ ♣❛✐r i = 0, 1, 2, ..., n − 1 ❛♥❞ i, j = 0, 1, 2, ..., m − 1✿

◮ ❛ ♣♦✐♥t Uij ✐♥ t❤❡ r❡❝t❛♥❣❧❡ Rij = [xi−1 , xi ] × [yj−1 , yj ]✳

❙✉❝❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❝t❛♥❣❧❡s ❛♥❞ t❡rt✐❛r② ♥♦❞❡s ✐♥ ✐ts ✐♥t❡r✈❛❧s ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ R✿

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✻

■♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ t❤❡ r✐❣❤t ✉♣♣❡r ❝♦r♥❡rs ✇❡r❡ ❝❤♦s❡♥✳ ❇❡❢♦r❡ ✇❡ ❛❞❞r❡ss ❤♦✇ t♦ ❝♦♠♣✉t❡ ✈♦❧✉♠❡s✱ ❧❡t✬s ❝♦♥s✐❞❡r ❛ s✐♠♣❧❡r ♣r♦❜❧❡♠✳ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (X) = f (x, y) ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ R ❛♥❞ ❣✐✈❡s ✉s t❤❡ ❛♠♦✉♥t ♦❢ s♦♠❡ ♠❛t❡r✐❛❧ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❡❧❧✳ ❚❤❡♥ t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡ ✇❤♦❧❡ r❡❝t❛♥❣❧❡ ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ f ✳

❉❡✜♥✐t✐♦♥ ✺✳✶✳✸✿ s✉♠ ❚❤❡ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ X

f=

R

m n X X

f (Uij )

i=1 j=1

❊①❛♠♣❧❡ ✺✳✶✳✹✿ ❞♦✉❜❧❡ s✉♠♠❛t✐♦♥ ❚❤❡ ❞♦✉❜❧❡ s✉♠♠❛t✐♦♥ ♦♥ t❤❡ r✐❣❤t ❢♦❧❧♦✇s t❤❡ s❛♠❡ ✐❞❡❛ ❛s t❤❡ s✐♥❣❧❡✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡s❡ ✈❛❧✉❡s aij ❛rr❛♥❣❡❞ ✐♥ ❛♥ ❛rr❛②✿ j\i

1

2 3

1

2

0 1

2

❚❤❡♥✱

2 3 X X

aij =

i=1 j=1

2 3 X X i=1

=

3  X

j=2

−1 2 3 aij

!

ai1 + ai2

i=1



= (a11 + a12 ) + (a21 + a22 ) + (a31 + a32 ) = (2 + (−1)) + (0 + 2) + (1 + 3) = 7.

❚❤❡r❡ ✐s ❛❧s♦ ❛ s✐♠♣❧❡ ✇❛② t♦ ❛❞❞ ❛❧❧ ♥✉♠❜❡rs ✐♥ ❛♥ ❛rr❛② ♣r❡s❡♥t❡❞ ❛s ❛ s♣r❡❛❞s❤❡❡t✿ ❂s✉♠✭❘✶❈✶✿❘✷❈✸✮

❊①❡r❝✐s❡ ✺✳✶✳✺ ❋♦r t❤❡ ❛❜♦✈❡ t❛❜❧❡ ❝♦♠♣✉t❡✿

3 2 X X

aij .

j=1 i=1

◆♦t❡ t❤❛t ✇❤❡♥ t❡rt✐❛r② ♥♦❞❡s ❛r❡♥✬t ♣r♦✈✐❞❡❞✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ 2✲❝❡❧❧s t❤❡♠s❡❧✈❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ Uij = Rij = [xi−1 , xi ] × [yj−1 , yj ]✳ ❚❤✐s ♠❛❦❡s f ❛ 2✲❢♦r♠✳ ❚❤❡ ❛r❡❛ ♦❢ ❡❛❝❤ 2✲❝❡❧❧ ✐s✿

∆Aij = ∆xi · ∆yj .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✐♥❝r❡♠❡♥ts ♦❢ x ❛♥❞ y ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❛r❡❛✳ ❙✉♣♣♦s❡ ♥❡①t ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = f (X) = f (x, y) ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✼

r❡❝t❛♥❣❧❡ R ❛♥❞ ❣✐✈❡s ✉s t❤❡ ❤❡✐❣❤t ♦❢ ❛ ❜❛r ♦♥ t♦♣ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❡❧❧✿ ❞❡♣t❤ ♦❢ ❜❛s❡ ✇✐❞t❤ ♦❢ ❜❛s❡

z}|{ ∆xi

t❤❡ ✈♦❧✉♠❡ ♦❢ ij ❜❛r = f (Uij ) · | {z }

❤❡✐❣❤t ♦❢ ❜❛r

z}|{ ∆yj

·

❲❡ t❤❡♥ ❛❞❞ ❛❧❧ ♦❢ t❤❡s❡ t♦❣❡t❤❡r ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ z = f (x, y) ♦✈❡r r❡❝t❛♥❣❧❡ R✳

❊①❛♠♣❧❡ ✺✳✶✳✻✿ ♣❛rt✐t✐♦♥s ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❤❡♥ t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ P ❝♦♠❡ ❢r♦♠ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s [a, b] ❛♥❞ [c, d]✿ Uij = (si , tj ) .

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿ y0

∆y1

y1

∆y2

y3 ... ym−1

∆ym

ym

−−

•

−−

•

−−

•

−−

•

−−

•

−−

•

x2 •

−−

•

−−

•

...

•

−−

•

xn−1 •

−−

•

•

−−

•

−−

•

−−

•

x0 •

∆x1 |

f (s1 , t1 )∆x1 ∆y1

∆x2 |

f (s2 , t1 )∆x2 ∆y1

x1 •

.. .

∆xn |

xn •

..

f (sn , t1 )∆xn ∆y1

|

f (s1 , t2 )∆x1 ∆y2

|

f (s2 , t2 )∆x2 ∆y2

.

..

|

...

|

...

|

...

.

...

−−

•

...

−−

•

f (sn , t2 )∆xn ∆y2

|

...

... ...

• |

f (s1 , tm )∆x1 ∆ym

|

f (s2 , tm )∆x2 ∆ym

.

..

•

|

f (sn , tm )∆xn ∆ym

•

| | .

|

❉❡✜♥✐t✐♦♥ ✺✳✶✳✼✿ ❘✐❡♠❛♥♥ s✉♠ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ X R

f (Uij ) ∆xi ∆yj =

n X m X

f (Uij )∆xi ∆yj

i=1 j=1

❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐s s❤♦✇♥ ❜❡❧♦✇✿

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✽

❚❤❡ ❛❜❜r❡✈✐❛t❡❞ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s✿ ❘✐❡♠❛♥♥ s✉♠

X

f ∆x∆y

R

❊①❛♠♣❧❡ ✺✳✶✳✽✿ ♣❧❛♥❡

▲❡t✬s ❝♦♥s✐❞❡r ❛ ✈❡r② s✐♠♣❧❡ ❡①❛♠♣❧❡✿ f (x, y) = x + y, R = [0, 1] × [0, 1] .

❲❡ ❝❤♦♦s❡

n = 2, m = 2 .

❚❤❡♥ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡✿ x0 = 0, x1 = .5, x2 = 1 ❛♥❞ y0 = 0, y1 = .5, y2 = 1 .

❚❤❡ ♥♦❞❡s ❛r❡ (0, 1) − (.5, 1) − (1, 1) |

|

|

|

|

|

(0, .5) − (.5, .5) − (1, .5) (0, 0) − (.5, 0) − (1, 0)

❚❤❡② ❛r❡ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

[0, .5] × [.5, 1] [.5, 1] × [.5, 1][0, .5] × [0, .5] [.5, 1] × [0, .5]

◆♦✇ ✇❡ ❝❤♦♦s❡ t❤❡ t❡rt✐❛r② ♥♦❞❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❧❡t✬s ❝❤♦♦s❡ t❤❡ ❜♦tt♦♠ ❧❡❢t ❝♦r♥❡rs✿ (0, .5) (.5, .5) (0, 0)

(.5, 0)

❧❡❛❞✐♥❣ t♦

f (0, .5) = .5 f (.5, .5) = 1 f (0, 0) = 0

f (.5, 0) = .5

❧❡❛❞✐♥❣ t♦

.5 · .52 = .125 1 · .52 = .25 0 · .52 = 0

.5 · .52 = .125

✺✳✶✳

❱♦❧✉♠❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠s

✸✺✾

❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡ ❜❛rs ❛r❡ s❤♦✇♥ ♦♥ r✐❣❤t✳ ❚❤❡♥ t❤❡ s✉♠ ♦❢ t❤♦s❡ ✐s✿

X

f ∆x∆y = .5 .

R

❊①❛♠♣❧❡ ✺✳✶✳✾✿ ♣❛r❛❜♦❧♦✐❞

❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ ♣❛r❛❜♦❧♦✐❞ ♦❢ r❡✈♦❧✉t✐♦♥✿

❏✉st ❛s ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ ❛r❡ ❛❧❧♦✇❡❞ t♦ ❤❛✈❡ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ♦❢ ✈♦❧✉♠❡s ♦❢ t❤❡ ❜❛rs✳ ❚❤❡s❡ ❛r❡ t❤❡ s✐❣♥❡❞ ❞✐st❛♥❝❡ ❛♥❞ t❤❡ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞

s✐❣♥❡❞ ✈♦❧✉♠❡

f✱

✇✐t❤ ♣♦ss✐❜❧② ♥❡❣❛t✐✈❡

r❡s♣❡❝t✐✈❡❧②✳ ❲❡ s♣❡❛❦ t❤❡♥ ♦❢

❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤ ♦❢ z = f (x, y) ❛♥❞ t❤❡ r❡❝t❛♥❣❧❡ R ✐♥ t❤❡ xy✲♣❧❛♥❡✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ❤❛✈❡ ♥❡❣❛t✐✈❡ ❧❡♥❣t❤s ❢♦r t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s t♦♦✳ ❙✉♣♣♦s❡

a dV Z (x/2 + y/2 + 1)y dV > =< (x/2 + y/2 + 1)x dV , =

RZ

R

R

= ... ❚❤❡r❡❢♦r❡✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s

Q=

M M = . m 14

❊①❡r❝✐s❡ ✺✳✶✵✳✽

❋✐♥✐s❤ t❤❡ ♣r♦❜❧❡♠✳ ❊①❛♠♣❧❡ ✺✳✶✵✳✾✿ ❡①♣❡❝t❡❞ ✈❛❧✉❡

▲❡t✬s r❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ❜❛❦❡r t❤❡ ❜r❡❛❞ ♦❢ ✇❤♦s❡ ✐s ♣r✐❝❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♣r✐❝❡s ♦❢ ✇❤❡❛t ❛♥❞ s✉❣❛r t❤❛t ❝❤❛♥❣❡ ❡✈❡r② ❞❛②✳ ❊✈❡r② ❞❛② ❢♦r ❛ ♠♦♥t❤ ❤❡ r❡❝♦r❞❡❞ t✇♦ ♥✉♠❜❡rs ✕ x ❛♥❞ y ✕ t❤❛t r❡♣r❡s❡♥t

✺✳✶✵✳

✹✵✶

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

❤♦✇ ♠✉❝❤ t❤❡ t✇♦ ♣r✐❝❡s ❞❡✈✐❛t❡❞ ❢r♦♠ s♦♠❡ ♠✐♥✐♠✉♠✳ ❆♥❞ ♥♦✇ ❤❡ ✇❛♥ts t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✇❛s t❤❡ ❛✈❡r❛❣❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣r✐❝❡s✳ ❋♦r ❡❛❝❤ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣r✐❝❡s ❤❡ r❡❝♦r❞s ❤♦✇ ♠❛♥② t✐♠❡s ✐s ❤❛s ♦❝❝✉rr❡❞✳ ❍❡ ♣✉ts t❤❡s❡ ♥✉♠❜❡rs ✐♥ ❛ t❛❜❧❡✱ ✇❤✐❝❤ ♠❛❦❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❚❤❡s❡ ♠❛② ❜❡ ✐ts ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✿

y\x

0

1

2

...

10

0

(0, 0)

(0, 1)

(0, 2)

...

(0, 10)

1

(1, 0)

(1, 1)

(2, 2)

...

(1, 10)

2

(2, 0)

(1, 1)

(2, 2)

...

(2, 10)

...

...

...

...

...

10

❧❡❛❞✐♥❣ t♦

(10, 0) (10, 1) (10, 2) ... (10, 10)

y\x

0

1

2 ... 10

0

1

3

5 ...

0

1

2

4

6 ...

0

2

3

5

7 ...

1

...

... ... ... ...

10

0

1

7 ...

0

❚❤✐s ♠❛② ❧♦♦❦ ❧✐❦❡ ❛ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❜✉t ❧❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t ❤♦✇ t❤❡ ❞❛t❛ ✐s ❝♦❧❧❡❝t❡❞✳ ■t✬s ♥♦t t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ ❡✐t❤❡r ♣r✐❝❡ t❤❛t ♠❛tt❡rs ❜✉t r❛t❤❡r ✐ts r❛♥❣❡✱ s❛② 2 ≤ x < 3✳ ❚❤✐s r❛♥❣❡ ✐s ❛♥ ✐♥t❡r✈❛❧ ♦❢ ✈❛❧✉❡s ❛♥❞ t♦❣❡t❤❡r t❤❡ r❛♥❣❡ ♦❢ ♣❛✐rs ♦❢ ♣r✐❝❡s ✐s ❛ r❡❝t❛♥❣❧❡✱ s❛② [2, 3] × [1, 2]✳ ❚❤❡ ❞❛t❛ t❤❡♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ t❛❜❧❡ t❤❛t ❧♦♦❦s ❛ ❜✐t ❞✐✛❡r❡♥t✿

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❲❡ ❛r❡ ❥✉st✐✜❡❞ t♦ ✈✐s✉❛❧✐③❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ❛s ❝♦❧✉♠♥s ♦✈❡r t❤❡s❡ r❡❝t❛♥❣❧❡s✿

❲❡ r❡❛❧✐③❡ t❤❛t t❤✐s ✐s♥✬t ❥✉st ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❀ ✐t✬s ❛ ❞✐s❝r❡t❡ 2✲❢♦r♠ ✦ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❛✈❡r❛❣❡ ♣r✐❝❡ ❝♦♠❜✐♥❛t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t ✕ ✐♥ t❤❡ ❞✐♠❡♥s✐♦♥s 1, 2, 3 ✕ t♦ ✜♥❞✐♥❣ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤✐s ❝♦❧❧❡❝t✐♦♥ ♦❢ ❜❛rs✳

✺✳✶✵✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✹✵✷

❊①❛♠♣❧❡ ✺✳✶✵✳✶✵✿ ❣r❛✈✐t②

❆ ❢❛♠✐❧✐❛r ♣r♦❜❧❡♠ ❛❜♦✉t ❛ ❜❛❧❧ t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ❤❛s ❛ s♦❧✉t✐♦♥✿ ✐ts tr❛❥❡❝t♦r② ✐s ❛

♣❛r❛❜♦❧❛✳

❍♦✇❡✈❡r✱

✇❡ ❛❧s♦ ❦♥♦✇ t❤❛t ✐❢ ✇❡ t❤r♦✇ r❡❛❧❧②✲r❡❛❧❧② ❤❛r❞ ✭❧✐❦❡ ❛ r♦❝❦❡t✮ t❤❡ ❜❛❧❧ ✇✐❧❧ st❛rt t♦ ♦r❜✐t t❤❡ ❊❛rt❤ ❢♦❧❧♦✇✐♥❣ ❛♥

❡❧❧✐♣s❡✳

❚❤❡ ♠♦t✐♦♥ ♦❢ t✇♦ ♣❧❛♥❡ts ✭♦r ❛ st❛r ❛♥❞ ❛ ♣❧❛♥❡t✱ ♦r ❛ ♣❧❛♥❡t ❛♥❞ ❛ s❛t❡❧❧✐t❡✱ ❡t❝✳✮ ✐s ❣♦✈❡r♥❡❞ ❜② ❛ s✐♥❣❧❡ ❢♦r❝❡✿ t❤❡

❣r❛✈✐t②✳

❘❡❝❛❧❧ ❤♦✇ t❤✐s ❢♦r❝❡ ♦♣❡r❛t❡s✳

◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②✿ ❚❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛✿

F =G

mM , r2

✇❤❡r❡✿

• • • • •

F ✐s t❤❡ ❢♦r❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝ts✳ G ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✳ m ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♦❜❥❡❝t✳ M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✳ r ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs

♦❢ t❤❡ ♠❛ss❡s✳

♦r✱ ✐♥ t❤❡ ✈❡❝t♦r ❢♦r♠ ✭✇✐t❤ t❤❡ ✜rst ♦❜❥❡❝t ✐s ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✮✿

F = −GmM

X . ||X||3

❚❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇✳

•

❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ✏❧❛r❣❡✑ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ tr❛❥❡❝t♦r②✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s

•

❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ✏s♠❛❧❧✑ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ tr❛❥❡❝t♦r②✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s

❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ♣❛r❛❧❧❡❧ ✐♥ ❛❧❧ ❧♦❝❛t✐♦♥s ✭t❤❡ ♦r❜✐t ✐s ❛ ♣❛r❛❜♦❧❛✮✳ ❛r❡ ❛ss✉♠❡❞ t♦ ♣♦✐♥t r❛❞✐❛❧❧② t♦✇❛r❞ t❤❛t ♣♦✐♥t ✭t❤❡ ♦r❜✐t✮ ♠❛② ❜❡ ❛♥ ❡❧❧✐♣s❡✱ ♦r ❛ ❤②♣❡r❜♦❧❛✱ ♦r ❛ ♣❛r❛❜♦❧❛✳

❲❤❡♥ t❤❡ s✐③❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❊❛rt❤ ♠❛tt❡r✱ t❤✐♥❣s ❣❡t ❝♦♠♣❧✐❝❛t❡❞✳✳✳

❈❤❛♣t❡r ✻✿ ❱❡❝t♦r ✜❡❧❞s

❈♦♥t❡♥ts

✻✳✶ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹ ❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✵ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵✸ ✹✵✽ ✹✶✸ ✹✶✽ ✹✷✹ ✹✸✶ ✹✸✽ ✹✹✺ ✹✺✷ ✹✺✾

✻✳✶✳ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄ ❚❤❡ ✜rst ♠❡t❛♣❤♦r ❢♦r ❛ ✈❡❝t♦r ✜❡❧❞ ✐s ❛ ❤②❞r❛✉❧✐❝ s②st❡♠✳ ❊①❛♠♣❧❡ ✻✳✶✳✶✿ ♣❧✉♠❜✐♥❣

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s②st❡♠ ♦❢ ♣✐♣❡s ✇✐t❤ ✇❛t❡r ✢♦✇✐♥❣ t❤r♦✉❣❤ t❤❡♠✳ ❲❡ ♠♦❞❡❧ t❤❡ ♣r♦❝❡ss ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✇✐t❤ ✐ts ❡❞❣❡s r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣✐♣❡s ❛♥❞ ♥♦❞❡s r❡♣r❡s❡♥t✐♥❣ t❤❡ ❥✉♥❝t✐♦♥s✳ ❚❤❡♥ ❛ ♥✉♠❜❡r ✐s ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ❡❞❣❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ✢♦✇ ✭✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦♥❡ ♦❢ t❤❡ ❛①❡s✮✳ ❙✉❝❤ ❛ s②st❡♠ ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❍❡r❡ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ✢♦✇ ✐s s❤♦✇♥ ❛s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❛rr♦✇✳ ❚❤✐s ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ 1✲❢♦r♠✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ♠❛② ❜❡ ❧❡❛❦❛❣❡✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r t❤❛t ❛❝t✉❛❧❧② ♣❛ss❡s ❛❧❧ t❤❡ ✇❛② t❤r♦✉❣❤ t❤❡ ♣✐♣❡✱ ✇❡ ❝❛♥ ♠❛❦❡ r❡❝♦r❞ ♦❢ t❤❡ ❛♠♦✉♥t t❤❛t ✐s ❧♦st✳

✻✳✶✳ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄

✹✵✹

❚❤❛t✬s ❛♥♦t❤❡r r❡❛❧✲✈❛❧✉❡❞ 1✲❢♦r♠✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❧❡❛❦❛❣❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣✐♣❡✱ t❤❡ t✇♦ ♥✉♠❜❡rs ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ❛ ✈❡❝t♦r✳ ❚❤❡ r❡s✉❧t ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠✿

❲❛r♥✐♥❣✦ ❚❤❡ t✇♦ r❡❛❧✲✈❛❧✉❡❞

1✲❢♦r♠s 1✲❢♦r♠

❢r♦♠ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞

❛r❡ r❡✲❝♦♥str✉❝t❡❞ ❜✉t

♥♦t

❛s ✐ts t✇♦

❝♦♠♣♦♥❡♥ts ❜✉t ❛s ✐ts ♣r♦❥❡❝t✐♦♥s ♦♥ t❤❡ ❝♦rr❡✲ s♣♦♥❞✐♥❣ ❡❞❣❡s✳

❚❤❡ s❡❝♦♥❞ ♠❡t❛♣❤♦r ❢♦r ❛ ✈❡❝t♦r ✜❡❧❞ ✐s ❛ ✢♦✇✲t❤r♦✉❣❤✳ ❊①❛♠♣❧❡ ✻✳✶✳✷✿ ❡①❝❤❛♥❣❡

❚❤❡ ❞❛t❛ ❢r♦♠ ❧❛st ❡①❛♠♣❧❡ ❝❛♥ ❜❡ ✉s❡❞ t♦ ✐❧❧✉str❛t❡ ❛ ✢♦✇ ♦❢ ❧✐q✉✐❞ ♦r ❛♥♦t❤❡r ♠❛t❡r✐❛❧ ❢r♦♠ ❝♦♠✲ ♣❛rt♠❡♥t t♦ ❝♦♠♣❛rt♠❡♥t t❤r♦✉❣❤ ✇❛❧❧s✳ ❆ ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠ ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❚❤❡ s✐t✉❛t✐♦♥ ✐s r❡✈❡rs❡❞ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ t❤❡ ❝♦♠♣♦♥❡♥t ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❡❞❣❡ ✐s t❤❡ r❡❧❡✈❛♥t ♦♥❡✳

✻✳✶✳ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄

✹✵✺

❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❝❤❛♥❣❡s ✐♥ ❞✐♠❡♥s✐♦♥ 3 ❤♦✇❡✈❡r✿ t❤❡ ❝♦♠♣♦♥❡♥t ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢❛❝❡ ✐s t❤❡ r❡❧❡✈❛♥t ♦♥❡✳ ❚❤❡ t❤✐r❞ ♠❡t❛♣❤♦r ❢♦r ❛ ✈❡❝t♦r ✜❡❧❞ ✐s ✈❡❧♦❝✐t✐❡s ♦❢ ♣❛rt✐❝❧❡s✳

❊①❛♠♣❧❡ ✻✳✶✳✸✿ ✜❡❧❞ ♦❢ ✢❛❣s ■♠❛❣✐♥❡ ❧✐tt❧❡ ✢❛❣s ♣❧❛❝❡❞ ♦♥ t❤❡ ❧❛✇♥❀ t❤❡♥ t❤❡✐r ❞✐r❡❝t✐♦♥s ❢♦r♠ ❛ ✈❡❝t♦r ✜❡❧❞✱ ✇❤✐❧❡ t❤❡ ❛✐r ✢♦✇ t❤❛t ♣r♦❞✉❝❡❞ ✐t r❡♠❛✐♥s ✐♥✈✐s✐❜❧❡✳

❊❛❝❤ ✢❛❣ s❤♦✇s t❤❡ ❞✐r❡❝t✐♦♥ ✭✐❢ ♥♦t t❤❡ ♠❛❣♥✐t✉❞❡✮ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ❛t t❤❛t ❧♦❝❛t✐♦♥✳ ❙✉❝❤ ✢❛❣s ❛r❡ ❛❧s♦ ♣❧❛❝❡❞ ♦♥ ❛ ♠♦❞❡❧ ❛✐r♣❧❛♥❡ ✐♥ ❛ ✇✐♥❞✲t✉♥♥❡❧✳ ❆ s✐♠✐❧❛r ✐❞❡❛ ✐s ✉s❡❞ t♦ ♠♦❞❡❧ ❛ ✢✉✐❞ ✢♦✇✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❡❛❝❤ ♣❛rt✐❝❧❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇✱ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ t❤❡ s❛♠❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✈❡❝t♦r ✜❡❧❞ s✉♣♣❧✐❡s ❛ ❞✐r❡❝t✐♦♥ t♦ ❡✈❡r② ❧♦❝❛t✐♦♥✳

❍♦✇ ❞♦ ✇❡ tr❛❝❡ t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛rt✐❝❧❡❄ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ✈❡❝t♦r ✜❡❧❞✿ V (x, y) =< y, −x > .

❊✈❡♥ t❤♦✉❣❤ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡ ♣❛t❤ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦✈❡r ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ❛s ❢♦❧❧♦✇s✳ ❆t ♦✉r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t t✐♠❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ ♠♦✈❡ ❛❝❝♦r❞✐♥❣❧② t♦ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✳ ❲❡ st❛rt ❛t t❤✐s ❧♦❝❛t✐♦♥✿ X0 = (2, 0) .

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s✿ V (0, 2) =< 2, 0 > .

❚❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿ X1 = (0, 2)+ < 2, 0 >= (2, 2) .

❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ V ✿ V (2, 2) =< 2, −2 > ,

✻✳✶✳ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄

✹✵✻

❧❡❛❞✐♥❣ t♦ t❤❡ ♥❡①t st❡♣✳ ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s✿ X2 = (2, 2)+ < 2, −2 >= (4, 0) .

❖♥❡ ♠♦r❡ st❡♣✿ X2 ✐s s✉❜st✐t✉t❡❞ ✐♥t♦ V ❛♥❞ ♦✉r ♥❡①t ❧♦❝❛t✐♦♥ ✐s✿ X3 = (4, 0)+ < 0, −4 >= (4, −4) .

❚❤❡ s❡q✉❡♥❝❡ ✐s s♣✐r❛❧✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ▲❡t✬s ♥♦✇ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ✭✇✐t❤ ❛ s♠❛❧❧❡r t✐♠❡ ✐♥❝r❡♠❡♥t✮✳ ❚❤❡ ❢♦r♠✉❧❛s ❢♦r xn ❛♥❞ yn ❛r❡ r❡s♣❡❝t✐✈❡❧②✿ ❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✶❪✯❘✸❈✶

❂❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✯❘✸❈✶

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿

■♥ ❣❡♥❡r❛❧✱ ❛ ✈❡❝t♦r ✜❡❧❞ V (x, y) =< f (x, y), g(x, y) > ✐s ✉s❡❞ t♦ ❝r❡❛t❡ ❛ s②st❡♠ ♦❢ t✇♦ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❖❉❊s✮✿ X ′ (t) = V (X(t))

♦r

< x′ (t), y ′ (t) >= V (x(t), y(t))

♦r

  x′ (t) = f (x(t), y(t)),  y ′ (t) = g(x(t), y(t)).

■ts s♦❧✉t✐♦♥ ✐s ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s x = x(t) ❛♥❞ y = y(t) t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥s ❢♦r ❡✈❡r② t✳

❚❤❡ ❡q✉❛t✐♦♥s ♠❡❛♥ t❤❛t t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡s❡ tr❛❥❡❝t♦r✐❡s✳ ❖❉❊s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ❱♦❧✉♠❡ ✺ ✭❈❤❛♣t❡r ✺❉❊✲✸✮✳ ❚❤❡ ❢♦✉rt❤ ♠❡t❛♣❤♦r ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐s ❛ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t ❢♦r❝❡✳

✻✳✶✳ ❲❤❛t ❛r❡ ✈❡❝t♦r ✜❡❧❞s❄

✹✵✼

❊①❛♠♣❧❡ ✻✳✶✳✹✿ ❣r❛✈✐t② ❘❡❝❛❧❧ t❤❛t ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t② st❛t❡s t❤❛t t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛✿ f (X) = G

❍❡r❡✿ • • • • •

mM . r2

f ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝ts✳ G ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✳ m ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♦❜❥❡❝t✳ M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✳ r ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs ♦❢ t❤❡ ♠❛ss❡s✳

◆♦✇✱ ❧❡t✬s ❛ss✉♠❡ t❤❛t t❤❡ ✜rst ♦❜❥❡❝t ✐s ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✳ ❚❤❡♥ t❤❡ ✈❡❝t♦r ♦❢ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t ✐s X ❛♥❞ t❤❡ ❢♦r❝❡ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ✈❡❝t♦r✳ ■❢ F (X) ✐s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❢♦r❝❡ ❛t t❤❡ ❧♦❝❛t✐♦♥ X ✱ t❤❡♥✿ F (X) = −GmM

X . ||X||3

❚❤❛t✬s t❤❡ ✈❡❝t♦r ❢♦r♠ ♦❢ t❤❡ ❧❛✇✦ ❲❡ ♣❧♦t t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

❆♥❞ t❤✐s ✐s t❤❡ r❡s✉❧t✐♥❣ ✈❡❝t♦r ✜❡❧❞✿

❚❤❡ ♠♦t✐♦♥ ✐s ❛♣♣r♦①✐♠❛t❡❞ ✐♥ t❤❡ ♠❛♥♥❡r ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ✇✐t❤ t❤❡ ❞❡t❛✐❧s ♣r♦✈✐❞❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❲❤❡♥ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ ❛♥ ♦❜❥❡❝t ✐s ③❡r♦✱ ✐t ✇✐❧❧ ❢♦❧❧♦✇ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r❝❡✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥ ♦❜❥❡❝t ✇✐❧❧ ❢❛❧❧ ❞✐r❡❝t❧② ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳ ❚❤✐s ✐❞❡❛ ❜r✐❞❣❡s t❤❡ ❣❛♣ ❜❡t✇❡❡♥ ✈❡❧♦❝✐t② ✜❡❧❞s ❛♥❞ ❢♦r❝❡ ✜❡❧❞s✳

❉❡✜♥✐t✐♦♥ ✻✳✶✳✺✿ ✈❡❝t♦r ✜❡❧❞ ❆ ✈❡❝t♦r ✜❡❧❞ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛ s✉❜s❡t ♦❢ Rn ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn ✳

❲❛r♥✐♥❣✦ ❚❤♦✉❣❤ ✉♥♥❡❝❡ss❛r② ♠❛t❤❡♠❛t✐❝❛❧❧②✱ ❢♦r t❤❡ ♣✉r✲ ♣♦s❡s ♦❢ ✈✐s✉❛❧✐③❛t✐♦♥ ❛♥❞ ♠♦❞❡❧✐♥❣ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ✐♥♣✉t ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛s ♣♦✐♥ts ❛♥❞ ♦✉t♣✉ts ❛s ✈❡❝✲

✻✳✷✳ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

✹✵✽ t♦rs✳

❇✉t ✇❤❛t ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s❄ ■t✬s ♥♦t ✈❡❝t♦r✲✈❛❧✉❡❞✦ ❙♦♠❡ ✈❡❝t♦r ✜❡❧❞s ❤♦✇❡✈❡r ♠✐❣❤t ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡❤✐♥❞ t❤❡♠✿ t❤❡ ♣r♦❥❡❝t✐♦♥ p ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ V ♦♥ ❛ ♣❛rt✐t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ ✈❡❝t♦rs ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦r✐❡♥t❡❞ ❡❞❣❡s✿ p(C) = V (C) · E ,

✇❤❡r❡ C ✐s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ❡❞❣❡ E ✳ ❲❤❡♥ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ V ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛❧❧ V ❣r❛❞✐❡♥t✳

❲❤❡♥ ♥♦ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ s♣❡❝✐✜❡❞✱ t❤❡ ❢♦r♠✉❧❛ p(E) = V (E) · E ♠❛❦❡s ❛ r❡❛❧✲✈❛❧✉❡❞ 1✲❢♦r♠ ❢r♦♠ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ♦♥❡✳

✻✳✷✳ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ t❤❡ ❢♦r❝❡s ❛✛❡❝t✐♥❣ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t✳ ❍♦✇ ❝❛♥ ✇❡ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s❄ ❲❡ s✐♠♣❧② ❣❡♥❡r❛❧✐③❡ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧②s✐s ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮ t♦ t❤❡ ✈❡❝t♦r ❝❛s❡✳ ❆ss✉♠✐♥❣ ❛ ✜①❡❞ ♠❛ss✱ t❤❡ t♦t❛❧ ❢♦r❝❡ ❣✐✈❡s ✉s ♦✉r ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❡ ❛r❡ t♦ ❝♦♠♣✉t❡✿ • t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ t❤❡♥ • t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳

❆ ✜①❡❞ t✐♠❡ ✐♥❝r❡♠❡♥t ∆t ✐s s✉♣♣❧✐❡❞ ❛❤❡❛❞ ♦❢ t✐♠❡ ❡✈❡♥ t❤♦✉❣❤ ✐t ❝❛♥ ❛❧s♦ ❜❡ ✈❛r✐❛❜❧❡✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ q✉❛♥t✐t✐❡s t❤❛t ❝♦♠❡ ❢r♦♠ t❤❡ s❡t✉♣ ♦❢ t❤❡ ♠♦t✐♦♥✿ • t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ✱

• t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② V0 ✱ ❛♥❞ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ P0 ✳

❚❤❡② ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ❝❡❧❧s ♦❢ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✿ ✐t❡r❛t✐♦♥ n t✐♠❡ tn ❛❝❝❡❧❡r❛t✐♦♥ An ✈❡❧♦❝✐t② Vn ❧♦❝❛t✐♦♥ Pn ✐♥✐t✐❛❧✿

0

3.5

−−

< 33, 44 >

< 22, 11 >

❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✳ ❚❤❡r❡ ✐s ❛ s❡t ♦❢ ❝♦❧✉♠♥s ❢♦r ❡❛❝❤ ✈❡❝t♦r✱ t✇♦ ♦r t❤r❡❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ r❡❧② ♦♥ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✳ ❚❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ A0 ❣✐✈❡♥ ✐♥ t❤❡ ✜rst ❝❡❧❧s ♦❢ t❤❡ s❡❝♦♥❞ r♦✇✳ ❚❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② V1 ✐s ❢♦✉♥❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ♣❛✐r ✭♦r tr✐♣❧❡✮ ♦❢ ❝❡❧❧s ♦❢ t❤❡ s❡❝♦♥❞ r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿

✻✳✷✳

✹✵✾

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

• ❝✉rr❡♥t ✈❡❧♦❝✐t② = ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② + ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ · t✐♠❡ ✐♥❝r❡♠❡♥t✳ ❚❤❡ s❡❝♦♥❞ q✉❛♥t✐t② ✇❡ ✉s❡ ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ P0 ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ♣❧❛❝❡❞ ✐♥ t❤❡ t❤✐r❞ s❡t ♦❢ ❝❡❧❧s ♦❢ t❤❡ s❡❝♦♥❞ r♦✇✿ • ❝✉rr❡♥t ❧♦❝❛t✐♦♥ = ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✳

❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐s s❤♦✇♥ ❜❡❧♦✇✿

✐t❡r❛t✐♦♥ n t✐♠❡ tn ❛❝❝❡❧❡r❛t✐♦♥ An ✐♥✐t✐❛❧✿

0

❝✉rr❡♥t✿

1

−−

3.6 t1

< 66, 77 >

✈❡❧♦❝✐t② Vn

❧♦❝❛t✐♦♥ Pn

< 33, 44 >

< 22, 11 >

↓

↓

→

V1

→

P1

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ r❡st ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥✉♠❜❡rs ❛♥❞ ✈❡❝t♦rs ❛r❡ s✉♣♣❧✐❡❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r s❡ts ♦❢ ❝♦❧✉♠♥s ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t ♦♥❡ r♦✇ ❛t ❛ t✐♠❡✿ tn , An , Vn , Pn , n = 1, 2, 3, .. .

❚❤❡ ✜rst q✉❛♥t✐t② ✐♥ ❡❛❝❤ r♦✇ ✇❡ ❝♦♠♣✉t❡ ✐s t❤❡ t✐♠❡✿ tn+1 = tn + ∆t .

❚❤❡ ♥❡①t ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ An+1 ✳ ❲❤❡r❡ ❞♦❡s ✐t ❝♦♠❡ ❢r♦♠❄ ■t ♠❛② ❝♦♠❡ ❛s ♣✉r❡ ❞❛t❛✿ t❤❡ ❝♦❧✉♠♥ ✐s ✜❧❧❡❞ ✇✐t❤ ♥✉♠❜❡r ❛❤❡❛❞ ♦❢ t✐♠❡ ♦r ✐t ✐s ❜❡✐♥❣ ✜❧❧❡❞ ❛s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✳ ❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡r❡ ✐s ❛♥ ❡①♣❧✐❝✐t✱ ❢✉♥❝t✐♦♥❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✭♦r t❤❡ ❢♦r❝❡✮ ♦♥ t❤❡ r❡st ♦❢ t❤❡ q✉❛♥t✐t✐❡s✳ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♠❛② ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • t❤❡ ❝✉rr❡♥t t✐♠❡✱ ❡✳❣✳✱ An+1 =< sin tn+1 , cos tn+1 >✱ s✉❝❤ ❛s ✇❤❡♥ ✇❡ s♣❡❡❞ ✉♣ t❤❡ ❝❛r✱ ♦r

• ❚❤❡ ❧❛st ❧♦❝❛t✐♦♥✱ s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❣r❛✈✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♣❧❛♥❡t ✭❜❡❧♦✇✮✱ ♦r

• ❚❤❡ ❧❛st ✈❡❧♦❝✐t②✱ ❡✳❣✳✱ An+1 = −Vn s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❛✐r r❡s✐st❛♥❝❡ ✇♦r❦s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥ ♦❢

t❤❡ ✈❡❧♦❝✐t②✱

♦r ❛❧❧ t❤r❡❡✳ ❚❤❡ nt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② Vn ✐s ❝♦♠♣✉t❡❞✿ • ❝✉rr❡♥t ✈❡❧♦❝✐t② = ❧❛st ✈❡❧♦❝✐t② + ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ · t✐♠❡ ✐♥❝r❡♠❡♥t✱

• Vn+1 = Vn + An · ∆t✳

❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ s❡❝♦♥❞ s❡t ♦❢ ❝♦❧✉♠♥s ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✳ ❚❤❡ nt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ Pn ✐s ❝♦♠♣✉t❡❞✿ • ❝✉rr❡♥t ❧♦❝❛t✐♦♥ = ❧❛st ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✱ • Pn+1 = Pn + Vn · ∆t✳

❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ t❤✐r❞ s❡t ♦❢ ❝♦❧✉♠♥s ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐t❡r❛t✐♦♥ n t✐♠❡ tn ✐♥✐t✐❛❧✿

❛❝❝❡❧❡r❛t✐♦♥ An

✈❡❧♦❝✐t② Vn

❧♦❝❛t✐♦♥ Pn

−−

< 33, 44 >

< 22, 11 >

0

3.5

1

3.6

< 66, 77 >

...

...

...

...

...

1000

103.5

< 666, 777 >

< 4, 1 >

< 336, 200 >

...

...

...

...

...

< 38.5, 45.1 > < 25.3, 13.0 >

✻✳✷✳

✹✶✵

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

❚❤❡ r❡s✉❧t ♠❛② ❜❡ s❡❡♥ ❛s ❢♦✉r s❡q✉❡♥❝❡s tn , An , Vn , Pn ♦r ❛s t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ♦❢ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦❢ t✳

✈❡❝t♦r✲✈❛❧✉❡❞

❊①❡r❝✐s❡ ✻✳✷✳✶

■♠♣❧❡♠❡♥t ❛ ✈❛r✐❛❜❧❡ t✐♠❡ ✐♥❝r❡♠❡♥t✿ ∆tn+1 = tn+1 − tn ✳ ❊①❛♠♣❧❡ ✻✳✷✳✷✿ r♦❧❧✐♥❣ ❜❛❧❧

❆ r♦❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s✳ ❚❤❡r❡❢♦r❡✱ An = 0 ❢♦r ❛❧❧ n✳ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ s✐♠♣❧✐❢② ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② Vn+1 = Vn + An · ∆t = Vn = V0 ✐s ❝♦♥st❛♥t✳ • ❚❤❡ ♣♦s✐t✐♦♥ Pn+1 = Pn + Vn · ∆t = Pn + V0 · ∆t ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s ❧✐♥❡❛r❧② ♦♥ t❤❡ t✐♠❡✳ ❊①❛♠♣❧❡ ✻✳✷✳✸✿ ❢❛❧❧✐♥❣ ❜❛❧❧

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✿ An = A ❢♦r ❛❧❧ n✳ ❚❤❡ ✜rst ♦❢ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡rt✐❝❛❧ ♠♦t✐♦♥ s✐♠♣❧✐✜❡s ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② Vn+1 = Vn + An · ∆t = Vn + A · ∆t ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ • ❚❤❡ ♣♦s✐t✐♦♥ Pn+1 = Pn + Vn · ∆t ❣r♦✇s ❛t ❧✐♥❡❛r❧② ✐♥❝r❡❛s✐♥❣ ✐♥❝r❡♠❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s q✉❛❞r❛t✐❝❛❧❧② ♦♥ t❤❡ t✐♠❡✳ ❊①❛♠♣❧❡ ✻✳✷✳✹✿ ✈❡❝t♦r ❛❧❣❡❜r❛

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✿ An =< 0, −g > .

◆♦✇ r❡❝❛❧❧ t❤❡ s❡t✉♣ ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧②✿ ❢r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡✿ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✱ P0 =< 0, 200 >✱ • t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✱ V0 =< 200, 0 >✳ ❚❤❡♥ ✇❡ ❤❛✈❡ r❡❝✉rs✐✈❡ ✈❡❝t♦r ❡q✉❛t✐♦♥s✿ Vn+1 = Vn + < 0, −g > ∆t ❛♥❞ Pn+1 = Pn + Vn ∆t .

■♠♣❧❡♠❡♥t❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✱ t❤❡ ❢♦r♠✉❧❛s ♣r♦❞✉❝❡ t❤❡s❡ r❡s✉❧ts✿

✻✳✷✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

✹✶✶

❊①❛♠♣❧❡ ✻✳✷✳✺✿ ♣❧❛♥❡ts

▲❡t✬s ❛♣♣❧② ✇❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ t♦ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✳ ❚❤❡ ♣r♦❜❧❡♠ ❛❜♦✈❡ ❛❜♦✉t ❛ ❜❛❧❧ t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ❤❛s ❛ s♦❧✉t✐♦♥✿ ✐ts tr❛❥❡❝t♦r② ✐s ❛ ♣❛r❛❜♦❧❛✳

❍♦✇❡✈❡r✱ ✇❡ ❛❧s♦ ❦♥♦✇ t❤❛t ✐❢ ✇❡ t❤r♦✇ r❡❛❧❧②✲r❡❛❧❧② ❤❛r❞ ✭❧✐❦❡ ❛ r♦❝❦❡t✮✱ t❤❡ ❜❛❧❧ ✇✐❧❧ st❛rt t♦ ♦r❜✐t t❤❡ ❊❛rt❤ ❢♦❧❧♦✇✐♥❣ ❛♥ ❡❧❧✐♣s❡✳

❚❤❡ ♠♦t✐♦♥ ♦❢ t✇♦ ♣❧❛♥❡ts ✭♦r t❤❡ s✉♠ ❛♥❞ ❛ ♣❧❛♥❡t✱ ♦r ❛ ♣❧❛♥❡t ❛♥❞ ❛ s❛t❡❧❧✐t❡✱ ❡t❝✳✮ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ◆❡✇t♦♥ ▲❛✇ ♦❢ ●r❛✈✐t②✳ ❋r♦♠ t❤✐s ❧❛✇✱ ❛♥♦t❤❡r ❧❛✇ ♦❢ ♠♦t✐♦♥ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✳ ❈♦♥s✐❞❡r t❤❡ ❑❡♣❧❡r✬s ▲❛✇s ♦❢ P❧❛♥❡t❛r② ▼♦t✐♦♥✿

• ❚❤❡ ♦r❜✐t ♦❢ ❛ ♣❧❛♥❡t ✐s ❛♥ ❡❧❧✐♣s❡ ✇✐t❤ t❤❡ ❙✉♥ ❛t ♦♥❡ ♦❢ t❤❡ t✇♦ ❢♦❝✐✳ • ❆ ❧✐♥❡ s❡❣♠❡♥t ❥♦✐♥✐♥❣ ❛ ♣❧❛♥❡t ❛♥❞ t❤❡ ❙✉♥ s✇❡❡♣s ♦✉t ❡q✉❛❧ ❛r❡❛s ❞✉r✐♥❣ ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢

t✐♠❡✳ • ❚❤❡ sq✉❛r❡ ♦❢ t❤❡ ♦r❜✐t❛❧ ♣❡r✐♦❞ ♦❢ ❛ ♣❧❛♥❡t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉❜❡ ♦❢ t❤❡ s❡♠✐✲♠❛❥♦r ❛①✐s ♦❢ ✐ts ♦r❜✐t✳ ❚♦ ❝♦♥✜r♠ t❤❡ ❧❛✇✱ ✇❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s ❛❜♦✈❡ ❜✉t t❤✐s t✐♠❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❧♦❝❛t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿

❚❤❡ r❡s✉❧t✐♥❣ tr❛❥❡❝t♦r② ❞♦❡s s❡❡♠ t♦ ❜❡ ❛♥ ❡❧❧✐♣s❡ ✭❝♦♥✜r♠❡❞ ❜② ✜♥❞✐♥❣ ✐ts ❢♦❝✐✮✿

✻✳✷✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧

✹✶✷

◆♦t❡ t❤❛t t❤❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇ ✐♠♣❧✐❡s t❤❛t t❤❡ ♠♦t✐♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ ♦♥❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡✳

❖✉r ❝♦♠♣✉t❛t✐♦♥ ❝❛♥ ♣r♦❞✉❝❡ ♦t❤❡r ❦✐♥❞s ♦❢ tr❛❥❡❝t♦r✐❡s s✉❝❤ ❛s ❛ ❤②♣❡r❜♦❧❛✿

❊①❛♠♣❧❡ ✻✳✷✳✻✿ ❙✉♥✲❊❛rt❤✲▼♦♦♥

❚❤❡ ❊❛rt❤ r❡✈♦❧✈❡s ❛r♦✉♥❞ t❤❡ ❙✉♥ ❛♥❞ t❤❡ ▼♦♦♥ r❡✈♦❧✈❡s ❛r♦✉♥❞ t❤❡ ❊❛rt❤✳ ❚❤❡ r❡s✉❧t ❞❡r✐✈❡❞ ❢r♦♠ s✉❝❤ ❛ ❣❡♥❡r✐❝ ❞❡s❝r✐♣t✐♦♥ s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ t❤❡ ♦♥❡ ♦♥ ❧❡❢t✳

✻✳✸✳

❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s

✹✶✸

◆♦✇✱ ❧❡t✬s ✉s❡ t❤❡ ❛❝t✉❛❧ ❞❛t❛✿

• • •

❚❤❡ ❛✈❡r❛❣❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❊❛rt❤ ❛♥❞ t❤❡ ❙✉♥ ✐s

149.60 ♠✐❧❧✐♦♥ ❦♠✳ 385, 000 ❦♠✳

❚❤❡ ❛✈❡r❛❣❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ▼♦♦♥ ❛♥❞ t❤❡ ❊❛rt❤ ✐s ❚❤❡ ▼♦♦♥ ♦r❜✐ts ❊❛rt❤ ♦♥❡ r❡✈♦❧✉t✐♦♥ ✐♥

27.323

❞❛②s✳

❚❤❡ ♣❛t❤s ❛r❡ ♣❧♦tt❡❞ ♦♥ r✐❣❤t✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ ♥♦t ♦♥❧② t❤❡ ▼♦♦♥ ♥❡✈❡r ❣♦❡s ❜❛❝❦✇❛r❞s ❜✉t ❛❧s♦ ✐ts ♦r❜✐t ✐s ✐♥ ❢❛❝t ❝♦♥✈❡①✦ ✭❇② ✏❝♦♥✈❡① ♦r❜✐t✑ ✇❡ ♠❡❛♥ ✏❝♦♥✈❡① r❡❣✐♦♥ ✐♥s✐❞❡ t❤❡ ♦r❜✐t✑✿ ❛♥② t✇♦ ♣♦✐♥ts ✐♥s✐❞❡ ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② t❤❡ s❡❣♠❡♥t t❤❛t ✐s ❛❧s♦ ✐♥s✐❞❡✳✮

❊①❛♠♣❧❡ ✻✳✷✳✼✿ s✐♠✉❧❛t✐♦♥s

❇❡❧♦✇ ✇❡ ❤❛✈❡✿ ❛ ❤②♣♦t❤❡t✐❝❛❧ st❛r ✭♦r❛♥❣❡✮ ✐s ♦r❜✐t❡❞ ❜② ❛ ♣❧❛♥❡t ✭❜❧✉❡✮ ✇❤✐❝❤ ✐s ❛❧s♦ ♦r❜✐t❡❞ ❜② ✐ts ♠♦♦♥ ✭♣✉r♣❧❡✮✳ ◆♦✇ ✇❡ ✈❛r② t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s ♣❡r ②❡❛r t❤❡ ♠♦♦♥ ♦r❜✐ts t❤❡ ♣❧❛♥❡t✱ ❢r♦♠

20

t♦

1/3✳

✻✳✸✳ ❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s ❱❡❝t♦r ✜❡❧❞s ❛♣♣❡❛r ✐♥ ❛❧❧ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡✿ t❤❡r❡ ✐s ❛ ✢♦✇ ♦❢ ❧✐q✉✐❞ ♦r ❣❛s ❛♥❞ ✇❡ r❡❝♦r❞ ❤♦✇ ❢❛st ❛ s✐♥❣❧❡ ♣❛rt✐❝❧❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ✐s ♠♦✈✐♥❣✳ ❊①❛♠♣❧❡ ✻✳✸✳✶✿ ❞✐♠❡♥s✐♦♥

1

❚❤❡ ✢♦✇ ✐s ✐♥ ❛ ♣✐♣❡✳ ❚❤❡ s❛♠❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❛ ❝❛♥❛❧ ✇✐t❤ t❤❡ ✇❛t❡r t❤❛t ❤❛s t❤❡ ❡①❛❝t s❛♠❡ ✈❡❧♦❝✐t② ❛t ❛❧❧ ❧♦❝❛t✐♦♥s ❛❝r♦ss ✐t✳

✻✳✸✳ ❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s

✹✶✹

❖❢ ❝♦✉rs❡ t❤❡s❡ ❛r❡ ❥✉st ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿

❚❤✐s ✐s ❥✉st ❛♥♦t❤❡r ✇❛② t♦ ✈✐s✉❛❧✐③❡ t❤❡♠✳ ❊①❛♠♣❧❡ ✻✳✸✳✷✿ ❞✐♠❡♥s✐♦♥

2

◆♦t ❡✈❡r② ✈❡❝t♦r ✜❡❧❞ ♦❢ ❞✐♠❡♥s✐♦♥ n > 1 ✐s ❣r❛❞✐❡♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ s♦♠❡ ♦❢ t❤❡♠ ❝❛♥♥♦t ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s ✢♦✇s ♦♥ ❛ s✉r❢❛❝❡ ✉♥❞❡r ♥♦t❤✐♥❣ ❜✉t ❣r❛✈✐t②✳ ❆ ✈❡❝t♦r ✜❡❧❞ ♦❢ ❞✐♠❡♥s✐♦♥ n = 2 ✐s t❤❡♥ s❡❡♥ ❛s ❛ ✢♦✇ ♦♥ t❤❡ ♣❧❛♥❡✿ ❧✐q✉✐❞ ✐♥ ❛ ♣♦♥❞ ♦r t❤❡ ❛✐r ♦✈❡r ❛ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳

❚❤❡ ♠❡t❛♣❤♦r ❛♣♣❧✐❡s ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❛✐r ♦r ✇❛t❡r ❤❛s t❤❡ ❡①❛❝t s❛♠❡ ✈❡❧♦❝✐t② ❛t ❡✈❡r② ❧♦❝❛t✐♦♥s r❡❣❛r❞❧❡ss ♦❢ t❤❡ ❡❧❡✈❛t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✸✳✸✿ ❞✐♠❡♥s✐♦♥

3

❚❤✐s t✐♠❡✱ ❛ ✈❡❝t♦r ✜❡❧❞ ✐s t❤♦✉❣❤t ♦❢ ❛s ❛ ✢♦✇ ✇✐t❤♦✉t ❛♥② r❡str✐❝t✐♦♥s ♦♥ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ♣❛rt✐❝❧❡s✳

✻✳✸✳

❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s

❆ ♠♦❞❡❧ ♦❢ st♦❝❦ ♣r✐❝❡s ❛s ❛ ✢♦✇ ✇✐❧❧ ❧❡❛❞ t♦ ♦❢

✈❡❝t♦r ♥♦t❛t✐♦♥✳

✹✶✺

10, 000✲❞✐♠❡♥s✐♦♥❛❧

✈❡❝t♦r ✜❡❧❞✳

❚❤✐s ♥❡❝❡ss✐t❛t❡s ♦✉r ✉s❡

❲❡ ❛❧s♦ st❛rt t❤✐♥❦✐♥❣ ♦❢ t❤❡ ✐♥♣✉t✱ ❥✉st ❛s t❤❡ ♦✉t♣✉t✱ t♦ ❜❡ ✈❡❝t♦rs ✭♦❢ t❤❡ s❛♠❡

❞✐♠❡♥s✐♦♥✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ t✇♦ ✏r❛❞✐❛❧✑ ✈❡❝t♦r ✜❡❧❞s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❤❛✈❡ t❤❡ s❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥✿

V (X) = X . ❆♥ ❡✈❡♥ s✐♠♣❧❡r ✈❡❝t♦r ✜❡❧❞ ✐s ❛ ❝♦♥st❛♥t✿

V (X) = V0 .

❊❛❝❤ ✈❡❝t♦r ✜❡❧❞ ✐s ❥✉st ❛ ✈❡❝t♦r ✕ ❛t ❛ ✜①❡❞ ❧♦❝❛t✐♦♥✳

❚❤❡♥ ✐t ✐s ❥✉st ❛ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t ✭❜✉t t✐♠❡✲

✐♥❞❡♣❡♥❞❡♥t✦✮ ✈❡❝t♦r ❜✉t st✐❧❧ ❛ ✈❡❝t♦r✳ ❚❤❛t ✐s ✇❤② ❛❧❧ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ❢♦r ✈❡❝t♦rs ❛r❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ✈❡❝t♦r ✜❡❧❞s✳ ❋✐rst✱

❛❞❞✐t✐♦♥✳

✈❡❝t♦r ✜❡❧❞

V

■♠❛❣✐♥❡ t❤❛t t❤❛t ✇❡ ❤❛✈❡ ❛ r✐✈❡r ✕ ✇✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ✇❛t❡r ♣❛rt✐❝❧❡s r❡♣r❡s❡♥t❡❞ ❜②

✕ ❛♥❞ t❤❡♥ ✇✐♥❞ st❛rts ✕ ✇✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ❛✐r ♣❛rt✐❝❧❡s r❡♣r❡s❡♥t❡❞ ❜② ✈❡❝t♦r ✜❡❧❞

U✳

❚❤✐s ✐s t❤❡✐r s✉♠✿

❖♥❡ ❝❛♥ ❛r❣✉❡ t❤❛t t❤❡ r❡s✉❧t✐♥❣ ❞②♥❛♠✐❝s ♦❢ ✇❛t❡r ♣❛rt✐❝❧❡s ✇✐❧❧ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ✈❡❝t♦r ✜❡❧❞ ❙❡❝♦♥❞✱

V

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

V + U✳

■❢ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ✇❛t❡r ♣❛rt✐❝❧❡s ✐♥ ❛ ♣✐♣❡ ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ✈❡❝t♦r ✜❡❧❞

❛♥❞ ✇❡ t❤❡♥ ❞♦✉❜❧❡ t❤❡ ♣r❡ss✉r❡ ✭✐✳❡✳✱ ♣✉♠♣ t✇✐❝❡ ❛s ♠✉❝❤ ✇❛t❡r✮✱ ✇❡ ❡①♣❡❝t t❤❡ ♥❡✇ ✈❡❧♦❝✐t✐❡s t♦ ❜❡

r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ✈❡❝t♦r ✜❡❧❞

2V ✳

❘❡✈❡rs✐♥❣ t❤❡ ✢♦✇ ✇✐❧❧ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ✈❡❝t♦r ✜❡❧❞

−V ✳

✻✳✸✳

❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s

✹✶✻

❋✉rt❤❡r♠♦r❡✱ t❤❡ s❝❛❧❛r ♠✐❣❤t ❛❧s♦ ❜❡ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t✱ ✐✳❡✳✱ ✇❡ ❛r❡ ♠✉❧t✐♣❧②✐♥❣ ♦✉r ✈❡❝t♦r ✜❡❧❞ ✐♥ ❛ ✭s❝❛❧❛r✮ ❢✉♥❝t✐♦♥ ♦❢

n

Rn

❜②

✈❛r✐❛❜❧❡s✳

❚❤❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ s♣❡❝✐✜❝ ✈❡❝t♦r ✜❡❧❞s ❛r❡ ❝❛rr✐❡❞ ♦✉t

•

♦♥❡ ❧♦❝❛t✐♦♥ ❛t ❛ t✐♠❡ ❛♥❞

•

♦♥❡ ❝♦♠♣♦♥❡♥t ❛t ❛ t✐♠❡✳

◆♦✇

❣❡♦♠❡tr②✳

❲❤❛t ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r❄ ❆s ❛ ❢✉♥❝t✐♦♥✱ ✐t t❛❦❡s ❛ ✈❡❝t♦r ❛s ❛♥ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s ❛ ♥✉♠❜❡r ❛s t❤❡ ♦✉t♣✉t✳ ■t✬s ❥✉st ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ♦❢

n

✈❛r✐❛❜❧❡s✳ ❲❡ ❝❛♥ ❛♣♣❧② ✐t t♦ ✈❡❝t♦r ✜❡❧❞s✱ ♣r♦❞✉❝✐♥❣ t❤✐s

❝♦♠♣♦s✐t✐♦♥✿

f (X) = ||V (X)|| ❚❤❡ r❡s✉❧t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢

n

✈❛r✐❛❜❧❡s t❤❛t ❣✐✈❡s ✉s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✈❡❝t♦r

V (X)

❛t ❧♦❝❛t✐♦♥

X✳

❚❤❡

❝♦♥str✉❝t✐♦♥ ✐s ❡①❡♠♣❧✐✜❡❞ ❜② t❤❡ ✏s❝❛❧❛r✑ ✈❡rs✐♦♥ ♦❢ t❤❡ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ✉s❡ t❤✐s ❢✉♥❝t✐♦♥ t♦ ♠♦❞✐❢② ✈❡❝t♦r ✜❡❧❞s ✐♥ ❛ s♣❡❝✐❛❧ ✇❛②✿

W (X) =

V (X) ||V (X)||

❚❤❡ r❡s✉❧t ✐s ❛ ♥❡✇ ✈❡❝t♦r ✜❡❧❞s ✇✐t❤ t❤❡ ❡①❛❝t❧② s❛♠❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rs ❜✉t ✇✐t❤ ✐t✬s ③❡r♦✮✳

❚❤✐s ❝♦♥str✉❝t✐♦♥ ✐s ❝❛❧❧❡❞

♥♦r♠❛❧✐③❛t✐♦♥✳

✉♥✐t ❧❡♥❣t❤

✭✉♥❧❡ss

✻✳✸✳

❚❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦r ✜❡❧❞s

✹✶✼

❲❛r♥✐♥❣✦ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♥❡✇ ✈❡❝t♦r ✜❡❧❞ ♠✐❣❤t ❝❤❛♥❣❡ ❛s ✐t ✐s ✉♥❞❡✜♥❡❞ ❛t t❤♦s❡ X ✇❤❡r❡ V (X) = 0✳

❊①❛♠♣❧❡ ✻✳✸✳✹✿ ♥♦r♠❛❧✐③❡❞ ♦✉t✢♦✇ ❚❤❡ ✏❛❝❝❡❧❡r❛t❡❞ ♦✉t✢♦✇✑ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✜rst s❡❝t✐♦♥ ✐s ♥♦ ❧♦♥❣❡r ❛❝❝❡❧❡r❛t❡❞ ❛❢t❡r ♥♦r♠❛❧✐③❛t✐♦♥✿

W (X) =

X . ||X||

❚❤❡ s♣❡❡❞ ✐s ❝♦♥st❛♥t✦

❚❤❡ ♣r✐❝❡ ✇❡ ♣❛② ❢♦r ♠❛❦✐♥❣ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✇❡❧❧✲❜❡❤❛✈❡❞ ✐s t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛ ❤♦❧❡ ✐♥ t❤❡ ❞♦♠❛✐♥✱

X 6= 0✳

❊①❡r❝✐s❡ ✻✳✸✳✺ ❙❤♦✇ t❤❛t t❤❡ ❤♦❧❡ ❝❛♥✬t ❜❡ r❡♣❛✐r❡❞✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ t❤❡r❡ ✐s ♥♦ s✉❝❤ ✈❡❝t♦r

U || → 0

❛s

X→0

✭✐✳❡✳✱ t❤✐s ✐s ❛ ♥♦♥✲r❡♠♦✈❛❜❧❡ ❞✐s❝♦♥t✐♥✉✐t②✮✳

U

t❤❛t

||W (X) −

❊①❡r❝✐s❡ ✻✳✸✳✻ ❲❤❛t ✐❢ ✇❡ ❞♦ t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t✇♦ ✈❡❝t♦r ✜❡❧❞s❄

■❢ ❝❛♥ ✇❡ r♦t❛t❡ ❛ ✈❡❝t♦r✱ ✇❡ ❝❛♥ r♦t❛t❡ ✈❡❝t♦r ✜❡❧❞s ✜❡❧❞

V =< u, v >

V❄

■♥ ❞✐♠❡♥s✐♦♥

2✱

t❤❡ ♥♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞ ♦❢ ❛ ✈❡❝t♦r

♦♥ t❤❡ ♣❧❛♥❡ ✐s ❣✐✈❡♥ ❜②

V ⊥ =< u, v >⊥ =< −v, u > .

❲❡ ❤❛✈❡ t❤❡♥ ❛ s♣❡❝✐❛❧ ♦♣❡r❛t✐♦♥ ♦♥ ✈❡❝t♦rs ✜❡❧❞s✳

❋♦r ❡①❛♠♣❧❡✱ r♦t❛t✐♥❣ ❛ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞ ✐s ❛❧s♦

❝♦♥st❛♥t✳ ❍♦✇❡✈❡r✱ t❤❡ ♥♦r♠❛❧ ♦❢ t❤❡ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ✐s t❤❡ r❛❞✐❛❧ ✈❡❝t♦r ✜❡❧❞✳

✻✳✹✳

❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

✹✶✽

✻✳✹✳ ❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦ ❊①❛♠♣❧❡ ✻✳✹✳✶✿ ♣✐♣❡s ❲❡ ❧♦♦❦ ❛t t❤✐s ❛s ❛ s②st❡♠ ♦❢

♣✐♣❡s

✇✐t❤ t❤❡ ♥✉♠❜❡rs ✐♥❞✐❝❛t✐♥❣ t❤❡ r❛t❡ ♦❢ t❤❡ ✢♦✇ ✐♥ ❡❛❝❤ ♣✐♣❡

✭❛❧♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❛①❡s✮✿

❲❤❛t ✐s t❤❡ t♦t❛❧ ✢♦✇ ❛❧♦♥❣ t❤✐s ✏st❛✐r❝❛s❡✑❄ ❲❡ s✐♠♣❧② ❛❞❞ t❤❡ ✈❛❧✉❡s ❧♦❝❛t❡❞ ♦♥ t❤❡s❡ ❡❞❣❡s✿

W = 1 + 0 + 0 + 2 + (−1) + 1 + (−2) . ❇✉t t❤❡s❡ ❡❞❣❡s ❥✉st ❤❛♣♣❡♥ t♦ ❜❡ ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞✳ ❲❤❛t ✐❢ ✇❡✱ ✐♥st❡❛❞✱ ❣♦ ❛r♦✉♥❞ t❤❡ ✜rst sq✉❛r❡❄ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

●♦✐♥❣

❛❣❛✐♥st

W = 1 + 0 − 2 − 0 = −1 . ♦♥❡ ♦❢ t❤❡ ♦r✐❡♥t❡❞ ❡❞❣❡s✱ ♠❛❦❡s ✉s ❝♦✉♥t t❤❡ ✢♦✇ ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥✳

Ei ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐♥ Rn ✐s ❛ ✈❡❝t♦r t❤❛t ❣♦❡s ✇✐t❤ C = {Ei : i = 0, 1, ..., n} ✐s s❡❡♥ ❛s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡✳

❘❡❝❛❧❧ t❤❛t ❛♥ ♦r✐❡♥t❡❞ ❡❞❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s✉❝❤ ❡❞❣❡s

♦r ❛❣❛✐♥st t❤❡ ❡❞❣❡ ❛♥❞ ❛♥②

❉❡✜♥✐t✐♦♥ ✻✳✹✳✷✿ s✉♠

n ✐s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ ✐♥ R t❤❛t ❝♦♥s✐sts ♦❢ ♦r✐❡♥t❡❞ ❡❞❣❡s Ei , i = n 1, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐♥ R ✳ ■❢ ❛ ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛t n t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐♥ R ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛t t❤❡ ❡❞❣❡s {Qi } ♦❢ t❤❡ ❝✉r✈❡✱ ❙✉♣♣♦s❡

t❤❡♥

C

t❤❡ s✉♠ ♦❢ G ❛❧♦♥❣ ❝✉r✈❡ C

X C

✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

G=

n X

G(Qi )

i=1

❲❤❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡♥✬t s♣❡❝✐✜❡❞✱ t❤✐s s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ r❡❛❧✲✈❛❧✉❡❞

1✲❢♦r♠ G✳

❯♥❧✐❦❡ t❤❡ ❛r❝✲❧❡♥❣t❤✱ t❤❡ s✉♠ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ tr✐♣✳

❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐s ❤♦✇❡✈❡r ✈❡r② s✐♠♣❧❡✿ t❤❡

s✐❣♥

✐s r❡✈❡rs❡❞ ✇❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ✐s r❡✈❡rs❡❞✳

✻✳✹✳

❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

✹✶✾

❚❤❡♦r❡♠ ✻✳✹✳✸✿ ◆❡❣❛t✐✈✐t② ♦❢ ❙✉♠

X −C

G=−

X

G

C

▼♦r❡ ❢❛♠✐❧✐❛r ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s✿

❚❤❡♦r❡♠ ✻✳✹✳✹✿ ▲✐♥❡❛r✐t② ♦❢ ❙✉♠

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s F ❛♥❞ G ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐♥ Rn ❛♥❞ ❛♥② t✇♦ ♥✉♠❜❡rs λ ❛♥❞ µ✱ ✇❡ ❤❛✈❡✿ X

(λF + µG) = λ

C

X

F +µ

C

X

G

C

❚❤❡ ❙✉♠ ❘✉❧❡ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✻✳✹✳✺✿ ❆❞❞✐t✐✈✐t② ♦❢ ❙✉♠

❋♦r ❛♥② t✇♦ ♦r✐❡♥t❡❞ ❝✉r✈❡s ♦❢ ❡❞❣❡s C ❛♥❞ K ✇✐t❤ ♥♦ ❡❞❣❡s ✐♥ ❝♦♠♠♦♥ ❛♥❞ t❤❛t t♦❣❡t❤❡r ❢♦r♠ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ ♦❢ ❡❞❣❡s C ∪ K ✱ ✇❡ ❤❛✈❡✿ X

C∪K

▲❡t✬s ❡①❛♠✐♥❡ ❛♥♦t❤❡r ♣r♦❜❧❡♠✿ t❤❡

✇♦r❦ ♦❢ ❛ ❢♦r❝❡✳

F =

X C

F+

X

F

K

❙✉♣♣♦s❡ ❛ ❜❛❧❧ ✐s t❤r♦✇♥✳

✻✳✹✳

❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

✹✷✵

❚❤✐s ❢♦r❝❡ ✐s ❞✐r❡❝t❡❞ ❞♦✇♥✱ ❥✉st ❛s t❤❡ ♠♦✈❡♠❡♥t ♦❢ t❤❡ ❜❛❧❧✳ ❚❤❡ ✇♦r❦ ❞♦♥❡ ♦♥ t❤❡ ❜❛❧❧ ❜② t❤✐s ❢♦r❝❡ ❛s ✐t ❢❛❧❧s ✐s ❡q✉❛❧ t♦ t❤❡ ✭s✐❣♥❡❞✮ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡✱ ✐✳❡✳✱ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❜❛❧❧✱ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ t♦ t❤❡ ❣r♦✉♥❞✱ ✐✳❡✳✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❆❧❧ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ ✐s ✐❣♥♦r❡❞ ❛s ✉♥r❡❧❛t❡❞ t♦ t❤❡ ❣r❛✈✐t②✳ ▼♦✈✐♥❣ ❛♥ ♦❜ ❥❡❝t ✉♣ ❢r♦♠ t❤❡ ❣r♦✉♥❞ t❤❡ ✇♦r❦ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡ ✐s ♥❡❣❛t✐✈❡✳ ❖❢ ❝♦✉rs❡✱ ✇❡ ❛r❡ s♣❡❛❦✐♥❣ ♦❢ ■♥ t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❧✐♥❡✳ ❚❤❡♥ t❤❡

✈❡❝t♦rs✳

❝❛s❡✱ s✉♣♣♦s❡ t❤❛t t❤❡ ❢♦r❝❡

✇♦r❦ W

F

✐s ❝♦♥st❛♥t ❛♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t

D

✐s ❛❧♦♥❣ ❛ str❛✐❣❤t

✐s ❡q✉❛❧ t♦ t❤❡✐r ♣r♦❞✉❝t✿

W = F ·D. ❚❤❡ ❢♦r❝❡ ♠❛② ✈❛r② ✇✐t❤ ❧♦❝❛t✐♦♥✱ ❤♦✇❡✈❡r✿ s♣r✐♥❣ t❡♥s✐♦♥✱ ❣r❛✈✐t❛t✐♦♥✱ ❛✐r ♣r❡ss✉r❡✳ ❘❡❝❛❧❧ ♦✉r ❛♣♣r♦❛❝❤✿

❉❡✜♥✐t✐♦♥ ✻✳✹✳✻✿ ✇♦r❦ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐♥t❡❣r❛❧

Z

F

♦♥ s❡❣♠❡♥t

b

F dx

✐s ❝❛❧❧❡❞ t❤❡

[a, b]

✇♦r❦

✐s ❝❛❧❧❡❞ ❛

❢♦r❝❡ ❢✉♥❝t✐♦♥

♦❢ t❤❡ ❢♦r❝❡ ♦✈❡r ✐♥t❡r✈❛❧

t❤❡♥ ✐ts ❘✐❡♠❛♥♥

[a, b]✳

a

❊①❛♠♣❧❡ ✻✳✹✳✼✿ ❍♦♦❦❡✬s ▲❛✇ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ♦❜❥❡❝t ❛tt❛❝❤❡❞ t♦ ❛

s♣r✐♥❣✱ t❤❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ ♦❢ t❤❡

♦❜❥❡❝t t♦ ✐ts ❡q✉✐❧✐❜r✐✉♠✿

F = −kx .

▲❡t✬s ♥♦✇ ♣r♦❝❡❡❞ t♦ t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡ ❜✉t st❛rt ✇✐t❤ ❛ ❝♦♥st❛♥t ❢♦r❝❡ ❛♥❞ ❧✐♥❡❛r ♠♦t✐♦♥✳

❚❤✐s t✐♠❡✱ t❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♠❛② ❜❡ ♠✐s❛❧✐❣♥❡❞✿

✻✳✹✳

❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

✹✷✶

■♥ ❛❞❞✐t✐♦♥ t♦ ♠♦t✐♦♥ ✏✇✐t❤ t❤❡ ❢♦r❝❡✑ ❛♥❞ ✏❛❣❛✐♥st t❤❡ ❢♦r❝❡✑✱ t❤❡ t❤✐r❞ ♣♦ss✐❜✐❧✐t② ❡♠❡r❣❡s✿ ❲❤❛t ✐❢ ✇❡ ♠♦✈❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢♦r❝❡❄

❚❤❡ ♠♦t✐♦♥ ✐s ✉♥❛✛❡❝t❡❞ ❜② t❤❡ ❢♦r❝❡✦

❚❤✐s ✐s t❤❡ ❝❛s❡ ♦❢ ❤♦r✐③♦♥t❛❧

♠♦t✐♦♥ ✉♥❞❡r ❣r❛✈✐t② ❢♦r❝❡✱ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ❝❧♦s❡ t♦ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳ ❲❡ ❝♦♥❝❧✉❞❡✿

◮

❲❤❡♥ t❤❡ ♠♦t✐♦♥ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢♦r❝❡✱ t❤❡ ✇♦r❦ ✐s ③❡r♦✳

❋✉rt❤❡r♠♦r❡✱ ✇❤❛t ✐❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦✉r ♣❛t❤ ✈❛r✐❡s ❜✉t ♦♥❧② ✇✐t❤✐♥ t❤❡ st❛♥❞❛r❞ sq✉❛r❡

❣r✐❞

♦♥ t❤❡ ♣❧❛♥❡❄

❲❡ r❡❛❧✐③❡ t❤❛t t❤❡r❡ ✐s ❛ ❢♦r❝❡ ✈❡❝t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ❡❞❣❡ ♦❢ ♦✉r tr✐♣ ❛♥❞ ♣♦ss✐❜❧② ✇✐t❤ ❡✈❡r② ❡❞❣❡ ♦❢ t❤❡ ❣r✐❞✳ ❍♦✇❡✈❡r✱ ♦♥❧② ♦♥❡ ♦❢ t❤❡s❡ ✈❡❝t♦r ❝♦♠♣♦♥❡♥ts ♠❛tt❡rs✿ t❤❡ ❤♦r✐③♦♥t❛❧ ✇❤❡♥ t❤❡ ❡❞❣❡ ✐s ❤♦r✐③♦♥t❛❧ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ✇❤❡♥ t❤❡ ❡❞❣❡ ✐s ✈❡rt✐❝❛❧✳ ■t ✐s t❤❡♥ s✉✣❝✐❡♥t t♦ ❛ss✐❣♥ t❤✐s

s✐♥❣❧❡

♥✉♠❜❡r t♦ ❡❛❝❤ ❡❞❣❡ t♦

✐♥❞✐❝❛t❡ t❤❡ ❢♦r❝❡ ❛♣♣❧✐❡❞ t♦ t❤✐s ♣❛rt ♦❢ t❤❡ tr✐♣✳ ❊①❛♠♣❧❡ ✻✳✹✳✽✿ s②st❡♠ ♦❢ ♣✐♣❡s

❆s ❛ ❢❛♠✐❧✐❛r ✐♥t❡r♣r❡t❛t✐♦♥✱ ✇❡ ❝❛♥ ❧♦♦❦ ❛t t❤✐s ❛s ❛ s②st❡♠ ♦❢

♣✐♣❡s

✇✐t❤ t❤❡ ♥✉♠❜❡rs ✐♥❞✐❝❛t✐♥❣

t❤❡ s♣❡❡❞ ♦❢ t❤❡ ✢♦✇ ✐♥ ❡❛❝❤ ♣✐♣❡ ✭❛❧♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❛①❡s✮✳ ■❢✱ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❛r❡ ♠♦✈✐♥❣ t❤r♦✉❣❤ ❛ ❣r✐❞ ✇✐t❤

∆x × ∆y

❝❡❧❧s✱ t❤❡ ✇♦r❦ ❛❧♦♥❣ t❤❡ ✏st❛✐r❝❛s❡✑ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

W = 1 · ∆x + 0 · ∆y + 0 · ∆x + 2 · ∆y + (−1) · ∆x + 1 · ∆y + (−2) · ∆x . ❲❤❡♥

∆x = ∆y = 1✱

t❤✐s ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ ✈❛❧✉❡s ♣r♦✈✐❞❡❞✿

W = 1 + 0 + 0 + 2 + (−1) + 1 + (−2) = 1 . ❲❤❛t ✐❢ ✇❡✱ ✐♥st❡❛❞✱ ❣♦ ❛r♦✉♥❞ t❤❡ ✜rst sq✉❛r❡❄ ❚❤❡♥

W = 1 + 0 − 2 − 0 = −1 . ●♦✐♥❣

❛❣❛✐♥st

♦♥❡ ♦❢ t❤❡ ♦r✐❡♥t❡❞ ❡❞❣❡s✱ ♠❛❦❡s ✉s ❝♦✉♥t t❤❡ ✇♦r❦ ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥✳ ■♥ ♦t❤❡r

✇♦r❞s✱ t❤❡ ❡❞❣❡ ❛♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳

❲❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r❝❡ ✐s♥✬t ❧✐♠✐t❡❞ t♦ t❤❡ ❣r✐❞ ❛♥②♠♦r❡✱ ✐t ❝❛♥ t❛❦❡✱ ♦❢ ❝♦✉rs❡✱ ♦♥❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥s✳ ■♥ ❢❛❝t✱ t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❝✐r❝❧❡ ♦❢ ♣♦ss✐❜❧❡ ❞✐r❡❝t✐♦♥s✿

❚❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❢♦r❝❡✱ t♦♦✱ ❝❛♥ t❛❦❡ ❛❧❧ ❛✈❛✐❧❛❜❧❡ ❞✐r❡❝t✐♦♥s✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ ❛♥❞ ❞✐s❝❛r❞ t❤❡ ✐rr❡❧❡✈❛♥t ✭♣❡r♣❡♥❞✐❝✉❧❛r✮ ♣❛rt ♦❢ t❤❡ ❢♦r❝❡

F✿

✻✳✹✳

❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

❲❡ ❞❡❝♦♠♣♦s❡ ✐t ✐♥t♦

✹✷✷

♣❛r❛❧❧❡❧ ❛♥❞ ♥♦r♠❛❧ ❝♦♠♣♦♥❡♥ts

r❡❧❛t✐✈❡ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✿

F = F⊥ + F|| ❚❤❡ r❡❧❡✈❛♥t ✭✏❝♦❧❧✐♥❡❛r✑✮ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢♦r❝❡

F

✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r✿

F|| = ||F || cos α , ✇❤❡r❡

α

✐s t❤❡ ❛♥❣❧❡ ♦❢

F

✇✐t❤

D✳

❖❢ ❝♦✉rs❡✱ ✇❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t t❤❡ ❚❤❡

❞♦t ♣r♦❞✉❝t

❤❡r❡✳

✇♦r❦ ♦❢ t❤❡ ❢♦r❝❡ ✈❡❝t♦r F ❛❧♦♥❣ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r D

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡✐r ❞♦t ♣r♦❞✉❝t✿

W =F ·D ❚❤❡ ✇♦r❦ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡ ❛♥❞ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ■t ✐s ❛❧s♦ ♣r♦♣♦rt✐♦♥❛❧ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠❡r ♦♥ t❤❡ ❧❛tt❡r ✭t❤❡ r❡❧❡✈❛♥t ♣❛rt ♦❢ t❤❡ ❢♦r❝❡✮ ❛♥❞ t❤❡ ❧❛tt❡r ♦♥ t❤❡ ❢♦r♠❡r ✭t❤❡ r❡❧❡✈❛♥t ♣❛rt ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✮✳ ■t ♠❛❦❡s s❡♥s❡✳ ■♥ ♦✉r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛s ❛ s②st❡♠ ♦❢

♣✐♣❡s

❤❛s t✇♦ ♥✉♠❜❡rs✱ t❤✐s ✐s ❛ ✈❡❝t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤

❡❛❝❤ ♣✐♣❡ ✐♥❞✐❝❛t✐♥❣ t❤❡ s♣❡❡❞ ♦❢ t❤❡ ✢♦✇ ✐♥ t❤❡ ♣✐♣❡ ✭❛❧♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦♥❡ ♦❢ t❤❡ ♣✐♣❡✮ ❛s ✇❡❧❧ ❛s t❤❡ ❧❡❛❦❛❣❡ ✭♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤✐s ❞✐r❡❝t✐♦♥✮✳ ❚❤❡♥✱ t❤❡ r❡❧❡✈❛♥t ♣❛rt ♦❢ t❤❡ ❢♦r❝❡ ✐s ❢♦✉♥❞ ❛s t❤❡ ✭s❝❛❧❛r✮

♣r♦❥❡❝t✐♦♥

♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❢♦r❝❡ ♦♥ t❤❡ ✈❡❝t♦r ♦❢ ❞✐s♣❧❛❝❡♠❡♥t✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❜❡t✇❡❡♥ r❡❛❧✲✈❛❧✉❡❞

❛♥❞ ✈❡❝t♦r✲✈❛❧✉❡❞

1✲❢♦r♠s✳

❚❤✉s✱ t❤❡ ✇♦r❦ ✐s r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✈❡❝t♦r ♦❢ ❞✐s♣❧❛❝❡♠❡♥t✳ ◆♦✇ t❤❡ ✇♦r❦ ♦✈❡r ❛ ✇❤♦❧❡ tr✐♣✿

❉❡✜♥✐t✐♦♥ ✻✳✹✳✾✿ ❘✐❡♠❛♥♥ s✉♠ n ✐s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ ✐♥ R t❤❛t ❝♦♥s✐sts ♦❢ ♦r✐❡♥t❡❞ ❡❞❣❡s Ei , i = 1, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐♥ Rn ✳ ■❢ ❛ ✈❡❝t♦r ✜❡❧❞ F ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s n ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐♥ R ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛t t❤❡ ❡❞❣❡s {Qi } ♦❢ t❤❡ ❙✉♣♣♦s❡

C

❝✉r✈❡✱ t❤❡♥

t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ F ❛❧♦♥❣ ❝✉r✈❡ C

✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ t♦ ❜❡

t❤❡ ❢♦❧❧♦✇✐♥❣✿

X C

F · ∆X =

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞✱

F · E✱

F✱

n X i=1

F (Qi ) · Ei

✐s t❤❡ s✉♠ ♦❢ ❛ ❝❡rt❛✐♥ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥✱

❛❧♦♥❣ ❛ ❝✉r✈❡ ❛s ❞❡✜♥❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳

✻✳✹✳ ❙✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❛ ❝✉r✈❡✿ ✢♦✇ ❛♥❞ ✇♦r❦

✹✷✸

❉❡✜♥✐t✐♦♥ ✻✳✹✳✶✵✿ ✇♦r❦ ❲❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ F ✐s ❝❛❧❧❡❞ ❛ ❢♦r❝❡ ✜❡❧❞✱ t❤❡♥ t❤❡ s✉♠ ♦❢ F ❛❧♦♥❣ C ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✇♦r❦ ♦❢ ❢♦r❝❡ F ❛❧♦♥❣ ❝✉r✈❡ C ✳

❲❛r♥✐♥❣✦ ❖♥❧② t❤❡ ♣❛rt ♦❢ t❤❡ ❢♦r❝❡ ✜❡❧❞ ♣❛ss❡❞ t❤r♦✉❣❤ ❜② t❤❡ ♦❜❥❡❝t ❛✛❡❝ts t❤❡ ✇♦r❦✳

❚❤❡ ♣r♦♣❡rt✐❡s ❢♦❧❧♦✇ t❤❡ ♦♥❡s ♣r❡s❡♥t❡❞ ❛❜♦✈❡✿

❚❤❡♦r❡♠ ✻✳✹✳✶✶✿ ◆❡❣❛t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠ X −C

F · ∆X = −

X C

F · ∆X

❚❤❡♦r❡♠ ✻✳✹✳✶✷✿ ▲✐♥❡❛r✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠ ❋♦r ❛♥② t✇♦ ✈❡❝t♦r ✜❡❧❞s F ❛♥❞ G ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐♥ Rn ❛♥❞ ❛♥② t✇♦ ♥✉♠❜❡rs λ ❛♥❞ µ✱ ✇❡ ❤❛✈❡✿ X C

(λF + µG) · ∆X = λ

X C

F · ∆X + µ

X C

G · ∆X

❚❤❡♦r❡♠ ✻✳✹✳✶✸✿ ❆❞❞✐t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠ ❋♦r ❛♥② t✇♦ ♦r✐❡♥t❡❞ ❝✉r✈❡s C ❛♥❞ K ✇✐t❤ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥ ❛♥❞ t❤❛t t♦❣❡t❤❡r ❢♦r♠ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ C ∪ K ✱ ✇❡ ❤❛✈❡✿ X

C∪K

F · ∆X =

X C

F · ∆X +

X K

F · ∆X

✻✳✺✳

✹✷✹

▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

✻✳✺✳ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

❆ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣ ✐s t❤❛t ♦❢ ❛ ♠♦t✐♦♥ t❤r♦✉❣❤ s♣❛❝❡✱ Rn ✱ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s❧② ❲❡ ✜rst ❛ss✉♠❡ t❤❛t ✇❡ ♠♦✈❡ ❢r♦♠ ♣♦✐♥t t♦ ♣♦✐♥t ❛❧♦♥❣ ❛ str❛✐❣❤t

✳

❝❤❛♥❣✐♥❣ ❢♦r❝❡

✳

❧✐♥❡

❊①❛♠♣❧❡ ✻✳✺✳✶✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②

❆✇❛② ❢r♦♠ t❤❡ ❣r♦✉♥❞✱ t❤❡ ❣r❛✈✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♦❜❥❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ♣❧❛♥❡t✿ F (X) = −

kX . ||X||3

❚❤❡ ♣r❡ss✉r❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♠❡❞✐✉♠✬s r❡s✐st❛♥❝❡ t♦ ♠♦t✐♦♥ ♠❛② ❝❤❛♥❣❡ ❛r❜✐tr❛r✐❧②✳ ▼✉❧t✐♣❧❡ s♣r✐♥❣s ❝r❡❛t❡ ❛ 2✲❞✐♠❡♥s✐♦♥❛❧ ✈❛r✐❛❜✐❧✐t② ♦❢ ❢♦r❝❡s✿

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✇♦r❦ ❛♣♣❧✐❡s t♦ str❛✐❣❤t tr❛✈❡❧✳✳✳ ♦r t♦ tr❛✈❡❧ ❛❧♦♥❣ ♠✉❧t✐♣❧❡ str❛✐❣❤t ❡❞❣❡s✿

■❢ t❤❡s❡ s❡❣♠❡♥ts ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦rs D1 , ..., Dn ❛♥❞ t❤❡ ❢♦r❝❡ ❢♦r ❡❛❝❤ ✐s ❣✐✈❡♥ ❜② t❤❡ ✈❡❝t♦rs F1 , ..., Fn ✱ t❤❡♥ t❤❡ ✇♦r❦ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s✐♠♣❧❡ s✉♠ ♦❢ t❤❡ ✇♦r❦ ❛❧♦♥❣ ❡❛❝❤✿ W = F1 · D1 + ... + Fn · Dn . ❊①❛♠♣❧❡ ✻✳✺✳✷✿ ❝♦♥st❛♥t ❢♦r❝❡

■❢ t❤❡ ❢♦r❝❡ ✐s ❝♦♥st❛♥t Fi = F ✱ ✇❡ s✐♠♣❧✐❢②✱ W = F · D1 + ... + F · Dn = F · (D1 + ... + Dn ) ,

❛♥❞ ❞✐s❝♦✈❡r t❤❛t t❤❡ t♦t❛❧ ✇♦r❦ ✐s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ t♦t❛❧

✳

❞✐s♣❧❛❝❡♠❡♥t

✻✳✺✳

▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

✹✷✺

❚❤✐s ♠❛❦❡s s❡♥s❡✳ ❚❤✐s ✐s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ ✏♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡✑✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ r♦✉♥❞ tr✐♣ ✇✐❧❧ r❡q✉✐r❡ ③❡r♦ ✇♦r❦✳✳✳ ✉♥❧❡ss ♦♥❡ ❤❛s t♦ ✇❛❧❦ t♦ s❝❤♦♦❧ ✏ 5 ♠✐❧❡s ✕ ✉♣❤✐❧❧ ❜♦t❤ ✇❛②s✦✑ ❚❤❡ ✐ss✉❡ ✐s♥✬t ❛s s✐♠♣❧❡ ❛s ✐t s❡❡♠s✿ ❡✈❡♥ t❤♦✉❣❤ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ♠❛❦❡ r♦✉♥❞ tr✐♣ ✇❤✐❧❡ ✇❛❧❦✐♥❣ ✉♣❤✐❧❧✱ ✐t ✐s ♣♦ss✐❜❧❡ ❞✉r✐♥❣ t❤✐s tr✐♣ t♦ ✇❛❧❦ ❛❣❛✐♥st t❤❡ ✇✐♥❞ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✇✐♥❞ ❞♦❡s♥✬t ❝❤❛♥❣❡✳ ■t ❛❧❧ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳

❊①❛♠♣❧❡ ✻✳✺✳✸✿ ❛❞❞✐♥❣ ✇♦r❦ ■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ✇♦r❦ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛❧♦♥❣ ❛ ❝✉r✈❡ ♠❛❞❡ ♦❢ str❛✐❣❤t ❡❞❣❡s✱ ❛❧❧ ✇❡ ♥❡❡❞ ✐s t❤❡ ❢♦r♠✉❧❛✿

W = F1 · D1 + ... + Fn · Dn . ■♥ ♦r❞❡r ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ t♦ ♠❛❦❡ s❡♥s❡✱ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛t❤ ❛♥❞ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ❢♦r❝❡ ❤❛✈❡ t♦ ❜❡ ♣❛✐r❡❞ ✉♣✦ ❍❡r❡✬s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡✿

❲❡ ♣✐❝❦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦r❝❡ ❢r♦♠ t❤❡

✐♥✐t✐❛❧

♣♦✐♥t ♦❢ ❡❛❝❤ ❡❞❣❡✿

W =< −1, 0 > · < 0, 1 > + < 0, 2 > · < 1, 0 > + < 1, 2 > · < 1, 1 >= 3 .

❊①❛♠♣❧❡ ✻✳✺✳✹✿ ❞❛t❛ ✢♦✇ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤❡r❡ ✐s

♥♦ ✈❡❝t♦r ✜❡❧❞

❛♥❞ t❤❡ ❢♦r❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❡♥t✐r❡❧② ❜② ♦✉r ♠♦t✐♦♥✳

❋♦r

❡①❛♠♣❧❡✱ t❤❡ ❛✐r ♦r ✇❛t❡r r❡s✐st❛♥❝❡ ✐s ❞✐r❡❝t❡❞ ❛❣❛✐♥st ♦✉r ✈❡❧♦❝✐t② ✭❛♥❞ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s♣❡❡❞✮✳

❚❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥s r❡♠❛✐♥ t❤❡ s❛♠❡✳

❚❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ❢♦r ❞❡✜♥✐♥❣ ❛♥❞ ❝♦♠♣✉t✐♥❣ ✇♦r❦ ❛❧♦♥❣ ❛ ❝✉r✈❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ ✇❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ s❡✈❡r❛❧ t✐♠❡s✳

✻✳✺✳ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

✹✷✻

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts Pi , i = 0, 1, ..., n✱ ✐♥ Rn ✳ ❲❡ ✇✐❧❧ tr❡❛t t❤✐s s❡q✉❡♥❝❡ ❛s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ C ❜② r❡♣r❡s❡♥t✐♥❣ ✐t ❛s t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛s ❢♦❧❧♦✇s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❛♠♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✿ a = t0 ≤ c1 ≤ t1 ≤ ... ≤ cn ≤ tn = b . ❲❡ ❞❡✜♥❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❜②✿

X(ti ) = Pi , i = 0, 1, ..., n .

❍♦✇❡✈❡r✱ ✐t ❞♦❡s♥✬t ♠❛tt❡r ❤♦✇ ❢❛st ✇❡ ❣♦ ❛❧♦♥❣ t❤✐s ♣❛t❤✳ ■t ✐s t❤❡ ♣❛t❤ ✐ts❡❧❢ ✕ t❤❡ ❧♦❝❛t✐♦♥s ✇❡ ✈✐s✐t ✕ t❤❛t ♠❛tt❡rs✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ tr✐♣ ♠❛tt❡rs t♦♦✳ ❚❤✐s ✐s t❤❡♥ ❛❜♦✉t ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ❛ ♥♦♥✲❝♦♥st❛♥t ✈❡❝t♦rs ❛❧♦♥❣ t❤❡ ♣❛t❤ t②♣✐❝❛❧❧② ❝♦♠❡ ❢r♦♠ ❛ ✈❡❝t♦r ✜❡❧❞✱ F = F (X)✳ ■❢ ✐ts ✈❡❝t♦rs ❝❤❛♥❣❡ ✐♥❝r❡♠❡♥t❛❧❧②✱ ♦♥❡ ♠❛② ❜❡ ❛❜❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ✇♦r❦ ❜② ❛ s✐♠♣❧❡ s✉♠♠❛t✐♦♥✱ ❛s ❛❜♦✈❡✳ ❲❡ t❤❡♥ ✜♥❞ ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❧❛tt❡r✿ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ X = X(t) ❞❡✜♥❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❞✐✈✐❞❡ t❤❡ ♣❛t❤ ✐♥t♦ s♠❛❧❧ s❡❣♠❡♥ts ✇✐t❤ ❡♥❞✲♣♦✐♥ts Xi = X(ti ) ❛♥❞ t❤❡♥ s❛♠♣❧❡ t❤❡ ❢♦r❝❡ ❛t t❤❡ ♣♦✐♥ts Qi = X(ci )✳

❚❤❡♥ t❤❡ ✇♦r❦ ❛❧♦♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ s❡❣♠❡♥ts ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ✇♦r❦ ✇✐t❤ t❤❡ ❢♦r❝❡ ❜❡✐♥❣ ❝♦♥st❛♥t❧② ❡q✉❛❧ t♦ F (Qi )✿ ✇♦r❦ ❛❧♦♥❣ it❤ s❡❣♠❡♥t ≈ ❢♦r❝❡ · ❧❡♥❣t❤ = F (Qi ) · ∆Xi ,

✇❤❡r❡ ∆Xi ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❛❧♦♥❣ t❤❡ it❤ s❡❣♠❡♥t✳ ❚❤❡♥✱ ❚♦t❛❧ ✇♦r❦ ≈

n X i=1

F (Qi ) · (Xi+1 − Xi ) =

n X i=1

F (X(ci )) · (X(ti+1 ) − X(ti )) .

✻✳✺✳ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

✹✷✼

❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ t❤❛t ✇❡ ❤❛✈❡ ✉s❡❞ ❛♥❞ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ◆♦t❡ t❤❛t t❤✐s ✐s ❥✉st t❤❡ s✉♠ ♦❢ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳

❊①❛♠♣❧❡ ✻✳✺✳✺✿ ❡st✐♠❛t✐♥❣ ▲❡t✬s ❡st✐♠❛t❡ t❤❡ ✇♦r❦ ♦❢ t❤❡ ❢♦r❝❡ ✜❡❧❞ F (x, y) =< xy, x − y >

❛❧♦♥❣ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❞✐r❡❝t❡❞ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✳ ❋✐rst✱ ✇❡ ♣❛r❛♠❡tr✐③❡ t❤❡ ❝✉r✈❡✿ X(t) =< cos t, sin t >, 0 ≤ t ≤ π .

❲❡ ❝❤♦♦s❡ n = 4 ✐♥t❡r✈❛❧s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤ ✇✐t❤ t❤❡ ❧❡❢t✲❡♥❞s ❛s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿ x0 = 0

x1 = π/4

c1 = 0

c2 = π/4 c3 = π/2 √ √ X1 = ( 2/2, 2/2) X2 = (0, 1) √ √ Q2 = ( 2/2, 2/2) Q3 = (0, 1)

X0 = (1, 0) Q1 = (1, 0)

F (Q1 ) =< 0, 1 > F (Q2 ) =< 1/2, 0 >

x2 = π/2

x3 = 3π/4

x4 = π

c4 = 3π/4 √ √ X3 = (− 2/2, 2/2) X4 = (−1, 0) √ √ Q4 = (− 2/2, 2/2) √ F (Q3 ) =< 0, −1 > F (Q4 ) =< −1/2, − 2 >

❚❤❡♥✱ √

√ √ √ 2/2 − 1, 2/2 > + < 1/2, 0 > · < − 2/2, 1 − 2/2 > √ √ √ √ √ + < 0, −1 > · < − 2/2, 2/2 − 1 > + < −1/2, − 2 > · < −1 + 2/2, − 2/2 >

W ≈< 0, 1 > · < = ...

❚♦ ❜r✐♥❣ t❤❡ ❢✉❧❧ ♣♦✇❡r ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♠❛❝❤✐♥❡r②✱ ✇❡✱ ♦♥❝❡ ❛❣❛✐♥✱ ♣r♦❝❡❡❞ t♦ ❝♦♥✈❡rt t❤❡ ❡①♣r❡ss✐♦♥ ✐♥t♦ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❝❡rt❛✐♥ ❢✉♥❝t✐♦♥ ♦✈❡r t❤✐s ♣❛rt✐t✐♦♥✿ t♦t❛❧ ✇♦r❦ ≈

n X i=1

 b  X X(ti+1 ) − X(ti ) ∆X F (X(ci )) · ∆t . (ti+1 − ti ) = (F ◦ X) · ti+1 − ti ∆t a

❚❤❡♥✱ ✇❡ ❞❡✜♥❡ t❤❡ ✇♦r❦ ♦❢ t❤❡ ❢♦r❝❡ ❛s t❤❡ ❧✐♠✐t✱ ✐❢ ✐t ❡①✐sts✱ ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳

❉❡✜♥✐t✐♦♥ ✻✳✺✳✻✿ ❧✐♥❡ ✐♥t❡❣r❛❧ ❙✉♣♣♦s❡ C ✐s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ ✐♥ Rn ✳ ❋♦r ❛ ✈❡❝t♦r ✜❡❧❞ F ✐♥ Rn ✱ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ♦❢ F ❛❧♦♥❣ C ✐s ❞❡♥♦t❡❞ ❛♥❞ ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ Z

C

F · dX =

Z

b a

F (X(t)) · X ′ (t) dt

✇❤❡r❡ X = X(t), a ≤ t ≤ b✱ ✐s ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ C ✳

❉❡✜♥✐t✐♦♥ ✻✳✺✳✼✿ ✇♦r❦ ❲❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ F ✐s ❝❛❧❧❡❞ ❛ ❢♦r❝❡ ✜❡❧❞✱ t❤❡♥ t❤❡ ✐♥t❡❣r❛❧ ♦❢ F ❛❧♦♥❣ C ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✇♦r❦ ♦❢ ❢♦r❝❡ F ❛❧♦♥❣ ❝✉r✈❡ C ✳ ❚❤❡ ✜rst t❡r♠ ✐♥ t❤❡ ✐♥t❡❣r❛❧ s❤♦✇s ❤♦✇ t❤❡ ❢♦r❝❡ ✈❛r✐❡s ✇✐t❤ t✐♠❡ ❞✉r✐♥❣ ♦✉r tr✐♣✳ ❏✉st ❛s ❛❧✇❛②s✱ t❤❡

✻✳✺✳ ▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

✹✷✽

▲❡✐❜♥✐③ ♥♦t❛t✐♦♥ r❡✈❡❛❧s t❤❡ ♠❡❛♥✐♥❣✿ Z

C

F · dX =

Z

b

(F ◦ X) ·

a

dX dt , dt

❖♥❝❡ ❛❧❧ t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ✐s ❞♦♥❡✱ ✇❡ ❛r❡ ❧❡❢t ✇✐t❤ ❥✉st ❛ ❢❛♠✐❧✐❛r ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮✳ ❋✉rt❤❡r♠♦r❡✱ ✇❤❡♥ n = 1✱ t❤❡ ✐♥t❡❣r❛❧ ✐s t❤❡ ❢❛♠✐❧✐❛r ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✳ ■♥❞❡❡❞✱ s✉♣♣♦s❡ x = F (t) ✐s ❥✉st ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ C ✐s t❤❡ ✐♥t❡r✈❛❧ [A, B] ✐♥ t❤❡ x✲❛①✐s✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

Z

C

F · dX =

Z

x=B

F dx = x=A

Z

t=b

F (x(t))x′ (t) dt , t=a

✇❤❡r❡ x = x(t) s❡r✈❡s ❛s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤✐s ✐♥t❡r✈❛❧ s♦ t❤❛t x(a) = A ❛♥❞ x(b) = B ✳ ❚❤✐s ✐s ❥✉st ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛✳

❊①❛♠♣❧❡ ✻✳✺✳✽✿ str❛✐❣❤t ♣❛t❤ ❈♦♠♣✉t❡ t❤❡ ✇♦r❦ ♦❢ ❛ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞✱ F =< −1, 2 >✱ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡✱ t❤❡ s❡❣♠❡♥t ❢r♦♠ (0, 0) t♦ (1, 3)✳ ❋✐rst✱ ♣❛r❛♠❡tr✐③❡ t❤❡ ❝✉r✈❡ ❛♥❞ ✜♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✿ X(t) =< 1, 3 > t, 0 ≤ t ≤ 1, =⇒ X ′ (t) =< 1, 3 > .

❚❤❡♥✱ W =

Z

C

F · dX =

Z

b ′

a

F (X(t)) · X (t) dt =

Z

1 0

< −1, 2 > · < 1, 3 > dt =

Z

1

5 dt = 5 . 0

❊①❛♠♣❧❡ ✻✳✺✳✾✿ r❛❞✐❛❧ ✈❡❝t♦r ✜❡❧❞ ❈♦♠♣✉t❡ t❤❡ ✇♦r❦ ♦❢ t❤❡ r❛❞✐❛❧ ✈❡❝t♦r ✜❡❧❞✱ F (X) = X =< x, y >✱ ❛❧♦♥❣ t❤❡ ✉♣♣❡r ❤❛❧❢✲❝✐r❝❧❡ ❢r♦♠ (1, 0) t♦ (−1, 0)✳ ❋✐rst ♣❛r❛♠❡tr✐③❡ t❤❡ ❝✉r✈❡ ❛♥❞ ✜♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✿ X(t) =< cos t, sin t >, 0 ≤ t ≤ π, =⇒ X ′ (t) =< − sin t, cos t > .

❚❤❡♥✱ W = = =

Z

F · dX =

ZCπ

Z0 π 0

= 0.

Z

b a

F (X(t)) · X ′ (t) dt

< cos t, sin t > · < − sin t, cos t > dt (cos t(− sin t) + sin t cos t) dt

✻✳✺✳

✹✷✾

▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

❚❤❡♦r❡♠ ✻✳✺✳✶✵✿ ■♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❲♦r❦ ❚❤❡ ✇♦r❦ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛r❛♠❡tr✐③❛t✐♦♥✳

❚❤✉s✱ ❥✉st ❛s ✇❡ ✉s❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s t♦ st✉❞② ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✱ ✇❡ ✉s❡ t❤❡♠ t♦ st✉❞② ❛ ✈❡❝t♦r ✜❡❧❞✳ ◆♦t❡ ❤♦✇❡✈❡r✱ t❤❛t ♦♥❧② t❤❡ ♣❛rt ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✈✐s✐t❡❞ ❜② t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛✛❡❝ts t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧✳ ❯♥❧✐❦❡ t❤❡ ❛r❝✲❧❡♥❣t❤✱ t❤❡ ✇♦r❦ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ tr✐♣✳

❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐s ❤♦✇❡✈❡r ✈❡r② s✐♠♣❧❡✿ t❤❡ s✐❣♥ ✐s r❡✈❡rs❡❞ ✇❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥ ✐s r❡✈❡rs❡❞✳ ❚❤❡♦r❡♠ ✻✳✺✳✶✶✿ ◆❡❣❛t✐✈✐t② ♦❢ ■♥t❡❣r❛❧

Z

−C

F · dX = −

Z

C

F · dX

❊①❛♠♣❧❡ ✻✳✺✳✶✷✿ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s

■s t❤❡ ✇♦r❦ ♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡❄

❲❤❡♥ ❛❧❧ t❤❡ ❛♥❣❧❡s ❛r❡ ❛❝✉t❡✱ ✐t✬s ♣♦s✐t✐✈❡✳ ❊①❡r❝✐s❡ ✻✳✺✳✶✸

❋✐♥✐s❤ t❤❡ ❡①❛♠♣❧❡✳ ❊①❡r❝✐s❡ ✻✳✺✳✶✹

❍♦✇ ♠✉❝❤ ✇♦r❦ ❞♦❡s ✐t t❛❦❡ t♦ ♠♦✈❡ ❛♥ ♦❜❥❡❝t ❛tt❛❝❤❡❞ t♦ ❛ s♣r✐♥❣ s ✉♥✐ts ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠❄ ❊①❡r❝✐s❡ ✻✳✺✳✶✺

❍♦✇ ♠✉❝❤ ✇♦r❦ ❞♦❡s ✐t t❛❦❡ t♦ ♠♦✈❡ ❛♥ ♦❜❥❡❝t s ✉♥✐ts ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ ❛ ♣❧❛♥❡t❄ ❊①❡r❝✐s❡ ✻✳✺✳✶✻

❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛❧♦♥❣ ♦♥❡ ♦❢ ✐ts ❧❡✈❡❧ ❝✉r✈❡s❄

✻✳✺✳

✹✸✵

▲✐♥❡ ✐♥t❡❣r❛❧s✿ ✇♦r❦

❚❤❡♦r❡♠ ✻✳✺✳✶✼✿ ▲✐♥❡❛r✐t② ♦❢ ■♥t❡❣r❛❧

❋♦r ❛♥② t✇♦ ✈❡❝t♦r ✜❡❧❞s F ❛♥❞ G ❛♥❞ ❛♥② t✇♦ ♥✉♠❜❡rs λ ❛♥❞ µ✱ ✇❡ ❤❛✈❡✿ Z

C

(λF + µG) · dX = λ

Z

C

F · dX + µ

Z

C

G · dX

❚❤❡♦r❡♠ ✻✳✺✳✶✽✿ ❆❞❞✐t✐✈✐t② ♦❢ ■♥t❡❣r❛❧

❋♦r ❛♥② t✇♦ ♦r✐❡♥t❡❞ ❝✉r✈❡s C ❛♥❞ K ✇✐t❤ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥ ❛♥❞ t❤❛t t♦❣❡t❤❡r ❢♦r♠ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ C ∪ K ✱ ✇❡ ❤❛✈❡✿ Z

▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❝♦♠♣♦♥❡♥t

C∪K

F · dX =

Z

C

F · dX +

Z

K

F · dX

✳ ❙t❛rt✐♥❣ ✇✐t❤ ❞✐♠❡♥s✐♦♥ n = 1✱ t❤❡ ❞❡✜♥✐t✐♦♥✱

r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧

Z

C

F · dX =

❜❡❝♦♠❡s ✭F = f, X = x, C = [A, B]✮✿ Z

Z

b a

B

f (x) dx = A

F (X(t)) · X ′ (t) dt , Z

b

f (x(t))x′ (t) dt , a

✇❤❡r❡ A = x(a) ❛♥❞ B = x(b)✳ ■♥ R ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣♦♥❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ F ❛♥❞ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ X ✿ F =< p, q > ❛♥❞ dX =< dx, dy > . 2

✻✳✻✳ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✹✸✶

❚❤❡♥ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ♦❢ F ❛❧♦♥❣ C ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ Z

C

Z

F · dX =

C

< p, q > · < dx, dy >=

❍❡r❡✱ t❤❡ ✐♥t❡❣r❛♥❞ ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 1✿

Z

p dx + q dy . C

p dx + q dy

❚❤❡ ♥♦t❛t✐♦♥ ♠❛t❝❤❡s t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■♥❞❡❡❞✱ t❤❡ ❝✉r✈❡✬s ♣❛r❛♠❡tr✐③❛t✐♦♥ X = X(t), a ≤

t ≤ b✱ ❤❛s ❛ ❝♦♠♣♦♥❡♥t r❡♣r❡s❡♥t❛t✐♦♥✿

X =< x, y > ,

t❤❡r❡❢♦r❡✱ Z

b ′

F (X(t))·X (t) dt = a

Z

b ′

′

F (x(t), y(t))· < x (t), y (t) > dt = a

Z

b ′

p(x(t), y(t))x (t) dt+ a

Z

b

q(x(t), y(t))y ′ (t) dt . a

❙✐♠✐❧❛r❧②✱ ✐♥ R3 ✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣♦♥❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ F ❛♥❞ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ X ✿ F =< p, q, r > ❛♥❞ dX =< dx, dy, dz > .

❚❤❡♥ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ♦❢ F ❛❧♦♥❣ C ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ Z

Z

F · dX =

C

p dx + q dy + r dz . C

▲❡t✬s r❡✈✐❡✇ t❤❡ r❡❝❡♥t ✐♥t❡❣r❛❧s t❤❛t ✐♥✈♦❧✈❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❙✉♣♣♦s❡ X = X(t) ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦♥ [a, b]✳ • ❚❤❡ ✜rst ✐s t❤❡ ✭❝♦♠♣♦♥❡♥t✇✐s❡✮

✐♥t❡❣r❛❧ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✿ Z

b

X(t) dt , a

♣r♦✈✐❞✐♥❣ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❦♥♦✇♥ ✈❡❧♦❝✐t②✱ ❛s ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✳ • ❚❤❡ s❡❝♦♥❞ ✐s t❤❡

❛r❝✲❧❡♥❣t❤ ✐♥t❡❣r❛❧ ✿ Z

f ds = C

Z

b

f (X(t))||X ′ (t)|| dt , a

♣r♦✈✐❞✐♥❣ t❤❡ ♠❛ss ♦❢ ❛ ❝✉r✈❡ ♦❢ ✈❛r✐❛❜❧❡ ❞❡♥s✐t②✳ • ❚❤❡ t❤✐r❞ ✐s t❤❡

❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ ✿ Z

♣r♦✈✐❞✐♥❣ t❤❡ ✇♦r❦ ♦❢ t❤❡ ❢♦r❝❡ ✜❡❧❞✳

C

F · dX =

Z

b a

F (X(t)) · X ′ (t) dt ,

❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✜rst ❛♥❞ t❤❡ ♦t❤❡r t✇♦ ✐s t❤❛t ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s t❤❡ ✐♥t❡❣r❛♥❞ ✭❛♥❞ t❤❡ ♦✉t♣✉t ✐s ❛♥♦t❤❡r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❛♥❞ ✐♥ t❤❡ ❧❛tt❡r ✐t ♣r♦✈✐❞❡s t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ✭❛♥❞ t❤❡ ♦✉t♣✉t ✐s ❛ ♥✉♠❜❡r✮✳

✻✳✻✳ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✻✳✻✳ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✹✸✷

❊①❛♠♣❧❡ ✻✳✻✳✶✿ s✐♥❣❧❡ ❝❡❧❧

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❝✉r✈❡ C t❤❛t ❣♦❡s ❛r♦✉♥❞ ❛ s✐♥❣❧❡ sq✉❛r❡ ♦❢ t❤❡ ❣r✐❞ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✳ ▲❡t G ❜❡ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✭1✲❢♦r♠✮ ♦♥ t❤❡ ♣❛rt✐t✐♦♥✿ ✐t ❤❛s s❛♠❡ ✈❛❧✉❡ ❢♦r ❡❛❝❤ ❤♦r✐③♦♥t❛❧ ❡❞❣❡ ❛♥❞ s❛♠❡ ❢♦r ❡❛❝❤ ✈❡rt✐❝❛❧ ❡❞❣❡ ✭❧❡❢t✮✿

❚❤❡♥ t❤❡ ✢♦✇ ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ✐s ③❡r♦ ✦ ◆♦t❡ t❤❛t G ✐s ❡①❛❝t✿ G = ∆f .

❚❤❡s❡ ❛r❡ G ✭❞❡✜♥❡❞ ♦♥ ❡❞❣❡s✮ ❛♥❞ f ✭❞❡✜♥❡❞ ♦♥ ♥♦❞❡s✮✿ • 1 •

G= 1

1

• 1 •

2 −− 3

f= |

|

1 −− 2

❙✉♣♣♦s❡ ♥♦✇ t❤❛t G ✐s r♦t❛t✐♦♥❛❧ ✭r✐❣❤t✮✿ • 1

G= 1

•

−1

f =?

• −1 •

❚❤❡♥ t❤❡ ✢♦✇ ✐s ♥♦t ③❡r♦ ✦ ◆♦t❡ t❤❛t G ✐s♥✬t ❡①❛❝t✱ ❛s ❞❡♠♦♥str❛t❡❞ ✐♥ t❤❡ ✜rst s❡❝t✐♦♥ ♦❢ t❤❡ ❝❤❛♣t❡r✳ ❚❤✐s ✐s♥✬t ❛ ❝♦✐♥❝✐❞❡♥❝❡✳ ❙✉♣♣♦s❡ C ✐s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ t❤❛t ❝♦♥s✐sts ♦❢ ♦r✐❡♥t❡❞ ❡❞❣❡s Qi , i = 1, ..., m✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✳

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s F ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ D✱ G = ∆f ✱ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❡ ❝❛rr② ♦✉t ❛ ❢❛♠✐❧✐❛r ❝♦♠♣✉t❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛❧❧ ♦❢ t❤❡s❡

✻✳✻✳

✹✸✸

❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

❛♥❞ ❝❛♥❝❡❧✐♥❣ t❤❡ r❡♣❡❛t❡❞ ♥♦❞❡s✿ X

G = G(Q1 )

+G(Q2 )

+... +G(Qm )

C

= G(P0 P1 ) +G(P1 P2 ) +... +G(Pm−1 Pm )       = f (P1 ) − f (P0 ) + f (P2 ) − f (P1 ) +... + f (Pm ) − f (Pm−1 ) = −f (P0 )

+f (Pm )

= f (B) − f (A) .

❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✻✳✻✳✷✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❉✐✛❡r❡♥❝❡s ■■ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s G ✐s ❡①❛❝t✱ ✐✳❡✳✱ G = ∆f ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ r❡❣✐♦♥ D✳ ■❢ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ C ✐♥ D st❛rts ❛t ♥♦❞❡ A ❛♥❞ ❡♥❞s ❛t ♥♦❞❡ B ✱ t❤❡♥ ✇❡ ❤❛✈❡✿ X C

G = f (B) − f (A)

◆♦✇✱ t❤❡ s✉♠ ♦♥ r✐❣❤t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦✉r ❝❤♦✐❝❡ ♦❢ C ❛s ❧♦♥❣ ❛s ✐t ✐s ❢r♦♠ A t♦ B ✦ ❲❡ ❢♦r♠❛❧✐③❡ t❤✐s ♣r♦♣❡rt② ❜❡❧♦✇✳

❉❡✜♥✐t✐♦♥ ✻✳✻✳✸✿ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✐s ❝❛❧❧❡❞ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ♦✈❡r D ✐❢ ✐ts s✉♠ ❛❧♦♥❣ ❛♥② ♦r✐❡♥t❡❞ ❝✉r✈❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ st❛rt✲ ❛♥❞ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡❀ ✐✳❡✳✱ X C

G=

X

G

K

❢♦r ❛♥② t✇♦ ❝✉r✈❡s ♦❢ ❡❞❣❡s C ❛♥❞ K ❢r♦♠ ♥♦❞❡ A t♦ ♥♦❞❡ B t❤❛t ❧✐❡ ❡♥t✐r❡❧② ✐♥

D✳

❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ s✉♠s ♦❢ s✉❝❤ ❢✉♥❝t✐♦♥s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s❄

❚❤❡ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ❛❧❧♦✇s ✉s t♦ ❝♦♠♣❛r❡ t❤❡ ❝✉r✈❡ t♦ ❛♥② ❝✉r✈❡ ✇✐t❤ t❤❡ s❛♠❡ ❡♥❞✲♣♦✐♥ts✳ ❲❤❛t ✐s t❤❡ s✐♠♣❧❡st ♦♥❡ t♦ ❝♦♠♣❛r❡ t♦❄ ❈♦♥s✐❞❡r t❤✐s✿ ■❢ t❤❡r❡ ❛r❡ ♥♦ ♣✐♣❡s✱ t❤❡r❡ ✐s ♥♦ ✢♦✇✦ ❲❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ♣❛t❤✱ ❛ ❝♦♥st❛♥t ❝✉r✈❡ ✿ K = {A}✳ ▲❡t✬s ❝♦♠♣❛r❡ ✐t t♦ ❛♥♦t❤❡r ❝✉r✈❡ C ✿

✻✳✻✳

❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✹✸✹

❚❤❡ ❝✉r✈❡ K ✐s tr✐✈✐❛❧❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ X

G=

C

X

G = 0.

K

❙♦✱ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ✐♠♣❧✐❡s ③❡r♦ s✉♠s ❛❧♦♥❣ ❛♥② ❝❧♦s❡❞ ❝✉r✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❝✉r✈❡s C ❛♥❞ K ❢r♦♠ A t♦ B ✳ ❲❡ ❝r❡❛t❡ ❛ ♥❡✇✱ ❝✉r✈❡ ❢r♦♠ t❤❡♠✳ ❲❡ ❣❧✉❡ C ❛♥❞ t❤❡ r❡✈❡rs❡❞ K t♦❣❡t❤❡r✿

❝❧♦s❡❞

Q = C ∪ −K .

■t ❣♦❡s ❢r♦♠ A t♦ A✳

❚❤❡♥✱ ❢r♦♠ ❆❞❞✐t✐✈✐t② ❛♥❞ ◆❡❣❛t✐✈✐t② ✇❡ ❤❛✈❡✿ 0=

X Q

❚❤❡r❡❢♦r❡✱

G=

X

G+

C

X

X −K

G=

C

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

G=

X

X C

G−

X

G.

K

G.

K

❚❤❡♦r❡♠ ✻✳✻✳✹✿ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡

❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥❞ ♦♥❧② ❛❧❧ ♦❢ ✐ts s✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ s♦♠❡ s❡t D ✐♥ R✳ ❲❡ ❦♥♦✇ ✐t t♦ ❜❡ ❡①❛❝t✱ ❜✉t ❤♦✇ ❞♦ ✇❡ ✜♥❞ f ✇✐t❤ ∆f = G❄ ❚❤❡ ✐❞❡❛ ❝♦♠❡s ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✳ ❋✐rst✱ ✇❡ ❝❤♦♦s❡ ❛♥ ❛r❜✐tr❛r② ♥♦❞❡ A ✐♥ D ❛♥❞ t❤❡♥ ❝❛rr② ♦✉t ❛ s✉♠♠❛t✐♦♥ ❛❧♦♥❣ ❡✈❡r② ♣♦ss✐❜❧❡ ❝✉r✈❡ ❢r♦♠ A✳ ❲❡ ❞❡✜♥❡ ❢♦r ❡❛❝❤ X ✐♥ D✿ X f (X) =

G,

C

✇❤❡r❡ C ✐s ❛♥② ❝✉r✈❡ ❢r♦♠ A t♦ X ✳ ❆ ❝❤♦✐❝❡ ♦❢ C ❞♦❡s♥✬t ♠❛tt❡r ❜❡❝❛✉s❡ G ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t✳

✻✳✻✳ ❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✹✸✺

❚♦ ❡♥s✉r❡ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ✐s ✇❡❧❧ ❞❡✜♥❡❞ ✇❡ ♥❡❡❞ ❛♥ ❡①tr❛ r❡q✉✐r❡♠❡♥t✳

❚❤❡♦r❡♠ ✻✳✻✳✺✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❉✐✛❡r❡♥❝❡s ■ ❖♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❛t❤✲❝♦♥♥❡❝t❡❞ r❡❣✐♦♥ D ✐♥ Rn ✱ ✐❢ G = ∆f ✱ t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❢♦r ❛ ✜①❡❞ A ✐♥ D✿ g(X) =

X

G,

C

✇❤❡r❡ C ✐s ❛♥② ❝✉r✈❡ ❢r♦♠ A t♦ X ✇✐t❤✐♥ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ D✱ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ∆g = G

Pr♦♦❢✳ ❇❡❝❛✉s❡ t❤❡ r❡❣✐♦♥ ✐♥ ♣❛t❤✲❝♦♥♥❡❝t❡❞✱ t❤❡r❡ ✐s ❛❧✇❛②s ❛ ❝✉r✈❡ ❢r♦♠ A t♦ ❛♥② X ✳ ❲❤❛t ❛❜♦✉t ✈❡❝t♦r ✜❡❧❞s❄ ■❢ F ✐s ❛ ✈❡❝t♦r ✜❡❧❞✱ ✇❡ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s t♦ ✐ts ♣r♦❥❡❝t✐♦♥ G = F · ∆X ✳ ❚❤❡ s✉♠s ❜❡❝♦♠❡ ❘✐❡♠❛♥♥ s✉♠s✳✳✳

❊①❛♠♣❧❡ ✻✳✻✳✻✿ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞ ❙✉♣♣♦s❡ C ✐s ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡ t❤❛t ❝♦♥s✐sts ♦❢ ♦r✐❡♥t❡❞ ❡❞❣❡s Qi , i = 1, ..., m✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✳ ▲❡t Qi = Pi−1 Pi ✇✐t❤ P0 = Pm = A . ■t ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❙✉♣♣♦s❡ F ✐s ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞ ✐♥ D❀ ✐✳❡✳✱ F (X) = G ❢♦r ❛❧❧ X ✐♥ D✿

✻✳✻✳

❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

❚❤❡♥ t❤❡ ✇♦r❦ ♦❢

G

❛❧♦♥❣

X C

C

✹✸✻

✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❘✐❡♠❛♥♥ s✉♠✿

F · ∆X = =

m X

i=1 m X i=1

=F· =F·

F (Qi ) · Qi F · Qi m X

Pi−1 Pi

i=1

m X i=1

(Pi − Pi−1 )

  = F · (P1 − P0 ) + (P2 − P1 ) + ... + (Pm − Pm−1 )   = F · − P0 + P m = 0.

❚❤❡ ✇♦r❦ ✐s

③❡r♦ ✦

❊①❛♠♣❧❡ ✻✳✻✳✼✿ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ❚❤❡ st♦r② ✐s t❤❡ ❡①❛❝t ♦♣♣♦s✐t❡ ❢♦r t❤❡

r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ✿ F =< −y, x > .

▲❡t✬s ❝♦♥s✐❞❡r ❛ s✐♥❣❧❡ sq✉❛r❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥❀ ❢♦r ❡①❛♠♣❧❡✱

S = [1, 2] × [1, 2]✳

✻✳✻✳

❙✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s r❡✈❡❛❧ ❡①❛❝t♥❡ss

✹✸✼

❙✉♣♣♦s❡ ❝✉r✈❡ C ❣♦❡s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ t❤❡ st❛rt✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❡❞❣❡s✳ ❚❤❡♥ t❤❡ ✇♦r❦ ♦❢ G ❛❧♦♥❣ C ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❘✐❡♠❛♥♥ s✉♠✿ X C

=

F · ∆X =

4 X i=1

F (Qi ) · Qi

= F (1, 1)· < 1, 0 > +F (2, 1)· < 0, 1 > +F (2, 2)· < −1, 0 > +F (1, 2)· < 0, −1 >

=< −1, 1 > · < 1, 0 > + < −1, 2 > · < 0, 1 > + < −2, 2 > · < −1, 0 > + < −2, 1 > · < 0, −1 > = −1 + 2 + 2 − 1 = 2.

❚❤❡ ✇♦r❦ ✐s ♥♦t

③❡r♦

✦

❚❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ ❢♦r ❞✐✛❡r❡♥❝❡s t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ X C

F · ∆X = f (B) − f (A) .

◆♦t ♦♥❧② t❤❡ ♣r♦♦❢ ❜✉t ❛❧s♦ t❤❡ ❢♦r♠✉❧❛ ✐ts❡❧❢ ❧♦♦❦s ❧✐❦❡ t❤❡ ❢❛♠✐❧✐❛r ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧s ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✳

❉❡✜♥✐t✐♦♥ ✻✳✻✳✽✿ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✜❡❧❞ ❆ ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✐s ❝❛❧❧❡❞ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✐❢ ✐ts ♣r♦❥❡❝t✐♦♥ F · ∆X ✐s❀ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛❧♦♥❣ ❛♥② ♦r✐❡♥t❡❞ ❝✉r✈❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ st❛rt✲ ❛♥❞ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡

✻✳✼✳

✹✸✽

P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

❝✉r✈❡✿ X C

F · ∆X =

X K

F · ∆X

❢♦r ❛♥② t✇♦ ❝✉r✈❡s ♦❢ ❡❞❣❡s C ❛♥❞ K ❢r♦♠ ♥♦❞❡ A t♦ ♥♦❞❡ B t❤❛t ❧✐❡ ❡♥t✐r❡❧② ✐♥ D✳ ❋♦r t❤❡ s✉♠ ❛❧♦♥❣ ❛ ❝❧♦s❡❞ ❝✉r✈❡✱ ✇❡ ♥♦t❡ ♦♥❝❡ ❛❣❛✐♥✿ ✐❢ ✇❡ st❛② ❤♦♠❡✱ ✇❡ ❞♦♥✬t ❞♦ ❛♥② ✇♦r❦✦ ❲❡ ❤❛✈❡ ❢♦r ❛ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✜❡❧❞ F ✿ X C

F · ∆X =

X K

F · ∆X = 0 .

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❝✉r✈❡s C ❛♥❞ K ❢r♦♠ A t♦ B ✳ ❲❡ ❝r❡❛t❡ ❛ ♥❡✇✱ ❝❧♦s❡❞ ❝✉r✈❡ ❢r♦♠ t❤❡♠✱ ❢r♦♠ A t♦ A✱ ❜② ❣❧✉✐♥❣ C ❛♥❞ t❤❡ r❡✈❡rs❡❞ K t♦❣❡t❤❡r✿ Q = C ∪ −K .

❋r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✉❧t ❢♦r ❞✐✛❡r❡♥❝❡s ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✻✳✻✳✾✿ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❱❡❝t♦r ❋✐❡❧❞s

❆ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ r❡❣✐♦♥ D ✐♥ Rn ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ♦❢ ✐ts ❘✐❡♠❛♥♥ s✉♠s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s ♦❢ ❡❞❣❡s ✐♥ D ❛r❡ ❡q✉❛❧ t♦ ③❡r♦✳

✻✳✼✳ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ❢♦r❝❡ ✜❡❧❞s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s✳

✻✳✼✳ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

✹✸✾

❉❡✜♥✐t✐♦♥ ✻✳✼✳✶✿ ❝❧♦s❡❞ ❝✉r✈❡ ❆ ❝❧♦s❡❞ ❝✉r✈❡ ✭❛ ❧♦♦♣✮ ✐s ❛ ❝✉r✈❡ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❛♥❞ t❤❡ ❡♥❞ ♣♦✐♥ts ♦❢ ✇❤✐❝❤ ❝♦✐♥❝✐❞❡♥❝❡❀ ✐✳❡✳✱ ✐t ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② s♦♠❡ X = X(t), a ≤ t ≤ b✱ ✇✐t❤ X(a) = X(b) = A✳

▲✐♥❡ ✐♥t❡❣r❛❧s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s ❤❛✈❡ ❛ s♣❡❝✐❛❧ ♥♦t❛t✐♦♥✿

▲♦♦♣ ✐♥t❡❣r❛❧ I

C

F · dX

❧❡t✬s ❝♦♥s✐❞❡r ❝♦♥st❛♥t ❢♦r❝❡ ✜❡❧❞s ❛❧♦♥❣

❊①❛♠♣❧❡ ✻✳✼✳✷✿ ❧♦♦♣ ✐♥ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞ ❖♥❝❡ ❛❣❛✐♥✱ ✇❤❛t ✐s t❤❡ ✇♦r❦ ♦❢ ❛ ❝♦♥st❛♥t ❢♦r❝❡ ✜❡❧❞ ❛❧♦♥❣ ❛ ❝❧♦s❡❞ ❝✉r✈❡ s✉❝❤ ❛s ❛ ❝✐r❝❧❡❄ ❈♦♥s✐❞❡r t✇♦ ❞✐❛♠❡tr✐❝❛❧❧② ♦♣♣♦s✐t❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✐r❝❧❡✳ ❚❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ t❛♥❣❡♥ts t♦ t❤❡ ❝✉r✈❡ ❛r❡ ♦♣♣♦s✐t❡ ✇❤✐❧❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s t❤❡ s❛♠❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t❡r♠s F · X ′ ✐♥ t❤❡ ✇♦r❦ ✐♥t❡❣r❛❧ ❛r❡ ♥❡❣❛t✐✈❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❙♦✱ ❜❡❝❛✉s❡ ♦❢ t❤✐s s②♠♠❡tr②✱ t✇♦ ♦♣♣♦s✐t❡ ❤❛❧✈❡s ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐❧❧ ❤❛✈❡ ✇♦r❦ ♥❡❣❛t✐✈❡ ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ ❝❛♥❝❡❧✳ ❚❤❡ ✇♦r❦ ♠✉st ❜❡ ③❡r♦ ✦ ▲❡t✬s ❝♦♥✜r♠ t❤✐s ❢♦r F =< p, q > ❛♥❞ t❤❡ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✿ W = = =

I

F · dX =

ZC2π

Z0 2π

Z0

Z

b a

F (X(t)) · X ′ (t) dt

< p, q > · < cos t, sin t >′ dt < p, q > · < − sin t, cos t > dt

2π

(−p sin t + q cos t) dt 2π 2π = (p cos t − q sin t) + (p cos t − q sin t) =

0

0

= 0 + 0 = 0.

❙♦✱ ✇♦r❦ ❝❛♥❝❡❧s ♦✉t ❞✉r✐♥❣ t❤✐s r♦✉♥❞ tr✐♣✳

❊①❛♠♣❧❡ ✻✳✼✳✸✿ ❧♦♦♣ ✐♥ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ❚❤❡ st♦r② ✐s t❤❡ ❡①❛❝t ♦♣♣♦s✐t❡ ❢♦r t❤❡ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ✿ F =< −y, x > .

0

✻✳✼✳

✹✹✵

P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

❈♦♥s✐❞❡r ❛♥② ♣♦✐♥t✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t❡r♠s F · X ′ ❝❛♥♥♦t ❝❛♥❝❡❧✳ ❚❤❡ ✇♦r❦ ✐s ♥♦t ③❡r♦✦ ▲❡t✬s ❝♦♥✜r♠ t❤✐s r❡s✉❧t✿ W = =

I

0

= = =

Z

Z

2π

b

F (X(t)) · X ′ (t) dt a < −y, x > · < cos t, sin t >′ dt

F · dX =

ZC2π

Z

x=cos t, y=sin t

< − sin t, cos t > · < − sin t, cos t > dt

0 2π

(sin2 t + cos2 t) dt

Z0 2π

1 dt

0

= 2π .

❲❡ ❤❛✈❡ ✇❛❧❦❡❞ ❛❣❛✐♥st ✇✐♥❞ ❛❧❧ t❤❡ ✇❛② ✐♥ t❤✐s r♦✉♥❞ tr✐♣✦ ❚❤❡ s❛♠❡ ❧♦❣✐❝ ❛♣♣❧✐❡s t♦ ❛♥② ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t ♠✉❧t✐♣❧❡ ♦❢ F ❛s ❧♦♥❣ ❛s t❤❡ s②♠♠❡tr② ✐s ♣r❡s❡r✈❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢❛♠✐❧✐❛r ♦♥❡ ❜❡❧♦✇ q✉❛❧✐✜❡s✿ G(X) =

F (X) . ||X||2

❊✈❡♥ t❤♦✉❣❤✱ ❛s ✇❡ ❦♥♦✇✱ t❤✐s ✈❡❝t♦r ✜❡❧❞ ♣❛ss❡s t❤❡ ●r❛❞✐❡♥t ❚❡st✱ ✐t ❤❛s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ♦✈❡r ❛ ❝✐r❝❧❡✿ Z b I W =

C

G · dX =

a

G(X(t)) · G′ (t) dt > 0 ,

❜❡❝❛✉s❡ t❤❡ ✐♥t❡❣r❛♥❞ ✐s ♣♦s✐t✐✈❡✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♦✉t❝♦♠❡s ♠❛② ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡ ❝♦♥st❛♥t ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t ✿ < p, q >= ∇f, ✇❤❡r❡ f (x, y) = px + qy ,

✇❤✐❧❡ t❤❡ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ✐s ♥♦t ✿

< −y, x >6= ∇f, ❢♦r ❛♥② z = f (x, y) .

■s t❤❡r❡ ❛♥②t❤✐♥❣ s♣❡❝✐❛❧ ❛❜♦✉t ❧✐♥❡ ✐♥t❡❣r❛❧s ♦❢ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞s ♦✈❡r ❝✉r✈❡s t❤❛t ❛r❡♥✬t ❝❧♦s❡❞❄ ❲❡ r❡❛❝❤ t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ❢♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✿ ❚❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ F ✳ ❇✉t t❤❡ ❧❛tt❡r ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ F ✦ ❲❡✱ t❤❡r❡❢♦r❡✱ s♣❡❛❦ ♦❢ ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■■ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭t❤❡r❡ ✇✐❧❧ ❜❡ ❋❚❈ ■ ❧❛t❡r✮✳

✻✳✼✳ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

✹✹✶

❚❤❡♦r❡♠ ✻✳✼✳✹✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ●r❛❞✐❡♥t ❱❡❝t♦r ❋✐❡❧❞s ■■ ■❢ ♦♥ ❛ s✉❜s❡t ♦❢ ♣♦✐♥t

A

Rn ✱

❛♥❞ ❡♥❞s ❛t

✇❡ ❤❛✈❡

B✱

t❤❡♥

Z

C

F = ∇f

❛♥❞ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡

C

✐♥

Rn

st❛rts ❛t

F · dX = f (B) − f (A)

Pr♦♦❢✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡✿

❛♥❞ ❛♥ ♦r✐❡♥t❡❞ ❝✉r✈❡

C

✐♥

❚❤❡♥✱ ❛❢t❡r ♣❛r❛♠❡tr✐③✐♥❣

Rn

C

F = ∇f , t❤❛t st❛rts ❛t ♣♦✐♥t

✇✐t❤

A

❛♥❞ ❡♥❞s ❛t

X = X(t), a ≤ t ≤ b✱

B✿

✇❡ ❤❛✈❡ ✈✐❛ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢

❈❛❧❝✉❧✉s ✭❈❤❛♣t❡r ✸■❈✲✶✮ ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✭❈❤❛♣t❡r ✷❉❈✲✹✮✿

W = = =

Z

ZCb

Za b

Z

a

F · dX F (X(t)) · X ′ (t) dt ∇f (X(t)) · X ′ (t) dt

b

d f (X(t)) dt a dt b = f (X(t)) =

❲❡ r❡❝♦❣♥✐③❡ t❤❡ ✐♥t❡❣r❛♥❞ ❛s ❛ ♣❛rt ♦❢ ❈❘✳ ❲❡ ❛♣♣❧② ♥♦✇ ❋❚❈ ■■✳

a

= f (X(b)) − f (X(a)) = f (B) − f (A) .

❋♦r ❞✐♠❡♥s✐♦♥ ✐♥t❡r✈❛❧

[A, B]

n = 1✱ ✐♥ t❤❡

✇❡ ❥✉st t❛❦❡

x✲❛①✐s✳

y = F (x)

t♦ ❜❡ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡

f

❛♥❞

C

✐s t❤❡

✻✳✼✳ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

✹✹✷

❲❡ ❛❧s♦ ❝❤♦♦s❡ x = x(t) t♦ ❜❡ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤✐s ✐♥t❡r✈❛❧ s♦ t❤❛t x(a) = A ❛♥❞ x(b) = B ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❢r♦♠ ❛❜♦✈❡✿ Z

F dX = C

Z

x=B

F dx = x=A

Z

t=b

t=b = f (x(b)) − f (x(a)) = f (B) − f (A) . F (x(t))x (t) dt = f (x(t)) ′

t=a

t=a

❲❡ ❤❛✈❡ ❛♥♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ s✉❜st✐t✉t✐♦♥ ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s✳

◆♦t ♦♥❧② t❤❡ ♣r♦♦❢ ❜✉t ❛❧s♦ t❤❡ ❢♦r♠✉❧❛ ✐ts❡❧❢ ❧♦♦❦s ❧✐❦❡ t❤❡ ❢❛♠✐❧✐❛r ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧s ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✳ ❇❡❝❛✉s❡ ✐t ✐s r❡str✐❝t❡❞ t♦ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞s✱ t❤✐s ✐s ❥✉st ❛ ♣r❡❧✐♠✐♥❛r② ✈❡rs✐♦♥✳ ❲❛r♥✐♥❣✦

❇❡❢♦r❡ ❛♣♣❧②✐♥❣ t❤❡ ❢♦r♠✉❧❛✱ ❝♦♥✜r♠ t❤❛t t❤❡ ✈❡❝✲ t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t ✦ ❚❤❡ ❡①❛♠♣❧❡ ♦❢ F =< −y, x > ✐s t♦ ❜❡ r❡♠❡♠❜❡r❡❞ ❛t ❛❧❧ t✐♠❡s✳

❙♦✱ ✐❢ F ✐s ❛ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞ t❤❡♥

I

C

F · dX = 0 .

❚❤❡r❡❢♦r❡✱ t❤❡ ✇♦r❦ ✐s ③❡r♦ ♦♥ ♥❡t s♦ t❤❛t t❤❡r❡ ✐s ♥♦ ❣❛✐♥ ♦r ❧♦ss ♦❢ ❡♥❡r❣②✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞s ❛r❡ ❛❧s♦ ❝❛❧❧❡❞ ❝♦♥s❡r✈❛t✐✈❡✳ ❊①❛♠♣❧❡ ✻✳✼✳✺✿ ❣r❛❞✐❡♥t❄

❈♦♥s✐❞❡r t❤✐s r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞✱ V =< −y, x >, ❛♥❞ ❡s♣❡❝✐❛❧❧② ✐ts ♠✉❧t✐♣❧❡✿ V 1 F = = 2 < y, −x >= 2 ||V || x + y2



x y , − 2 2 2 x +y x + y2



=< p, q > .

❲❡ ♣r❡✈✐♦✉s❧② ❞❡♠♦♥str❛t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∂ y 1 · (x2 + y 2 ) − y · 2y x2 − y 2 = = ∂y x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )2 2 2 ∂ −x 1 · (x + y ) − x · 2x y 2 − x2 qx = = − = − ∂x x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )2 py =

=⇒ rot F = qx − py = 0 .

✻✳✼✳

✹✹✸

P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

❙♦✱ t❤❡ r♦t♦r ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ③❡r♦ ❛♥❞ ✐t ♣❛ss❡s t❤❡ ●r❛❞✐❡♥t ❚❡st ❀ ❤♦✇❡✈❡r✱ ✐s ✐t ❣r❛❞✐❡♥t❄ ❲❡ ❞❡♠♦♥str❛t❡ ♥♦✇ t❤❛t ✐t ✐s ♥♦t✳ ■♥❞❡❡❞✱ s✉♣♣♦s❡ X = X(t) ✐s ❛ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✳ ❚❤❡♥ F (X(t)) ✐s ♣❛r❛❧❧❡❧ t♦ X ′ (t)✳ ❚❤❡r❡❢♦r❡✱ F (X(t)) · X ′ (t) > 0✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡ ✐s ♣♦s✐t✐✈❡✿ 0=

I

C

F · dX =

Z

b a

F (X(t)) · X ′ (t) dt > 0 .

■t ✐s ❛s ✐❢ ✇❡ ❤❛✈❡ ❝❧✐♠❜❡❞ ❛ s♣✐r❛❧ st❛✐r❝❛s❡✦ ❆ ❝♦♥tr❛❞✐❝t✐♦♥✳ ◆♦t ♦♥❧② t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ r✐❣❤t

Z

C

F · dX = f (B) − f (A)

✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ C ✱ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦✉r ❝❤♦✐❝❡ ♦❢ C ❛s ❧♦♥❣ ❛s ✐t ✐s ❢r♦♠ A t♦ B ✦

❉❡✜♥✐t✐♦♥ ✻✳✼✳✻✿ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✜❡❧❞ ❆ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ ❛ s✉❜s❡t D ♦❢ Rn ✐s ❝❛❧❧❡❞ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥② ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ ❛ ❝✉r✈❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ st❛rt✲ ❛♥❞ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡❀ ✐✳❡✳✱ Z

C

F · dX =

Z

K

F · dX

❢♦r ❛♥② t✇♦ ❝✉r✈❡s C ❛♥❞ K ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B t❤❛t ❧✐❡ ❡♥t✐r❡❧② ✐♥ D✳ ❲❤❛t ✐❢ A = B ❄ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ ❛ ❝❧♦s❡❞ ❝✉r✈❡ C ❄ ❆s ❛♥ ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤✐s✿ ✐❢ ✇❡ st❛② ❤♦♠❡✱ ✇❡ ❞♦♥✬t ❞♦ ❛♥② ✇♦r❦✦ ❲❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t ❛ ❝♦♥st❛♥t ❝✉r✈❡✱ K = {A}✳ ▲❡t✬s ❝♦♠♣❛r❡ ✐t t♦ ❛♥♦t❤❡r ❝✉r✈❡ C ✳

❚❤❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ K ✐s tr✐✈✐❛❧✿ X(t) = A ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡r❡❢♦r❡✱ X ′ (t) = 0 ❛♥❞ ✇❡ ❤❛✈❡✿ Z b Z b Z Z C

F · dX =

K

F · dX =

a

F (X(t)) · X ′ (t) dt =

a

F (X(t)) · 0 dt = 0 .

❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❝✉r✈❡s C ❛♥❞ K ❢r♦♠ A t♦ B ✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱

✻✳✼✳

✹✹✹

P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥t❡❣r❛❧s

✇❡ ❝r❡❛t❡ ❛ ♥❡✇✱

❝❧♦s❡❞

❝✉r✈❡ ❢r♦♠ t❤❡♠✳ ❲❡ ❣❧✉❡ C ❛♥❞ t❤❡ r❡✈❡rs❡❞ K t♦❣❡t❤❡r✿ Q = C ∪ −K .

■t ❣♦❡s ❢r♦♠ A t♦ A✳

❚❤❡♥✱ ❢r♦♠ ❆❞❞✐t✐✈✐t② ❛♥❞ ◆❡❣❛t✐✈✐t② ✇❡ ❤❛✈❡✿ 0=

❚❤❡r❡❢♦r❡✱

Z

Q

F · dX =

Z

C

F · dX +

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

Z

C

Z

−K

F · dX =

F · dX = Z

K

Z

C

F · dX −

Z

K

F · dX .

F · dX .

❚❤❡♦r❡♠ ✻✳✼✳✼✿ P❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❖✈❡r ▲♦♦♣s

❆ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ ❛ s✉❜s❡t D ♦❢ Rn ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❤❛s ❛❧❧ ♦❢ ✐ts ❧✐♥❡ ✐♥t❡❣r❛❧s ❛❧♦♥❣ ❝❧♦s❡❞ ❝✉r✈❡s ✐♥ D ❡q✉❛❧ t♦ ③❡r♦✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✻✳✼✳✽✿ ●r❛❞✐❡♥t ❂❃ P❛t❤✲✐♥❞❡♣❡♥❞❡♥t

❆❧❧ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞s ❛r❡ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t✳ Pr♦♦❢✳

• F I ✐s ❛ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞✱ t❤❡♥ • F · dX = 0, t❤❡♥ C

• F ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t✳

❘❡❝❛❧❧ t❤❛t ✇❡ ❝♦♥s✐❞❡r❡❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✭t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤✮ ❜✉t ✇✐t❤ ❛ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿ x r✉♥s ❢r♦♠ a t♦ b ❛♥❞ ❜❡②♦♥❞✳

✈❛r✐❛❜❧❡ ✉♣♣❡r ❧✐♠✐t✳

■t

✻✳✽✳

❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

❚❤❡♥ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜②

✹✹✺

❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■■

st❛t❡s t❤❛t ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ F ♦♥ [a, b]✱ t❤❡ Z

x

F dx a

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ F ♦♥ (a, b)✳ ■♥ t❤❡ ♥❡✇ s❡tt✐♥❣✱ ✇❡ ❤❛✈❡ ❛ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ s♦♠❡ s❡t D ✐♥ R ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ✐ts ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✐✳❡✳✱ ❛ ❢✉♥❝t✐♦♥ t❤❡ ❣r❛❞✐❡♥t ♦❢ ✇❤✐❝❤ ✐s F ✱ ∇f = F ✳ ❋✐rst✱ ✇❡ ❝❤♦♦s❡ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t A ✐♥ D ❛♥❞ t❤❡♥ ❞♦ ❛ ❧♦t ♦❢ ❧✐♥❡ ✐♥t❡❣r❛t✐♦♥✳ ❲❡ ❞❡✜♥❡ ❢♦r ❡❛❝❤ X ✐♥ D✿ Z f (X) =

C

F · dX ,

✇❤❡r❡ C ✐s ❛♥② ❝✉r✈❡ ❢r♦♠ A t♦ X ✳ ❆ ❝❤♦✐❝❡ ♦❢ C ❞♦❡s♥✬t ♠❛tt❡r ❜❡❝❛✉s❡ F ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❜② ❛ss✉♠♣t✐♦♥✳

❚❤❡r❡ ✐s ❛♥ ❡①tr❛ r❡q✉✐r❡♠❡♥t✳ ❚❤❡♦r❡♠ ✻✳✼✳✾✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ●r❛❞✐❡♥t ❱❡❝t♦r ❋✐❡❧❞s ■

❋♦r ❛♥② ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ ❛ ♣❛t❤✲❝♦♥♥❡❝t❡❞ r❡❣✐♦♥ ✐♥ Rn ✱ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❢♦r ❛ ✜①❡❞ A ✐♥ D ❜②✿ f (X) =

Z

C

F · dX

✇❤❡r❡ C ✐s ❛♥② ❝✉r✈❡ ❢r♦♠ A t♦ X ✇✐t❤✐♥ D✱ ✐s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ F ♦♥ D✳

✻✳✽✳ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ✜❡❧❞ t❤❛t ❞❡s❝r✐❜❡s t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ♦❢ ❛ ✢✉✐❞ ✢♦✇✳ ▲❡t✬s ♣❧❛❝❡ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ✇✐t❤✐♥ t❤❡ ✢♦✇✳ ❲❡ ♣✉t ✐t ♦♥ ❛ ♣♦❧❡ s♦ t❤❛t t❤❡ ❜❛❧❧ r❡♠❛✐♥s ✜①❡❞ ✇❤✐❧❡ ✐t ❝❛♥ ❢r❡❡❧② r♦t❛t❡✿

✻✳✽✳ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✹✻

❲❡ s❡❡ t❤❡ ♣❛rt✐❝❧❡s ❜♦♠❜❛r❞✐♥❣ t❤❡ ❜❛❧❧ ❛♥❞ t❤✐♥❦ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ✢♦✇ ❛s ❛ ❢♦r❝❡ ✜❡❧❞✳ ❉✉❡ t♦ t❤❡ ❜❛❧❧✬s r♦✉❣❤ s✉r❢❛❝❡✱ t❤❡ ✢✉✐❞ ✢♦✇✐♥❣ ♣❛st ✐t ✇✐❧❧ ♠❛❦❡ ✐t s♣✐♥ ❛r♦✉♥❞ t❤❡ ♣♦❧❡✳ ■t ✐s ❝❧❡❛r t❤❛t ❛ ❝♦♥st❛♥t ✈❡❝t♦r ✭❧❡❢t✮ ✜❡❧❞ ✇✐❧❧ ♣r♦❞✉❝❡ ♥♦ s♣✐♥ ♦r r♦t❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♥♦t t❤❡ r♦t❛t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦rs t❤❛t ✇❡ ✇✐❧❧ s♣❡❛❦ ♦❢✳ ❲❡ ❛r❡ ♥♦t ❛s❦✐♥❣✿ ■s ❛ s♣❡❝✐✜❝ ♣❛rt✐❝❧❡ ♦❢ ✇❛t❡r ♠❛❦✐♥❣ ❛ ❝✐r❝❧❡❄ ❘❛t❤❡r✿ ◮ ❉♦❡s t❤❡ ❝♦♠❜✐♥❡❞ ♠♦t✐♦♥ ♦❢ t❤❡ ♣❛rt✐❝❧❡s ♠❛❦❡ t❤❡ ❜❛❧❧ s♣✐♥❄

❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ✇❡ s❡❡ ✐♥ t❤❡ ✐♠❛❣❡ ♦♥ r✐❣❤t✿ • ❚❤❡ ❜❛❧❧ ✐♥ t❤❡ ❝❡♥t❡r ✐s ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ❛ ✇❤✐r❧ ❛♥❞ ✇✐❧❧ ❜❡ ❝❧❡❛r❧② s♣✉♥ ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡

❞✐r❡❝t✐♦♥✳

• ❚❤❡ ❜❛❧❧ ❛t t❤❡ ❜♦tt♦♠ ✐s ✐♥ t❤❡ ♣❛rt ♦❢ t❤❡ str❡❛♠ ✇✐t❤ ❛ ❝♦♥st❛♥t ❞✐r❡❝t✐♦♥ ❜✉t ♥♦t ♠❛❣♥✐t✉❞❡✳ ❲✐❧❧

✐t s♣✐♥❄

• ❚❤❡ ❜❛❧❧ ❛t t❤❡ t♦♣ ✐s ❜❡✐♥❣ ♣✉s❤❡❞ ✐♥ ✈❛r✐♦✉s ❞✐r❡❝t✐♦♥s ❛t t❤❡ s❛♠❡ t✐♠❡ ❛♥❞ ✐ts s♣✐♥ s❡❡♠s ✈❡r②

✉♥❝❡rt❛✐♥✳

❍♦✇ ❞♦ ✇❡ ♣r❡❞✐❝t ❛♥❞ ❤♦✇ ❞♦ ✇❡ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ r♦t❛t✐♦♥❄ ❊①❛♠♣❧❡ ✻✳✽✳✶✿ ♣✐♥❣✲♣♦♥❣

❚❤❡ ❛♥s✇❡r ✐s s✐♠♣❧❡ ✇❤❡♥ t❤❡ ❢♦r❝❡ ✐s ❛♣♣❧✐❡❞ t♦ ❥✉st ♦♥❡ s✐❞❡ ♦❢ t❤❡ ❜❛❧❧ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛❧❧ r❛❝❦❡t s♣♦rts✿

❍♦✇❡✈❡r✱ t❤❡ ✐❞❡❛ ♠✐❣❤t ❜❡ ✉t✐❧✐③❡❞ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❛ ❜❛❧❧ ✐♥ t❤❡ str❡❛♠✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ❧❡t✬s ❛ss✉♠❡ t❤❛t ✇❡ ❝❛♥ ❞❡t❡❝t ♦♥❧② ❢♦✉r ❞✐st✐♥❝t ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ♦♥ t❤❡ ❢♦✉r s✐❞❡s ✭♦♥ t❤❡ ❣r✐❞✮ ♦❢ t❤❡ ❜❛❧❧✳

❲❡ ❛❧s♦ ❛ss✉♠❡ ❛t ✜rst t❤❛t t❤❡s❡ ❢♦✉r ✈❡❝t♦rs ❛r❡ t❛♥❣❡♥t t♦ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❜❛❧❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❥✉st ❛ ✈❡❝t♦r ✜❡❧❞✳ ❲❤❛t ✐s t❤❡ ♥❡t ❡✛❡❝t ♦❢ t❤❡s❡ ❢♦r❝❡s ♦♥ t❤❡ ❜❛❧❧❄ ❚❤✐♥❦ ♦❢ t❤❡ ❜❛❧❧ ❛s ❛ t✐♥② ✇✐♥❞✲♠✐❧❧ ✇✐t❤ t❤❡ ❢♦✉r ❢♦r❝❡s ♣✉s❤✐♥❣ ✭♦r ♣✉❧❧✐♥❣✮ ✐ts ❢♦✉r ❜❧❛❞❡s✳ ❲❡ ❥✉st ❣♦ ❛r♦✉♥❞ t❤❡ ❜❛❧❧ ✭❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ st❛rt✐♥❣ ❛t t❤❡ ❜♦tt♦♠✮ ❛❞❞✐♥❣ t❤❡s❡ ♥✉♠❜❡rs✿ 1 + 1 − 2 + 1 = 1 > 0.

✻✳✽✳ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✹✼

❚❤❡ ❜❛❧❧ ✇✐❧❧ s♣✐♥ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✦ ■♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ s♣✐♥✱ ❧❡t✬s ❛ss✉♠❡ t❤❛t t❤✐s ✐s ❛ ✉♥✐t sq✉❛r❡✳ ❚❤❡♥✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s✉♠ ❛❜♦✈❡ ✐s ❥✉st ❛ ❧✐♥❡ s✉♠ ❢r♦♠ ❈❤❛♣t❡r ✻ r❡♣r❡s❡♥t✐♥❣ t❤❡ ✇♦r❦ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ t❤❡ ✢♦✇ t♦ s♣✐♥ t❤❡ ❜❛❧❧✳ ▲❡t✬s ❧♦♦❦ ❛t t❤✐s q✉❛♥t✐t② ❢r♦♠ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ❢♦r❝❡s ✇✐t❤ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ❜✉t ♦♥ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ❝❛♥❝❡❧❧❡❞✳ ❲❡ s❡❡ t❤✐s ❡✛❡❝t ✐❢ ✇❡ r❡✲❛rr❛♥❣❡ t❤❡ t❡r♠s✿ W = ❤♦r✐③♦♥t❛❧✿ 1 − 2 + ✈❡rt✐❝❛❧✿ 1 + 1 .

❲❡ t❤❡♥ r❡♣r❡s❡♥t ❡❛❝❤ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ ✐ts x ❛♥❞ y ❝♦♠♣♦♥❡♥ts✿ •− →→ −•

❢♦r❝❡ =

|

|

↓ |

•−

↑ →

•− 2 −• =

|

−•

|

−1 |

|

1 |

•− 1 −•

❚❤❡ ❡①♣r❡ss✐♦♥ ❝❛♥ t❤❡♥ ❜❡ s❡❡♥ ❛s✿ ◮ W = −✭t❤❡ ✈❡rt✐❝❛❧ ❝❤❛♥❣❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❛❧✉❡s✮ + ✭t❤❡ ❤♦r✐③♦♥t❛❧ ❝❤❛♥❣❡ ♦❢ t❤❡ ✈❡rt✐❝❛❧ ✈❛❧✉❡s✮✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❊①❛❝t♥❡ss ❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 2✱ ❛ ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ♥♦t ❡①❛❝t ✇❤❡♥ ∆y p 6= ∆x q ✳ ❲❡ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ t♦ st✉❞② t❤✐s ❢✉rt❤❡r✳

❉❡✜♥✐t✐♦♥ ✻✳✽✳✷✿ ❞✐✛❡r❡♥❝❡

❋♦r ❛ ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ G ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ❛t t❤❡ 2✲❝❡❧❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆G = ∆x q − ∆y p

✇❤❡r❡ p ❛♥❞ q ❛r❡ t❤❡ x✲ ❛♥❞ y ✲❝♦♠♣♦♥❡♥ts ♦❢ G ✭✐✳❡✳✱ ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❡❞❣❡s r❡s♣❡❝t✐✈❡❧②✮✳ ■t ✐s ❛s ✐❢ ✇❡ ❝♦✈❡r t❤❡ ✇❤♦❧❡ str❡❛♠ ✇✐t❤ t❤♦s❡ ❧✐tt❧❡ ❜❛❧❧s ❛♥❞ st✉❞② t❤❡✐r r♦t❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✻✳✽✳✸✿ ❝❧♦s❡❞ ❢✉♥❝t✐♦♥ ■❢ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ③❡r♦✱ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ❝❛❧❧❡❞ ❝❧♦s❡❞✳ ❚❤❡ ♥❡❣❛t✐✈❡ r♦t❛t✐♦♥ s✐♠♣❧② ♠❡❛♥s r♦t❛t✐♦♥ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳

❊①❛♠♣❧❡ ✻✳✽✳✹✿ r♦t❛t❡ ✇✐t❤♦✉t r♦t❛t✐♦♥ ❆❧❧ ✈❡❝t♦r ✜❡❧❞s ❤❛✈❡ ✈❡❝t♦rs t❤❛t ❝❤❛♥❣❡ ❞✐r❡❝t✐♦♥s✱ ✐✳❡✳✱ r♦t❛t❡✳ ❲❤❛t ✐❢ t❤❡② ❞♦♥✬t❄ ▲❡t✬s ❝♦♥s✐❞❡r ❛

✻✳✽✳

❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✹✽

✢♦✇ ✇✐t❤ ❛ ❝♦♥st❛♥t ❞✐r❡❝t✐♦♥ ❜✉t ✈❛r✐❛❜❧❡ ♠❛❣♥✐t✉❞❡✿ •− →→ −•

❋♦r❝❡ =

|

|

|

|

·

•− 2 −• =

·

•−

→

−•

|

|

0

0

|

|

•− 1 −•

❚❤❡ r♦t♦r ✐s −1✱ ❜✉t ✇❤❡r❡ ✐s r♦t❛t✐♦♥❄ ❲❡❧❧✱ t❤❡ s♣❡❡❞ ♦❢ t❤❡ ✇❛t❡r ♦♥ ♦♥❡ s✐❞❡ ✐s ❢❛st❡r t❤❛♥ ♦♥ t❤❡ ♦t❤❡r ❛♥❞ t❤✐s ❞✐✛❡r❡♥❝❡ ✐s t❤❡ ❝❛✉s❡ ♦❢ t❤❡ ❜❛❧❧✬s s♣✐♥♥✐♥❣✳ ❲✐t❤ t❤✐s ♥❡✇ ❝♦♥❝❡♣t✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤❡ ❊①❛❝t♥❡ss ❚❡st✳ ❚❤❡♦r❡♠ ✻✳✽✳✺✿ ❊①❛❝t♥❡ss ❚❡st ❉✐♠❡♥s✐♦♥

2

■❢ G ✐s ❡①❛❝t✱ ✐t ✐s ❝❧♦s❡❞❀ ❜r✐❡✢②✿ ∆(∆h) = 0

❢♦r ❛❧❧ h✳

▲❡t✬s tr② ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ✈❡❝t♦r ✜❡❧❞s✳ ❊①❛♠♣❧❡ ✻✳✽✳✻✿ ❝♦♠♣✉t❛t✐♦♥

❲❡ r❡♣r❡s❡♥t ❡❛❝❤ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ ✐ts x✲ ❛♥❞ y ✲❝♦♠♣♦♥❡♥ts✿ •− →→ −•

❋♦r❝❡ =

|

|

↓ |

•−

↑ →

|

−•

•− =

< 2, 0 >

|

|

< 0, −1 > |

•−

−• < 0, 1 >

< 1, 0 >

|

−•

❚❤❡ ❡①♣r❡ss✐♦♥ ❝❛♥ t❤❡♥ ❜❡ s❡❡♥ ❛s✿ ◮ W = −✭t❤❡ ✈❡rt✐❝❛❧ ❝❤❛♥❣❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❝t♦rs✮ + ✭t❤❡ ❤♦r✐③♦♥t❛❧ ❝❤❛♥❣❡ ♦❢ t❤❡ ✈❡rt✐❝❛❧ ✈❡❝t♦rs✮❀ ♦r✿ W = ❤♦r✐③♦♥t❛❧✿ − (2 − 1) + ✈❡rt✐❝❛❧✿ 1 − (−1) .

❖❢ ❝♦✉rs❡✱ ♦♥❧② t❤❡ ✈❡rt✐❝❛❧✴❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦rs ❛❝t✐♥❣ ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧✴❤♦r✐③♦♥t❛❧ ❡❞❣❡s ♠❛tt❡r✦ ❙♦ t❤❡ r❡s✉❧t s❤♦✉❧❞ r❡♠❛✐♥ t❤❡ s❛♠❡ ✐❢ ✇❡ ♠♦❞✐❢② ♠❛❦❡ t❤❡ ♦t❤❡r ❝♦♠♣♦♥❡♥ts ♥♦♥✲③❡r♦✿

✻✳✽✳

❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✹✾

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ •−

−•

< 2, 0 >

|

|

F = < 1/2, −1 > |

•−

< 1, 1 > |

< 1, −1 >

−•

❚❤❡ ✈❛❧✉❡ ♦❢ W ❛❜♦✈❡ r❡♠❛✐♥s t❤❡ s❛♠❡ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ❢♦r❝❡s ❛r❡ ❞✐r❡❝t❡❞ ♦✛ t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❜❛❧❧✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❜❡t✇❡❡♥ ❛ r❡❛❧✲✈❛❧✉❡❞ 1✲❢♦r♠ ❛♥❞ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠✳ ■❢ F =< p, q >✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣♦♥❡♥t✇✐s❡✿ •−

2 −•

p = 1/2

1

| |

•−

•−

|

=⇒ ∆y p = 2 − 1 = 1,

|

0

|

−• |

q = −1

1

|

1 −•

=⇒ ∆x y = 1 − (−1) = 2 .

|

•− −1 −•

❚❤❡♥✱ W = −∆y p + ∆x q = −1 + 2 = 1 .

❚❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r r♦t♦r ❢r♦♠ ❜❡❢♦r❡✦

❍❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❛rr✐✈❡ t♦ t❤✐s q✉❛♥t✐t②✳ ■❢ C ✐s t❤❡ ❜♦r❞❡r ♦❢ t❤❡ sq✉❛r❡ ♦r✐❡♥t❡❞ ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✱ t❤❡ ❧✐♥❡ s✉♠ ❛❧♦♥❣ C ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ W =

X

F

C

= < 1/2, −1 > · < 0, −1 > +

< 2, 0 > · < −1, 0 >

+

+

< 1, 1 > · < 0, 1 >

< 1, −1 > · < 1, 0 >

−2 + = 1

+

+

1

1

= 1.

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ●r❛❞✐❡♥t

❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 2✱ ❛ ✈❡❝t♦r ✜❡❧❞ F =< p, q > ✐s ♥♦t ❣r❛❞✐❡♥t ✇❤❡♥ ∆p ∆q 6= . ∆y ∆x

❲❡ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s t♦ st✉❞② t❤✐s ❢✉rt❤❡r✳

✻✳✽✳

❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✺✵

❉❡✜♥✐t✐♦♥ ✻✳✽✳✼✿ r♦t♦r ❋♦r ❛ ✈❡❝t♦r ✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✭t❤❡ 1✲❝❡❧❧s✮ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ r❡❣✐♦♥ ✐♥ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ r♦t♦r ♦❢ F ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❡rt✐❛r② ♥♦❞❡s ✭t❤❡ 2✲❝❡❧❧s✮ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ rot F =

∆p ∆q − ∆y ∆x

✇❤❡r❡ p ❛♥❞ q ❛r❡ t❤❡ x✲ ❛♥❞ y ✲❝♦♠♣♦♥❡♥ts ♦❢ V ✭✐✳❡✳✱ ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❡❞❣❡s r❡s♣❡❝t✐✈❡❧②✮✳

❉❡✜♥✐t✐♦♥ ✻✳✽✳✽✿ ✐rr♦t❛t✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞

■❢ t❤❡ r♦t♦r ✐s ③❡r♦✱ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❝❛❧❧❡❞ ✐rr♦t❛t✐♦♥❛❧✳ ❖♥❡ ❝❛♥ s❡❡ ❛ ❤✐❣❤ ✈❛❧✉❡ ♦❢ t❤❡ r♦t♦r ✐♥ t❤❡ ❝❡♥t❡r ❛♥❞ ③❡r♦ ❛r♦✉♥❞ ✐t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✿

❊①❛♠♣❧❡ ✻✳✽✳✾✿ r♦t♦r ♦❢ ❣r❛❞✐❡♥t ❋r♦♠ t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ♠✐①❡❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ r♦t♦r ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡s ✈❛❧✉❡s ❡①❛❝t❧② ❡q✉❛❧ t♦ 0✿

❲✐t❤ t❤✐s ♥❡✇ ❝♦♥❝❡♣t✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤❡ ●r❛❞✐❡♥t ❚❡st✳

❈♦r♦❧❧❛r② ✻✳✽✳✶✵✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ 2 ■❢ ❛ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ ✐t✬s ✐rr♦t❛t✐♦♥❛❧✳

❲❤❛t ❛❜♦✉t t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡❄ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♣❧❛❝❡ ❛ s♠❛❧❧ ❜❛❧❧ ✇✐t❤✐♥ t❤❡ ✢♦✇ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❜❛❧❧ r❡♠❛✐♥s ✜①❡❞ ✇❤✐❧❡ ❜❡✐♥❣ ❛❜❧❡ t♦ r♦t❛t❡✳ ■❢ t❤❡ ❜❛❧❧ ❤❛s ❛ r♦✉❣❤ s✉r❢❛❝❡✱ t❤❡ ✢✉✐❞ ✢♦✇✐♥❣ ♣❛st ✐t ✇✐❧❧ ♠❛❦❡ ✐t s♣✐♥✳ ■♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ ❡❛❝❤ ❢❛❝❡✱ ✐✳❡✳✱ ❛ 2✲❝❡❧❧✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐s s✉❜❥❡❝t t♦ t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧②s✐s ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❜❛❧❧ ❧♦❝❛t❡❞ ✇✐t❤✐♥ ❛ ❢❛❝❡ r♦t❛t❡s ❛r♦✉♥❞ t❤❡ ❛①✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢❛❝❡✿

✻✳✽✳ ❍♦✇ ❛ ❜❛❧❧ ✐s s♣✉♥ ❜② t❤❡ str❡❛♠

✹✺✶

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❊①❛❝t♥❡ss ❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 3✱ G =< p, q, r > ✐s ♥♦t ❡①❛❝t ✇❤❡♥ ♦♥❡ ♦❢ t❤❡s❡ ❢❛✐❧s✿ ∆y p = ∆x q, ∆z q = ∆y r, ∆x r = ∆z p .

❲❡ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r ✜❡❧❞ t♦ st✉❞② t❤✐s ❢✉rt❤❡r✳

❉❡✜♥✐t✐♦♥ ✻✳✽✳✶✶✿ ❞✐✛❡r❡♥❝❡ ❋♦r ❛ ❢✉♥❝t✐♦♥ F ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✭❡❞❣❡s✮ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ xyz ✲s♣❛❝❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ G ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ✭2✲❝❡❧❧s✮ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧ ✐♥ t❤❡ xyz ✲s♣❛❝❡ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

∆G =

    ∆y r − ∆z q   

∆z p − ∆x r

∆x q − ∆y p

♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡ ♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡ ♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡

✇❤❡r❡ p✱ q ✱ ❛♥❞ r ❛r❡ t❤❡ x✲✱ y ✲✱ ❛♥❞ z ✲❝♦♠♣♦♥❡♥ts ♦❢ G r❡s♣❡❝t✐✈❡❧②✳ ■❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ③❡r♦✱ G ✐s ❝❛❧❧❡❞ ❝❧♦s❡❞✳ ❖❢ ❝♦✉rs❡✱ t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ ❞✐✛❡r❡♥❝❡ ✐s ♠❛❞❡ ♦❢ t❤❡ t❤r❡❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ♦♥❡s ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♣❛✐rs ♦❢ ❝♦♦r❞✐♥❛t❡s✳

❚❤❡♦r❡♠ ✻✳✽✳✶✷✿ ❊①❛❝t♥❡ss ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ 3 ■❢ G ✐s ❡①❛❝t✱ ✐t ✐s ❝❧♦s❡❞❀ ❜r✐❡✢②✿ ∆(∆h) = 0

❢♦r ❛❧❧ h✳

❙❛♠❡ st❛t❡♠❡♥t ❛s ❢♦r ❞✐♠❡♥s✐♦♥ 2✦ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ●r❛❞✐❡♥t ❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 3✱ ❛ ✈❡❝t♦r ✜❡❧❞ V =< p, q, r > ✐s ♥♦t ❣r❛❞✐❡♥t ✇❤❡♥ ♦♥❡ ♦❢ t❤❡s❡ ❢❛✐❧s✿ ∆p ∆q ∆q ∆r ∆r ∆p = , = , = . ∆y ∆x ∆z ∆y ∆x ∆z

❉❡✜♥✐t✐♦♥ ✻✳✽✳✶✸✿ ❝✉r❧ ❋♦r ❛ ❢✉♥❝t✐♦♥ F ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ xyz ✲s♣❛❝❡✱ t❤❡ ❝✉r❧✱ ♦❢ F ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ❛t t❤❡ 2✲❝❡❧❧s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❡❧❧

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✷

✐♥ t❤❡ xyz ✲s♣❛❝❡ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿  ∆r ∆q    −   ∆y ∆z   ∆p ∆r curl F = −  ∆z ∆x    ∆q ∆p    − ∆x ∆y

♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡ ♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡ ♦♥ t❤❡ ❢❛❝❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡

✇❤❡r❡ p✱ q ✱ ❛♥❞ r ❛r❡ t❤❡ x✲✱ y ✲✱ ❛♥❞ z ✲❝♦♠♣♦♥❡♥ts ♦❢ F r❡s♣❡❝t✐✈❡❧②✳ ■❢ t❤❡ ❝✉r❧ ✐s ③❡r♦✱ F ✐s ❝❛❧❧❡❞ ✐rr♦t❛t✐♦♥❛❧✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❝✉r❧ ✐s ♠❛❞❡ ♦❢ t❤❡ t❤r❡❡ r♦t♦rs ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❤r❡❡ ♣❛✐rs ♦❢ ❝♦♦r❞✐♥❛t❡s✳ ❈♦r♦❧❧❛r② ✻✳✽✳✶✹✿ ●r❛❞✐❡♥t ✐s ■rr♦t❛t✐♦♥❛❧

■❢ ❛ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t✱ t❤❡♥ ✐t✬s ✐rr♦t❛t✐♦♥❛❧✳ ❙❛♠❡ st❛t❡♠❡♥t✦ ❚❤❡ t✇♦ t❤❡♦r❡♠s ❝❛♥ ❜❡ r❡st❛t❡❞ ✐♥ ❛♥ ❡✈❡♥ ♠♦r❡ ❝♦♥❝✐s❡ ❢♦r♠✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✿ ∆∆ = 0

❲❤❡♥ ♥♦ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ s♣❡❝✐✜❡❞✱ ✇❡ ❞❡❛❧ ✇✐t❤ ❞✐s❝r❡t❡ ❢♦r♠s✳ ❚❤❡♥✱ ✐❢ tr❛✈❡❧ ❛❧♦♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠✱ ✇❡ ❡♥❞ ✉♣ ❛t ③❡r♦ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s✿ 0✲❢♦r♠s

∆

−−−−→

1✲❢♦r♠s

∆

−−−−→

2✲❢♦r♠s.

✻✳✾✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡✲ ❣r❡❡

2

❙✉♣♣♦s❡ ❝✉r✈❡ C ✐s t❤❡ ❜♦r❞❡r ♦❢ t❤❡ r❡❝t❛♥❣❧❡ R ♦r✐❡♥t❡❞ ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✳

❙✉♣♣♦s❡ t❤❡ ✢♦✇ ✐s ❣✐✈❡♥ ❜② t❤❡s❡ ♥✉♠❜❡rs ❛s ❞❡✜♥❡❞ ♦♥ ❡❛❝❤ ♦❢ t❤❡ ❡❞❣❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡✿ • p3

G = q4

• p1

•

q2 •

• ← •

C= ↓

↑

• → •

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✸

❚❤❡♥ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ C ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

W =

X

G

C

=

+ −q4

−p3

+

+

q2

p1

= −p3 − q4 + q2 + p1

= (q2 − q4 ) − (p3 − p1 ) ❚❤✐s ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❝❤❛♥❣❡ ♦❢ q ❛♥❞ ✈❡rt✐❝❛❧ ❝❤❛♥❣❡ ♦❢ p.

= ∆x G − ∆y G

= ∆G .

❆s ②♦✉ ❝❛♥ s❡❡✱ r❡❛rr❛♥❣✐♥❣ t❤❡ ❢♦✉r t❡r♠s ♦❢ t❤❡ ✇♦r❦ t❤❛t ❝♦♠❡ ❢r♦♠ t❤❡ tr✐♣ ❛r♦✉♥❞ t❤❡ sq✉❛r❡ ❝r❡❛t❡s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❋✐rst✱ ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❡rt✐❝❛❧ ✢♦✇ ♦♥ t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ❜❛❧❧ ❛♥❞✱ s❡❝♦♥❞✱ ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✢♦✇ ♦♥ t❤❡ ♦t❤❡r t✇♦ s✐❞❡s✳ ❋✐♥❛❧❧②✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s ❛♣♣❡❛rs ❛♥❞ ✐t ✐♥❞✐❝❛t❡s t❤❡ t♦t❛❧ ✢♦✇✳ ■t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ G✳ ❲❡ ❤❛✈❡ ❛ ♣r❡❧✐♠✐♥❛r② r❡s✉❧t ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✻✳✾✳✶✿ ❙✉♠ ❆r♦✉♥❞ ❈❡❧❧

■♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❣✐♦♥ R✱ ✐❢ C ✐s ❛ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡ t❤❛t ❝♦♥st✐t✉t❡s t❤❡ ❜♦✉♥❞❛r② ♦❢ ❛ s✐♥❣❧❡ 2✲❝❡❧❧ D ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜② ❣♦✐♥❣ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❛r♦✉♥❞ D✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ X

G = ∆G

C

❲❤❛t ✐❢ ✇❡ ❤❛✈❡ ❛ ♠♦r❡ ❝♦♠♣❧❡① ♦❜❥❡❝t ✐♥ t❤❡ str❡❛♠❄ ❋♦r ❡①❛♠♣❧❡✱ ❛ ❜♦❛t✿

❍♦✇ ❞♦ ✇❡ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ ✢♦✇ ❛r♦✉♥❞ ✐t❄ ❲❡ ❛♣♣r♦❛❝❤ t❤❡ ♣r♦❜❧❡♠ ❛s ❢♦❧❧♦✇s✿ ❲❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡r❡ ❛r❡ ♠❛♥② ❧✐tt❧❡ ❜❛❧❧s ✐♥ t❤❡ ✢♦✇ ❢♦r♠✐♥❣ t❤❡ s❤❛♣❡ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ✢♦✇ ❛r♦✉♥❞ t❤❡ ❜❛❧❧s✳ ◆♦t❡ t❤❛t ❡✈❡r② ❜❛❧❧ ✇✐❧❧ tr② t♦ r♦t❛t❡ ❛❧❧ ♦❢ ✐ts ❛❞❥❛❝❡♥t ❜❛❧❧s ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ❛t t❤❡ s❛♠❡ s♣❡❡❞ ✇✐t❤ ♥♦ ♠♦r❡ ✢♦✇ r❡q✉✐r❡❞✳ ❚❤✐s ✐❞❡❛ ♦❢ ❝❛♥❝❡❧❧❛t✐♦♥ ♦❢ s♣✐♥ t❛❦❡s ❛♥ ❛❧❣❡❜r❛✐❝ ❢♦r♠ ❜❡❧♦✇✳ ❲❡ ✇✐❧❧ st❛rt ✇✐t❤ ❛ s✐♥❣❧❡ r❡❝t❛♥❣❧❡ ❛♥❞ t❤❡♥ ❜✉✐❧❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ❝♦♠♣❧❡① r❡❣✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡ ❢r♦♠ t❤❡ r❡❝t❛♥❣❧❡s ♦❢ ♦✉r ❣r✐❞ ✕ ❛s ✐❢ ❡❛❝❤ ❝♦♥t❛✐♥s ❛ ❜❛❧❧ ✕ ✇❤✐❧❡ ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❢♦r♠✉❧❛✿

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✹

▲❡t✬s ♣✉t t✇♦ r❡❝t❛♥❣❧❡s t♦❣❡t❤❡r✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❛❞❥❛❝❡♥t ♦♥❡s✱ R1 ❛♥❞ R2 ✱ ❜♦✉♥❞❡❞ ❜② ❝✉r✈❡s C1 ❛♥❞ C2 ✳ ❲❡ ✇r✐t❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ❢♦r ❡✐t❤❡r ❛♥❞ t❤❡♥ ❛❞❞ t❤❡ t✇♦✿ X

G

=

C1

+

X

X

∆G

R1

G

=

C2

X

X

∆G

R2

G =

C1 ∪C2

X

∆G

R1 ∪R2

■♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ s✉♠ ❛❝❝♦r❞✐♥❣ t♦ ❆❞❞✐t✐✈✐t② ♦❢ s✉♠s ❛♥❞ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ s✉♠s ❛❝❝♦r❞✐♥❣ t♦ ❆❞❞✐t✐✈✐t②✳ ❍❡r❡ C1 ∪ C2 ✐s t❤❡ ❝✉r✈❡ t❤❛t ❝♦♥s✐sts ♦❢ C1 ❛♥❞ C2 tr❛✈❡❧❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✳ ◆♦✇✱ t❤✐s ✐s ❛♥ ✉♥s❛t✐s❢❛❝t♦r② r❡s✉❧t ❜❡❝❛✉s❡ C1 ∪ C2 ❞♦❡s♥✬t ❜♦✉♥❞ R1 ∪ R2 ✳ ❋♦rt✉♥❛t❡❧②✱ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞✿ t❤❡ t✇♦ ❝✉r✈❡s s❤❛r❡ ❛♥ ❡❞❣❡ ❜✉t tr❛✈❡❧ ✐t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✳

❲❡ ❤❛✈❡ ❛ ❝❛♥❝❡❧❧❛t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ ◆❡❣❛t✐✈✐t② ❢♦r s✉♠s✳ ❚❤❡ r❡s✉❧t ✐s✿ X ∂D

∆G =

X

∆G ,

D

✇❤❡r❡ D ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s ❛♥❞ ∂D ✐s ✐ts ❜♦✉♥❞❛r②✳ ❲❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ❢♦r t❤✐s ♠♦r❡ ❝♦♠♣❧❡① r❡❣✐♦♥✦ ❲❡ ❝♦♥t✐♥✉❡ ♦♥ ❛❞❞✐♥❣ ♦♥❡ r❡❝t❛♥❣❧❡ ❛t ❛ t✐♠❡ t♦ ♦✉r r❡❣✐♦♥ D ❛♥❞ ❝❛♥❝❡❧❧✐♥❣ t❤❡ ❡❞❣❡s s❤❛r❡❞ ✇✐t❤ ♦t❤❡rs ♣r♦❞✉❝✐♥❣ ❜✐❣❣❡r ❛♥❞ ❜✐❣❣❡r ❝✉r✈❡ C = ∂D t❤❛t ❜♦✉♥❞s D✿

❲❡ ❝❛♥ ❛❞❞ ❛s ♠❛♥② r❡❝t❛♥❣❧❡s ❛s ✇❡ ❧✐❦❡ ❛♥❞ ♣r♦❞✉❝✐♥❣ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r r❡❣✐♦♥ ♠❛❞❡ ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❛♥❞ ❜♦✉♥❞❡❞ ❜② ❛ s✐♥❣❧❡ ❝❧♦s❡❞ ❝✉r✈❡ ♠❛❞❡ ♦❢ ❡❞❣❡s✳✳✳ ✉♥❧❡ss ✇❡ ❝✐r❝❧❡ ❜❛❝❦✦

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡

✹✺✺

2

❚❤❡♥ t❤❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ♠✐❣❤t ❜r❡❛❦ ✐♥t♦ t✇♦✳✳✳ ❲❡ ✇✐❧❧ ✐❣♥♦r❡ t❤✐s ♣♦ss✐❜✐❧✐t② ❢♦r ♥♦✇ ❛♥❞ st❛t❡ t❤❡ s❡❝♦♥❞ ♣r❡❧✐♠✐♥❛r② ✈❡rs✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✻✳✾✳✷✿ ❙✉♠ ❆r♦✉♥❞ ❘❡❣✐♦♥

■♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❣✐♦♥ R✱ ✐❢ C ✐s ❛ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡ t❤❛t ❝♦♥st✐t✉t❡s t❤❡ ❜♦✉♥❞❛r② ♦❢ R ❜② ❣♦✐♥❣ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❛r♦✉♥❞ R✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ X

G=

X

∆G

R

C=∂R

❲❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡❄ ❚❤❡♥ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ③❡r♦ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♦✉r ❢♦r♠✉❧❛ t❛❦❡s t❤✐s ❢♦r♠✿ X

∆G =

X

∆G =

D

∂D

X

0 = 0.

D

❚❤❡ s✉♠ ❛❧♦♥❣ ❛♥② ❝❧♦s❡❞ ❝✉r✈❡ ✐s t❤❡♥ ③❡r♦ ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡♥✱ X C

✱ G ✐s ♣❛t❤✲

❚❤❡♦r❡♠

∆G = f (B) − f (A) ,

❢♦r ❛♥② ❝✉r✈❡ C ❢r♦♠ A t♦ B ✱ ✇❤❡r❡ f ✐s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ G✳ ❲❡ ❤❛✈❡ ❛rr✐✈❡❞ ❛t t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❞✐✛❡r❡♥❝❡s✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✐s ✐ts ❣❡♥❡r❛❧✐③❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ❛s t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ✐✳❡✳✱ ❞❡❣r❡❡ 1✱ ✐♥❞✐❝❛t❡s✱ t❤❡r❡ ❛r❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ❢♦r ❡❛❝❤ ❞✐♠❡♥s✐♦♥ ✦ ❲❤❛t ✐s ♦✉r ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ y ✱ ✐✳❡✳✱ G(x, y) = q(x)✱ ✇❤✐❧❡ R ✐s ❛ r❡❝t❛♥❣❧❡ [a, b] × [c, d]❄ ■♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛✱ t❤❡ s✉♠s ❛❧♦♥❣ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ s✐❞❡s ♦❢ R ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✿ X C

G · dX = q(b) − q(b) .

■♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ❤❛✈❡✿ X R

❲❡ ❤❛✈❡ ❛rr✐✈❡❞ ❛t t❤❡ ♦r✐❣✐♥❛❧ ✭❈❤❛♣t❡r ✸■❈✲✶✮✿

∆G =

X

∆x G =

X

∆q .

[a,b]

[a,b]×[c,d]

❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s

q(b) − q(a) =

X

✭❞❡❣r❡❡ 1✮ ❢r♦♠ ❱♦❧✉♠❡ ✷

∆q .

[a,b]

◆♦t ♦♥❧② ❤❛✈❡ ✇❡ ❞❡r✐✈❡❞ t❤❡ ❞❡❣r❡❡ 1 ❢r♦♠ ❞❡❣r❡❡ 2✱ ❜✉t ❛❧s♦ ❜♦t❤ t❤❡♦r❡♠s ❤❛✈❡ t❤❡ s❛♠❡ ❢♦r♠✦ ❲❡ r❡❛❧✐③❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✱ X X q=

{a,b}

∆q ,

[a,b]

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✻

t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❛♥ ✐♥t❡❣r❛❧ ♦❢ ❛ 1✲❢♦r♠ ♦✈❡r ❛ ✭1✲❞✐♠❡♥s✐♦♥❛❧✮ r❡❣✐♦♥✱ R = [a, b]✱ ✇❤✐❧❡ t❤❡ ❧❡❢t ❤❛♥❞✲s✐❞❡ ✐s ❛ 0✲❢♦r♠ ♦✈❡r t❤❡ ❜♦✉♥❞❛r②✱ ∂R = {a, b}✱ ♣r♦♣❡r❧② ♦r✐❡♥t❡❞✱ ♦❢ t❤❛t r❡❣✐♦♥✳

◆♦✇✱ ✇❤❛t ✐❢ t❤❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ❞♦❡s ❜r❡❛❦ ✐♥t♦ t✇♦ ✇❤❡♥ ✇❡ ❛❞❞ ❛ ♥❡✇ sq✉❛r❡❄ ■♥ t❤❡ ❡①❛♠♣❧❡ ❜❡❧♦✇ t❤❡ sq✉❛r❡ ✐s ❛❞❞❡❞ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♦❢ ✐ts ❡❞❣❡s✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❛❞❞ t❤❡ t✇♦ ✈❡rt✐❝❛❧ ❡❞❣❡s ✇❤✐❧❡ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ ❛r❡ ❝❛♥❝❡❧ ❛s ❜❡❢♦r❡✳ ❚❤✉s ❛ ♥❡✇ sq✉❛r❡ ✐s s❡❛♠❧❡ss❧② ❛❞❞❡❞ ❜✉t ✇❡ ❛❧s♦ s❡❡ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛ ❤♦❧❡ ✿

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❞r❛♠❛t✐❝✿ ♥♦t ♦♥❧② t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ r❡❣✐♦♥ ✐s ♥♦✇ ♠❛❞❡ ♦❢ t✇♦ ❝✉r✈❡s ❜✉t ❛❧s♦ t❤❡ ♦♥❡ ♦✉ts✐❞❡ ❣♦❡s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭❛s ❜❡❢♦r❡✮ ✇❤✐❧❡ t❤❡ ♦♥❡ ✐♥s✐❞❡ ❣♦❡s ❝❧♦❝❦✇✐s❡✦ ❍♦✇❡✈❡r✱ ❡✐t❤❡r ❝✉r✈❡ ❤❛s t❤❡ r❡❣✐♦♥ t♦ ✐ts ❧❡❢t✳ ❖✉r ❢♦r♠✉❧❛✱

X C

G=

X

∆G ,

R

❞♦❡s♥✬t ✇♦r❦ ❛♥②♠♦r❡✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s st✐❧❧ ❝❧❡❛r✳ ❇✉t ✇❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡❄ ■t s❤♦✉❧❞ ❜❡ t❤❡ t♦t❛❧ s✉♠ ♦❢ G ♦✈❡r ❛❧❧ ❜♦✉♥❞❛r② ❝✉r✈❡s ♦❢ R✱ ❝♦rr❡❝t❧② ♦r✐❡♥t❡❞✦

❚❤✉s✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ r❡❣✐♦♥ R ✐♥ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ❜♦✉♥❞❛r② ∂R✳ ❚❤❡♦r❡♠ ✻✳✾✳✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡❣r❡❡

2

■♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❣✐♦♥ R✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ G ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ X ∂R

G=

X

∆G

R

❊①❛♠♣❧❡ ✻✳✾✳✹✿ ❧♦♦♣ ✐s ❜♦✉♥❞❛r②

❲❡ ❦♥♦✇ t❤❛t ❢♦r ❛ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② ❛ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡✱ t❤❡ s✉♠ ❛❧♦♥❣ ❛♥② ❝❧♦s❡❞ ❝✉r✈❡ ✐s 0✳ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ✇❤❛t ❤❛♣♣❡♥s ✐♥ r❡❣✐♦♥s ✇✐t❤ ❤♦❧❡s✳ ❈♦♥s✐❞❡r t❤✐s r♦t❛t✐♦♥ ❢✉♥❝t✐♦♥ G✿

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✼

■ts ✈❛❧✉❡s ❛r❡ ±1 ✇✐t❤ ❞✐r❡❝t✐♦♥s ✐♥❞✐❝❛t❡❞ ❡①❝❡♣t ❢♦r t❤❡ ❢♦✉r ❡❞❣❡s ✐♥ t❤❡ ♠✐❞❞❧❡ ✇✐t❤ ✈❛❧✉❡s ♦❢ ±3✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ 3 × 3 r❡❣✐♦♥ R t❤❛t ❡①❝❧✉❞❡s t❤❡ ♠✐❞❞❧❡ sq✉❛r❡✳ ❇② ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ G ✐s ③❡r♦ ❛t ❡✈❡r② ❢❛❝❡ ♦❢ R✿

∆G = 0 . ❙♦✱ G ♣❛ss❡s t❤❡ ❊①❛❝t♥❡ss ❚❡st ❀ ❤♦✇❡✈❡r✱ ✐s ✐t ❡①❛❝t❄ ❲❡ ❞❡♠♦♥str❛t❡ ♥♦✇ t❤❛t ✐t ✐s ♥♦t✳ ■♥❞❡❡❞✱ t❤❡ s✉♠ ♦❢ G ❛❧♦♥❣ t❤❡ ♦✉t❡r ❜♦✉♥❞❛r② ♦❢ R ✐s♥✬t ③❡r♦✿ X G = 12 . C

❍♦✇ ❞♦❡s ✐t ✇♦r❦ ✇✐t❤ ♦✉r t❤❡♦r❡♠✿

X

G=

C

X

∆G ?

R

■t s❡❡♠s t❤❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ♣♦s✐t✐✈❡ ✇❤✐❧❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ③❡r♦✳✳✳ ❲❤❛t ✇❡ ❤❛✈❡ ♦✈❡r❧♦♦❦❡❞ ✐s t❤❛t G ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ❞✐✛❡r❡♥❝❡ ❛r❡ ✉♥❞❡✜♥❡❞ ❛t t❤❡ ♠✐❞❞❧❡ sq✉❛r❡✦ ❙♦✱ C ❞♦❡s♥✬t ❜♦✉♥❞ R✳ ■♥ ❢❛❝t✱ t❤❡ ❜♦✉♥❞❛r② ♦❢ R ✐♥❝❧✉❞❡s ❛♥♦t❤❡r ❝✉r✈❡✱ C ′ ✱ ❣♦✐♥❣ ❝❧♦❝❦✇✐s❡✳ ❚❤❡♥✱ X X X G= ∆G = 0 . G+ C

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

R

C′

X C

G=

X

G.

−C ′

❙♦✱ ♠♦✈✐♥❣ ❢r♦♠ t❤❡ ❧❛r❣❡r ♣❛t❤ t♦ t❤❡ s♠❛❧❧❡r ✭♦r ✈✐❝❡ ✈❡rs❛✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ t❤❡ s✉♠✦ ❆❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ s✉♠s ❢r♦♠ ♦♥❡ ❝♦r♥❡r t♦ t❤❡ ♦♣♣♦s✐t❡ ❛r❡ 6 ❛♥❞ −6✳ ❚❤❡r❡ ✐s ♥♦ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡✦ ❚♦ s✉♠♠❛r✐③❡✱ ❡✈❡♥ ✇❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✕ ✇✐t❤✐♥ t❤❡ r❡❣✐♦♥ ✕ ③❡r♦✱ t❤❡ s✉♠ ❛❧♦♥❣ ❛ ♣❛t❤ t❤❛t ❣♦❡s t❤❡ ❤♦❧❡ ♠❛② ❜❡ ♥♦♥✲③❡r♦✿

❛r♦✉♥❞

❋✉rt❤❡r♠♦r❡✱ t❤❡ s✉♠ r❡♠❛✐♥s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❝❧♦s❡❞ ❝✉r✈❡s ❛s ❧♦♥❣ ❛s t❤❡② ♠❛❦❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ♥✉♠❜❡r

✻✳✾✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❞❡❣r❡❡ 2

✹✺✽

♦❢ t✉r♥s ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✦ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ❝❤❛♥❣❡s ❛❝❝♦r❞✐♥❣❧②❀ ✐t ❛❧❧ ❞❡♣❡♥❞s ♦♥ ❤♦✇ t❤❡ ❝✉r✈❡ ❣♦❡s ❜❡t✇❡❡♥ t❤❡ ❤♦❧❡s✿

◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ r❡❧❛t✐♦♥ ♦❢ t❤✐s ❧✐♥❡ ✐♥t❡❣r❛❧ t❤❛t r❡♣r❡s❡♥ts t❤❡ t❤❡ ❜❛❧❧ ❛♥❞ t❤❡ r♦t♦r ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳

✇♦r❦

♣❡r❢♦r♠❡❞ ❜② t❤❡ ✢♦✇ t♦ s♣✐♥

❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ✜❡❧❞ F =< p, q > ♦❢ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ♦❢ ❛ ✢✉✐❞ ✢♦✇ ✇✐t❤ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ✇✐t❤✐♥ ✐t t❤❛t ❝❛♥ ❢r❡❡❧② r♦t❛t❡ ❜✉t ♥♦t ♠♦✈❡✳ ❲❡ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ r♦t❛t✐♦♥ ❛s t❤❡ ✇♦r❦ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ t❤❡ ✢♦✇ r♦t❛t✐♥❣ t❤❡ ❜❛❧❧✳ ▲❡t✬s ✜rst s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❣r✐❞ ♦♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ r❡❝t❛♥❣❧❡s✿ ∆x × ∆y .

❙✉♣♣♦s❡ t❤❛t t❤❡ ✢♦✇ r♦t❛t❡s t❤✐s r❡❝t❛♥❣❧❡ ❥✉st ❧✐❦❡ t❤❡ ❜❛❧❧ ❜❡❢♦r❡✳

❲❡ ❝❛♥ ❛❧s♦ ❧♦♦❦ ❛t ❛ ✈❡❝t♦r ✜❡❧❞ F =< p, q > ♦❢ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ♦❢ ❛ ✢✉✐❞ ✢♦✇✳

❚❤✉s✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❛❧♦♥❣ t❤❡ ❜♦✉♥❞❛r② ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐s ❡q✉❛❧ t♦ t❤❡ ✭❞♦✉❜❧❡✮ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ r♦t♦r ♦✈❡r t❤✐s r❡❝t❛♥❣❧❡ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ♦✈❡r ❛♥② r❡❣✐♦♥ ♠❛❞❡ ♦❢ s✉❝❤ r❡❝t❛♥❣❧❡s✳ ❚❤❡♦r❡♠ ✻✳✾✳✺✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ❢♦r ❱❡❝t♦r ❋✐❡❧❞s ■♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❣✐♦♥

R✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② ✈❡❝t♦r

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✺✾

✜❡❧❞ F ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ X ∂R

F · ∆X =

X

rot F ∆A

R

Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ ❧❛st t❤❡♦r❡♠✳ ❙✉♣♣♦s❡ ❝✉r✈❡ C ✐s t❤❡ ❜♦r❞❡r ♦❢ t❤❡ r❡❝t❛♥❣❧❡ R ♦r✐❡♥t❡❞ ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✳ ❙✉♣♣♦s❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❣✐✈❡♥ ❜② t❤❡s❡ ✈❡❝t♦rs ❛s ❞❡✜♥❡❞ ♦♥ ❡❛❝❤ ♦❢ t❤❡ ❡❞❣❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡✱ ✇❤✐❝❤ ✐s s❤♦✇♥ ♦♥ r✐❣❤t✿ •

< p3 , q3 >

•

< p1 , q1 >

F = < p4 , q4 >

•

< p2 , q2 > •

•

< −∆x, 0 >

•

< ∆x, 0 >

∆X = < 0, −∆y >

•

< 0, ∆y > •

❚❤❡♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛❧♦♥❣ C ✐s✿ W =

X C

F · ∆X < p3 , q3 > · < −∆x, 0 >

= + < p4 , q4 > · < 0, −∆y >

+

< p2 , q2 > · < 0, ∆y >

< p1 , q1 > · < ∆x, 0 >

+

−p3 ∆x

= +

−q4 ∆y

+

+

q2 ∆y

p1 ∆x

= −p3 ∆x − q4 ∆y + q2 ∆y + p1 ∆x = (q2 − q4 )∆y − (p3 − p1 )∆x p3 − p1 q 2 − q4 ∆x∆y − ∆y∆x = ∆y  ∆x  ∆q ∆p = − ∆x∆y . ∆x ∆y

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧✲ ❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2 ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ●r❛❞✐❡♥t ❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 2✱ ❛ ✈❡❝t♦r ✜❡❧❞ F =< p, q > ✐s ♥♦t ❣r❛❞✐❡♥t ✇❤❡♥ py 6= qx ✳ ❲❡ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s t♦ st✉❞② t❤✐s ❢✉rt❤❡r ✭❛s ✐❢ ✇❡ ❝♦✈❡r t❤❡ ✇❤♦❧❡ str❡❛♠ ✇✐t❤ t❤♦s❡ ❧✐tt❧❡ ❜❛❧❧s✮✳

✻✳✶✵✳

●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✻✵

❉❡✜♥✐t✐♦♥ ✻✳✶✵✳✶✿ r♦t♦r ❚❤❡

r♦t♦r

♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ r❡❣✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡ ✈❡❝t♦r ✜❡❧❞

F =
✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ t❤❡ r❡❣✐♦♥ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s rot F = qx − py

❉❡✜♥✐t✐♦♥ ✻✳✶✵✳✷✿ ✐rr♦t❛t✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞ ■❢ t❤❡ r♦t♦r ✐s ③❡r♦✱ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❝❛❧❧❡❞

✐rr♦t❛t✐♦♥❛❧✳

❖♥❡ ❝❛♥ s❡❡ ❛ ❤✐❣❤ ✈❛❧✉❡ ♦❢ t❤❡ r♦t♦r ✐♥ t❤❡ ❝❡♥t❡r ❛♥❞ ③❡r♦ ❛r♦✉♥❞ ✐t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✿

❚❤❡ ♥❡❣❛t✐✈❡ r♦t❛t✐♦♥ s✐♠♣❧② ♠❡❛♥s r♦t❛t✐♦♥ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳

❊①❛♠♣❧❡ ✻✳✶✵✳✸✿ r♦t❛t✐♥❣ ✇✐t❤♦✉t r♦t❛t✐♦♥ ❆❧❧ ✈❡❝t♦r ✜❡❧❞s ❤❛✈❡ ✈❡❝t♦rs t❤❛t ❝❤❛♥❣❡ ❞✐r❡❝t✐♦♥s✱ ✐✳❡✳✱ ✏r♦t❛t❡✑✳ ❲❤❛t ✐❢ t❤❡② ❞♦♥✬t❄ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ❛ ❝♦♥st❛♥t ❞✐r❡❝t✐♦♥ ❜✉t ✈❛r✐❛❜❧❡ ♠❛❣♥✐t✉❞❡✳ ▲❡t✬s tr②✿

F (x, y) =< y 2 , 0 > .

❚❤❡♥

rot V = qx − py = 0 − 2y 6= 0 . ❚❤❡ r♦t❛t✐♦♥ ✐s ❛❣❛✐♥ ♥♦♥✲③❡r♦✳ ■♥ ❢❛❝t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ r♦t♦r s❤♦✇s t❤❛t t❤❡ r♦t❛t✐♦♥ ✇✐❧❧ ❜❡ ❝♦✉♥✲ t❡r❝❧♦❝❦✇✐s❡ ♦♥ r✐❣❤t ❛♥❞ ❝❧♦❝❦✇✐s❡ ♦♥ ❧❡❢t✳ ❚❤❡ ❡✛❡❝t ✐s s❡❡♥ ✇❤❡♥ ❛ ♣❡rs♦♥ ❧✐❡s ♦♥ t❤❡ t♦♣ ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✕ ✉♣ ❛♥❞ ❞♦✇♥ ✕ ❡s❝❛❧❛t♦rs✿

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✻✶

❊①❛♠♣❧❡ ✻✳✶✵✳✹✿ r♦t♦r ♦❢ ❣r❛❞✐❡♥t

❋r♦♠ t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ♠✐①❡❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ r♦t♦r ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡s ✈❛❧✉❡s ❡①❛❝t❧② ❡q✉❛❧ t♦ 0✿

❲✐t❤ t❤✐s ♥❡✇ ❝♦♥❝❡♣t✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤❡ ●r❛❞✐❡♥t ❚❡sts ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✳ ❚❤❡♦r❡♠ ✻✳✶✵✳✺✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

2

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ♦♥ ❛♥ ♦♣❡♥ r❡❣✐♦♥ ✐♥ R2 ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r✲ ❡♥t✐❛❜❧❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✳ ■❢ F ✐s ❣r❛❞✐❡♥t ✭✐✳❡✳✱ F = grad h✮✱ t❤❡♥ ✐t✬s ✐rr♦t❛t✐♦♥❛❧✿ rot F = 0❀ ❜r✐❡✢②✿ rot(grad h) = 0

❲❤❛t ❛❜♦✉t 3✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞s❄ ❖♥❝❡ ❛❣❛✐♥✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ✜❡❧❞ t❤❛t ❞❡s❝r✐❜❡s t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ♦❢ ❛ ✢✉✐❞ ✢♦✇✳ ❲❡ ♣❧❛❝❡ ❛ s♠❛❧❧ ❜❛❧❧ ✇✐t❤✐♥ t❤❡ ✢♦✇ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❜❛❧❧ r❡♠❛✐♥s ✜①❡❞ ✇❤✐❧❡ ❜❡✐♥❣ ❛❜❧❡ t♦ r♦t❛t❡✳ ■❢ t❤❡ ❜❛❧❧ ❤❛s ❛ r♦✉❣❤ s✉r❢❛❝❡✱ t❤❡ ✢✉✐❞ ✢♦✇✐♥❣ ♣❛st ✐t ✇✐❧❧ ♠❛❦❡ ✐t s♣✐♥✳ ❚❤❡ ❜❛❧❧ ❝❛♥ r♦t❛t❡ ❛r♦✉♥❞ ❛♥② ❛①✐s✳ ❲❡ ❝❛♥ r❡st❛t❡ t❤❡ ●r❛❞✐❡♥t ❚❡st ❢♦r ❞✐♠❡♥s✐♦♥ 3 ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✻✳✶✵✳✻✿ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥

3

❙✉♣♣♦s❡ F ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ♦♥ ❛♥ ♦♣❡♥ r❡❣✐♦♥ ✐♥ R3 ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r✲ ❡♥t✐❛❜❧❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✳ ■❢ F ✐s ❣r❛❞✐❡♥t ✭✐✳❡✳✱ F = grad h✮✱ t❤❡♥ ✐t✬s ✐rr♦t❛t✐♦♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ❛❧❧ t❤r❡❡ ♣❛✐rs ♦❢ ❝♦♦r❞✐♥❛t❡s✿ roty,z < q, r >= 0, rotz,x < r, p >= 0, rotx,y < p, q >= 0

❚❤❡ s✉❜s❝r✐♣ts ✐♥❞✐❝❛t❡ ✇✐t❤ r❡s♣❡❝t t♦ ✇❤✐❝❤ t✇♦ ✈❛r✐❛❜❧❡s ✇❡ ❞✐✛❡r❡♥t✐❛t❡ ✇❤✐❧❡ t❤❡ t❤✐r❞ t♦ ❜❡ ❦❡♣t ✜①❡❞✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r ✜❡❧❞ ❝❛❧❧❡❞ t❤❡ ❝✉r❧ ♦❢ ❋ t❤❛t t❛❦❡ ❝❛r❡ ♦❢ ❛❧❧ t❤r❡❡ r♦t♦rs✿ curl F = roty,z < q, r > i + rotz,x < r, p > j + rotx,y < p, q > k =< ry − qz , pz − rx , qx − py > .

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✻✷

■♥ ♣❛rt✐❝✉❧❛r✱ ✇❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ V = pi + qj + rk ❤❛s ❛ ③❡r♦ z ✲❝♦♠♣♦♥❡♥t✱ r = 0✱ ✇❤✐❧❡ p ❛♥❞ q ❞♦♥✬t ❞❡♣❡♥❞ ♦♥ z ✱ t❤❡ ❝✉r❧ ✐s r❡❞✉❝❡❞ t♦ t❤❡ r♦t♦r✿

curl(pi + qj) = rot < p, q > k . ❊①❡r❝✐s❡ ✻✳✶✵✳✼

❉❡✜♥❡ ❛ 4✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣ ♦❢ t❤❡ r♦t♦r✳ ❚❤❡ t✇♦ t❤❡♦r❡♠s ❝❛♥ ❜❡ r❡st❛t❡❞ ✐♥ ❛♥ ❡✈❡♥ ♠♦r❡ ❝♦♥❝✐s❡ ❢♦r♠✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✿ rot grad = 0 ❛♥❞ curl grad = 0 ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❡♥❞ ✉♣ ❛t ③❡r♦ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s✿ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❢✉♥❝t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s

grad

−−−−−−→ grad

−−−−−−→

rot

❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s,

curl

✈❡❝t♦r ✜❡❧❞s

✈❡❝t♦r ✜❡❧❞s ✐♥ R2 −−−−−→

✈❡❝t♦r ✜❡❧❞s ✐♥ R3 −−−−−→

❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ✇♦r❦ ✐♥t❡❣r❛❧ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❢♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✳ ❍♦✇ ❞♦ ✇❡ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ ✐ts r♦t❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ ✇♦r❦ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ t❤❡ ✢♦✇ r♦t❛t✐♥❣ ✐t❄

❲❡ s✉♣♣♦s❡ t❤❛t t❤❡r❡ ❛r❡ ♠❛♥② ❧✐tt❧❡ ❜❛❧❧s ✐♥ t❤❡ ✢♦✇ ❢♦r♠✐♥❣ s♦♠❡ s❤❛♣❡ ❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡✐r t♦t❛❧ r♦t❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ ✇♦r❦ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ t❤❡ ✢♦✇ r♦t❛t✐♥❣ t❤❡ ❜❛❧❧s✳

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ st❛rt ✇✐t❤ ❛ s✐♥❣❧❡ r❡❝t❛♥❣❧❡ ❛♥❞ t❤❡♥ ❜✉✐❧❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ❝♦♠♣❧❡① r❡❣✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡ ❢r♦♠ t❤❡ r❡❝t❛♥❣❧❡s ♦❢ ♦✉r ❣r✐❞ ✕ ❛s ✐❢ ❡❛❝❤ ❝♦♥t❛✐♥s ❛ ❜❛❧❧ ✕ ✇❤✐❧❡ ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❢♦r♠✉❧❛✳ ■❢ ✇❡ ❤❛✈❡ ❥✉st t✇♦ ❛❞❥❛❝❡♥t sq✉❛r❡s✱ R1 ❛♥❞ R2 ✱ ❜♦✉♥❞❡❞ ❜② ❝✉r✈❡s C1 ❛♥❞ C2 ✱ ✇❡ ✇r✐t❡ ●r❡❡♥✬s ❢♦r♠✉❧❛ ❢♦r ❡✐t❤❡r ❛♥❞

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2 t❤❡♥ ❛❞❞ t❤❡ t✇♦✿

I

C1

F · dX

=

ZZ

✹✻✸

rot F dA R1

+ I

I

C2

F · dX

C1 ∪C2

=

F · dX =

ZZ

ZZ

rot F dA R2

rot F dA R1 ∪R2

■♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ✐♥t❡❣r❛❧ ❛❝❝♦r❞✐♥❣ t♦ ❆❞❞✐t✐✈✐t② ♦❢ ❞♦✉❜❧❡ ✐♥t❡❣r❛❧s ❛♥❞ ✐♥ t❤❡ ❧❡❢t✲ ❤❛♥❞ s✐❞❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ✐♥t❡❣r❛❧ ❛❝❝♦r❞✐♥❣ t♦ ❆❞❞✐t✐✈✐t② ♦❢ ❧✐♥❡ ✐♥t❡❣r❛❧s✳ ❍❡r❡ C1 ∪ C2 ✐s t❤❡ ❝✉r✈❡ t❤❛t ❝♦♥s✐sts ♦❢ C1 ❛♥❞ C2 tr❛✈❡❧❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✳ ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s s✐♠♣❧✐✜❡❞✿ t❤❡ t✇♦ ❝✉r✈❡s s❤❛r❡ ❛♥ ❡❞❣❡ ❜✉t tr❛✈❡❧ ✐t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✳

❲❡ ❤❛✈❡ ❛ ❝❛♥❝❡❧❧❛t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ ◆❡❣❛t✐✈✐t② ❢♦r ❧✐♥❡ ✐♥t❡❣r❛❧s✳ ❚❤❡ r❡s✉❧t ✐s✿ I

∂D

F · dX =

ZZ

rot F dA , D

✇❤❡r❡ D ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s ❛♥❞ ∂D ✐s ✐ts ❜♦✉♥❞❛r②✳ ❲❡ ❝♦♥t✐♥✉❡ ♦♥ ❛❞❞✐♥❣ ♦♥❡ r❡❝t❛♥❣❧❡ ❛t ❛ t✐♠❡ t♦ ♦✉r r❡❣✐♦♥ D ❛♥❞ ❝❛♥❝❡❧❧✐♥❣ t❤❡ ❡❞❣❡s s❤❛r❡❞ ✇✐t❤ ♦t❤❡rs ♣r♦❞✉❝✐♥❣ ❜✐❣❣❡r ❛♥❞ ❜✐❣❣❡r ❝✉r✈❡ C = ∂D t❤❛t ❜♦✉♥❞s D✿

❖r ✇❡ ❝❛♥ ❛❞❞ ✇❤♦❧❡ r❡❣✐♦♥s✳✳✳ ■t ✐s ♣♦ss✐❜❧❡✱ ❤♦✇❡✈❡r✱ t❤❛t t❤❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ♠✐❣❤t s❡✐③❡ t♦ ❜❡ ❛ s✐♥❣❧❡ ❝❧♦s❡❞ ❝✉r✈❡✦ ❚❤❡♦r❡♠ ✻✳✶✵✳✽✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❱❡❝t♦r ❋✐❡❧❞s

❙✉♣♣♦s❡ ❛ ♣❧❛♥❡ r❡❣✐♦♥ R ✐s ❜♦✉♥❞❡❞ ❜② ♣✐❡❝❡✇✐s❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❝✉r✈❡ C ✭♣♦s✲ s✐❜❧② ♠❛❞❡ ♦❢ s❡✈❡r❛❧ ❞✐s❝♦♥♥❡❝t❡❞ ♣✐❡❝❡s✮✳ ❚❤❡♥ ❢♦r ❛♥② ✈❡❝t♦r ✜❡❧❞ F ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❝♦♠♣♦♥❡♥ts ♦♥ ❛♥ ♦♣❡♥ s❡t ❝♦♥t❛✐♥✐♥❣ R✱ ✇❡ ❤❛✈❡✿ I

C

F · dX =

ZZ

rot F dA R

Pr♦♦❢✳

❲❡ ♦♥❧② ❞❡♠♦♥str❛t❡ t❤❡ ♣r♦♦❢ ❢♦r ❛ r❡❣✐♦♥ R t❤❛t ❤❛s ❛ ♣❛rt✐t✐♦♥ t❤❛t ❛❧s♦ ♣r♦❞✉❝❡s ❛ ♣❛rt✐t✐♦♥ ♦❢ C ✳ ❲❡ s❛♠♣❧❡ F ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ C ❛♥❞ rot F ❛t t❤❡ t❡rt✐❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ R✳ ❲❡ t❤❡♥ ✉s❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✿ X ∂R

F · ∆X =

X R

rot F ∆A .

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✻✹

❲❡ t❛❦❡ t❤❡ ❧✐♠✐ts ♦❢ t❤❡s❡ t✇♦ ❘✐❡♠❛♥♥ s✉♠s ♦✈❡r t❤❡ ♣❛rt✐t✐♦♥s ✇✐t❤ t❤❡ ♠❡s❤ ❛♣♣r♦❛❝❤✐♥❣ ③❡r♦✳ ❚❤✐s ✐s ❛❧s♦ ❦♥♦✇♥ ❛s ●r❡❡♥✬s ❋♦r♠✉❧❛✳ ❲r✐tt❡♥ ❝♦♠♣♦♥❡♥t✇✐s❡✱ ✐t t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ Z

p dx + q dy = C

ZZ

R

(qx − py ) dxdy .

▲❡t✬s tr❛❝❡ t❤❡ t❤❡♦r❡♠ ❜❛❝❦ t♦ s♦♠❡ ❢❛♠✐❧✐❛r t❤✐♥❣s✳ ❲❤❛t ✐❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ❣r❛❞✐❡♥t❄ ❚❤❡♥ ✐ts r♦t♦r ✐s ③❡r♦ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♦✉r ❢♦r♠✉❧❛ t❛❦❡s t❤✐s ❢♦r♠✿ I

∂D

F · dX =

ZZ

rot F dA = D

ZZ

0 dA = 0 . D

❚❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ ❛♥② ❝❧♦s❡❞ ❝✉r✈❡ ✐s t❤❡♥ ③❡r♦ ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ❚❤❡♦r❡♠✱ F ✐s ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡♥✱ Z C

F · dX = f (B) − f (A) ,

❢♦r ❛♥② ❝✉r✈❡ C ❢r♦♠ A t♦ B ✱ ✇❤❡r❡ f ✐s ❛ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ F ✳ ❲❡ ❤❛✈❡ ❛rr✐✈❡❞ ❛t t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞s✳ ■t ❢♦❧❧♦✇s t❤❛t ●r❡❡♥✬s ❚❤❡♦r❡♠ ✐s ✐ts ❣❡♥❡r❛❧✐③❛t✐♦♥✳ ❚❤✐s ❝♦♥✜r♠s t❤❡ r♦❧❡ ♦❢ ●r❡❡♥✬s ❚❤❡♦r❡♠ ❛s t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❛❧❧ ✈❡❝t♦r ✜❡❧❞s ❢♦r ❞✐♠❡♥s✐♦♥ 2✳ ❲❤❛t ✐s t❤❡ ✈❡❝t♦r ✜❡❧❞ ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ y ✱ ✐✳❡✳✱ F (x, y) = F (x) =< p(x), q(x) >✱ ✇❤✐❧❡ R ✐s ❛ r❡❝t❛♥❣❧❡ [a, b] × [c, d]❄ ❋✐rst t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛✳✳✳ ❚❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧s ❛❧♦♥❣ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ s✐❞❡s ♦❢ R ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤✿ I

∂D

F · dX = F (b) · A − F (a) · A ,

✇❤❡r❡ A ✐s t❤❡ ✈❡❝t♦r t❤❛t r❡♣r❡s❡♥ts t❤❡ ✈❡rt✐❝❛❧ s✐❞❡s ♦❢ R ✭♦r✐❡♥t❡❞ ✈❡rt✐❝❛❧❧②✮✳ ❚❤❡♥✱ I

∂D

F · dX = (q(b) − q(a))(d − c) .

◆♦✇ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛✳✳✳ ❚❤❡ r♦t♦r ✐s s✐♠♣❧② q ′ (x)✳ ❚❤❡♥✱ ZZ

rot F dA = R

ZZ

′

q (x) dxdy =

Z bZ a

[a,b]×[c,d]

d ′

q (x) dxdy = c

Z

b a

q ′ (x) dx (d − c) .

❲❡ ❤❛✈❡ ❛rr✐✈❡❞ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢r♦♠ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✶✮✿ q(b) − q(a) =

Z

b

q ′ (x) dx . a

❊①❛♠♣❧❡ ✻✳✶✵✳✾✿ r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ▲❡t✬s✱ ❛❣❛✐♥✱ ❝♦♥s✐❞❡r t❤✐s r♦t❛t✐♦♥ ✈❡❝t♦r ✜❡❧❞ ✿ 1 F = 2 < y, −x >= x + y2



x y , − 2 2 2 x +y x + y2



=< p, q > ,

✇❤✐❝❤ ✐s ✐rr♦t❛t✐♦♥❛❧✿ rot F = qx − py = 0 .

❊✈❡♥ t❤♦✉❣❤ ✐t ♣❛ss❡s t❤❡ ●r❛❞✐❡♥t ❚❡st✱ ✐t ✐s ♥♦t ❣r❛❞✐❡♥t✳ ■♥❞❡❡❞✱ ✐❢ X = X(t) ✐s ❛ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✱ F (X(t)) ✐s ♣❛r❛❧❧❡❧ t♦ X ′ (t)✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ t❤❡

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2 ❝✐r❝❧❡ ✐s ♣♦s✐t✐✈❡✿

Z

✹✻✺

b a

F (X(t)) · X ′ (t) dt > 0 .

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿ ZZ I rot F dA = 0 . F · dX = R

C

❙♦✱ C ❞♦❡s♥✬t ❜♦✉♥❞ R✳

❆ ❤♦❧❡ ✐s ✇❤❛t ♠❛❦❡s ❛ s♣✐r❛❧ st❛✐r❝❛s❡ ♣♦ss✐❜❧❡ ❜② ♣r♦✈✐❞✐♥❣ ❛ ♣❧❛❝❡ ❢♦r t❤❡ ♣♦❧❡✳ ◆♦✇✱ ✇❡✬❞ ♥❡❡❞ R t♦ ❜❡ ❛ r✐♥❣ s♦ t❤❛t t❤❡ ❜♦✉♥❞❛r② ♦❢ R ✇♦✉❧❞ ✐♥❝❧✉❞❡ ❛♥♦t❤❡r ❝✉r✈❡✱ ♠❛②❜❡ ❛ s♠❛❧❧❡r ❝✐r❝❧❡✱ C ′ ✱ ❣♦✐♥❣ ❝❧♦❝❦✇✐s❡✱ ❚❤❡♥✱ I I ZZ C

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

F · dX + I

C

C′

F · dX =

F · dX =

I

−C ′

rot F dA = 0 .

R

F · dX .

❙♦✱ ♠♦✈✐♥❣ ❢r♦♠ t❤❡ ❧❛r❣❡r ❝✐r❝❧❡ t♦ t❤❡ s♠❛❧❧❡r ✭♦r ✈✐❝❡ ✈❡rs❛✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧✱ ✐✳❡✳ t❤❡ ✇♦r❦✳ ❆ r❡♠❛r❦❛❜❧❡ r❡s✉❧t✦ ■t ✐s s❡❡♥ ❛s ❡✈❡♥ ♠♦r❡ r❡♠❛r❦❛❜❧❡ ♦♥❝❡ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ✐♥t❡❣r❛❧ r❡♠❛✐♥s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❝❧♦s❡❞ ❝✉r✈❡s ❛s ❧♦♥❣ ❛s t❤❡② ♠❛❦❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ t✉r♥s ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✦ ❚♦ s✉♠♠❛r✐③❡✱ ❡✈❡♥ ✇❤❡♥ t❤❡ r♦t♦r ✐s ✕ ✇✐t❤✐♥ t❤❡ r❡❣✐♦♥ ✕ ③❡r♦✱ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ ❛ ❝✉r✈❡ t❤❛t ❣♦❡s ❛r♦✉♥❞ t❤❡ ❤♦❧❡ ♠❛② ❜❡ ♥♦♥✲③❡r♦✳

❋✉rt❤❡r♠♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ r❡♠❛✐♥s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❝❧♦s❡❞ ❝✉r✈❡s ❛s ❧♦♥❣ ❛s t❤❡② ♠❛❦❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ t✉r♥s ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✦ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ❝❤❛♥❣❡s ❛❝❝♦r❞✐♥❣❧②❀ ✐t ❛❧❧ ❞❡♣❡♥❞s ♦♥ ❤♦✇ t❤❡ ❝✉r✈❡ ❣♦❡s ❜❡t✇❡❡♥ t❤❡ ❤♦❧❡s✿

✻✳✶✵✳ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✈❡❝t♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ 2

✹✻✻

❊①❛♠♣❧❡ ✻✳✶✵✳✶✵✿ ✇❛❧❦ ❛r♦✉♥❞

■♠❛❣✐♥❡ t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ ❛ ♣✐❡❝❡ ♦❢ ❧❛♥❞ ✇❡ ❤❛✈❡ ♥♦ ❛❝❝❡ss t♦✱ s✉❝❤ ❛s ❛ ❢♦rt✐✜❝❛t✐♦♥ ♦r ❛ ♣♦♥❞✳ ❈♦♥✈❡♥✐❡♥t❧②✱ ●r❡❡♥✬s ❋♦r♠✉❧❛ ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ ❛r❡❛ ♦❢ ❛ r❡❣✐♦♥ ✇✐t❤♦✉t ✈✐s✐t✐♥❣ t❤❡ ✐♥s✐❞❡ ❜✉t ❜② ❥✉st t❛❦✐♥❣ ❛ tr✐♣ ❛r♦✉♥❞ ✐t✳ ❲❡ ❥✉st ♥❡❡❞ t♦ ♣✐❝❦ ❛♥ ❛♣♣r♦♣r✐❛t❡ ✈❡❝t♦r ✜❡❧❞✿ F =< 0, x > =⇒ p = 0, q = x =⇒ py = 0, qx = 1 .

❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ZZ

Z ZR

(qx − py ) dxdy = 1 dxdy

=

Z

p dx +q dy

ZC C

R

❆r❡❛ ♦❢ R

=

0 dx +x dy Z x dy C

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❞✐s❦ R ♦❢ r❛❞✐✉s r ✐s ❛ ❝❡rt❛✐♥ ❧✐♥❡ ✐♥t❡❣r❛❧ ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ C ✳ ❲❡ t❛❦❡ C t♦ ❜❡ ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉s✉❛❧ ✇❛②✿ x = r cos t, y = r sin t .

❚❤❡♥✱ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ =

Z

x dy ZC2π r cos t(r sin t)′ dt = 0Z 2π = r2 cos2 t dt 0  2π 2 = r x/2 + sin 2x 0

= r2 2π/2 = πr2 .

❈❤❛♣t❡r ✿

❈♦♥t❡♥ts ✶ ✷ ✸ ✹ ✺ ✻ ✼

❊①❡r❝✐s❡s✿ ❊①❡r❝✐s❡s✿ ❊①❡r❝✐s❡s✿ ❊①❡r❝✐s❡s✿ ❊①❡r❝✐s❡s✿ ❊①❡r❝✐s❡s✿ ❊①❛♠♣❧❡s

❊①❡r❝✐s❡s

❇❛s✐❝ ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ■♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱❡❝t♦r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✹✻✼ ✹✼✻ ✹✼✽ ✹✽✸ ✹✾✵ ✹✾✸ ✹✾✻

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

❊①❡r❝✐s❡ ✶✳✶

❊①❡r❝✐s❡ ✶✳✹

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x)✱ ✇❤❡r❡ x ✐s t❤❡ ✐♥❝♦♠❡ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❛♥❞ f (x) ✐s t❤❡ t❛① ❜✐❧❧ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❢♦r t❤❡ ✐♥❝♦♠❡ ♦❢ x✱ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ♥♦ t❛① ♦♥ t❤❡ ✜rst $10, 000✱ t❤❡♥ 5% ❢♦r t❤❡ ♥❡①t $10, 000✱ ❛♥❞ 10% ❢♦r t❤❡ r❡st ♦❢ t❤❡ ✐♥❝♦♠❡✳ ■♥✈❡st✐❣❛t❡ ✐ts ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐✲ t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x3 6= 3✳ x→0

❊①❡r❝✐s❡ ✶✳✺

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x3 = +∞✳ x→+∞

❊①❡r❝✐s❡ ✶✳✷

❊①❡r❝✐s❡ ✶✳✻

❊①♣❧❛✐♥ ✇❤② t❤❡ ❧✐♠✐t lim sin x→0

1 ❞♦❡s ♥♦t ❡①✐st✳ x

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t✇♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿ x = 0 ❛♥❞ x = 2✳ ❊①❡r❝✐s❡ ✶✳✼

❊①❡r❝✐s❡ ✶✳✸

✭❛✮ ❙t❛t❡ t❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x2 = 0✳ x→0

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✿ y = −1✱ ❛♥❞ ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✿ x = 2✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✻✽

❊①❡r❝✐s❡ ✶✳✽

■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ♦❢ t❤✐s ❣r❛♣❤✿

❊①❡r❝✐s❡ ✶✳✶✻ ❊①❡r❝✐s❡ ✶✳✾

❆ ❤♦✉s❡ ❤❛s

❊①♣r❡ss t❤❡ ❛s②♠♣t♦t❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❧✐♠✐ts

❲❤❛t ✇❛s t❤❡ ②❡❛r ✇❤❡♥ t❤❡ ❞♦♦r♠❛♥✬s ❣r❛♥❞✲

❛♥❞ ✐❞❡♥t✐❢② ♦t❤❡r ♦❢ ✐ts ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✿

♠♦t❤❡r ❞✐❡❞❄

4

✢♦♦rs ❛♥❞ ❡❛❝❤ ✢♦♦r ❤❛s

7

✇✐♥❞♦✇s✳

❊①❡r❝✐s❡ ✶✳✶✼

■❧❧✉str❛t❡ ✇✐t❤ ♣❧♦ts ✭s❡♣❛r❛t❡❧②✮ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡

f (x) → +∞ ❛s x → 1❀ ✭❜✮ x → 2+ ❀ ✭❝✮ f (x) → 3 ❛s x → −∞✳

❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦r✿ ✭❛✮

f (x) → −∞

❛s

❊①❡r❝✐s❡ ✶✳✶✽

●✐✈❡♥

❋✐♥❞ t❤❡ ❧❡❛❞✐♥❣

t❡r♠ ❛♥❞ ✉s❡ ✐t t♦ ❞❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r

❊①❡r❝✐s❡ ✶✳✶✵

f ✐s (a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

f (x) = −(x − 3)4 (x + 1)3 ✳

(a, b)✱

❝♦♥t✐♥✉♦✉s ♦♥

t❤❡♥

f

♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✶✾ ❊①❡r❝✐s❡ ✶✳✶✶

✭❛✮ ❙t❛t❡ t❤❡ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✳

f ✐s [a, b]✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ❜♦✉♥❞❡❞ ♦♥

❝♦♥t✐♥✉♦✉s ♦♥

[a, b]✱

t❤❡♥

f

✐s

✭❜✮

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✷✵ ❊①❡r❝✐s❡ ✶✳✶✷

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❲❤❛t ✐s s♦

f ✐s [a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

[a, b)✱

t❤❡♥

f

[a, ∞)✱

t❤❡♥

f

❝♦♥t✐♥✉♦✉s ♦♥

s♣❡❝✐❛❧ ❛❜♦✉t t❤❡♠❄ ❋✐♥❞ t❤❡✐r s❧♦♣❡s✳

❊①❡r❝✐s❡ ✶✳✶✸

f ✐s ❝♦♥t✐♥✉♦✉s [a, ∞)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

♦♥

❊①❡r❝✐s❡ ✶✳✶✹

❚r✉❡ ♦r ❢❛❧s❡✿

✏❊✈❡r② ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ♦♥ ❛

❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✑❄

❊①❡r❝✐s❡ ✶✳✷✶

✭❛✮ ❙✉♣♣♦s❡ ❞✉r✐♥❣ t❤❡ ✜rst

❊①❡r❝✐s❡ ✶✳✶✺

❚❤❡ ❣r❛♣❤ ♦❢

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ■t ❤❛s ❛s②♠♣t♦t❡s✳

❉❡s❝r✐❜❡ t❤❡♠ ❛s ❧✐♠✐ts✳ ❍✐♥t✿ ❯s❡ ❜♦t❤

−∞✳

2

s❡❝♦♥❞s ♦❢ ✐ts ✢✐❣❤t

❛♥ ♦❜❥❡❝t ♣r♦❣r❡ss❡❞ ❢r♦♠ ♣♦✐♥t

+∞

❛♥❞

(2, 0)✳

(0, 0)

t♦

(1, 0)

t♦

❲❤❛t ✇❛s ✐ts ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❛♥❞ ❛✈❡r❛❣❡

❛❝❝❡❧❡r❛t✐♦♥❄ ✐♥st❡❛❞❄

✭❜✮ ❲❤❛t ✐❢ t❤❡ ❧❛st ♣♦✐♥t ✐s

(1, 1)

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✻✾

❊①❡r❝✐s❡ ✶✳✷✷

❊①❡r❝✐s❡ ✶✳✷✻

❙✉♣♣♦s❡ t ✐s t✐♠❡ ❛♥❞ x ✐s t❤❡ ♣r✐❝❡ ♦❢ ❜r❡❛❞✳ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t ✐ts ❞②♥❛♠✐❝s❄ ❇❡ ❛s s♣❡❝✐✜❝ ❛s ♣♦ss✐❜❧❡✳

❊❛❝❤ ♦❢ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ ❞r♦✇♥ t❤r♦✉❣❤ t✇♦ ♣♦✐♥t ♦❢ t❤❡ ❣r❛♣❤✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥❄

❊①❡r❝✐s❡ ✶✳✷✸

❋✐♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛✿ x

y = f (x)

−1

2

1

2

3

3

5

3

7

−2

9

5

❊①❡r❝✐s❡ ✶✳✷✹

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥✿

❊①❡r❝✐s❡ ✶✳✷✼

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 + 1 ❛t a = 2 ✇✐t❤ h = 0.2 ❛♥❞ h = 0.1✳ ❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡✳ ❊①❡r❝✐s❡ ✶✳✷✽

✭❛✮ ❈♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x2 − x ❛t a = 1 ❛♥❞ h = .5✳ ✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝❛♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤✐s ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❊①❡r❝✐s❡ ✶✳✷✾

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊s✲ t✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆f ∆x ❢♦r x = 0, 4, ❛♥❞ 6 ❛♥❞ ∆x = .5✳

❊①❡r❝✐s❡ ✶✳✷✺

❚❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❛r❡ ♣❧♦tt❡❞ ❜❡❧♦✇✳ P❧♦t t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❊①❡r❝✐s❡ ✶✳✸✵

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊s✲ t✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r x = 2, 4, 9 ❛♥❞ ∆x = 1✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✵

❊①❡r❝✐s❡ ✶✳✸✶

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡

t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞

∆x = 2, 1, .5✳

∆f ∆x

❢♦r

x=1

❊①❡r❝✐s❡ ✶✳✸✼

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝✲

❊①❡r❝✐s❡ ✶✳✸✷

t✐♦♥ ❛t ♣♦✐♥t

❚❤❡ s❡❝❛♥t ❧✐♥❡ ♦❢ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❛r❡ s❤♦✇♥ ❜❡✲

a✳

✭❜✮ Pr♦✈✐❞❡ ❛ ❣r❛♣❤✐❝❛❧ ✐♥t❡r♣r❡t❛✲

t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✳

❧♦✇✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ②♦✉ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥t✐❛✲ ❜✐❧✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

x = 0❄ ❊①❡r❝✐s❡ ✶✳✸✽

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱

f (x) = x2 + 1

❛t

❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

a = 2✳

❊①❡r❝✐s❡ ✶✳✸✾ ❊①❡r❝✐s❡ ✶✳✸✸

❨♦✉ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❛✐❧ ❢r♦♠ ②♦✉r ❜♦ss✿ ✏❚✐♠✱ ▲♦♦❦ ❛t t❤❡ ♥✉♠❜❡rs ✐♥ t❤✐s s♣r❡❛❞✲ s❤❡❡t✳ ❚❤✐s st♦❝❦ s❡❡♠s t♦ ❜❡ ✐♥❝❤✐♥❣ ✉♣✳✳✳ ❉♦❡s

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞

f ′ (x) ❢♦r x = 0, 4,

6.

✐t❄ ■❢ ✐t ❞♦❡s✱ ❤♦✇ ❢❛st❄ ❚❤❛♥❦s✳ ✕ ❚♦♠✑✳ ❉❡s❝r✐❜❡ ②♦✉r ❛❝t✐♦♥s✳

❊①❡r❝✐s❡ ✶✳✸✹

■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ ❞♦ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❤❛✈❡ t♦ ❜❡ ❡q✉❛❧ t♦♦❄

❊①❡r❝✐s❡ ✶✳✹✵ ❊①❡r❝✐s❡ ✶✳✸✺

❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

(2, 1)

t♦

t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s

e

x2

✳

❊①❡r❝✐s❡ ✶✳✸✻

❲❤❛t ❞♦ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝✲ t✐♦♥❄

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞

9✳

f ′ (x) ❢♦r x = 2, 4,

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✶

❊①❡r❝✐s❡ ✶✳✹✶

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✶✳✹✼

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡

t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

f′

❢♦r

x=0

❛♥❞

x = 4✳

❙❤♦✇ ②♦✉r ❝♦♠♣✉t❛t✐♦♥s✳

✭✶✮ ❙t❛t❡ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❛♥❞ ✐❧❧✉str❛t❡ ✐t ✇✐t❤ ❛ s❦❡t❝❤✳ ✭❜✮ ◗✉♦t❡ ❛♥❞ st❛t❡ t❤❡ t❤❡♦r❡♠✭s✮ ♥❡❝❡s✲ s❛r② t♦ ♣r♦✈❡ ✐t✳

✭❝✮ ❲❤❛t t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠

✐t❄

❊①❡r❝✐s❡ ✶✳✹✽

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜❡❧♦✇✳ Pr♦✲ ✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✿

f (x) =

x2 + 7x + 3 . x

❊①❡r❝✐s❡ ✶✳✹✾

✭❛✮ ❙t❛t❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ✭❜✮ ❱❡r✐❢② t❤❛t

❊①❡r❝✐s❡ ✶✳✹✷

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞

f ′ (x) ❢♦r x = 1, 3✱

t❤❡ ❢✉♥❝t✐♦♥

f (x) =

x x+2 s❛t✐s✜❡s t❤❡ ❤②♣♦t❤❡s❡s ♦❢

t❤❡ t❤❡♦r❡♠ ♦♥ t❤❡ ✐♥t❡r✈❛❧

[1, 4]✳

6✳ ❊①❡r❝✐s❡ ✶✳✺✵

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) = x4 − x2 ✳

Pr♦✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✶✳✺✶

❋✐♥❞ ❣❧♦❜❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱

f (x) = x3 − 3x

♦♥ t❤❡ ✐♥t❡r✈❛❧

[−2, 10]✳

❊①❡r❝✐s❡ ✶✳✺✷

❋✐♥❞ t❤❡ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) = x3 − 3x✳

❊①❡r❝✐s❡ ✶✳✺✸ ❊①❡r❝✐s❡ ✶✳✹✸

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

❋✐♥❞ ❛❧❧ ❧♦❝❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) = x3 − 3x − 1✳

✇❤❛t ♣♦✐♥ts ✐s

f

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳

❝♦♥t✐♥✉♦✉s❄

❞♦❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

f

✭❛✮ ❆t

✭❜✮ ❆t ✇❤❛t ♣♦✐♥ts

❡①✐st❄

❊①❡r❝✐s❡ ✶✳✹✹

✭❛✮ ❆♥❛❧②③❡ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡

f (x) = x4 −2x2 ✳ ❣r❛♣❤ ♦❢ f ✳

❢✉♥❝t✐♦♥ ✐ts

✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ s❦❡t❝❤

❊①❡r❝✐s❡ ✶✳✹✺

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❢♦❧❧♦✇ ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ✭❛✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢

h′ (x) = 0

❢♦r ❛❧❧

(a, b)✱ t❤❡♥✳✳✳✑✳ ✭❜✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ′ f (x) = g ′ (x) ❢♦r ❛❧❧ x ✐♥ (a, b)✱ t❤❡♥✳✳✳✑✳

✐♥

x ✏■❢ ❊①❡r❝✐s❡ ✶✳✺✹

■♥❞✐❝❛t❡ ✇❤✐❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❜❡❧♦✇ ✐s tr✉❡ ♦r ❢❛❧s❡ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿

❊①❡r❝✐s❡ ✶✳✹✻

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❏✉st✐❢② t❤❡ ❣r❛♣❤ ❜② st✉❞②✐♥❣ t❤❡

√

xe−x ✳ ❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ f (x) =

✶✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥

f

✐s ✐♥❝r❡❛s✐♥❣✱ t❤❡♥ s♦ ✐s

f −1 ✳

✷✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❤❛s ❛♥ ❛s②♠♣t♦t❡✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✷

✸✳ ■❢ f ′ (c) = 0✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦r ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♦❢ f ✳ ✹✳ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ t❤❡♥ ✐t ✐s ❝♦♥✲ t✐♥✉♦✉s✳ ✺✳ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳ ✻✳ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ t❤❡✐r ❛♥t✐❞❡r✐✈❛✲ t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳

s❦❡t❝❤✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ❥✉st✐❢② t❤❛t Z

Z

1

sin −1

cf (x) dx = c a

❢♦r ❛ ❝♦♥st❛♥t c✳

Z

b

f (x) dx a

❊①❡r❝✐s❡ ✶✳✻✷

❙✉♣♣♦s❡

❊①❡r❝✐s❡ ✶✳✺✺

✭❛✮ ❙t❛t❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳ ✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ ❡✈❛❧✉❛t❡

b

❋✐♥❞

x dx . 3

Z

1

f dx = 2, 0

Z

3

f dx, 1

Z

Z

4

f dx = 0, 0

Z

2

f dx = 2 . 1

Z

1

(f (x) + 3) dx, 0

4

f dx . 2

❊①❡r❝✐s❡ ✶✳✺✻

✭❛✮ ❙t❛t❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳ ✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ ❡✈❛❧✉❛t❡ d dx

Z

x

t2

❊①❡r❝✐s❡ ✶✳✻✸

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②✿

e dt . 0

❊①❡r❝✐s❡ ✶✳✺✼

✭❛✮ Z▼❛❦❡ ❛ s❦❡t❝❤ ♦❢ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠s 1√

x dx ✇✐t❤ n = 4 ✐♥t❡r✈❛❧s✳ ✭❜✮ ❙t❛t❡ t❤❡ ❢♦r 0 ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳

F (x) =

Z

x

f dx . 2

❋✐♥❞✱ ✐♥ t❡r♠s ♦❢ F ✱ t❤❡ ❢♦❧❧♦✇✐♥❣✿ Z

4

f dx, 0

Z

2

f dx, 1

Z

−1 0

f dx,

Z

2 1

(f (x) − 1) dx .

❊①❡r❝✐s❡ ✶✳✺✽

●✐✈❡♥ f (x) = x2 + 1, ✇r✐t❡ ✭❜✉t ❞♦ ♥♦t ❡✈❛❧✉❛t❡✮ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢♦r t❤❡ ✐♥t❡❣r❛❧ ♦❢ f ❢r♦♠ −1 t♦ 2 ✇✐t❤ n = 6 ❛♥❞ ❧❡❢t ❡♥❞s ❛s s❛♠♣❧❡ ♣♦✐♥ts✳ ▼❛❦❡ ❛ s❦❡t❝❤✳

❊①❡r❝✐s❡ ✶✳✻✹

❊✈❛❧✉❛t❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ❜❡❧♦✇ ♦♥ t❤❡ ✐♥✲ t❡r✈❛❧ [−1, 1.5] ✇✐t❤ n = 5✳ ❲❤❛t ❛r❡ ✐ts s❛♠♣❧❡ ♣♦✐♥ts❄ ❲❤❛t ❞♦❡s ✐t ❡st✐♠❛t❡❄

❊①❡r❝✐s❡ ✶✳✺✾

Pr♦✈✐❞❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✈✐❛ ✐ts ❘✐❡♠❛♥♥ s✉♠s✳ ▼❛❦❡ ❛ s❦❡t❝❤✳ ❊①❡r❝✐s❡ ✶✳✻✵

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✐♥❝❧✉❞❡s Z b t❤❡ ❢♦r♠✉❧❛ f (x) dx = F (b) − F (a)✳ ✭❛✮ ❙t❛t❡ a

t❤❡ ✇❤♦❧❡ t❤❡♦r❡♠✳ ✭❜✮ Pr♦✈✐❞❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ✐t❡♠s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❊①❡r❝✐s❡ ✶✳✻✶

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ Z b

a

f (x) dx ❛♥❞ ✐❧❧✉str❛t❡ t❤❡ ❝♦♥str✉❝t✐♦♥ ✇✐t❤ ❛

❊①❡r❝✐s❡ ✶✳✻✺

❲r✐t❡ ✭❞♦♥✬tZ❡✈❛❧✉❛t❡✮ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ♦❢ 5

f (x) dx ❢♦r ❢✉♥❝t✐♦♥ f s❤♦✇♥ ❜❡❧♦✇ t❤❡ ✐♥t❡❣r❛❧ 0 ✇✐t❤ n = 5 ✐♥t❡r✈❛❧s✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✸

❊①❡r❝✐s❡ ✶✳✼✵

❊①❡❝✉t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ✭❞♦♥✬t ❡✈❛❧✉❛t❡ t❤❡ r❡s✉❧t✐♥❣ ✐♥t❡❣r❛❧✮✿ Z

√

cos x + sin x dx,

u = sin x .

❊①❡r❝✐s❡ ✶✳✼✶

❙✉♣♣♦s❡ s(t) r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ❛♥❞ v(t) ✐ts ✈❡❧♦❝✐t②✳ ■❢ v(t) = sin t − cos t ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ✐s s(0) = 0, ✜♥❞ t❤❡ ♣♦s✐t✐♦♥ s(1). ❊①❡r❝✐s❡ ✶✳✼✷

❊①❡r❝✐s❡ ✶✳✻✻

❲r✐t❡ t❤❡ ♠✐❞✲♣♦✐♥t Z ❘✐❡♠❛♥♥ s✉♠ t❤❛t ❛♣♣r♦①✐✲ ♠❛t❡s t❤❡ ✐♥t❡❣r❛❧

1

0

❊✈❛❧✉❛t❡

Z

sin x dx ✇✐t❤✐♥ .01✳

e3x dx .

❊①❡r❝✐s❡ ✶✳✼✸

❊✈❛❧✉❛t❡

❊①❡r❝✐s❡ ✶✳✻✼

▲❡t I =

Z

8 2

Z

f dx✳ ✭❛✮ ❯s❡ t❤❡ ❣r❛♣❤ ♦❢ y = f (x)

❜❡❧♦✇ t♦ ❡st✐♠❛t❡ L4 , M4 , R4 ✳ ✭❜✮ ❈♦♠♣❛r❡ t❤❡♠ t♦ I ✳

2

ex 2x dx .

❊①❡r❝✐s❡ ✶✳✼✹

❊✈❛❧✉❛t❡

Z

2x sin 5x dx .

❊①❡r❝✐s❡ ✶✳✼✺

❊✈❛❧✉❛t❡

Z

3

et+1 dx .

1

❍✐♥t✿ ❲❛t❝❤ t❤❡ ✈❛r✐❛❜❧❡s✳ ❊①❡r❝✐s❡ ✶✳✼✻

❊①❡r❝✐s❡ ✶✳✻✽

❈♦♠♣❧❡t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts✿ 2 ′

′

2

• (f (x) · x ) = f (x) · x + ... Z • x−1 dx = ... • •

Z

Z

❈❛❧❝✉❧❛t❡✿

esin x

2 +77

′

dx .

❊①❡r❝✐s❡ ✶✳✼✼

❊✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧ ❜② s✉❜st✐t✉t✐♦♥

′

f (x) dx = ...

Z

u dv = uv ...

• u = cos t =⇒ du = ...

Z 

2

xex dx .

❊①❡r❝✐s❡ ✶✳✼✽

❋✐♥❞ ❛❧❧ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ f (x) = e−x .

❊①❡r❝✐s❡ ✶✳✻✾

❙✉♣♣♦s❡ t❤❛t F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ✳ ■❢ F ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ [a, b]✱ ✇❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t f ❄

❊①❡r❝✐s❡ ✶✳✼✾

❋✐♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ F ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x2 − 1 s❛t✐s❢②✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ F (1) = 0✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✹

❊①❡r❝✐s❡ ✶✳✽✵

❊①❡r❝✐s❡ ✶✳✽✾

❊✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧

❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

1

x dx . 0

π

sin x cos2 x dx .

0

❊①❡r❝✐s❡ ✶✳✾✵

❊①❡r❝✐s❡ ✶✳✽✶

❊✈❛❧✉❛t❡✿

Z

3

Z

2

x dx −

Z

❊✈❛❧✉❛t❡ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧✿ 2

Z

x dx .

❊①❡r❝✐s❡ ✶✳✽✷

∞ 1

1 dx . 2x

❊①❡r❝✐s❡ ✶✳✾✶

■♥t❡❣r❛t❡ ❜② ♣❛rts✿

❋✐♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡

Z

3xe−x dx .

x

e +x

F

f (x) = F (0) = 1✳

♦❢ t❤❡ ❢✉♥❝t✐♦♥

s❛t✐s❢②✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥

❊①❡r❝✐s❡ ✶✳✾✷ ❊①❡r❝✐s❡ ✶✳✽✸

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

❯s❡ t❤❡ t❛❜❧❡ ♦❢ ✐♥t❡❣r❛❧s t♦ ❡✈❛❧✉❛t❡✿

y=

0, ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ x✲❛①✐s✳

√

x, y =

❋✐♥❞ t❤❡

s✉r❢❛❝❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳

Z

sin

−1

2x dx . ❊①❡r❝✐s❡ ✶✳✾✸

❆ ❝❤♦r❞ ♦❢ ❛ ❝✐r❝❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ✇❤♦s❡

❊①❡r❝✐s❡ ✶✳✽✹

❡♥❞✲♣♦✐♥ts ❧✐❡ ♦♥ t❤❡ ❝✐r❝❧❡✳ ❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ ❧❡♥❣t❤

❊✈❛❧✉❛t❡✿

Z

1 0

1 dx . 2x

♦❢ ❛ ❝❤♦r❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐❛♠❡t❡r✳

❲❤❛t

❛❜♦✉t ♣❛r❛❧❧❡❧❄

❊①❡r❝✐s❡ ✶✳✾✹ ❊①❡r❝✐s❡ ✶✳✽✺

❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

π

sin x cos2 x dx .

0

❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ ❧❡♥❣t❤ ♦❢ ❛ s❡❣♠❡♥t ✐♥ ❛ sq✉❛r❡ ♣❛r❛❧❧❡❧ t♦ ✭❛✮ t❤❡ ❜❛s❡✱ ✭❜✮ t❤❡ ❞✐❛❣♦♥❛❧✳

❊①❡r❝✐s❡ ✶✳✾✺

❋✐♥❞ ✭❜② ✐♥t❡❣r❛t✐♦♥✮ t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

r✳

❊①❡r❝✐s❡ ✶✳✽✻

❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

❊①❡r❝✐s❡ ✶✳✾✻

1 0

√

1 dx . 4 − x2

❋✐♥❞ t❤❡ ❛r❡❛ ❡♥❝❧♦s❡❞ ❜② t❤❡ ❝✉r✈❡s ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✶✳✽✼

❯s❡ t❤❡ t❛❜❧❡ ♦❢ ✐♥t❡❣r❛❧s t♦ ❡✈❛❧✉❛t❡✿

Z

x2 (

p

x2 − 4 −

p x2 + 9) dx .

❊①❡r❝✐s❡ ✶✳✽✽

❯s❡ s✉❜st✐t✉t✐♦♥

u = 1 + x2 t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧ Z p 1 + x2 x5 dx .

❊①❡r❝✐s❡ ✶✳✾✼

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② ❛♥❞

y = 3✳

y = x2 − 1

✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ❝❛❧❝✉❧✉s

✹✼✺

❊①❡r❝✐s❡ ✶✳✾✽

f

❙✉♣♣♦s❡

f

❊①❡r❝✐s❡ ✶✳✶✵✾

✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ✭❛✮ ❙❤♦✇ t❤❛t Z a

f dx = 0✳

✐s ❛❧s♦ ♦❞❞ t❤❡♥

−a

r❡❧❛t❡❞ ❢♦r♠✉❧❛ ❢♦r ❛♥ ❡✈❡♥

✭❜✮ ❙✉❣❣❡st ❛

f✳

y = x2 , y = 1✳

f (x) = ex

x✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r 2 3 ❜❡t✇❡❡♥ y = x ❛♥❞ y = x ✳

2

❜② ❛♥② ♠❡t❤♦❞ ②♦✉ ❧✐❦❡✳

1✳

❊①❡r❝✐s❡ ✶✳✶✵✸

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ r❡❣✐♦♥ ❜❡❧♦✇

y = 2x

0 ≤ x ≤ 1✳

❊①❡r❝✐s❡ ✶✳✶✵✹

❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♦❧✐❞ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❛r❡❛s ♦❢ ✐ts ❝r♦ss✲s❡❝t✐♦♥s✳ ❊①♣❧❛✐♥ ❛♥❞ ❥✉st✐❢② ✉s✐♥❣ ❘✐❡✲ ♠❛♥♥ s✉♠s✳ ❊①❡r❝✐s❡ ✶✳✶✵✺

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

y = x2 +

1, y = 0, x = 0 ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ x✲❛①✐s✳ ❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳ ❊①❡r❝✐s❡ ✶✳✶✵✻

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

y = x2 +

1, y = 0, x = 0, ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ y ✲❛①✐s✳ ❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳ ❊①❡r❝✐s❡ ✶✳✶✵✼

♠ ❧♦♥❣✱

1

♠ ✇✐❞❡✱ ❛♥❞

1

♠ ❞❡❡♣ ✐s

❢✉❧❧ ♦❢ ✇❛t❡r✳ ❋✐♥❞ t❤❡ ✇♦r❦ ♥❡❡❞❡❞ t♦ ♣✉♠♣ ❤❛❧❢ ♦❢ t❤❡ ✇❛t❡r ♦✉t ♦❢ t❤❡ ❛q✉❛r✐✉♠ ✭t❤❡ ❞❡♥s✐t② ♦❢

1000

3

❦❣✴♠ ✮✳

❊①❡r❝✐s❡ ✶✳✶✵✽

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ♦❢ r❡✈♦❧✉t✐♦♥ ❛r♦✉♥❞ t❤❡

x✲❛①✐s

2x − 3

♦♥ t❤❡ ✐♥t❡r✈❛❧

f (x) =

[1, 3]✳

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜②

t❤❡ s♣❤❡r❡ ♦❢ r❛❞✐✉s

✇❛t❡r ✐s

x = 1✳

❊①❡r❝✐s❡ ✶✳✶✶✸

❈♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ ❛r❡❛ ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢

2

t♦

❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✶✳✶✵✷

❆♥ ❛q✉❛r✐✉♠

x = −1

❊①❡r❝✐s❡ ✶✳✶✶✷

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ r✐❣❤t ❝✐r❝✉❧❛r ❝♦♥❡ ♦❢ r❛❞✐✉s

❢♦r

❢r♦♠

♦❢ ♠❛ss ♦❢ t❤❡

❊①❡r❝✐s❡ ✶✳✶✵✶

h

y = 3✳

❋✐♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❋✐♥❞ t❤❡

❛♥❞ ❤❡✐❣❤t

❛♥❞

y = x2 − 1

❊①❡r❝✐s❡ ✶✳✶✶✶

❊①❡r❝✐s❡ ✶✳✶✵✵

R

y = 4 − x 2 , y = x + 2✳

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜②

❋✐♥❞ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡

r❡❣✐♦♥

❝✉r✈❡s

❊①❡r❝✐s❡ ✶✳✶✶✵

❊①❡r❝✐s❡ ✶✳✾✾

❝✉r✈❡s

❋✐♥❞ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡

♦❜t❛✐♥❡❞ ❢r♦♠

y=

√

x, 4 ≤ x ≤ 9✳

t❤❡

x✲❛①✐s✱

❛♥❞ t❤❡ ❧✐♥❡s

x=1

❛♥❞

y =

x = 4✳

√

x✱

✷✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

✹✼✻

✷✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❊①❡r❝✐s❡ ✷✳✶

❙❡t ✉♣✱ ❜✉t ❞♦ ♥♦t s♦❧✈❡✱ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛✲ t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ♣♦rt❢♦❧✐♦ ✐s ✇♦rt❤ st♦❝❦s

A

A $2.1

❛♥❞

B✳

✏❙✉♣♣♦s❡ ②♦✉r

$1, 000, 000 ❛♥❞ ✐t ❝♦♥s✐sts ♦❢ t✇♦ ❚❤❡ st♦❝❦s ❛r❡ ♣r✐❝❡❞ ❛s ❢♦❧❧♦✇s✿

♣❡r s❤❛r❡✱

B $1.5

♣❡r s❤❛r❡✳ ❙✉♣♣♦s❡ ❛❧s♦

t❤❛t ②♦✉ ❤❛✈❡ t✇✐❝❡ ❛s ♠✉❝❤ ♦❢ st♦❝❦

A

t❤❛♥

B✳

❍♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ❤❛✈❡❄✑

❊①❡r❝✐s❡ ✷✳✾

❋✐♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs ❊①❡r❝✐s❡ ✷✳✷

t❤❡

●✐✈❡ t❤❡ ♥✉♠❜❡r

Y =< 2, t, t >

t

t❤❛t ♠❛❦❡s

X =< 3, 2, 1 >

x✲❛①✐s✳

< 1, 1, 1 >

❛♥❞

❉♦♥✬t s✐♠♣❧✐❢②✳

❛♥❞

♣❡r♣❡♥❞✐❝✉❧❛r✳

❊①❡r❝✐s❡ ✷✳✶✵

❋✐♥❞ t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❊①❡r❝✐s❡ ✷✳✸

t❤❡ ❧✐♥❡ ❢r♦♠

xyz ✲❡q✉❛t✐♦♥s ❢♦r t✇♦ ♣❧❛♥❡s✿ x+y−z = 0 x − y + z = 0✳ ❊①♣❧❛✐♥ ❤♦✇ ②♦✉ ❝❛♥ t❡❧❧ t❤❛t

(1, 0, 0)

❛♥❞

(0, 1, 1)✳

❍❡r❡ ❛r❡ ❛♥❞

t❤❡s❡ ♣❧❛♥❡s ❝✉t ❡❛❝❤ ♦t❤❡r ◆❖❚ ❛t r✐❣❤t ❛♥❣❧❡s✳

❊①❡r❝✐s❡ ✷✳✶✶

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤ ♣❛r❛❧❧❡❧ t♦

❊①❡r❝✐s❡ ✷✳✹

❆ ♣❧❛♥❡ ❤❛s ❛♥

xyz ✲❡q✉❛t✐♦♥ x + y = 2✳

●✐✈❡ ❛

✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡✳

❛♥❞

❊①❡r❝✐s❡ ✷✳✶✷

✭❛✮ ❋✐♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ♣❧❛♥❡s ❛♥❞

❊①❡r❝✐s❡ ✷✳✺

(2, 1, 0)

x + 4y − 3z = 1✳

x − 2y + 3z = 1✳

x+y +z = 1

✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡

❧✐♥❡ ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ ♣❧❛♥❡s✳

■♥ ❛♥ ❡✛♦rt t♦ ✜♥❞ t❤❡ ❧✐♥❡ ✐♥ ✇❤✐❝❤ t❤❡ ♣❧❛♥❡s

2x − y − z = 2

−4x + 2y + 2z = 1 ✐♥t❡rs❡❝t✱ ❛ 2 ❛♥❞ t❤❡♥ ❛❞❞❡❞ t❤❡ r❡s✉❧t t♦ t❤❡ s❡❝♦♥❞✳ ❍❡ ❣♦t 0 = 5✳ ❊①♣❧❛✐♥ t❤❡ ❛♥❞

st✉❞❡♥t ♠✉❧t✐♣❧✐❡❞ t❤❡ ✜rst ♦♥❡ ❜②

❊①❡r❝✐s❡ ✷✳✶✸

❋✐♥❞ t❤❡ ✈❡❝t♦r ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ ❜♦t❤

xy ✲ ❛♥❞ xz ✲ (2, 3, 1)✳

r❡s✉❧t✳

❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ❛♥ ♣❛ss✐♥❣ t❤r♦✉❣❤

❊①❡r❝✐s❡ ✷✳✻

❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡s❡ ♣♦✐♥ts ❧✐❡ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✿

❊①❡r❝✐s❡ ✷✳✶✹

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿

  x −y = −1 ,  2x +y = 0 .

A = (0, −5, 5), B = (1, −2, 4), C = (3, 4, 2) . ❊①❡r❝✐s❡ ✷✳✼

P = (−1, 6, −5) A =< 1, 1, 0 > ❛♥❞

❋✐♥❞ t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ❛♥❞ ♣❛r❛❧❧❡❧ t♦ t❤❡ ✈❡❝t♦rs

❊①❡r❝✐s❡ ✷✳✶✺

❋✐♥❞ t❤❡ r❡❞✉❝❡❞ r♦✇ ❡❝❤❡❧♦♥ ❢♦r♠ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣

B =< 0, 1, 1 > .

s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❲❤❛t ❦✐♥❞ ♦❢ s❡t ✐s ✐ts s♦❧✉t✐♦♥ s❡t❄

❊①❡r❝✐s❡ ✷✳✽

❱❡❝t♦rs

A ❛♥❞ B

❛r❡ ❣✐✈❡♥ ❜❡❧♦✇✳ ❈♦♣② t❤❡ ♣✐❝t✉r❡

A + B ✱ ✭❜✮ A − B ✱ ✭❝✮ A ♦♥ B ✱ ✭❡✮ t❤❡ ♣r♦❥❡❝✲

❛♥❞ ✐❧❧✉str❛t❡ ❣r❛♣❤✐❝❛❧❧② ✭❛✮

||A||✱

✭❞✮ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢

t✐♦♥ ♦❢

B

♦♥

A✳

    −x −2y +z = 0 , 3x +z = 2 ,    x −y +z = 1 .

✷✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

✹✼✼

❊①❡r❝✐s❡ ✷✳✶✻

❘❡♣r❡s❡♥t t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❛s ❛ ♠❛✲ tr✐① ❡q✉❛t✐♦♥✿     x   

−y +z = −1 ,

3x

+z = 2 ,

2x +y +z = 1 .

❊①❡r❝✐s❡ ✷✳✶✼

❘❡♣r❡s❡♥t t❤✐s ♠❛tr✐① ❡q✉❛t✐♦♥ ❛s ❛ s②st❡♠ ♦❢ ❧✐♥✲ ❡❛r ❡q✉❛t✐♦♥s✿ 

1

3 2



x





1



      0  y  =  2 . 2 0      −1 0 3 z 3 ❊①❡r❝✐s❡ ✷✳✶✽

●✐✈❡ ❡①♣❧✐❝✐t❧② t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ s②st❡♠ ♦❢ ❧✐♥✲ ❡❛r ❡q✉❛t✐♦♥s r❡♣r❡s❡♥t❡❞ ❜② ✐ts ❛✉❣♠❡♥t❡❞ ♠❛tr✐①✿  

1 0 2 −1 0 1 1 2



.

❊①❡r❝✐s❡ ✷✳✶✾

❋✐♥❞ s❝❛❧❛rs a ❛♥❞ b s✉❝❤ t❤❛t a < 1, 2 > +b < −1, 3 >=< 1, 7 > . ❊①❡r❝✐s❡ ✷✳✷✵

■s ✐t ♣♦ss✐❜❧❡ ❢♦r ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t♦ ❤❛✈❡✿ ✭❛✮ ♥♦ s♦❧✉t✐♦♥s✱ ✭❜✮ ❡①❛❝t❧② ♦♥❡ s♦❧✉t✐♦♥✱ ✭❝✮ ❡①❛❝t❧② t✇♦ s♦❧✉t✐♦♥s✱ ✭❞✮ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s❄ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦r ❡①♣❧❛✐♥ ✇❤② ✐t✬s ♥♦t ♣♦ss✐❜❧❡✳ ❊①❡r❝✐s❡ ✷✳✷✶

❋✐♥❞ t❤❡ s❡t ♦❢ ❛❧❧ ✈❡❝t♦rs ✐♥ R2 t❤❛t ❛r❡ ♦rt❤♦❣♦♥❛❧ t♦ < −1, 3 >✳ ❲r✐t❡ t❤❡ s❡t ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳ ❊①❡r❝✐s❡ ✷✳✷✷

❈♦♠♣✉t❡✿ 

1

3 2



1



    0 .  0 2 0    2 −1 0 3

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✹✼✽

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❊①❡r❝✐s❡ ✸✳✶

❊①❡r❝✐s❡ ✸✳✼

❉❡s❝r✐❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✇✐t❤ ♣♦s✐t✐♦♥

✭✶✮ ❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

(x, y)✱

sin 2t✳

✇❤❡r❡

❣❧❡ ♦❢ t❤✐s ✐♥t❡rs❡❝t✐♦♥✳

x = 2 + t cos t, y = 1 + t sin t , ❛s

t

✈❛r✐❡s ✇✐t❤✐♥

x = cos t, y =

✭✷✮ ❚❤❡ ❝✉r✈❡ ✐♥t❡rs❡❝ts ✐ts❡❧❢✳ ❋✐♥❞ t❤❡ ❛♥✲

[0, ∞)✳

❊①❡r❝✐s❡ ✸✳✽

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s♣✐r❛❧ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ♦r✐❣✐♥ ❛s ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✸✳✷

❙✉♣♣♦s❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜②

x = cos 3t, y = 2 sin t . ❙❡t ✉♣✱ ❜✉t ❞♦ ♥♦t ❡✈❛❧✉❛t❡✱ t❤❡ ✐♥t❡❣r❛❧s t❤❛t r❡♣✲ r❡s❡♥t ✭❛✮ t❤❡ ❛r❝✲❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✱ ✭❜✮ t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ♦❜t❛✐♥❡❞ ❜② r♦t❛t✐♥❣ t❤❡ ❝✉r✈❡ ❛❜♦✉t t❤❡

x✲❛①✐s✳

❊①❡r❝✐s❡ ✸✳✸

❙✉♣♣♦s❡ ❝✉r✈❡

C

✐s t❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

y = f (x)✳

✭❛✮ ❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ C ✳ ✭❜✮ ❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

C

t❤❛t ❣♦❡s ❢r♦♠

r✐❣❤t t♦ ❧❡❢t✳

❊①❡r❝✐s❡ ✸✳✾

P❧♦t t❤✐s ❡♥t✐r❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = sin t, y =

cos 2t✳ ❊①❡r❝✐s❡ ✸✳✹

❋✐♥❞ ❛❧❧ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡

x = cos 3t, y = 2 sin t , ✇❤❡r❡ t❤❡ t❛♥❣❡♥t ✐s ❡✐t❤❡r ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧✳

❊①❡r❝✐s❡ ✸✳✶✵

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❜❡❧♦✇✱ ❛ s♣✐r❛❧ ✇r❛♣♣✐♥❣ ❛r♦✉♥❞ ❛ ❝✐r❝❧❡✳ ❲❤❛t ❛❜♦✉t ♦♥❡ t❤❛t ✐s ✇r❛♣♣✐♥❣ ❢r♦♠ t❤❡ ✐♥s✐❞❡❄ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿

❊①❡r❝✐s❡ ✸✳✺

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = | cos t|, y = | sin t|, −∞ < t < +∞ . ❉❡s❝r✐❜❡ t❤❡ ❝✉r✈❡ ❛♥❞ t❤❡ ♠♦t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✻

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✿

• x(t) =

1 , y(t) = sin t , t > 0 t

• x = cos t , y = 2

❊①❡r❝✐s❡ ✸✳✶✶

●✐✈❡♥ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ 2

• x = 1/t, y = 1/t , t > 0

x = sin t, y = t2 ✳

t❤❡ ❧✐♥❡✭s✮ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❛t t❤❡ ♦r✐❣✐♥✳

❋✐♥❞

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✹✼✾

❊①❡r❝✐s❡ ✸✳✶✷

❊①❡r❝✐s❡ ✸✳✷✷

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ t❤❛t ❧♦♦❦s ❧✐❦❡ t❤❡ ✜❣✉r❡ ❡✐❣❤t ♦r ❛ ✢♦✇❡r ✭♥♦ ♣r♦♦❢ ♥❡❝✲ ❡ss❛r②✮✳

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s❄

❊①❡r❝✐s❡ ✸✳✶✸

❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ x = t2 − 1, y = 2t2 + 3 . ❊①❡r❝✐s❡ ✸✳✶✹

❙✉♣♣♦s❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡✲ s❡♥ts t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ t❤❡ ♣❧❛♥❡✿ x = 3t − 1, y = t2 − 1 .

✭❛✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ x✲❛①✐s❄ ✭❜✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ y ✲❛①✐s❄

❊①❡r❝✐s❡ ✸✳✷✸

❙✉♣♣♦s❡ h(t) = (sin et , cos et ) ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ✭❛✮ ❲❤❛t ✐s ✐ts ♣❛t❤❄ ✭❜✮ ❙❤♦✇ ❤♦✇ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✐s ✉s❡❞ t♦ ❝♦♠♣✉t❡ ✐ts ❞❡r✐✈❛t✐✈❡✳ ❊①❡r❝✐s❡ ✸✳✷✹

❊①❡r❝✐s❡ ✸✳✶✺

❘❡♣r❡s❡♥t ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❡ r♦t❛t✐♦♥ ♦❢ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ 2 t❤❛t ♠❛❦❡s ♦♥❡ ❢✉❧❧ t✉r♥ ❡✈❡r② 3 s❡❝♦♥❞s✳

❙✉♣♣♦s❡ ②♦✉ ❛r❡ t♦✇✐♥❣ ❛ tr❛✐❧❡r✲❤♦♠❡✳ ❉✉r✐♥❣ t❤❡ ✜rst ❢❡✇ ♠✐♥✉t❡s✱ ❡✈❡r② t✐♠❡ ②♦✉ ❧♦♦❦ ❛t t❤❡ r❡❛r ✈✐❡✇ ♠✐rr♦r ②♦✉ ❝❛♥ s❡❡ ♦♥❧② t❤❡ ❧♦✇❡r ♣❛rt ♦❢ t❤❡ ❤♦♠❡✳ ▲❛t❡r✱ ❡✈❡r② t✐♠❡ ②♦✉ ❧♦♦❦ ②♦✉ ❝❛♥ s❡❡ ♦♥❧② t❤❡ t♦♣ ♣❛rt✳ ❉✐s❝✉ss t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r♦❛❞✳

❊①❡r❝✐s❡ ✸✳✶✻

❖♥❡ ❝✐r❝❧❡ ✐s ❝❡♥t❡r❡❞ ❛t (0, 0) ❛♥❞ ❤❛s r❛❞✐✉s 1✳ ❚❤❡ s❡❝♦♥❞ ✐s ❝❡♥t❡r❡❞ ❛t (3, 3)✳ ❲❤❛t ✐s t❤❡ r❛✲ ❞✐✉s ♦❢ t❤❡ s❡❝♦♥❞ ✐❢ t❤❡ t✇♦ ❝✐r❝❧❡s t♦✉❝❤❄ ❊①❡r❝✐s❡ ✸✳✶✼

❘❡♣r❡s❡♥t ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ (1, 2)❀ ✭❜✮ (−1, −1)❀ ✭❝✮ (0, 0)✳ ❊①❡r❝✐s❡ ✸✳✶✽

❘❡♣r❡s❡♥t ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ θ = 0, r = −1❀ ✭❜✮ θ = π/4, r = 2❀ ✭❝✮ θ = 1, r = 0✳

❊①❡r❝✐s❡ ✸✳✷✺

❙✉♣♣♦s❡ ❞✉r✐♥❣ t❤❡ ✜rst 2 s❡❝♦♥❞s ♦❢ ✐ts ✢✐❣❤t ❛♥ ♦❜❥❡❝t ♣r♦❣r❡ss❡❞ ❢r♦♠ ♣♦✐♥t (0, 0) t♦ (1, 0) t♦ (1, 1)✳ ❲❤❛t ✇❛s ✐ts ✭❛✮ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t②✱ ✭❜✮ ❛✈❡r✲ ❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥❄ ❊①❡r❝✐s❡ ✸✳✷✻

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ ✶✳ x(t) =

1 , y(t) = sin t, t > 0 t

✷✳ < x , y >=< cos t , 2 > ✸✳ r = 2t i + 22t j ✹✳ x = 1/t , y = 1/t2 , t > 0

❊①❡r❝✐s❡ ✸✳✶✾

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿ (2 + 3i)(−1 + 2i)✳ ■♥❞✐❝❛t❡ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ✭❜✮ ❋✐♥❞ ✐ts ♠♦❞✉❧❡ ❛♥❞ ❛r✲ ❣✉♠❡♥t✳ ❊①❡r❝✐s❡ ✸✳✷✵

❙✐♠♣❧✐❢② (1 + i)2 ✳

❊①❡r❝✐s❡ ✸✳✷✼

❋✐♥❞ ❛❧❧ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ x = t cos t , y = t sin t , t > 0 ,

✇❤❡r❡ t❤❡ t❛♥❣❡♥t ✈❡❝t♦r ♣♦✐♥ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐r❡❝t✐♦♥✿ ✶✳ ✉♣✱ ✷✳ ❧❡❢t✱ ✸✳ ❞♦✇♥✱ ✹✳ r✐❣❤t✳ ❊①❡r❝✐s❡ ✸✳✷✽

❊①❡r❝✐s❡ ✸✳✷✶

✭❛✮ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ x2 + 2x + 2✳ ✭❜✮ ❋✐♥❞ ✐ts x✲✐♥t❡r❝❡♣ts✳ ✭❝✮ ❋✐♥❞ ✐ts ❢❛❝t♦rs✳

❙✉♣♣♦s❡ ❛ ❜❛❧❧ ✐♥ t❤r♦✇♥ ❤♦r✐③♦♥t❛❧❧② ❛t s♣❡❡❞ v ❢❡❡t ♣❡r s❡❝♦♥❞ ❜② ❛ ♣❡rs♦♥ h ❢❡❡t t❛❧❧✳ ❘❡♣r❡✲ s❡♥t t❤❡ ♠♦t✐♦♥ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ t❤❡ 3✲

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✹✽✵

❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳

❣❧❡ ♦❢ t❤✐s ✐♥t❡rs❡❝t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✷✾

❊①❡r❝✐s❡ ✸✳✸✽

❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

x=

■❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛♥ ♦❜❥❡❝t ❛t t✐♠❡

2

1 1 cos t , y = sin t , t > 0 . t t

√

V (t) =< 1 + t , t >✱ ✇❤❛t t = 3 ✐❢ t❤❡ ♦❜❥❡❝ts st❛rts ❛t

t

✐s ❣✐✈❡♥ ❜②

✐s ✐ts ♣♦s✐t✐♦♥ ❛t t✐♠❡ t❤❡ ♦r✐❣✐♥❄

❊①❡r❝✐s❡ ✸✳✸✵ ❊①❡r❝✐s❡ ✸✳✸✾

❙✉♣♣♦s❡ ❛♥ ♦❜❥❡❝t ✐s ❞r♦♣♣❡❞ ❢r♦♠ ❛ ✶✵✵ ❢❡❡t ❜✉✐❧❞✲ ✐♥❣✳ ❨♦✉ ❛r❡ st❛♥❞✐♥❣ ✶✵✵ ❢❡❡t ❛✇❛② ❢r♦♠ t❤❡ ❜✉✐❧❞✲

❋✐♥❞ t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r ♦❢ t❤❡ ❝✉r✈❡

✐♥❣ ❛♥❞ tr❛❝✐♥❣ t❤❡ ♦❜❥❡❝t ✇✐t❤ ❛ ❧❛s❡r✳ ❊①♣r❡ss t❤❡

F (t) = ti + tj + (1 + t2 )k

❛♥❣❧❡ ♦❢ t❤❡ ❧❛s❡r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❛t t❤❡ ♣♦✐♥t

(0, 0, 1)✳

❊①❡r❝✐s❡ ✸✳✸✶

❆t t✐♠❡

t

0 ≤ t✱ ❛♥ ♦❜❥❡❝t ✐s ❛t t❤❡   1 2 2t P (t) = t + 1 , ,e . t+1

✇✐t❤

❈❛❧❝✉❧❛t❡ ✐ts ✈❡❧♦❝✐t②✱

A(t)✱

V (t)✱

♣♦s✐t✐♦♥

❊①❡r❝✐s❡ ✸✳✹✵

P❧♦t ❛♥❞ ❞❡s❝r✐❜❡ t❤❡ ❝✉r✈❡

❛♥❞ ✐ts ❛❝❝❡❧❡r❛t✐♦♥✱

x(t) = | sin t| , y(t) = | cos t| , z = t .

t✳

❛s ❢✉♥❝t✐♦♥s ♦❢

❊①❡r❝✐s❡ ✸✳✹✶ ❊①❡r❝✐s❡ ✸✳✸✷

❋✐♥❞ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡

P❛r❛♠❡tr✐❝❛❧❧② ❞❡s❝r✐❜❡ t❤❡ ❧✐♥❡ s❡❣♠❡♥t ✇✐t❤ ❡♥❞✲ ♣♦✐♥ts

(−1, −1, −1)

❛♥❞

F (t) = (t5 , t4 , t3 )

(1, 1, 1)✳ ❛t ♣♦✐♥t

(1, 1, 1)✳

❊①❡r❝✐s❡ ✸✳✸✸

✏■❢

f

✐s ❝♦♥t✐♥✉♦✉s ❛t

x0

lim

x→x0

t❤❡♥ ❊①❡r❝✐s❡ ✸✳✹✷

f (x) ex

❋✐♥❞ t❤❡ ✉♥✐t ♥♦r♠❛❧ ✈❡❝t♦r ♦❢ t❤❡ ❝✉r✈❡

G(t) = 3ti + 2t2 j .

❡①✐sts✳✑ ❚r✉❡ ♦r ❢❛❧s❡❄ ❊①♣❧❛✐♥✳

❊①❡r❝✐s❡ ✸✳✸✹

❋✐♥❞

❛

♣❛r❛♠❡tr✐❝

(et , sin t)

❛♥❞

❊①❡r❝✐s❡ ✸✳✹✸

❝✉r✈❡

s✉❝❤

t❤❛t

′

F (t)

=

F (0) = (0, 1)✳

< t, t2 >✳

❊①❡r❝✐s❡ ✸✳✹✹

❊①❡r❝✐s❡ ✸✳✸✺

❙❤♦✇ t❤❛t ❢♦r ❛♥② ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♠♣❧✐❡s

❋✐♥❞ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✉r✈❡

lim 1 F (t) t→t0 t

=0

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡

lim F (t) = 0✳

x = t5 , y = t 4 , z = t3 ,

t→t0

❛t t❤❡ ♣♦✐♥t

(1, 1, 1).

❊①❡r❝✐s❡ ✸✳✸✻

X = F (t) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ■❢ F (t) ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ F (t) ❢♦r ❛❧❧ t✱ s❤♦✇ t❤❛t ||F (t)|| ✐s ❝♦♥st❛♥t✳ ▲❡t

′

sin 2t✳

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = | cos t| , y = | sin t| , −∞ < t < +∞ .

❊①❡r❝✐s❡ ✸✳✸✼

✭✶✮ ❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

❊①❡r❝✐s❡ ✸✳✹✺

x = cos t , y =

✭✷✮ ❚❤❡ ❝✉r✈❡ ✐♥t❡rs❡❝ts ✐ts❡❧❢✳ ❋✐♥❞ t❤❡ ❛♥✲

❉❡s❝r✐❜❡ t❤❡ ❝✉r✈❡ ❛♥❞ t❤❡ ♠♦t✐♦♥✳

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✹✽✶

❊①❡r❝✐s❡ ✸✳✹✻

❋✐♥❞ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ F (t) =< t5 , t4 , t3 >

❛t t❤❡ ♣♦✐♥t (1, 1, 1)✳ ❊①❡r❝✐s❡ ✸✳✹✼

❙✉♣♣♦s❡ ❛ ❜❛❧❧ ✐♥ t❤r♦✇♥ ❤♦r✐③♦♥t❛❧❧② ❛t s♣❡❡❞ w ❢❡❡t ♣❡r s❡❝♦♥❞ ❜② ❛ ♣❡rs♦♥ h ❢❡❡t t❛❧❧✳ ❆t ✇❤❛t s♣❡❡❞ ✇✐❧❧ t❤❡ ❜❛❧❧ ❤✐t t❤❡ ❣r♦✉♥❞❄ ❊①❡r❝✐s❡ ✸✳✺✺

❊①❡r❝✐s❡ ✸✳✹✽

●✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝✉r✈❛t✉r❡✳ ●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ❝✉r✈❡s ✇✐t❤ ✈❛r✐♦✉s ❝✉r✈❛t✉r❡s✳ ❊①❡r❝✐s❡ ✸✳✹✾

❋✐♥❞ ❛❧❧ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ x = 3 cos t, y = 2 sin t ✇❤❡r❡ t❤❡ t❛♥❣❡♥t ✈❡❝t♦r ♣♦✐♥ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐r❡❝t✐♦♥✿ ✭❛✮ ✉♣✱ ✭❜✮ ❧❡❢t✱ ✭❝✮ ❞♦✇♥✱ ✭❞✮ r✐❣❤t✳

❆♥ ❛rt✐❧❧❡r② ❣✉♥ ✇✐t❤ ❛ ♠✉③③❧❡ ✈❡❧♦❝✐t② ♦❢ 1000 f t/s ✐s ❧♦❝❛t❡❞ ❛t♦♣ ❛ s❡❛s✐❞❡ ❝❧✐✛ 500 ❢t ❤✐❣❤✳ ❆t ✇❤❛t ✐♥✐t✐❛❧ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡ s❤♦✉❧❞ ✐t ✜r❡ ❛ ♣r♦❥❡❝t✐❧❡ ✐♥ ♦r❞❡r t♦ ❤✐t ❛ s❤✐♣ ❛t s❡❛ 20, 000 ❢t ❢r♦♠ t❤❡ ❢♦♦t ♦❢ t❤❡ ❝❧✐✛❄ ❆ss✉♠❡ g = 32 f t/s2 . ❊①❡r❝✐s❡ ✸✳✺✻

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ❛t t❤❡ ♣♦✐♥t (1, 1)✳

❊①❡r❝✐s❡ ✸✳✺✵

❙❦❡t❝❤ ❛ ❝✉r✈❡ ♦♥ t❤❡ ♣❧❛♥❡ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ✇❤✐❝❤ ✐s✿ ✭❛✮ ✐♥❝r❡❛s✐♥❣✱ ✭❜✮ ❞❡❝r❡❛s✐♥❣✱ ✭❝✮ ❝♦♥st❛♥t ♥♦♥✲ ③❡r♦✱ ✭❞✮ ③❡r♦✳

❊①❡r❝✐s❡ ✸✳✺✼

❋✐♥❞ t❤❡ ❛r❝✲❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ x = 2et ✱ y = e−t ✱ z = 2t✱ ❢r♦♠ t = 0 t♦ t = 1✳ ❊①❡r❝✐s❡ ✸✳✺✽

❊①❡r❝✐s❡ ✸✳✺✶

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ x = t3 , y = t5 , −1 ≤ t ≤ 1 .

■s t❤✐s ❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥❄ ❊①❡r❝✐s❡ ✸✳✺✷

✭❛✮ ●✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ✭❛s ❛ ❝❡r✲ t❛✐♥ ❞❡r✐✈❛t✐✈❡✮✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ❝♦♠♣✉t❡ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R✳

❉❡s❝r✐❜❡ ❛♥❞ s❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ f (t) = (t3 − t, 1 − t2 )✳ ❊①❡r❝✐s❡ ✸✳✺✾

❘❡♣r❡s❡♥t ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✇✐t❤ ❞♦♠❛✐♥ (−∞, ∞) ❛ ♣❧❛♥ s♣✐r❛❧ ❛♣♣r♦❛❝❤✐♥❣ 0 ❜✉t ♥❡✈❡r r❡❛❝❤✐♥❣ ✐t✳ ❊①❡r❝✐s❡ ✸✳✻✵

❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ t❤❡ ❝✉r✈❡ f (t) = (t, t2 , t3 ) ❛t t❤❡ ♣♦✐♥t (1, 1, 1)✳

❊①❡r❝✐s❡ ✸✳✺✸

❋✐♥❞ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ r(t) =< t5 , t4 , t3 >

❛t t❤❡ ♣♦✐♥t (1, 1, 1)✳

❊①❡r❝✐s❡ ✸✳✻✶

❙❤♦✇ t❤❛t t❤❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ f (t) = (3 sin t , 4 cos t) ✐s♥✬t ♥❛t✉r❛❧✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r✳ ❉♦ ♥♦t s♦❧✈❡✳ ❊①❡r❝✐s❡ ✸✳✻✷

❊①❡r❝✐s❡ ✸✳✺✹

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s♣✐r❛❧ ✐♥ s♣❛❝❡ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ♦r✐❣✐♥ ❛s ❜❡❧♦✇ ✭✈✐❡✇ ❢r♦♠ ❛❜♦✈❡✮✿

❋✐♥❞ t❤❡ ❢✉♥❝t✐♦♥ f : R → R2 s✉❝❤ t❤❛t f ′′ (t) = (cos t , sin 3t) ❛♥❞ f (0) = (1, 0), f ′ (0) = (0, 0)✳

✸✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❊①❡r❝✐s❡ ✸✳✻✸

f (t) = (t3 − 3t, t2 )✳ ✭❛✮ ❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ f ✳ ✭❜✮ ❯s❡ f ′ t♦ ♣❧♦t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ f ✳

✹✽✷

❊①❡r❝✐s❡ ✸✳✻✹

▲❡t

❯s❡ t❤❡ ❛r❝✲❧❡♥❣t❤ ❢♦r♠✉❧❛ t♦ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤

♦❢

♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

3

❝❡♥t❡r❡❞ ❛t

(1, 1)✳

❊①❡r❝✐s❡ ✸✳✻✺

✭❛✮ ❋✐♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ❤❡❧✐①

F (t) = (cos t , sin t, t)✳

✭❜✮ ❋✐♥❞ ✐ts ❝✉r✈❛t✉r❡✳ ✭❝✮

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ♦s❝✉❧❛t✐♥❣ ❝✐r❝❧❡ ❛t

(1, 0, 0)✳

❊①❡r❝✐s❡ ✸✳✻✻

❚❤❡ ❝✉r✈❡

0✳

(t3 − t , t2 − 1)

❤❛s ❛ s❡❧❢✲✐♥t❡rs❡❝t✐♦♥ ❛t

❈♦♠♣✉t❡ t❤❡ ❛♥❣❧❡✳

❊①❡r❝✐s❡ ✸✳✻✼

❍❡r❡

❡q✉❛t✐♦♥s

❢♦r

(x, y, z) = (1 + 2t, 3t, 2 + 5t) (−1, −1, −1) + t < 1, 0, 1 >✳ ❙❛②

❛r❡

♣❛r❛♠❡tr✐❝

❛♥❞

t❤❛t t❤❡s❡ ❧✐♥❡s ❛r❡ ◆❖❚ ♣❛r❛❧❧❡❧✳

t✇♦

❧✐♥❡s✿

(x, y, z) =

❤♦✇ ②♦✉ ❝❛♥ t❡❧❧

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✽✸

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✹✳✶

❊①❡r❝✐s❡ ✹✳✻

❉r❛✇ ❛ ❢❡✇ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

❙❤♦✇ t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✿

2

x + y✳

xy . 2 x + y2 (x,y)→(0,0) lim

❊①❡r❝✐s❡ ✹✳✷ ❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

y = g(x)

s❤♦✇♥ ❜❡❧♦✇✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥

x✱

t✇♦

♦❢

♦♥❡

✈❛r✐❛❜❧❡ ✐s

z = f (x, y) = g(x)

✈❛r✐❛❜❧❡s✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥❧②

❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✳

✇❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ♦❢

f

✐s ❡q✉❛❧ t♦

❋✐♥❞ ❛❧❧ ♣♦✐♥ts

0✳

❊①❡r❝✐s❡ ✹✳✼ ❉r❛✇ t❤❡ ❝♦♥t♦✉r ♠❛♣ ✭❧❡✈❡❧ ❝✉r✈❡s✮ ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥

f (x, y) = ey/x ✳

❊①♣❧❛✐♥ ✇❤❛t t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

❛r❡✳

❊①❡r❝✐s❡ ✹✳✽ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❢♦❧✲

❧♦✇✐♥❣ s✐❣♥s✿

fx > 0 , fxx > 0 , fy < 0 , fyy < 0 . ❊①❡r❝✐s❡ ✹✳✾ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ ❙❦❡t❝❤ t❤❡ ❣r❛❞✐❡♥t ❢♦r ❡❛❝❤ ♦♥ ❛ s❡♣❛r❛t❡

xy ✲♣❧❛♥❡✿

❊①❡r❝✐s❡ ✹✳✸ ❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

2x3 − 6x + y 2 − 2y + 7✳ ❊①❡r❝✐s❡ ✹✳✹ ❙❦❡t❝❤ t❤❡ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ❛❧♦♥❣ ✇✐t❤ ♣♦✐♥ts

A, B, C, D✱

♦♥ t❤❡

xy ✲♣❧❛♥❡✿

❊①❡r❝✐s❡ ✹✳✶✵ ❋✐♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = x2 y −3

(1, 1)✳ ❯s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤✐s ♣♦✐♥t✳ ❊①♣❧❛✐♥✳ ❛t t❤❡ ♣♦✐♥t

❊①❡r❝✐s❡ ✹✳✶✶ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ Pr♦✈✐❞❡ t❤❡ s✐❣♥s ✭♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡✮ ♦❢ t❤❡ ♣❛r✲

❊①❡r❝✐s❡ ✹✳✺

f (x, y) = z = −1, 0, 1, 2 .

❙❦❡t❝❤ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

2xy + 1

❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛❧✉❡s ♦❢

t✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢

∂f ✳

♦❢

t❤❡

(1, 1)

❢✉♥❝t✐♦♥

✐♥ t❤❡ ❞✐✲

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✽✺

❊①❡r❝✐s❡ ✹✳✷✷

❙❦❡t❝❤ t❤❡ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ❛❧♦♥❣ ✇✐t❤ ♣♦✐♥ts

A, B, C, D✱

♦♥ t❤❡

xy ✲♣❧❛♥❡✿

❊①❡r❝✐s❡ ✹✳✸✵

▼❛❦❡ ❛ s❦❡t❝❤ ♦❢ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ❢♦r t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ ❊①❡r❝✐s❡ ✹✳✷✸

❋✐♥❞ t❤❡ p ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ s✉r✲ ❢❛❝❡

z=

4 − x2 − 2y 2

❛t t❤❡ ♣♦✐♥t

(1, −1, 1).

❊①❡r❝✐s❡ ✹✳✷✹

❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ 3

f (x, y) =

2

2x − 6x + y − 2y + 7✳

❉r❛✇ t❤❡ ❝♦♥t♦✉r ♠❛♣ ✭❧❡✈❡❧ ❝✉r✈❡s✮ ♦❢ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

❊①❡r❝✐s❡ ✹✳✷✺

❋✐♥❞ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = x/(y + z)

(4, 1, 1) v =< 0, 2, −1 >✳

❛t t❤❡ ♣♦✐♥t

❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r

✐♥ t❤❡

❊①❡r❝✐s❡ ✹✳✷✻

❋✐♥❞ t❤❡ ♠❛①✐♠✉♠ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = sin(xy)

❛t t❤❡ ♣♦✐♥t

(1, 0)

g(x, y) = ln(x + y)

✷✳

f (u, v) = uv

✸✳

h(x, y) = 2x − 3y + 7

✹✳

z = x2 + y 2

❊①❡r❝✐s❡ ✹✳✸✷

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✹✳✷✼

❋✐♥❞

✶✳

❛♥❞ t❤❡ ❞✐r❡❝✲

t✐♦♥ ✐♥ ✇❤✐❝❤ ✐t ♦❝❝✉rs✳

dw ✱ dt

❊①❡r❝✐s❡ ✹✳✸✶

z = f (x, y)

t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❢♦❧✲

❧♦✇✐♥❣ s✐❣♥s✿

✇❤❡r❡

1

w = xy + yz 2 , x = et , y = et sin t , z = et cos t . fx ❊①❡r❝✐s❡ ✹✳✷✽

3

4

+ + − +

fxx − + + −

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

2

t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❢♦❧✲

❧♦✇✐♥❣ s✐❣♥s✿

fy

− − + +

fyy

− + − +

fx > 0, fxx > 0, fy < 0, fyy < 0 . ❊①❡r❝✐s❡ ✹✳✸✸ ❊①❡r❝✐s❡ ✹✳✷✾

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❛ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✿ ✶✳

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❆✱ ✷✳ ❇✱ ✸✳ ❈✱ ✹✳ ❉✳ ❉❡t❡r♠✐♥❡ t❤❡ s✐❣♥s ♦❢ t❤❡

❙❦❡t❝❤ t❤❡ ❣r❛❞✐❡♥t ❢♦r ❡❛❝❤ ♦♥ ❛ s❡♣❛r❛t❡ xy ✲♣❧❛♥❡✿

❞❡r✐✈❛t✐✈❡s

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

fx , fxx , fy , fyy

❛t t❤❛t ♣♦✐♥t✿

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✽✻

❊①❡r❝✐s❡ ✹✳✹✵

❙t❛t❡ t❤❡ ❝❤❛✐♥ r✉❧❡✱ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✳

❊①❡r❝✐s❡ ✹✳✹✶

❋✐♥❞ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

f (x, y) = exy

(0, 1)✳

❊①❡r❝✐s❡ ✹✳✹✷

❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥

3

f (x, y) =

2

2x z − 6x + z − 2y + 7✳ ❊①❡r❝✐s❡ ✹✳✸✹

❋✐♥❞ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝✲ t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✿

❊①❡r❝✐s❡ ✹✳✹✸

❋✐♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥

✶✳

√ g(x, y, z) = z x + y

✷✳

u=1

✸✳

w = ex+y+z

✹✳

f (u, v, w) = uv + ew

❛t t❤❡ ♣♦✐♥t t❤❡ ❣r❛♣❤ ♦❢

f (x, y) = x2 y −3

(1, 1)✳ ❯s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ s❦❡t❝❤ f ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤✐s ♣♦✐♥t✳ ❊①♣❧❛✐♥✳

❊①❡r❝✐s❡ ✹✳✹✹

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ Pr♦✈✐❞❡ t❤❡ s✐❣♥s ♦❢ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ ♣♦✐♥ts✳ ❋♦r ❡①❛♠♣❧❡✱

❊①❡r❝✐s❡ ✹✳✸✺

∂f ✱

✹✳

(1, 1) < 3, 0 >✱

❛t t❤❡ ♣♦✐♥t

< −1, −1 >✳ ❊①❡r❝✐s❡ ✹✳✹✺

❊①❡r❝✐s❡ ✹✳✸✼

❋✐♥❞ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

x + sin(xy) + xe2y . y

f (x, y) = f ❛r♦✉♥❞ ❣r❛♣❤ ♦❢ f ❛s

❆ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐s ❣✐✈❡♥ ❜②

xy 2 ✳

✭❛✮ ❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

t❤❡ ♣♦✐♥t

(1, 1)✳

✭❜✮ ❘❡♣r❡s❡♥t t❤❡

❛ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡ ❛♥❞ ✜♥❞ t❤❡ ♣❧❛♥❡ t❛♥❣❡♥t t♦ t❤✐s s✉r❢❛❝❡ ❛t t❤❡ ♣♦✐♥t

(1, 1, 1)✳

❊①❡r❝✐s❡ ✹✳✸✽

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ s✉r✲ ❢❛❝❡

z = y cos(x − y)

❛t t❤❡ ♣♦✐♥t

(2, 2, 2).

❊①❡r❝✐s❡ ✹✳✹✻

❙t❛t❡ t❤❡ ❝❤❛✐♥ r✉❧❡✱ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✳

❊①❡r❝✐s❡ ✹✳✸✾

❙❡t ✉♣ ❛s ❛ ♠❛①✴♠✐♥ ♣r♦❜❧❡♠✱ ❜✉t ❞♦ ♥♦t s♦❧✈❡✱

❊①❡r❝✐s❡ ✹✳✹✼

t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✏❋✐♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ ❛ r❡❝t❛♥❣✉✲

❚❤❡ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ❢♦r ❛ ❢✉♥❝t✐♦♥ ❛r❡ ❣✐✈❡♥

❧❛r ❜♦① ♦❢ ♠❛①✐♠❛❧ ✈♦❧✉♠❡ s✉❝❤ t❤❛t t❤❡ s✉♠ ♦❢

❜❡❧♦✇✳ ❚❤❡② ❛r❡ ❡q✉❛❧❧② s♣❛❝❡❞✳ ❙❦❡t❝❤ ❛ ♣♦ss✐❜❧❡

❧❡♥❣t❤s ♦❢ ✐ts ❡❞❣❡s ✐s ❡q✉❛❧ t♦ ✶✵✑✳

❣r❛♣❤ t❤❛t ♣r♦❞✉❝❡❞ ✐t ❛♥❞ ❞❡s❝r✐❜❡ ✐t✳

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✽✼

✸✳ w = ex+y+z ✹✳ f (u, v, w) = uv + ew

❊①❡r❝✐s❡ ✹✳✺✹

❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = xey ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✐♥t✿ ✶✳ (0, 0)✱ ✷✳ (0, 1)✱ ✸✳ (1, 1)✱ ✹✳ (1, 0)✳

❊①❡r❝✐s❡ ✹✳✹✽

p

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ h(x, y) = x2 y − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ❋✐♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s ✉s✐♥❣ t❤❡ ❈❤❛✐♥ ❘✉❧❡✳ ❊①❡r❝✐s❡ ✹✳✹✾

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ y = g(x) ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ✐s s❤♦✇♥ ❜❡❧♦✇✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t z = f (x, y) = g(x) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥❧② ♦♥ x✱ ❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✳ ❋✐♥❞ ❛❧❧ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ f ✐s ❡q✉❛❧ t♦ 0✳

❊①❡r❝✐s❡ ✹✳✺✺

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✜♥❞ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = 2x − 3y ❛t t❤❡ ♣♦✐♥t (1, 1) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r✿ ✶✳ < 3, 0 >✱ ✷✳ −2j ✱ ✸✳ < 1, 1 >✱ ✹✳ < −1, −1 >✳

❊①❡r❝✐s❡ ✹✳✺✻

❆ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐s ❣✐✈❡♥ ❜② f (x, y) = xy 2 ✳ ✭❛✮ ❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ❛r♦✉♥❞ t❤❡ ♣♦✐♥t (1, 1)✳ ✭❜✮ ❘❡♣r❡s❡♥t t❤❡ ❣r❛♣❤ ♦❢ f ❛s ❛ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡ ❛♥❞ ✜♥❞ t❤❡ ♣❧❛♥❡ t❛♥❣❡♥t t♦ t❤✐s s✉r❢❛❝❡ ❛t t❤❡ ♣♦✐♥t (1, 1, 1)✳

❊①❡r❝✐s❡ ✹✳✺✼

❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = xy 2 ❛t t❤❡ ♣♦✐♥t (0, 1)✳ ❊①❡r❝✐s❡ ✹✳✺✵

❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = 2x3 − 6x + y 2 − 2y + 7✳ ❊①❡r❝✐s❡ ✹✳✺✶

❋✐♥❞ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = 2x2 − 3y ❛t t❤❡ ♣♦✐♥t (1, 1) ✐♥ t❤❡ ❞✐✲ r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < 1, 0 >✳

❊①❡r❝✐s❡ ✹✳✺✽

❋✐♥❞ t❤❡ ♠❛①✐♠✉♠ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = sin(xy) ❛t t❤❡ ♣♦✐♥t (1, 0) ❛♥❞ t❤❡ ❞✐r❡❝✲ t✐♦♥ ✐♥ ✇❤✐❝❤ ✐t ♦❝❝✉rs✳

❊①❡r❝✐s❡ ✹✳✺✾

❊①❡r❝✐s❡ ✹✳✺✷

p

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ h(x, y) = x2 y − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ❋✐♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s ✉s✐♥❣ t❤❡ ❈❤❛✐♥ ❘✉❧❡✳

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ s✉r✲ ❢❛❝❡ z = 4x2 − y 2 + 2y ❛t t❤❡ ♣♦✐♥t (−1, 2, 4).

❊①❡r❝✐s❡ ✹✳✺✸

❊①❡r❝✐s❡ ✹✳✻✵

❋✐♥❞ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝✲ t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✿

❋✐♥❞ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ √ f (x, y) = 1 + 2x y ❛t t❤❡ ♣♦✐♥t (3, 4) ✐♥ t❤❡ ❞✐✲ r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < 4, −1 >✳

√

✶✳ g(x, y, z) = z x + y ✷✳ u = 1

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✽✽

❊①❡r❝✐s❡ ✹✳✻✶

❊①❡r❝✐s❡ ✹✳✻✾

▼❛❦❡ ❛ s❦❡t❝❤ ♦❢ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ❢♦r t❤❡ ❢✉♥❝✲ t✐♦♥ ❜❡❧♦✇✿

▲❡t f (x, y) =

p

x2 + (y − 2)2 − 4 + 1 .

✶✳ ❋✐♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ f ✳ ✷✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f ✳ ✸✳ ❲❤❛t ✐s ✐t❄ ❊①❡r❝✐s❡ ✹✳✼✵

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ z = f (x, y) s✉❝❤ t❤❛t ∂f ∂f (0, 0) ❡①✐sts ❜✉t (0, 0) ❞♦❡s ♥♦t✳ ∂x

∂y

❊①❡r❝✐s❡ ✹✳✻✷

❋✐♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ ❛ r❡❝t❛♥❣✉❧❛r ❜♦① ♦❢ ♠❛①✐✲ ♠❛❧ ✈♦❧✉♠❡ s✉❝❤ t❤❛t t❤❡ t♦t❛❧ s✉r❢❛❝❡ ❛r❡❛ ✐s ❡q✉❛❧ t♦ 64✳

❊①❡r❝✐s❡ ✹✳✼✶

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ❝♦♠♣✉t❡ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = xy + y 2

❊①❡r❝✐s❡ ✹✳✻✸

❆ ♣❧❛♥❡ ❤❛s ❛♥ xyz ✲❡q✉❛t✐♦♥ 2(x − 1) + 3(y − 2) + 4(z − 3) = 0✳ ●✐✈❡ ❛ ✈❡❝t♦r P❆❘❆▲▲❊▲ t♦ t❤❡ ♣❧❛♥❡✳

❛t t❤❡ ♣♦✐♥t a = (2, 1) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ v = (1, 1)✳ ❊①❡r❝✐s❡ ✹✳✼✷

❊①❡r❝✐s❡ ✹✳✻✹

❈❛❧❝✉❧❛t❡

t❤❡

❣r❛❞✐❡♥t

x f (x, y, z) = xyz + ✳ y

grad f (x, y, z)

♦❢

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ♣r♦✈❡ t❤❛t T (x) = −2x2 − 1 ✐s t❤❡ ❜❡st ❛✣♥❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x41 + x22 ,

✇❤❡r❡ x = (x1 , x2 )✱ ❛t t❤❡ ♣♦✐♥t a = (0, −1)✳

❊①❡r❝✐s❡ ✹✳✻✺

❉♦❡s f (x, y) = −x2 ex +y ❤❛✈❡ ❛ ♠❛①✐♠♠✉ ♦r ♠✐♥✲ ✐♠✉♠❄ ❍♦✇ ❞♦ ②♦✉ ❦♥♦✇❄ 2

2

❊①❡r❝✐s❡ ✹✳✼✸

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ f (t) = (sin et , cos et )

❊①❡r❝✐s❡ ✹✳✻✻

∂f

❈❛❧❝✉❧❛t❡ ❢♦r f (x, y, z) = exyz + x ❛t t❤❡ ♣♦✐♥t ∂z (1, 2, 3)✳

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡♥ ✉s❡ t❤❡ ❈❤❛✐♥ ❘✉❧❡ t♦ ❝♦♠♣✉t❡ ✐ts ❞❡r✐✈❛t✐✈❡ ❛t t0 = 0✳ ❊①❡r❝✐s❡ ✹✳✼✹

❊①❡r❝✐s❡ ✹✳✻✼

❨♦✉ ❛r❡ ❛t t❤❡ ♣♦✐♥t (0, 0)✳ ■♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ✇❤❛t ✈❡❝t♦r s❤♦✉❧❞ ②♦✉ st❡♣ ♦✛ (0, 0) ✐♥ ♦r❞❡r t♦ ❣❡t t❤❡ ❣r❡❛t❡st ✐♥✐t✐❛❧ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❢✉♥❝t✐♦♥ 1 f (x, y) = x2 + y 2 ✳ ❊①♣❧❛✐♥✳ 4

❊①❡r❝✐s❡ ✹✳✻✽

▲❡t f (x, y) = sin(x − y)✳ ●✐✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ h(x) ❞❡✜♥❡❞ ❜② h(x) =

Z

x

f (s, y) ds . 0

❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y, z) = xz + 5y 2 .

❊①❡r❝✐s❡ ✹✳✼✺

❋✐♥❞ ❛♥❞ ❝❧❛ss✐❢② t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = x4 + y 4 − 4xy + 1✳ ❊①❡r❝✐s❡ ✹✳✼✻

❋✐♥❞ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ s✉r❢❛❝❡ f (x, y) =

sin(x − y) ❛t ( π2 , 0, 1)✳

✹✳ ❊①❡r❝✐s❡s✿ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✹✳✼✼

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ✶✳ ■ts ✐♠❛❣❡ ✐s t❤❡ ❝✐r❝❧❡ x2 + y 2 = 1✳ ✷✳ ■ts ✐♠❛❣❡ ✐s t❤❡ s♣❤❡r❡ x2 + y 2 + z 2 = 1✳ ✸✳ ■ts ❣r❛♣❤ ✐s t❤❡ s♣❤❡r❡ x2 + y 2 + z 2 = 1✳ ✹✳ ■ts ❞♦♠❛✐♥ ✐s t❤❡ ❞✐s❦ x2 + y 2 ≤ 1✳ ✺✳ ■t ✐s ❡q✉❛❧ t♦ ✐ts ♦✇♥ ❜❡st ❧✐♥❡❛r ✭❛✣♥❡✮ ❛♣✲ ♣r♦①✐♠❛t✐♦♥✳ ❊①❡r❝✐s❡ ✹✳✼✽

❋✐♥❞ t❤❡ ❜❡st ❛✣♥❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x21 + x22 + x23 ,

✇❤❡r❡ x = (x1 , x2 , x3 )✱ ❛t t❤❡ ♣♦✐♥t a = (0, −1, 1)✳ ❊①❡r❝✐s❡ ✹✳✼✾

❋✐♥❞ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y, z) = x2 z 2 + y 2 + y . ❊①❡r❝✐s❡ ✹✳✽✵

❆ ♠♦✉♥t❛✐♥ r✐❞❣❡ ❤❛s t❤r❡❡ ♣❡❛❦s ✇✐t❤ t✇♦ ♣❛ss❡s ❜❡t✇❡❡♥ t❤❡♠✳ ❙❦❡t❝❤ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ t❡rr❛✐♥✳ ❊①❡r❝✐s❡ ✹✳✽✶

❉❡✜♥❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❲❤❛t ❞♦❡s ✐t t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥❄

✹✽✾

✺✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛❧s

✹✾✵

✺✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛❧s

❊①❡r❝✐s❡ ✺✳✶

t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧

❊st✐♠❛t❡ t❤❡ ✐♥t❡❣r❛❧

ZZ

ZZ

2

x y dA , B

✇❤❡r❡

✇❤❡r❡

sin(x + y) dA , D

R = [0, π] × [0, π]✳

❈❤♦♦s❡ ②♦✉r ♦✇♥ s❛♠♣❧❡

♣♦✐♥ts✳

B = {(x, y) : 0 ≤ x ≤ 2, 0 ≤ y ≤ 3} , ❜② ♣r♦✈✐❞✐♥❣ ❛ ❘✐❡♠❛♥♥ s✉♠ ✇✐t❤

6

❊①❡r❝✐s❡ ✺✳✽

sq✉❛r❡s✳

❋✐♥❞ t❤❡ ♠❛ss ♦❢ t❤❡ ❧❛♠✐♥❛ t❤❛t ♦❝❝✉♣✐❡s t❤❡ r❡✲ ❣✐♦♥

❯s❡ ❛ ❘✐❡♠❛♥♥ s✉♠ ✇✐t❤

8

t❡r♠s t♦ ❡st✐♠❛t❡ t❤❡

❊①❡r❝✐s❡ ✺✳✾

✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧

ZZZ

y = ex , ②❂✵✱ x = 0, x = 1✱ ❢✉♥❝t✐♦♥ ✐s ρ(x, y) = y.

❜♦✉♥❞❡❞ ❜②

❛♥❞ ✐ts ❞❡♥s✐t②

❊①❡r❝✐s❡ ✺✳✷

♦✈❡r t❤❡ ❝✉❜❡

D

❊①♣r❡ss t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

(x + y + z) dV ,

r

❛s ❛♥

❛r❝ ❧❡♥❣t❤ ✐♥t❡❣r❛❧ ❛♥❞ ❡✈❛❧✉❛t❡ ✐t✳

D

D = [0, 1] × [0, 1] × [0, 1]✳

❈❤♦♦s❡

❊①❡r❝✐s❡ ✺✳✶✵

②♦✉r ♦✇♥ s❛♠♣❧❡ ♣♦✐♥ts✳

❊✈❛❧✉❛t❡ t❤❡ ✐t❡r❛t❡❞ ✐♥t❡❣r❛❧

Z1 Zz Zy

❊①❡r❝✐s❡ ✺✳✸

: R → R

f

▲❡t

Z

b

y ≤ d}

❜❡ ❛

❛

❢✉♥❝t✐♦♥

s✉❝❤

t❤❛t

B = {(x, y) : a ≤ x ≤ b, c ≤ ZZ r❡❝t❛♥❣❧❡✳ ❋✐♥❞ f dA✳

f (x) dx = 1✳ a

❜❡

0

B

❊①❡r❝✐s❡ ✺✳✶✶

D

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ s✉r✲

❛♥❞

t❤❡

xy ✲♣❧❛♥❡

❛♥❞ t❤❡ ♣❧❛♥❡s

y=0

❊①❡r❝✐s❡ ✺✳✶✷ ❋✐♥❞

❊①❡r❝✐s❡ ✺✳✺ ❋✐♥❞

t❤❡

✈♦❧✉♠❡

D ✐s y = 0, y = x2 , x = 1, z = 0, z = 2✿ ZZZ (y + z) dV .

❈♦♠♣✉t❡ t❤✐s ✐♥t❡❣r❛❧ ❜❡❧♦✇ ✇❤❡r❡ r❡❣✐♦♥

❊①❡r❝✐s❡ ✺✳✹

z = 1 − x2 ✱ y = 1✳

0

▲❡t

❜♦✉♥❞❡❞ ❜②✿

❢❛❝❡

0

2

ze−y dxdydz .

♦❢

x + 2y − z = 0 ❛♥❞ y = x ❛♥❞ y = x4 ✳

t❤❡

s♦❧✐❞

✉♥❞❡r

t❤❡

♣❧❛♥❡

t❤❡

✈♦❧✉♠❡

♦❢

x + 2y − z = 0 ❛♥❞ y = x ❛♥❞ y = x4 ✳

t❤❡

s♦❧✐❞

✉♥❞❡r

t❤❡

♣❧❛♥❡

❛❜♦✈❡ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜②

❛❜♦✈❡ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜②

❊①❡r❝✐s❡ ✺✳✶✸ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❛t ♣❛rt ♦❢ t❤❡ ♣❧❛♥❡

❊①❡r❝✐s❡ ✺✳✻ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣❛rt ♦❢ t❤❡ s✉r❢❛❝❡

3x + 2y 2 t❤❛t ❧✐❡s ❛❜♦✈❡ (0, 0), (0, 1), ❛♥❞ (2, 1)✳

z = 1+

t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s

2x + 2y t❤❛t ❧✐❡s ❞✐r❡❝t❧② ❛❜♦✈❡ t❤❡ r❡❣✐♦♥ ✐♥ t❤❡ xy ✲♣❧❛♥❡ ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❛r❛❜♦❧❛s y = x2 − 1 ❛♥❞ y = −x2 + 1✳ ❊①❡r❝✐s❡ ✺✳✶✹ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣❛rt ♦❢ t❤❡ s✉r❢❛❝❡

❊①❡r❝✐s❡ ✺✳✼ ❯s❡ ❛ ❘✐❡♠❛♥♥ s✉♠ ✇✐t❤

z = 1+

m=n=2

t♦ ❡st✐♠❛t❡

3x + 2y 2 t❤❛t ❧✐❡s ❛❜♦✈❡ (0, 0), (0, 1), ❛♥❞ (2, 1).

z = 1+

t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s

✺✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛❧s

✹✾✶

❊①❡r❝✐s❡ ✺✳✶✺

❊✈❛❧✉❛t❡ ✇❤❡r❡

R ✐♥ t❡r♠s ♦❢ u, v ✳ ZZZ

2x dV , D

❊①❡r❝✐s❡ ✺✳✷✸

❇② ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥✱ ❡✈❛❧✉❛t❡ D = {(x, y, z) : 0 ≤ y ≤ 2 , p 0 ≤ x ≤ 4 − y2 , 0 ≤ z ≤ y}.

❊①❡r❝✐s❡ ✺✳✶✻

❊✈❛❧✉❛t❡ t❤❡ ❞♦✉❜❧❡ ✐♥t❡❣r❛❧ ZZ

1 dxdy , R

✇❤❡r❡ R ✐s ❣✐✈❡♥ ✐♥ u, v ❝♦♦r❞✐♥❛t❡s ❛s 0 ≤ u ≤ 1, 1 ≤ v ≤ 3✱ ❛♥❞ x = u2 , y = v + u2 ✳ ❊①❡r❝✐s❡ ✺✳✷✹

❨♦✉ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥ ♦❢

(x + y) dA , D

✇❤❡r❡ D ✐s ❜♦✉♥❞❡❞ ❜② y =

ZZ

√

x ❛♥❞ y = x ✳ 2

❊①❡r❝✐s❡ ✺✳✶✼

❉❡✜♥❡ t❤❡ ❞♦✉❜❧❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐✲ ❛❜❧❡s ♦✈❡r ❛ r❡❝t❛♥❣❧❡✳ ▼❛❦❡ ❛ s❦❡t❝❤ ❛♥❞ ❡①♣❧❛✐♥✳ ❊①❡r❝✐s❡ ✺✳✶✽

ZZZ

f (x, y, z) dxdydz , R

✇❤❡r❡ R ✐s t❤❡ ✸❉ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡ ❜② t❤❡ ✉♥✐t s♣❤❡r❡ ❛♥❞ ❢r♦♠ ❜❡❧♦✇ ❜② t❤❡ xy ✲♣❧❛♥❡✳ ❉❡s❝r✐❜❡ R ✐♥ ❛ ✇❛② ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥t❡❣r❛t✐♦♥✳

❊①❡r❝✐s❡ ✺✳✷✺

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❧❛♥❡s x + 2y + z = 2, x = 2y, x = 0, ❛♥❞ z = 0✳ ❊①❡r❝✐s❡ ✺✳✶✾

❉❡✜♥❡ t❤❡ ❞♦✉❜❧❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐✲ ❛❜❧❡s ♦✈❡r ❛ r❡❝t❛♥❣❧❡✳ ▼❛❦❡ ❛ s❦❡t❝❤ ❛♥❞ ❡①♣❧❛✐♥✳ ❊①❡r❝✐s❡ ✺✳✷✵

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❧❛♥❡s x + 2y + z = 2, x = 2y, x = 0, ❛♥❞ z = 0. ❊①❡r❝✐s❡ ✺✳✷✶

❨♦✉ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ZZZ

f (x, y, z) dxdydz , R

✇❤❡r❡ R ✐s t❤❡ ✸❉ r❡❣✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❡✈❡r②t❤✐♥❣ ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❧❛♥❡s y = x, y = x + 1, y = −x + 1, y = −x + 2, z = 0, z = 2✳ ❙✇✐t❝❤ t♦ ♥❡✇✱ ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥t❡❣r❛t✐♦♥✱ ❝♦♦r❞✐♥❛t❡s u, v, w ❜② ✐♥❞✐❝❛t✐♥❣ ✇❤❛t u, v, w ❛r❡ ✐♥ t❡r♠s ♦❢ x, y, z ❛♥❞ ❞❡s❝r✐❜❡ R ✐♥ t❡r♠s ♦❢ u, v, w✳

❊①❡r❝✐s❡ ✺✳✷✻

❈♦♠♣✉t❡ t❤✐s ✐♥t❡❣r❛❧ ZZZ

(y + z) dV , D

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❝♦♥✈❡rs✐♦♥ ❢❛❝t♦r ♦❢ t❤❡ tr❛♥s❢♦r✲ ♠❛t✐♦♥ x(u, v, w) = u2 v , y(u, v, w) = v 2 , z(u, v, w) = w2 eu .

✇❤❡r❡ r❡❣✐♦♥ D ✐s ❜♦✉♥❞❡❞ ❜②✿ y = 0, y = x2 , x = 1, z = 0, z = 2✳ ❊①❡r❝✐s❡ ✺✳✷✼

❨♦✉ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥ ♦❢

❊①❡r❝✐s❡ ✺✳✷✷

❨♦✉ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ZZ

f (x) dxdy , R

✇❤❡r❡ R ✐s t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ r❡❣✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❡✈❡r②t❤✐♥❣ ❜♦✉♥❞❡❞ ❜② t❤❡ ❝✉r✈❡s y = x2 , y = x2 + 1 ❛♥❞ t❤❡ ❧✐♥❡s x = 0, x = 1✳ ❙✇✐t❝❤ t♦ ♥❡✇✱ ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥t❡❣r❛t✐♦♥✱ ❝♦♦r❞✐♥❛t❡s u, v ❜② ✐♥✲ ❞✐❝❛t✐♥❣ ✇❤❛t u, v ❛r❡ ✐♥ t❡r♠s ♦❢ x, y ❛♥❞ ❞❡s❝r✐❜❡

ZZ

f (x, y) dxdy , R

✇❤❡r❡ R ✐s t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ r❡❣✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❡✈❡r②t❤✐♥❣ ❜♦✉♥❞❡❞ ❜② t❤❡ ❝✉r✈❡s y = x2 , y = x2 + 2 ❛♥❞ t❤❡ ❧✐♥❡s x + y = 1, x + y = 2✳ ✭❛✮ ❙❦❡t❝❤ R ✐♥ xy ✲♣❧❛♥❡✳ ✭❜✮ ❙✇✐t❝❤ t♦ ♥❡✇✱ ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥t❡❣r❛t✐♦♥✱ ❝♦♦r❞✐♥❛t❡s u, v ❜② ✐♥❞✐❝❛t✐♥❣ ✇❤❛t u, v ❛r❡ ✐♥ t❡r♠s ♦❢ x, y ✳ ✭❝✮ ❙❦❡t❝❤ R ✐♥ uv ✲♣❧❛♥❡✳

✺✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛❧s

✹✾✷

❊①❡r❝✐s❡ ✺✳✷✽

❊①❡r❝✐s❡ ✺✳✸✻

❇② ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥✱ ❡✈❛❧✉❛t❡

❊✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧

ZZ

ZZZ

2d xdy , R

xyz dV , B

R ✐s ❣✐✈❡♥ ✐♥ u, v ❝♦♦r❞✐♥❛t❡s ❛s 1 ≤ u ≤ 2, 0 ≤ v ≤ 1✱ ❛♥❞ x = u2 + 2v, y = uev ✳

✇❤❡r❡

❊①❡r❝✐s❡ ✺✳✷✾

❊①❡r❝✐s❡ ✺✳✸✼

❨♦✉ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥ ♦❢

❇② ✉s✐♥❣ ♦♥❧② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ❝♦♠✲

✇❤❡r❡

ZZZ ✇❤❡r❡

R

♣✉t❡

R

✐s t❤❡ ✏✐❝❡✲❝r❡❛♠ ❝♦♥❡✑✱ ✐✳❡✳✱ t❤❡ ✸❉ r❡❣✐♦♥

❛♥❞ t❤❡ s♣❤❡r❡

ZZ

f (x, y, z) dxdydz ,

z 2 = x2 + y 2 , z ≥ x2 + y 2 + z 2 = 2✳ ❉❡s❝r✐❜❡ R ✐♥

♦❜t❛✐♥❡❞ ❜② ✐♥t❡rs❡❝t✐♥❣ t❤❡ ❝♦♥❡

0✱

B = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} .

f (x, y) dA , D

D ✐s t❤❡ ❞✐s❦ x2 + y 2 ≤ 4 ❛♥❞ f (x, y) = 2 ✐❢ −2 ≤ x ≤ 0✱ f (x, y) = −1 ✐❢ 0 ≤ x ≤ 2✱ f (x, y) = 55 ❢♦r ❛❧❧ ♦t❤❡r ✈❛❧✉❡s ♦❢ (x, y)✳ ✇❤❡r❡

❛ ✇❛② ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥t❡❣r❛t✐♦♥✳ ❊①❡r❝✐s❡ ✺✳✸✽

❇② ✉s✐♥❣ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✱ ❝♦♠♣✉t❡ t❤❡ ✈♦❧✉♠❡ ♦❢

❊①❡r❝✐s❡ ✺✳✸✵

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❝♦♥✈❡rs✐♦♥ ❢❛❝t♦r ♦❢ t❤❡ tr❛♥s❢♦r✲ ♠❛t✐♦♥

t❤❡ s♦❧✐❞ ❜♦✉♥❞❡❞ ❜② t❤❡ ❝②❧✐♥❞❡r t❤❡ ♣❧❛♥❡s

z=0

❛♥❞

x2 + y 2 = 1

❛♥❞

z = 1✳

x(u, v, w) = u2 + v + 1 ❊①❡r❝✐s❡ ✺✳✸✾

y(u, v, w) = v 2 + w + 2

❈♦♠♣✉t❡ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ❧②✐♥❣ ❛❜♦✈❡ t❤❡

z(u, v, w) = w2 + u + 3

r❡❝t❛♥❣❧❡

(x, y, 0)

R

❢r♦♠ ❛❜♦✈❡ ❊①❡r❝✐s❡ ✺✳✸✶

❋✐♥❞ ❜② ✐♥t❡❣r❛t✐♦♥ t❤❡ ❛r❡❛ ♦❢ t❤❛t ♣❛rt ♦❢ t❤❡ ♣❧❛♥❡

2x + 3y + z = 6

t❤❛t ❧✐❡s ✐♥ t❤❡ ✜rst ♦❝t❛♥t✳

❊①❡r❝✐s❡ ✺✳✸✷

R3 ❜♦✉♥❞❡❞ ❜② t❤❡ z = x2 + y 2 , z = 0, y = x2 ✱

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ✐♥ ❢♦❧❧♦✇✐♥❣ s✉r❢❛❝❡s✿

y = 2x✳

❊①❡r❝✐s❡ ✺✳✸✸

❇② ♠❡❛♥s ♦❢ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱ Z

xy dA,

❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧ ❜♦✉♥❞❡❞ ❜② t❤❡ ❧✐♥❡s

✇❤❡r❡

D

D ⊂ R2

✐s

x = 0✱ x = 1✱ y = x✱ y = x+1✳

❊①❡r❝✐s❡ ✺✳✸✹

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♣❤❡r❡ ♦❢ r❛❞✐✉s

a✳

❊①❡r❝✐s❡ ✺✳✸✺

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ s✉r✲ ❢❛❝❡s

z = y ✱ y = 4✱ z = 0✱ y = x2 ✳

xy ✲♣❧❛♥❡ ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♣♦✐♥ts −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 ❛♥❞ ❜♦✉♥❞❡❞ 2 ❜② t❤❡ s✉r❢❛❝❡ ❣✐✈❡♥ ❜② z = x y ✳ ✐♥ t❤❡

✇✐t❤

✻✳

❊①❡r❝✐s❡s✿ ❱❡❝t♦r ✜❡❧❞s

✹✾✸

✻✳ ❊①❡r❝✐s❡s✿ ❱❡❝t♦r ✜❡❧❞s ❊①❡r❝✐s❡ ✻✳✶

❊①❡r❝✐s❡ ✻✳✻

❙❦❡t❝❤ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❣✐✈❡♥ ❜❡❧♦✇ ❛♥❞ ❡st✐♠❛t❡ ✐ts

❋✐♥❞ t❤❡ ✇♦r❦ ❞♦♥❡ ❜② ❢♦r❝❡ ✜❡❧❞

❧✐♥❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ sq✉❛r❡ ♦r✐✲

V (x, y) =< xy , y 2 > ,

❡♥t❡❞ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭♠✉❧t✐♣❧❡ ❛♥s✇❡rs ❛r❡ ♣♦s✲ s✐❜❧❡✮✿

✐♥ ♠♦✈✐♥❣ ❛♥ ♦❜❥❡❝t ❛❧♦♥❣ t❤❡ ♣❛r❛❜♦❧❛

F (0, 0) =< 1, 1 >

F (.5, 0) =< 0, 1 >

t2 , 0 ≤ t ≤ 1.

F (1, 0) =< 1, 1 >

F (1, .5) =< −1, 1 >

❊①❡r❝✐s❡ ✻✳✼

F (1, 1) =< −1, 0 > F (.5, 1) =< 0, 0 >

❱❡❝t♦r ✜❡❧❞

F (0, .5) =< −1, −1 >

F (0, 1) =< 2, 1 >

V

✐s s❦❡t❝❤❡❞ ❜❡❧♦✇✳

x = t, y =

✭❛✮ ❙✉♣♣♦s❡

C

✐s t❤❡ ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ❝✉r✈❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡✳ ■s

Z

❊①❡r❝✐s❡ ✻✳✷

■♥ t❤❡ ❢♦r♠✉❧❛ ♦❢ ●r❡❡♥✬s ❚❤❡♦r❡♠ s❤♦✇♥ ❜❡❧♦✇✱ ✐❞❡♥t✐❢② ❛❧❧ ♦❢ ✐ts ♣❛rts ✭s✉❝❤ ❛s ✏ F ✐s✳✳✳✑✮✿

I

C

F · dP =

ZZ

D

C

V · dR

0❄ div V (P )

♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r

P = (4, 4)✳

■s

❊①♣❧❛✐♥✳

✭❜✮ ❙✉♣♣♦s❡

♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r ✵❄

❊①♣❧❛✐♥✳

(qx − py ) dA .

❊①❡r❝✐s❡ ✻✳✸

❙❦❡t❝❤ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ✜❡❧❞

F (x, y) =< x, −y >

✐❞❡♥t✐❢②✐♥❣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✳ ❉❡s❝r✐❜❡ t❤❡ ♠♦t✐♦♥ ✐♥ ❞❡t❛✐❧✳

❊①❡r❝✐s❡ ✻✳✽

❊①❡r❝✐s❡ ✻✳✹

❙t❛t❡

t❤❡

♣❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡

♣r♦♣❡rt②✳

❉♦❡s

❊✈❛❧✉❛t❡ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧

t❤❡ ✈❡❝t♦r ✜❡❧❞ s❤♦✇♥ ❜❡❧♦✇ s❛t✐s❢② t❤❡ ♣❛t❤✲

Z

✐♥❞❡♣❡♥❞❡♥❝❡ ♣r♦♣❡rt②❄ ❊①♣❧❛✐♥✳

✇❤❡r❡

C

C

√ x2 y 3 dx − y x dy ,

✐s ♣❛r❛♠❡tr✐③❡❞ ❜②

t ≤ 1.

x = t2 , y = −t3 , 0 ≤

❊①❡r❝✐s❡ ✻✳✾

✭❛✮ ❊①♣❧❛✐♥ ✇❤❛t ✐t ♠❡❛♥s ❢♦r ❛ ❧✐♥❡ ✐♥t❡❣r❛❧

dR

t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛t❤✳

✭❜✮ ■s

✇❤❡r❡

Z

C

Z

C

V·

V · dR,

V =< 1 − ye−x , e−x >

❊①❡r❝✐s❡ ✻✳✺

✭✶✮ ❘❡♣r❡s❡♥t t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

1

❝❡♥t❡r❡❞ ❛t

0

✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛t❤❄

❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ✭✷✮ ❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ t❤✐s ❝✐r❝❧❡ ❛t t❤❡ ♣♦✐♥t t❤❡ ✢✉① ♦❢ t❤❡ ✈❡❝t♦r

√ √ ( 2/2, 2/2)✳ ✭✸✮ ❈♦♠♣✉t❡ ✜❡❧❞ F =< 2, 1 > ❛❝r♦ss t❤❡

♣❛rt ♦❢ t❤❡ ❝✐r❝❧❡ t❤❛t ❧✐❡s ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t✳

❊①❡r❝✐s❡ ✻✳✶✵

✭❛✮ ❙t❛t❡ ●r❡❡♥✬s ❚❤❡♦r❡♠✳

✭❜✮ ❱❡r✐❢② ✐t ❢♦r t❤❡

✻✳

❊①❡r❝✐s❡s✿ ❱❡❝t♦r ✜❡❧❞s

✹✾✹

❊①❡r❝✐s❡ ✻✳✶✽

✈❡❝t♦r ✜❡❧❞

❆ ✈❡❝t♦r ✜❡❧❞ V ✐s s❦❡t❝❤❡❞ ❜❡❧♦✇✳ ❙✉♣♣♦s❡ C ✐s t❤❡ ❝❧♦❝❦✇✐s❡ ♦r✐❡♥t❡❞ sq✉❛r❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐✲ Z ❣✐♥✳ ■s V · dR ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r 0❄ ❊①♣❧❛✐♥✳

V (x, y) = yi − xj

❛♥❞ ❛ ✉♥✐t sq✉❛r❡✳

C

❊①❡r❝✐s❡ ✻✳✶✶

❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ♦❢ ●r❡❡♥✬s ❚❤❡♦r❡♠✿ I

C

V · dR =

ZZ  D

∂L ∂M − ∂x ∂y



dA .

❊①♣❧❛✐♥ ✐ts ♣❛rts ❛♥❞ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡♠✳ Pr♦✲ ✈✐❞❡ ❛ s❦❡t❝❤✳ ❊①❡r❝✐s❡ ✻✳✶✷

❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡ ✈❡❝t♦r ✜❡❧❞ V (x, y, z) = ez i + j + xet k ✐s ❝♦♥s❡r✈❛t✐✈❡✳ ❊①❡r❝✐s❡ ✻✳✶✸

❊①❡r❝✐s❡ ✻✳✶✾

Pr♦✈❡ t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ V (x, y, z) = zj − yk ✐s ♥♦t ❝♦♥s❡r✈❛t✐✈❡✳

❙❦❡t❝❤ t❤❡ ✈❡❝t♦r ✜❡❧❞ 1 V (x, y) = p (yi − xj) . x2 + y 2

❊①❡r❝✐s❡ ✻✳✷✵

❙❦❡t❝❤ t❤❡ ✈❡❝t♦r ✜❡❧❞ V (x, y) =

❊①❡r❝✐s❡ ✻✳✶✹

x2

1 (yi − xj) . + y2

❋✐♥❞ t❤❡ ✇♦r❦ ❞♦♥❡ ❜② ❢♦r❝❡ ✜❡❧❞ V (x, y) = xyi + y 2 j

✐♥ ♠♦✈✐♥❣ ❛♥ ♦❜❥❡❝t ❛❧♦♥❣ t❤❡ ♣❛r❛❜♦❧❛ x = t, y =

t2 , 0 ≤ t ≤ 1.

❊①❡r❝✐s❡ ✻✳✶✺

✭❛✮ ●✐✈❡♥ ❛ ✈❡❝t♦r ✜❡❧❞ V (x, y) =< 3 + 2xy, x2 − 3y 2 >, ✜♥❞ ❛ ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t Z ∇f = V. ✭❜✮ ❯s❡

✭❛✮ t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧✐♥❡ ✐♥t❡❣r❛❧ ✐s t❤❡ ❝✉r✈❡ ❣✐✈❡♥ ❜②

C

V · dR, ✇❤❡r❡ C

R(t) =< et sin t , et cos t > , 0 < t < π .

❊①❡r❝✐s❡ ✻✳✷✶

✭✶✮ ❘❡♣r❡s❡♥t t❤❡ ❝②❧✐♥❞❡r ♦❢ r❛❞✐✉s 1 ❛♥❞ ❤❡✐❣❤t 1 ❝❡♥t❡r❡❞ ♦♥ t❤❡ z ✲❛①✐s ❛s ❛ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡✳ ✭✷✮ ❋✐♥❞ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ ❝②❧✐♥❞❡r ❛t t❤❡ √ t❤❡ √ ♣♦✐♥t ( 2/2, 2/2, 1/2)✳ ✭✸✮ ❈♦♠♣✉t❡ t❤❡ ✢✉① ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ F =< 2, 1, 1 > ❛❝r♦ss t❤❡ ♣❛rt ♦❢ t❤❡ ❝②❧✐♥❞❡r t❤❛t ❧✐❡s ✐♥ t❤❡ ✜rst ♦❝t❛♥t✳

❊①❡r❝✐s❡ ✻✳✷✷

❆ ✈❡❝t♦r ✜❡❧❞ V ✐s s❦❡t❝❤❡❞ ❜❡❧♦✇✳ ❙✉♣♣♦s❡ C ✐s t❤❡ ❝❧♦❝❦✇✐s❡ ♦r✐❡♥t❡❞ sq✉❛r❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐✲ Z ❣✐♥✳ ■s V · dR ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r 0❄ ❊①♣❧❛✐♥✳ C

❊①❡r❝✐s❡ ✻✳✶✻

❱❡r✐❢② ●r❡❡♥✬s ❚❤❡♦r❡♠ ❢♦r t❤❡ ✈❡❝t♦r ✜❡❧❞ V (x, y) = (x − y)i + xj

❛♥❞ t❤❡ r❡❣✐♦♥ D ❜♦✉♥❞❡❞ ❜② t❤❡ ✉♥✐t ❝✐r❝❧❡ C : R(t) = cos ti + sin tj, 0 ≤ t ≤ 2π . ❊①❡r❝✐s❡ ✻✳✶✼

❙❦❡t❝❤ t❤❡ ✈❡❝t♦r ✜❡❧❞ x2

1 < x, y > . + y2

❊①❡r❝✐s❡ ✻✳✷✸

Pr♦✈❡ t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ V (x, y, z) = zj − yk ✐s ♥♦t ❝♦♥s❡r✈❛t✐✈❡✳

✻✳

❊①❡r❝✐s❡s✿ ❱❡❝t♦r ✜❡❧❞s

✹✾✺

❊①❡r❝✐s❡ ✻✳✷✹

❊①❡r❝✐s❡ ✻✳✸✷

❙❦❡t❝❤ t❤❡ ✈❡❝t♦r ✜❡❧❞

▼❡❛s✉r❡ t❤❡ ✢♦✇ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ G(x, y) =< ex + y, ey > ❆▲❖◆● t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✇✐t❤ ❝♦r♥❡rs ❛t (0, 0), (1, 0), (0, 1), (1, 1)✳

V (x, y) =

x2

1 (yi − xj) . + y2

❊①❡r❝✐s❡ ✻✳✸✸

❊①❡r❝✐s❡ ✻✳✷✺

❋✐♥❞ t❤❡ ✇♦r❦ ❞♦♥❡ ❜② ❢♦r❝❡ ✜❡❧❞ V (x, y) =< xy, y 2 >

✐♥ ♠♦✈✐♥❣ ❛♥ ♦❜❥❡❝t ❛❧♦♥❣ t❤❡ ♣❛r❛❜♦❧❛ x = t, y =

t2 , 0 ≤ t ≤ 1.

❙✉♣♣♦s❡ ❛ ❝❧♦s❡❞ ❝✉r✈❡ ✐s ❧♦❝❛t❡❞ ✇✐t❤✐♥ t❤❡ ✉♥✐t x3

y3

− 2x, > ❝✐r❝❧❡✳ ■s t❤❡ ✢♦✇ ♦❢ H(x, y) =< 3 3 ❆❈❘❖❙❙ t❤✐s ❝✉r✈❡ ♥❡❣❛t✐✈❡ ♦r ♣♦s✐t✐✈❡❄ ❊①♣❧❛✐♥✳ ❊①❡r❝✐s❡ ✻✳✸✹

❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ F (x, y) = (xy, x2 + y 2 ) ❛t t❤❡ ♣♦✐♥t ✇❤❡r❡ x = 1 ❛♥❞ y = 1✳

❊①❡r❝✐s❡ ✻✳✷✻



 1 ●✐✈❡♥ ✈❡❝t♦r ✜❡❧❞ F (x, y) = x, y ✳ ❈❤♦♦s❡ ❛ 2 ❢❡✇ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ❞r❛✇ t❤❡ ✈❡❝t♦r F (x, y) ✇✐t❤ t❛✐❧ ❛t (x, y)✳ ❚❤❡r❡ ❛r❡ ❛ ❢❡✇ ❢❛♠✐❧✐❡s ♦❢ tr❛✲

❥❡❝t♦r✐❡s ✐♥ t❤✐s ✈❡❝t♦r ✜❡❧❞✳ P❡♥❝✐❧ ✐♥ ❛ ❢❡✇ tr❛❥❡❝✲ t♦r✐❡s ♦❢ ❡❛❝❤ t②♣❡✳

❊①❡r❝✐s❡ ✻✳✸✺

❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ F (x, y, z) = (xz + y, x2 + zy 2 ) ❛t t❤❡ ♣♦✐♥t ✇❤❡r❡ x = 1✱ y = 1✱ ❛♥❞ z = 0✳ ❊①❡r❝✐s❡ ✻✳✸✻

❊①❡r❝✐s❡ ✻✳✷✼

❙✉♣♣♦s❡ t❤❛t ❛ ♠❛ss M ✐s ✜①❡❞ ❛t t❤❡ ♦r✐❣✐♥ ✐♥ s♣❛❝❡✳ ❲❤❡♥ ❛ ♣❛rt✐❝❧❡ ♦❢ ✉♥✐t ♠❛ss ✐s ♣❧❛❝❡❞ ❛t t❤❡ ♣♦✐♥t (x, y) ♦t❤❡r t❤❛♥ t❤❡ ♦r✐❣✐♥✱ ✐t ✐s s✉❜❥❡❝t❡❞ t♦ ❛ ❢♦r❝❡ G(x, y) ♦❢ ❣r❛✈✐t❛t✐♦♥❛❧ ❛ttr❛❝t✐♦♥✳ P❧♦t t❤❡ ✈❡❝t♦r ✜❡❧❞ G(x, y)✱ ✐❢ t❤❡ ♠❛❣♥✐t✉❞❡ ✭❧❡♥❣t❤✮ ♦❢ G(x, y) ✐s

p kM x2 + y 2 ✳ ✱ ✇❤❡r❡ r = r2

❊①❡r❝✐s❡ ✻✳✷✽

❍❡r❡ ✐s ❛ ♣❧♦t ♦❢ ❛ ❢❡✇ tr❛❥❡❝t♦r✐❡s ✭❛♥❞ ✈❡❝t♦rs✮ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞✳ ❖♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ♣❧♦t✱ ❞❡t❡r♠✐♥❡ ✐❢ ✐t ✐s ❛ ❣r❛❞✐❡♥t ✜❡❧❞ ♦r ♥♦t✳ ❊①♣❧❛✐♥✳ ❊①❡r❝✐s❡ ✻✳✷✾

❯s❡ ●❛✉ss✬s ❢♦r♠✉❧❛ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✢♦✇ ♦❢ t❤❡ ✈❡❝✲ t♦r ✜❡❧❞ F (x, y, z) =< z 2 , y, x2 > ❛❝r♦ss t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ♣②r❛♠✐❞ ❜♦✉♥❞❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ❛♥❞ t❤❡ ✜rst ♦❝t❛♥t ♣❛rt ♦❢ t❤❡ ♣❧❛♥❡ ✇✐t❤ ❡q✉❛t✐♦♥ x + y + z = 1✳ ❊①❡r❝✐s❡ ✻✳✸✵

❯s❡ ●❛✉ss✬s ❢♦r♠✉❧❛ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✢♦✇ ♦❢ t❤❡ ✈❡❝✲ t♦r ✜❡❧❞ F (x, y, z) =< z + y, x + y, z > ❛❝r♦ss t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ✸❉ ❜♦① ✇✐t❤ ❛ s❧❛♥t❡❞ t♦♣ ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♣♦✐♥ts (x, y, z) ✇✐t❤ 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ x + 1✳ ❊①❡r❝✐s❡ ✻✳✸✶

❋✐♥❞ t❤❡ ❞✐✈❡r❣❡♥❝❡ ❛♥❞ t❤❡ r♦t❛t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ F (x, y) =< x2 y, xy sin y >✳

❊✈❛❧✉❛t❡ Z

(y 2 + 2xy) dx + (x2 + 2xy) dy , C

✇❤❡r❡ C ✐s t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ y = 2x2 ❢r♦♠ (0, 0) t♦ (1, 2)✳ ❊①❡r❝✐s❡ ✻✳✸✼

 x −y ❜❡ ❛ ✈❡❝✲ , ▲❡t F (x, y) = x2 + y 2 x2 + y 2 t♦r ✜❡❧❞✱ ❛♥❞ ❧❡t C ❛ s✐♠♣❧❡ ✭✐✳❡✳✱ ✇✐t❤♦✉t s❡❧❢✲ 

✐♥t❡rs❡❝t✐♦♥s✮ ❝❧♦s❡❞ ♣❛t❤ t❤❛t ❡♥❝❧♦s❡s t❤❡ ♦r✐❣✐♥✳ ❋✐♥❞ t❤❡ ✇♦r❦ ♦❢ F ❛❧♦♥❣ C ✳ ❍✐♥t✿ ■t ✐s ❡q✉❛❧ t♦ t❤❡ ✇♦r❦ ❛❧♦♥❣ ❛ ❝❡rt❛✐♥ ❝✐r❝❧❡✳

✼✳

❊①❛♠♣❧❡s

✹✾✻

✼✳ ❊①❛♠♣❧❡s

❊①❛♠♣❧❡ ✼✳✶✿ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❈♦♥s✐❞❡r

1

f (x, y) = (x2 + (y − 2)2 − y) 2 + 1 .

✭❛✮ ❉♦♠❛✐♥✿

x2 + (y − 2)2 − y ≥ 0 , x2 + (y − 2)2 ≥ 4 ,

✐✳❡✳✱ t❤❡ r❡❣✐♦♥ ✇✐t❤ ❜♦✉♥❞❛r②✿

x2 + (y − 2)2 = 22 .

❚❤✐s ✐s t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 2 ✇✐t❤ ❝❡♥t❡r ❛t (0, 2)✳ ✭❜✮ ●r❛♣❤✿

1

f (0, y) = ((y − 2)2 − y) 2 + 1 = z ,

❤❡♥❝❡

1

❚❤❡♥ ✇✐t❤ x = 0✿

z = (x2 + (y − 2)2 − y) 2 + 1 =⇒ (z − 1)2 = x2 + (y − 2)2 .

❛♥❞ ✇✐t❤ y = 0✿

(z − 1)2 = (y − 2)2 =⇒ z − 1 = ±(y − 2) , (z − 1)2 = x2 =⇒ z − 1 = ±x .

❚❤❡♥ z = y − 1 ❛♥❞ z = −y + 3 r❡s✉❧t ✐♥ t✇♦ ❧✐♥❡s✳ ❚❤❡s❡ ❝r♦ss✲s❡❝t✐♦♥s ❣✐✈❡ ✉s t❤❡ ❣r❛♣❤✳ ✭❝✮ ❚❤✐s ✐s ❢✉♥♥❡❧✱ ❛ tr✉♥❝❛t❡❞ ❝♦♥❡✳ ❊①❛♠♣❧❡ ✼✳✷✿ ♥♦r♠

❚♦ ♣r♦✈❡ t❤❛t ||x|| ✐s ❝♦♥t✐♥✉♦✉s✱ ✉s❡ ❛♥❞ s❤♦✇ t❤❛t ❜② t❤❡ ❚r✐❛♥❣❧❡

■♥❡q✉❛❧✐t②✳

1

||x|| = (x21 + ... + x2n ) 2 , ||x − a|| < δ =⇒ ||x|| − ||a|| < δ ,

❊①❛♠♣❧❡ ✼✳✸✿ ✉♥✐♦♥ ♦❢ ♣❛t❤✲❝♦♥♥❡❝t❡❞

❙✉♣♣♦s❡ A ❛♥❞ B ❛r❡ ♣❛t❤✲❝♦♥♥❡❝t❡❞ ❛♥❞ a ✐s ✐♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥✳ ■❢ P ❛♥❞ Q ❜❡❧♦♥❣ t♦ t❤❡ ✉♥✐♦♥✱ ✜♥❞ ❛ ♣❛t❤ ❢r♦♠ P t♦ a✱ ❢r♦♠ a t♦ Q✳ ❚❤✐s ❣✐✈❡s ②♦✉ ❛ ♣❛t❤ ❢r♦♠ P t♦ Q✳ ❊①❛♠♣❧❡ ✼✳✹✿ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❙❤♦✇ t❤❛t

T (x) = x22 + 2x2 + 1

✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x41 + x22 .

✼✳

❊①❛♠♣❧❡s ❈♦♥s✐❞❡r

✹✾✼

f (x) − T (x) x→a ||x − a|| lim

x41 + x22 + 2x2 + 1 x→(0,−1) ||(x1 , x2 ) − (0, −1)|| x41 + x22 + 2x2 + 1 = lim 1 x→(0,−1) ((x − 0)2 + (x + 1)2 ) 2 1 2 2 x2 + 2x2 + 1 = lim |x2 + 1| x2 →(−1) (x2 + 1)2 = lim x2 →−1 |x2 + 1| = lim |x2 + 1| =

lim

❜② ❝❛♥❝❡❧✐♥❣ (x2 + 1)

x2 →−1

= 0.

❊①❛♠♣❧❡ ✼✳✺✿ ❈❤❛✐♥ ❘✉❧❡

▲❡t

h(t) = (sin et , cos et ) .

❚❤❡♥ h ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ✶✳ f : R → R ❣✐✈❡♥ ❜② f (t) = et ✱ ❛♥❞ ✷✳ g : R → R2 , g(x) = (sin x, cos x)✳ ❚❤❡♥ ❛♥❞ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡✱

f ′ (t) = et , ∇g = (cos x, − sin x) , h′ = (g◦f )′ = ∇g · f ′ = (cos et , − sin et )et .

❊①❛♠♣❧❡ ✼✳✻✿ ❞❡r✐✈❛t✐✈❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

❉❡s❝r✐❜❡ t❤❡ ❝✉r✈❡ ✇❤✐❝❤ r❡s✉❧ts ❢r♦♠ t❤❡ ✈❡❝t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ r(t) = (cos 2t, sin 2t, t) ,

✇❤❡r❡ t ∈ R✳ ❙♦❧✉t✐♦♥✿ ❚❤❡ ✜rst t✇♦ ❝♦♠♣♦♥❡♥ts ✐♥❞✐❝❛t❡ t❤❛t ❢♦r r(t) = (x(t), y(t), z(t)) ,

t❤❡ ♣❛✐r (x(t), y(t)) tr❛❝❡s ♦✉t ❛ ❝✐r❝❧❡✳ ❲❤✐❧❡ ✐t ✐s ❞♦✐♥❣ s♦✱ z(t) ✐s ♠♦✈✐♥❣ ❛t ❛ st❡❛❞② r❛t❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝✉r✈❡ ✇❤✐❝❤ r❡s✉❧ts ✐s ❛ ❝♦r❦ s❝r❡✇ s❤❛♣❡✱ ✐✳❡✳ ❛ ❤❡❧✐①✳ ❊①❛♠♣❧❡ ✼✳✼✿ ♣❛rt✐❝❧❡

❚❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ✐s (x, y)✱ ✇❤❡r❡ x = sin t , y = sin2 t .

❉❡s❝r✐❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ♣❛rt✐❝❧❡ ❛s t ✈❛r✐❡s ♦✈❡r t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [a, b]✳ ❙♦❧✉t✐♦♥✿ ❲❡ ❝❛♥ ❡❧✐♠✐♥❛t❡ t t♦ s❡❡ t❤❛t t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ♦❜❥❡❝t t❛❦❡s ♣❧❛❝❡ ♦♥ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ✳ ❚❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ✐s ❢r♦♠ (sin a, sin2 a) t♦ (sin b, sin2 b)✳ ❊①❛♠♣❧❡ ✼✳✽✿ ❧✐♠✐t

❋✐♥❞ t❤❡ ❧✐♠✐t

x2 − x . x→1 x − 1 lim

✼✳

❊①❛♠♣❧❡s

✹✾✽

❙♦❧✉t✐♦♥✿ ■t ✐s

x · (x − 1) = x for x 6= 1 . x−1

❍❡♥❝❡

x · (x − 1) = lim x = 1 . x→1 x→1 x−1 lim

❊①❛♠♣❧❡ ✼✳✾✿ ❧✐♠✐t ❛t ✐♥✜♥✐t② ❋✐♥❞ t❤❡ ❧✐♠✐t lim

x→∞

❙♦❧✉t✐♦♥✿ ❘❡✇r✐t❡

x . 1+x

x 1 = 1+x 1+

◆♦✇ ✐t ✐s

1 x

.

1 = 1 6= 0 . x→∞ 1 + x lim

❚❤❡r❡❢♦r❡✱

1 x = lim x→∞ 1 + x→∞ 1 + x lim

1 x

=

1 = 1. 1

❊①❛♠♣❧❡ ✼✳✶✵✿ ❝♦♥t✐♥✉✐t② ♦♥ ♣❧❛♥❡ ▲❡t f : R → R2 ✳ ■s

1

f (t) = ((t + 1) 2 , tan(t))

❝♦♥t✐♥✉♦✉s❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ ✜rst ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r t ≥ −1,

✇❤✐❧❡ t❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r t 6=

❍❡♥❝❡ ✐t ✐s ❝♦♥t✐♥✉♦✉s ♦♥ R ❢♦r

π + kπ, k ∈ Z . 2

t ≥ −1 ❛♥❞ t 6=

π + kπ, k ∈ Z . 2

❊①❛♠♣❧❡ ✼✳✶✶✿ ❝♦♥t✐♥✉✐t② ✐♥ s♣❛❝❡ ▲❡t f : R → R3 ✱ f (t) =

■❢ f ❝♦♥t✐♥✉♦✉s❄



1 1 1 , (1 − t2 ) 2 , t2 − 1 t

❙♦❧✉t✐♦♥✿ ❚❤❡ ✜rst ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r t 6= ±1 .

❚❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r t ∈ [−1, 1] ,

✇❤✐❧❡ t❤❡ t❤✐r❞ ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r t 6= 0 .

❍❡♥❝❡✱ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ R ❢♦r t ∈ (−1, 0) ∪ (0, 1) .



.

✼✳

❊①❛♠♣❧❡s

✹✾✾

❊①❛♠♣❧❡ ✼✳✶✷✿ ❞✐♠❡♥s✐♦♥

▲❡t g : R → R4 ✱ ■❢ g ❝♦♥t✐♥✉♦✉s❄

4 1

g(t) = (cos(4t), 1 − (3t + 1) 2 , sin(5t), sec(t)) .

❙♦❧✉t✐♦♥✿ ❚❤❡ ✜rst ❛♥❞ t❤❡ t❤✐r❞ ❝♦♠♣♦♥❡♥t ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ R✳ ❚❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ 1 1 ✐♥ t❤❡ ❢♦✉rt❤ ❝♦♠♣♦♥❡♥t ✐s ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢♦r ❞❡✜♥❡❞ ♦♥ (− , ∞)✱ ✇❤✐❧❡ t❤❡ s❡❝❛♥t sec(t) = 3

cos(t)

t 6=

π + kπ, k ∈ Z . 2

❍❡♥❝❡✱ g(t) ✐s ❝♦♥t✐♥✉♦✉s ❛s ❧♦♥❣ ❛s t≥−

π 1 ❛♥❞ t 6= + lπ, l ∈ Z . 3 2

❊①❛♠♣❧❡ ✼✳✶✸✿ t❛♥❣❡♥t ✈❡❝t♦r

▲❡t

f (t) = (sin t, t2 , t + 1) ❢♦r t ∈ [0, 5] .

❋✐♥❞ ❛ t❛♥❣❡♥t ❧✐♥❡ t♦ t❤❡ ❝✉r✈❡ ♣❛r❛♠❡tr✐③❡❞ ❜② f ❛t t❤❡ ♣♦✐♥t t = 2✳

❙♦❧✉t✐♦♥✿ ❆ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ❤❛s t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ❛s f ′ (2)✳ ❚❤❡r❡❢♦r❡✱ ✐t s✉✣❝❡s t♦ s✐♠♣❧② ✉s❡ f ′ (2) ❛s ❛ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ❢♦r t❤❡ ❧✐♥❡✳ ❋✉rt❤❡r✱ f ′ (2) = (cos 2, 4, 1) .

❍❡♥❝❡✱ ❛ ♣❛r❛♠❡tr✐③❡❞ ❡q✉❛t✐♦♥ ❢♦r t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s (x, y, z) = (sin 2, 4, 3) + t(cos 2, 4, 1) . ❊①❛♠♣❧❡ ✼✳✶✹✿ ✈❡❧♦❝✐t② ✈❡❝t♦r

▲❡t ❋✐♥❞ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❢♦r t = 1✳

f (t) = (sin t, t2 , t + 1) ❢♦r t ∈ [0, 5] .

❙♦❧✉t✐♦♥✿ ❚❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ✐s s✐♠♣❧② f ′ (1) = (cos 1, 2, 1)✳ ❊①❛♠♣❧❡ ✼✳✶✺✿ ✈❡❧♦❝✐t②✲s♣❡❡❞✲❛❝❝❡❧❡r❛t✐♦♥

❈♦♥s✐❞❡r g : R → R2 ✱

g(t) = (t, t2 ) .

❲❤❛t ✐s ✐ts ✈❡❧♦❝✐t②✱ s♣❡❡❞✱ ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥ ❛s ✐t ♣❛ss❡s t❤r♦✉❣❤ (2, 4)❄ ❙♦❧✉t✐♦♥✿ ■t ✐s ❛♥❞

g ′ (t) = (1, 2t) g ′ ‘(t) = (0, 2) .

❋✉rt❤❡r✱ ❛t (2, 4)✱ ✇❡ ❦♥♦✇ t = 2✳ ❋♦r t❤❡ ✈❡❧♦❝✐t② ❛t t = 2✱ ✇❡ ❤❛✈❡ g ′ (2) = (1, 4)✳ ❚❤❡ s♣❡❡❞ ❡q✉❛❧s 1

||g ′ (2)|| = (12 + 42 ) 2 =

✇❤❡r❡❛s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❡q✉❛❧ t♦ g ′ ‘(2) = (0, 2)✳

√

17 ,

✼✳

❊①❛♠♣❧❡s

✺✵✵

❊①❛♠♣❧❡ ✼✳✶✻✿ ✈❡❧♦❝✐t②

❆♥ ♦❜❥❡❝t ❤❛s ♣♦s✐t✐♦♥ r(t) = (t3 ,

1 t , (t2 + 2) 2 ) km . 1+t

✇❤❡r❡ t ✐s ❣✐✈❡♥ ✐♥ ❤♦✉rs✳ ❋✐♥❞ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♦❜❥❡❝t ✐♥ ❦✐❧♦♠❡t❡rs ♣❡r ❤♦✉r ✇❤❡♥ t = 1✳ ❙♦❧✉t✐♦♥✿ ❙✐♥❝❡ ✈❡❧♦❝✐t② ❛t t✐♠❡ t ❡q✉❛❧s r′ (t)✱ ✇❡ ❝❛❧❝✉❧❛t❡ 

1 1 1(1 + t) − t , 2t · (t2 + 2)− 2 r (t) = 3t , 2 (1 + t) 2 ! 1 1 2 = 3t , , t . (1 + t)2 (t2 + 2) 12

′

2

❋♦r t = 1✱ t❤❡ ✈❡❧♦❝✐t② ✐s ′

r (1) =



1 1 3, , √ 4 3





km/hour .

❊①❛♠♣❧❡ ✼✳✶✼✿ ❤②♣❡r❜♦❧❛

❈♦♥s✐❞❡r g : R → R2 ✱

1 g(t) = ( , 2t) t

❞❡✜♥❡❞ ♦♥ (0, ∞)✳ ■ts ✐♠❛❣❡ ✐s ♦♥❡ ❜r❛♥❝❤ ♦❢ ❛ ❤②♣❡r❜♦❧❛ ✭t❤✐s ❝❛♥ ❜❡ s❡❡♥ ❜② ✇r✐t✐♥❣ g(t) = (x, y)✱ ✐✳❡✳ 2 1 x = , y = 2t✱ ✇❤✐❝❤ ②✐❡❧❞s y = )✳ ❋✐♥❞ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ s♣❡❡❞ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛t t✐♠❡ t✳ t x

❙♦❧✉t✐♦♥✳ ❚❤❡ ✈❡❧♦❝✐t② ❡q✉❛❧s g ′ (t) = (−

t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❡q✉❛❧ t♦ g ′ ‘(t) = (

❛♥❞ t❤❡ s♣❡❡❞ ✐s ||g ′ (t)|| = ||(−

1 , 2) , t2

2 , 0) , t3

1 1 1 , 2)|| = ( 4 + 4) 2 . 2 t t

❊①❛♠♣❧❡ ✼✳✶✽✿ ❝♦♥❥❡❝t✉r❡

❋♦r t❤❡ ❢✉♥❝t✐♦♥ f (t) = (3t cos(2t), 4t sin(2t)) ,

s❤♦✇ t❤❛t

❙♦❧✉t✐♦♥✿ ■t ✐s

d ||f (t)|| = 6 ||f ′ (t)|| . dt

1

||f (t)|| = (9t2 cos2 (2t) + 16t2 sin2 (2t)) 2

❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❡q✉❛❧s

❋✉rt❤❡r ❛♥❞ t❤✉s

1

= (9t2 + 7t2 sin}2 (2t)) 2 ,

d 9t + 14t2 sin(2t) cos(2t) + 7t sin2 (2t) . ||f (t)|| = 1 dt 9t2 + 7t2 sin2 (2t)) 2 f ′ (t) = (3 cos(2t) − 6t sin(2t), 4 sin(2t) + 8t cos(2t)) , 1

||f ′ (t)|| = ((3 cos(2t) − 6t sin(2t))2 + (4 sin(2t) + 8t cos(2t))2 ) 2 1

= (9 + 7 sin2 (2t) + 36t2 + 28t2 cos2 (2t) + 14t sin(4t)) 2 .

✼✳

❊①❛♠♣❧❡s

✺✵✶

◆♦✇ ❡✈❛❧✉❛t✐♥❣ ❜♦t❤ ❢✉♥❝t✐♦♥s ❛t t = π ✭❢♦r ❡①❛♠♣❧❡✮ ②✐❡❧❞s (

✇❤❡r❡❛s ❤❡♥❝❡

d 9π = 3, ||f (t)||)t=π = √ dt 9π 2

||f ′ (π)|| =

d ||f (t)|| = 6 ||f ′ (t)||✳ dt

p

9 + 64π 2 ≈ 25.31 ,

❊①❛♠♣❧❡ ✼✳✶✾✿ ✢♦✇ ❧✐♥❡ ▲❡t ❜❡ ❛ ✈❡❝t♦r ✜❡❧❞✳ ❱❡r✐❢② t❤❛t t❤❡ ♣❛t❤

F = yi − xj + 2k =< y, −x, 2 > g(t) = (sin t, cos t, 2t)

✐s ❛ ✢♦✇ ❧✐♥❡ ✭✐✳❡✳ ❛ ♣❛t❤ s✉❝❤ t❤❛t t❤❡ ✈❡❧♦❝✐t② ❛❧♦♥❣ t❤❡ ♣❛t❤ ✐s ❛ ✈❡❝t♦r ✐♥ t❤❡ ✈❡❝t♦r ✜❡❧❞✱ g ′ (t) = F (g(t))) ❢♦r t❤❡ ✈❡❝t♦r ✜❡❧❞ F ✳ ❙♦❧✉t✐♦♥✿ ❲✐t❤ (x, y, z) = (sin t, cos t, 2t)

✐t ❢♦❧❧♦✇s t❤❛t

g ′ (t) = (cos t, − sin t, 2) ,

❛♥❞

F (g(t)) = (y, −x, 2t) = (cos t, − sin t, 2t) .

❊①❛♠♣❧❡ ✼✳✷✵✿ ❛str♦✐❞ ❈♦♥s✐❞❡r t❤❡ ♣❛t❤ g : R → R2 ✱

g(t) = (cos3 t, sin3 t) , π 2

✇❤✐❝❤ ❞❡s❝r✐❜❡s ❛♥ ❛str♦✐❞✳ ❋♦r t ∈ [0, ]✱ ♦♥❡ ❢♦✉rt❤ ♦❢ t❤❡ ❛str♦✐❞ ✐s ❞❡s❝r✐❜❡❞✳ ❈❛❧❝✉❧❛t❡ ✐ts ❛r❝✲❧❡♥❣t❤✳ ❙♦❧✉t✐♦♥✿ ❲✐t❤

g ′ (t) = (−3 sin t cos2 t, 3 sin2 t cos t) , 3 ||g ′ (t)|| = 3 sin t cos t = sin2t , 2

✐t ✐s

Z

π 2

0

′

||g (t)|| dt

=

Z

π 2

0

3 sin 2t dt 2

π 3 = − cos}(2t)|02 4 3 3 3 = − cos π + cos 0 = . 4 4 2

❊①❛♠♣❧❡ ✼✳✷✶✿ ❛r❝✲❧❡♥❣t❤ ❈♦♥s✐❞❡r t❤❡ ♣❛t❤ f : R → R2 ✱

3 2 f (t) = (t2 , (2t + 1) 2 ), 0 ≤ t ≤ 4 . 3

❈❛❧❝✉❧❛t❡ ✐ts ❛r❝✲❧❡♥❣t❤✳ ❙♦❧✉t✐♦♥✿ ■t ✐s Z

4 0

′

||f (t)||dt =

Z

4

2

1 2

(4t + 4(2t + 1)) dt = 0

Z

4 0

2(t + 1)dt = t2 + 2t|40 = 24.

✼✳

❊①❛♠♣❧❡s

✺✵✷

❊①❛♠♣❧❡ ✼✳✷✷✿ ❤❡❧✐① ❋✐♥❞ t❤❡ ❛r❝✲❧❡♥❣t❤ ♦❢ t❤❡ ❤❡❧✐①

g(t) = (cos t, sin t, t) ❢r♦♠

t=0

t = 2π ✳

t♦

❙♦❧✉t✐♦♥✿ ❚❤❡ ❛r❝✲❧❡♥❣t❤ ❡q✉❛❧s

Z

2π 0

′

||g (t)||dt =

Z

2π 0Z

1

(−(sin t)2 + cos2 t + 1) 2 dt 2π

=

0

1

(sin2 t + cos2 t + 1) 2 dt Z 2π √ 2dt = √0 = 2(2π − 0) √ = 2 2π.

❊①❛♠♣❧❡ ✼✳✷✸✿ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥ P❛r❛♠❡tr✐③❡ t❤❡ ❤❡❧✐①

g(t) = (cos t, sin t, t) ❢♦r

t ∈ [0, 2π]

❜② ❛r❝✲❧❡♥❣t❤✳

❙♦❧✉t✐♦♥✿ ■t ✐s

❢♦r ❛❧❧

t ∈ [0, 2π]✳

❲❡ ❝❛♥ s♦❧✈❡ ❢♦r

t

Z

t 0

′

||g (t)||dt =

✐♥ t❡r♠s ♦❢

s✿

Z t√

2dt =

√

2t

0

s t = α(s) = √ , 2 ❛♥❞ ❤❡♥❝❡

g(s) = ❢♦r ❛❧❧

√ s ∈ [0, 2 2π]✳



s s s cos √ , sin √ , √ 2 2 2



❊①❛♠♣❧❡ ✼✳✷✹✿ ❝②❧✐♥❞r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ❋✐♥❞ t❤❡ ❛r❝✲❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ✇❤♦s❡ ❝②❧✐♥❞r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ❛r❡ ❣✐✈❡♥ ❜②

r = et , θ = t, z = et ❢♦r

t ∈ [0, 1]✳

❙♦❧✉t✐♦♥✿ ❲✐t❤

r′ (t) = et , θ′ (t) = 1, z ′ (t) = et , ✇❡ ♦❜t❛✐♥

Z

1 0

1

(r′ (t)2 + r(t)2 θ′ (t)2 + z ′ (t)2 ) 2 dt =

Z

1 0

1

(e2t + e2t (1) + e2t ) 2 dt Z 1 √ et 3 dt = √0 = 3(e − 1).

✼✳

❊①❛♠♣❧❡s

✺✵✸

❊①❛♠♣❧❡ ✼✳✷✺✿ ❣r❛❞✐❡♥t ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t✳ ❈♦♥s✐❞❡r ❛ r♦♦♠ ✐♥ ✇❤✐❝❤ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❣✐✈❡♥ ❜② ❛ s❝❛❧❛r ✜❡❧❞ T ✱ s♦ t❤❛t ❛t ❡❛❝❤ ♣♦✐♥t (x, y, z) t❤❡ t❡♠♣❡r❛t✉r❡ ❡q✉❛❧s T (x, y, z), ❛ss✉♠✐♥❣ t❤❡ t❡♠♣❡r❛t✉r❡ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✐♥ t✐♠❡✳ ❍♦✇ ❝❛♥ t❤❡ ❣r❛❞✐❡♥t ❜❡ ✐♥t❡r♣r❡t❡❞❄ ❙♦❧✉t✐♦♥✿ ■♥ t❤✐s ❝❛s❡✱ ❛t ❡❛❝❤ ♣♦✐♥t ✐♥ t❤❡ r♦♦♠✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ T ❛t t❤❛t ♣♦✐♥t ✇✐❧❧ s❤♦✇ t❤❡ ❞✐r❡❝t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡ t❡♠♣❡r❛t✉r❡ r✐s❡s ♠♦st q✉✐❝❦❧②✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t ✇✐❧❧ ❞❡t❡r♠✐♥❡ ❤♦✇ ❢❛st t❤❡ t❡♠♣❡r❛t✉r❡ r✐s❡s ✐♥ t❤❛t ❞✐r❡❝t✐♦♥✳

❊①❛♠♣❧❡ ✼✳✷✻✿ ❣r❛❞✐❡♥t ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t✳ ❈♦♥s✐❞❡r ❛ ❤✐❧❧ ✇❤♦s❡ ❤❡✐❣❤t ❛❜♦✈❡ s❡❛ ❧❡✈❡❧ ❛t ❛ ♣♦✐♥t (x, y) ✐s H(x, y)✳ ❍♦✇ ❝❛♥ t❤❡ ❣r❛❞✐❡♥t ♦❢ H ❜❡ ✐♥t❡r♣r❡t❡❞❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ H ❛t ❛ ♣♦✐♥t ✐s ❛ ✈❡❝t♦r ♣♦✐♥t✐♥❣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ st❡❡♣❡st s❧♦♣❡ ❛t t❤❛t ♣♦✐♥t✳ ❚❤❡ st❡❡♣♥❡ss ♦❢ t❤❡ s❧♦♣❡ ❛t t❤❛t ♣♦✐♥t ✐s ❣✐✈❡♥ ❜② t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t ✈❡❝t♦r✳

❊①❛♠♣❧❡ ✼✳✷✼✿ ✢② ❈♦♥s✐❞❡r ❛ r♦♦♠ ✐♥ ✇❤✐❝❤ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❣✐✈❡♥ ❜② f (x, y, z) = xy 2 z 3 .

❈♦♥s✐❞❡r ❢✉rt❤❡r t❤❛t ❛ ✢② ✐s ❝r❛✇❧✐♥❣ ❛t ✉♥✐t s♣❡❡❞ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r v =< −1, 1, 0 >

st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t s = (2, 1, 1) .

❈♦♠♣✉t❡ t❤❛t r❛t❡ ♦❢ t❡♠♣❡r❛t✉r❡ ❝❤❛♥❣❡ t❤❡ ✢② ✐s ❛❜♦✉t t♦ ❡①♣❡r✐❡♥❝❡✳ ❙♦❧✉t✐♦♥✿ ■t ✐s ❤❡♥❝❡

∇f (x, y, z) =< y 2 z 3 , 2xyz 3 , 3xy 2 z 2 > , ∇f (2, 1, 1) =< 1, 4, 6 > .

❋✉rt❤❡r✱ ❛♥❞ t❤✉s

1 1 v =< − √ , √ , 0 > , ||v|| 2 2 1 1 3 df =< 1, 4, 6 > · < − √ , √ , 0 >= √ . ds 2 2 2

❊①❛♠♣❧❡ ✼✳✷✽✿ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ▲❡t f : R3 → R✱

f (x, y, z) = x + sin(xy) + z .

❋✐♥❞ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ Dv f (1, 0, 1)✱ ✇❤❡r❡ 1 1 1 v =< √ , √ , √ > . 3 3 3

❙♦❧✉t✐♦♥✿ ◆♦t❡ t❤❛t v ✐s ❛❧r❡❛❞② ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♦♥❧② ♥❡❝❡ss❛r② t♦ ✜♥❞ ∇f (1, 0, 1) ❛♥❞ t❛❦❡ t❤❡ ❞♦t ♣r♦❞✉❝t✳ ■t ✐s ❛♥❞ ❤❡♥❝❡

∇f (x, y, z) = (2x + cos(xy)y, cos(xy)x, 1) , ∇f (1, 0, 1) = (2, 1, 1) .

✼✳

❊①❛♠♣❧❡s

✺✵✹

❚❤❡ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ✐s ♦❜t❛✐♥❡❞ ❛s 1 1 1 4√ < 2, 1, 1 > · < √ , √ , √ >= 3. 3 3 3 3 ❊①❛♠♣❧❡ ✼✳✷✾✿ t❛♥❣❡♥t ♣❧❛♥❡

▲❡t f : R3 → R✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ ❧❡✈❡❧ s✉r❢❛❝❡ f (x, y, z) = 6 ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x, y, z) = x2 + 2y 2 + 3z 2

❛t t❤❡ ♣♦✐♥t (1, 1, 1)✳ ❙♦❧✉t✐♦♥✿ ❋✐rst ♥♦t❡ t❤❛t (1, 1, 1) ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ ❧❡✈❡❧ s✉r❢❛❝❡ ✭✐✳❡✳ f (1, 1, 1) = 6 ✮✳ ❚♦ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ♣❧❛♥❡ ✐t s✉✣❝❡s t♦ ✜♥❞ t❤❡ ♥♦r♠❛❧ ✈❡❝t♦r t♦ t❤❡ ♣r♦♣♦s❡❞ ♣❧❛♥❡✳ ❇✉t ✇❡ s❡❡ ∇f (x, y, z) =< 2x, 4y, 6z > ,

❛♥❞ ❤❡♥❝❡ ∇f (1, 1, 1) =< 2, 4, 6 > .

❚❤❡r❡❢♦r❡✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ♣❧❛♥❡ ✐s✿

< 2, 4, 6 > +(x − 1, y − 1, z − 1) = 0 ,

♦r 2x + 4y + 6z − 12 = 0 . ❊①❛♠♣❧❡ ✼✳✸✵✿ ❣r❛❞✐❡♥t

❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = x2 sin(xy)

❛t (π, 0)✳ ❙♦❧✉t✐♦♥✿ ■t ✐s 

∇f (π, 0) = 

fx (π, 0) fy (π, 0)





=

2x sin(xy) + x2 y cos(xy) 3

x cos(xy)

 

=< 0, π 3 > . (π,0)

◆❡①t✳✳✳

■♥❞❡①

❆❞❞✐t✐✈✐t② ❋♦r ■♥t❡❣r❛❧s✱ ✸✾✹ ❆❞❞✐t✐✈✐t② ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✽ ❆❞❞✐t✐✈✐t② ♦❢ ❉♦✉❜❧❡ ■♥t❡❣r❛❧✱ ✸✻✼ ❆❞❞✐t✐✈✐t② ♦❢ ■♥t❡❣r❛❧✱ ✹✸✵ ❆❞❞✐t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠✱ ✹✷✸ ❆❞❞✐t✐✈✐t② ♦❢ ❙✉♠✱ ✹✶✾ ❆❞❞✐t✐✈✐t② ❘✉❧❡✱ ✶✾✷ ❆❞❞✐t✐✈✐t② ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✸✻✵ ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥✱ ✻✾ ❛✣♥❡ s♣❛❝❡✱ ✸✾ ❆❧❣❡❜r❛ ♦❢ ❈♦♥t✐♥✉✐t②✱ ✷✺✽ ❆❧❣❡❜r❛ ♦❢ ●r❛❞✐❡♥ts✱ ✸✹✵ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✷ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s✱ ✶✹✺ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✷✺✷ ❆❧t❡r♥❛t✐✈❡ ❉❡✜♥✐t✐♦♥ ♦❢ ▲✐♠✐t✱ ✶✹✸ ❆❧t❡r♥❛t✐✈❡ ❋♦r♠✉❧❛ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✹ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✶✱ ✾✷ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✾✹ ❆♥❣❧❡s ❢♦r ❉✐♠❡♥s✐♦♥ ✸✱ ✾✺ ❛♥❣❧❡s ❢♦r ❞✐♠❡♥s✐♦♥ ♥✱ ✶✵✵ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✱ ✶✽✶ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✹✷ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✹✷ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✸✹✷ ❛♥t✐❞❡r✐✈❛t✐✈❡✱ ✶✾✵ ❆♥t✐❞❡r✐✈❛t✐✈❡ P❧✉s ❈♦♥st❛♥t✱ ✶✾✶ ❆r❝✲❧❡♥❣t❤ P❛r❛♠❡tr✐③❛t✐♦♥✱ ✷✶✹ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❛t✐♦♥✱ ✷✶✷ ❛r❡❛ ♦❢ r❡❣✐♦♥✱ ✸✽✻ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❢♦r♠✱ ✸✽✹ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✽✺ ❆①✐♦♠s ♦❢ ❆✣♥❡ ❙♣❛❝❡✱ ✹✺ ❆①✐♦♠s ♦❢ ■♥♥❡r Pr♦❞✉❝t ❙♣❛❝❡✱ ✶✵✵ ❆①✐♦♠s ♦❢ ▼❡tr✐❝ ❙♣❛❝❡✱ ✸✹ ❆①✐♦♠s ♦❢ ◆♦r♠❡❞ ❙♣❛❝❡✱ ✼✽ ❆①✐♦♠s ♦❢ ❱❡❝t♦r ❙♣❛❝❡✱ ✻✸ ❜❛s✐s ✈❡❝t♦rs✱ ✺✺ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥✱ ✸✶✸ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✱ ✷✼✼✱ ✷✼✾✱ ✸✶✵ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ❋♦r ❖♥❡ ❱❛r✐❛❜❧❡✱ ✷✼✼ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ❋♦r ❚✇♦ ❱❛r✐❛❜❧❡s✱ ✷✼✾ ❜♦✉♥❞❛r② ♦❢ ❛ ✵✲❝❡❧❧✱ ✶✷✾ ❜♦✉♥❞❛r② ♦❢ ❛ ❝✉r✈❡✱ ✶✷✾ ❜♦✉♥❞❡❞ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✶✺✵ ❜♦✉♥❞❡❞ s❡t✱ ✶✶✷ ❇♦✉♥❞❡❞♥❡ss✱ ✶✶✺✱ ✶✺✵✱ ✷✻✷ ❈❛✉❝❤② ■♥❡q✉❛❧✐t②✱ ✶✵✵ ❝❡❧❧✱ ✶✷✷

❈❡♥t❡r ♦❢ ▼❛ss ❱✐❛ ■♥t❡❣r❛❧s✱ ✸✾✾ ❈❡♥t❡r ♦❢ ▼❛ss ❱✐❛ ❙✉♠s✱ ✸✾✽ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✼✻ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✷✸ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ■■✱ ✸✷✺ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✼✻ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✷✷ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ■■✱ ✸✷✹ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✶✼✻ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✷✷ ❈❤❛✐♥ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s ■■✱ ✸✷✹ ❈❧❛✐r❛✉t ❚❤❡♦r❡♠✱ ✷✾✺ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠✱ ✸✹✼ ❝❧♦s❡❞ ❝✉r✈❡✱ ✹✸✾ ❝❧♦s❡❞ ❢✉♥❝t✐♦♥✱ ✹✹✼ ❝❧♦s❡❞ s❡t✱ ✶✶✶ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❋♦r ❉♦✉❜❧❡ ■♥t❡❣r❛❧s✱ ✸✻✽ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s✱ ✸✾✹ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✸✻✵ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✽ ❈♦♠♣♦♥❡♥t✇✐s❡ ❉✐✛❡r❡♥t✐❛t✐♦♥✱ ✶✻✽✱ ✷✵✶ ❈♦♠♣♦♥❡♥t✇✐s❡ ■♥t❡❣r❛t✐♦♥✱ ✷✵✶ ❝♦♠♣♦s✐t✐♦♥✱ ✸✷✷✱ ✸✷✸ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s✱ ✷✻✵ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✱ ✶✹✾ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■✱ ✷✺✽ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■■✱ ✷✺✾ ❈♦♥st❛♥t ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✽✶ ❈♦♥st❛♥t ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✽✵ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❘✉❧❡✱ ✶✽✺✱ ✸✺✾ ❈♦♥st❛♥t ■♥t❡❣r❛❧ ❘✉❧❡✱ ✶✽✻✱ ✸✻✻✱ ✸✼✼ ❈♦♥st❛♥t ■♥t❡❣r❛❧ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s✱ ✸✾✹ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ ✶✶✺✱ ✶✶✼✱ ✷✵✵ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✼✵ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✶✽ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✼✵ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✶✽ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✶✼✵ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✶✼ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉♦✉❜❧❡ ■♥t❡❣r❛❧s✱ ✸✻✾ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s✱ ✶✾✹✱ ✸✾✺ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✸ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s✱ ✶✹✹ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✶✾✹✱ ✸✻✶ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❙✉♠s✱ ✶✾✹ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✾ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✹✶ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✹✵ ❈♦♥st❛♥t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✸✹✵ ❈♦♥st❛♥t ❙♣❡❡❞ P❛r❛♠❡tr✐③❛t✐♦♥✱ ✷✶✸

✺✵✽

❈♦♥t✳ ❂❃ ❇♦✉♥❞❡❞✱ ✷✻✶ ❈♦♥t✳ ❂❃ ■♥t❡❣r✳✱ ✶✽✻✱ ✶✾✸✱ ✸✻✻✱ ✸✻✼✱ ✸✼✼✱ ✸✼✽ ❈♦♥t✐♥✉✐t② ❛♥❞ ❆❧❣❡❜r❛✱ ✶✹✾ ❈♦♥t✐♥✉✐t② ♦❢ ❋✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✻✵ ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✶✺✵ ❝♦♥t✐♥✉♦✉s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✶✹✼ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡✱ ✶✵✽ ❝♦♥✈❡r❣❡s✱ ✶✶✹ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥✱ ✻✾ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥t✐♥✉✐t② ❖❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡✱ ✶✺✹ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡✱ ✶✶✸ ❈♦♦r❞✐♥❛t❡✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ❋♦r P❛r❛♠❡tr✐❝ ❈✉r✈❡✱ ✶✺✷ ❝✉r❧✱ ✹✺✶ ❈✉r✈❛t✉r❡✱ ✷✶✼ ❝✉r✈❛t✉r❡✱ ✷✶✷✱ ✷✶✻ ❈✉r✈❛t✉r❡ ♦❢ ❍❡❧✐①✱ ✷✷✺ ❈✉r✈❡ ♦♥ ❙♣❤❡r❡✱ ✷✵✹ ❞❡r✐✈❛t✐✈❡✱ ✶✻✺✱ ✸✶✵✱ ✸✶✻✱ ✸✶✽✱ ✸✷✸ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧s✱ ✶✻✺ ❉✐✛ ❂❃ ❈♦♥t✱ ✶✻✻ ❞✐✛❡r❡♥❝❡✱ ✶✸✵✱ ✶✻✵✱ ✷✻✽✱ ✸✵✻✱ ✹✹✼✱ ✹✺✶ ❉✐✛❡r❡♥❝❡ ❆♥❞ P❛rt✐❛❧ ❉✐✛❡r❡♥❝❡s✱ ✷✻✾ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✶✺✱ ✸✷✷✱ ✸✷✻ ❞✐✛❡r❡♥❝❡ ♦❢ s❡q✉❡♥❝❡✱ ✸✶✼✱ ✸✷✽ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✶✻✵✱ ✶✽✾✱ ✷✼✶✱ ✸✵✽✱ ✸✶✺✱ ✸✶✽✱ ✸✷✷ ❉✐r❡❝t✐♦♥❛❧ ❉❡r✐✈❛t✐✈❡✱ ✸✸✾ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡✱ ✸✸✽ ❉✐s❝r❡t❡ ❈❧❛✐r❛✉t ❚❤❡♦r❡♠✱ ✷✾✹ ❉✐s❝r❡t❡ ❈❧❛✐r❛✉t✬s ❚❤❡♦r❡♠✱ ✸✹✹ ❉✐s❝r❡t❡ ❈✉r✈❡ ♦♥ ❙♣❤❡r❡✱ ✷✵✹ ❞✐s❝r❡t❡ ❢♦r♠✱ ✶✷✺✱ ✶✸✵ ❞✐s♣❧❛❝❡♠❡♥t✱ ✸✽ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✶✱ ✸✷ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✸✷ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❋♦r ❉✐♠❡♥s✐♦♥ ✸✱ ✷✺ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✸✱ ✸✷ ❞✐✈❡r❣❡♥t s❡q✉❡♥❝❡✱ ✶✵✽ ❞♦t ♣r♦❞✉❝t✱ ✾✹✱ ✾✺ ❞♦t ♣r♦❞✉❝t ✐♥ ♥✲s♣❛❝❡✱ ✾✼ ❉♦t Pr♦❞✉❝t ❘✉❧❡✱ ✶✶✼✱ ✶✼✹ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✼✸ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✼✸ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✶✼✷ ❉♦t Pr♦❞✉❝t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s✱ ✶✹✺ ❊st✐♠❛t❡ ❘✉❧❡✱ ✶✾✸ ❊st✐♠❛t❡ ❘✉❧❡ ❋♦r ❉♦✉❜❧❡ ■♥t❡❣r❛❧s✱ ✸✻✽ ❊st✐♠❛t❡ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s✱ ✸✾✹ ❊st✐♠❛t❡ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✸✻✶ ❊st✐♠❛t❡ ❘✉❧❡ ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✽ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡✱ ✸✹✱ ✸✺ ❊✉❝❧✐❞❡❛♥ ♠❡tr✐❝✱ ✸✹✱ ✸✺ ❊✉❝❧✐❞❡❛♥ ♥♦r♠✱ ✼✽ ❡①❛❝t ❢✉♥❝t✐♦♥✱ ✸✹✸ ❊①❛❝t♥❡ss ❚❡st ❉✐♠❡♥s✐♦♥ ✷✱ ✹✹✽ ❊①❛❝t♥❡ss ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✸✹✹

❊①❛❝t♥❡ss ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ ✸✱ ✸✹✺✱ ✹✺✶ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✶✺✷✱ ✷✻✸ ❢❛❝❡✱ ✶✷✸ ❋❡r♠❛t ❚❤❡♦r❡♠✱ ✷✽✹ ❋✉❜✐♥✐✬s ❚❤❡♦r❡♠ ❋♦r ❚❤r❡❡ ❱❛r✐❛❜❧❡s✱ ✸✾✷ ❋✉❜✐♥✐✬s ❚❤❡♦r❡♠ ❋♦r ❚✇♦ ❱❛r✐❛❜❧❡s✱ ✸✽✽ ❋✉❜✐♥✐✬s ❚❤❡♦r❡♠✿ ■♥t❡❣r❛❧ ♦❢ ■♥t❡❣r❛❧✱ ✸✼✶ ❋✉❜✐♥✐✬s ❚❤❡♦r❡♠✿ ■♥t❡❣r❛❧ ♦❢ ■♥t❡❣r❛❧ ♦❢ ■♥t❡❣r❛❧✱ ✸✽✵ ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ✐♥✜♥✐t②✱ ✷✺✺ ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ✈❛❧✉❡ ❛t ✐♥✜♥✐t②✱ ✷✺✻ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t✱ ✷✺✼ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❣♦❡s t♦ ✐♥✜♥✐t②✱ ✷✺✻ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ✶✾✶ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❉✐✛❡r❡♥❝❡s ■✱ ✹✸✺ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❉✐✛❡r❡♥❝❡s ■■✱ ✹✸✸ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ❱❡❝t♦r ❋✐❡❧❞s✱ ✹✻✸ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ●r❛❞✐❡♥t ❱❡❝t♦r ❋✐❡❧❞s ■✱ ✹✹✺ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ●r❛❞✐❡♥t ❱❡❝t♦r ❋✐❡❧❞s ■■✱ ✹✹✶ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s✱ ✶✾✵ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ❢♦r ❱❡❝t♦r ❋✐❡❧❞s✱ ✹✺✽ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡❣r❡❡ ✶✱ ✶✸✶✱ ✶✸✷✱ ✶✽✼ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ♦❢ ❉❡❣r❡❡ ✷✱ ✹✺✻ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✱ ✷✻✸ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✈❛❧✉❡✱ ✷✻✸ ❣❧✉✐♥❣✱ ✶✷✶ ❣r❛❞✐❡♥t✱ ✷✼✽✱ ✸✶✵✱ ✸✹✸ ●r❛❞✐❡♥t ❂❃ P❛t❤✲✐♥❞❡♣❡♥❞❡♥t✱ ✹✹✹ ●r❛❞✐❡♥t ✐s ■rr♦t❛t✐♦♥❛❧✱ ✹✺✷ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✸✹✺✱ ✸✹✽✱ ✹✺✵✱ ✹✻✶ ●r❛❞✐❡♥t ❚❡st ❋♦r ❉✐♠❡♥s✐♦♥ ✸✱ ✸✹✺✱ ✸✹✾✱ ✹✻✶ ❣r❛❞✐❡♥t ✈❡❝t♦r ✜❡❧❞✱ ✷✼✷✱ ✸✹✻ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥✱ ✷✹✶✱ ✷✹✻ ❍♦♠♦❣❡♥❡✐t② ♦❢ ◆♦r♠✱ ✼✹ ■♠❛❣❡ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥✱ ✷✻✶ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥✱ ✷✹✺ ■♥❝r❡♠❡♥t❛❧ ❋✉❜✐♥✐✬s ❚❤❡♦r❡♠✱ ✸✾✷ ■♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❲♦r❦✱ ✹✷✾ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡✱ ✶✵✼ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✶✺✶✱ ✷✻✵ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❢♦r ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✻✶ ✐rr♦t❛t✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞✱ ✹✺✵✱ ✹✻✵ ▲❡♥❣t❤ ■s ■♥t❡❣r❛❧ ♦❢ ❙♣❡❡❞✱ ✷✶✺ ▲❡♥❣t❤ ♦❢ ❈✉r✈❡✱ ✷✷✶ ❧❡♥❣t❤ ♦❢ ❝✉r✈❡✱ ✷✷✶ ❧❡♥❣t❤ ♦❢ ✈❡❝t♦r✱ ✼✶ ▲❡✈❡❧ ❈✉r✈❡s ❆r❡ P❡r♣❡♥❞✐❝✉❧❛r t♦ ●r❛❞✐❡♥t✱ ✸✸✸✱ ✸✸✹

✺✵✾

❧❡✈❡❧ s❡t✱ ✷✹✷ ▲❡✈❡❧ ❙❡ts✱ ✷✹✸ ▲❡✈❡❧ ❙❡ts ❛♥❞ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✸✵ ▲❡✈❡❧ ❙❡ts ❛♥❞ ❉✐✛❡r❡♥❝❡s✱ ✸✸✵ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ✷✹✾ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥✱ ✷✻✹ ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✺ ❧✐♠✐t ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ✶✹✶ ❧✐♥❡ ✐♥t❡❣r❛❧✱ ✹✷✼ ❧✐♥❡ t❤r♦✉❣❤ ♣♦✐♥t ✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r✱ ✷✸✹ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡✱ ✷✵✵ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✱ ✻✾✱ ✼✵ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇❛s✐s ❱❡❝t♦rs✱ ✼✵ ▲✐♥❡❛r✐t② ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥✱ ✸✶✾ ▲✐♥❡❛r✐t② ♦❢ ■♥t❡❣r❛❧✱ ✹✸✵ ▲✐♥❡❛r✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠✱ ✹✷✸ ▲✐♥❡❛r✐t② ♦❢ ❙✉♠✱ ✹✶✾ ▲♦❝❛❧ ■♥t❡❣r❛❜✐❧✐t②✱ ✶✾✸ ▲♦❝❛❧ ■♥t❡❣r❛❜✐❧✐t② ❋♦r ❉♦✉❜❧❡ ■♥t❡❣r❛❧s✱ ✸✻✼ ▲♦❝❛❧ ■♥t❡❣r❛❜✐❧✐t② ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✽ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✱ ✷✽✷ ❧♦♦♣✱ ✹✸✾

♣❛rt✐t✐♦♥✱ ✶✷✹ ♣❛t❤✲❝♦♥♥❡❝t❡❞ s✉❜s❡t✱ ✶✺✶ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡✱ ✹✸✹ P❛t❤✲✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❱❡❝t♦r ❋✐❡❧❞s✱ ✹✸✽ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥✱ ✹✸✸ P❛t❤✲✐♥❞❡♣❡♥❞❡♥t ❖✈❡r ▲♦♦♣s✱ ✹✹✹ ♣❛t❤✲✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦r ✜❡❧❞✱ ✹✸✼✱ ✹✹✸ ♣❧❛♥❡ t❤r♦✉❣❤ ♣♦✐♥t ✇✐t❤ ♥♦r♠❛❧ ✈❡❝t♦r✱ ✷✸✼ P❧❛♥❡s P❛r❛❧❧❡❧ t♦ ❈♦♦r❞✐♥❛t❡ P❧❛♥❡s✱ ✷✸ ♣r♦❞✉❝t ♦❢ t❤❡ ♣❛rt✐t✐♦♥s✱ ✶✷✸ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✷✼ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✷✼ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✷✻ Pr♦❞✉❝t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✸ Pr♦❥❡❝t✐♦♥ ❱✐❛ ❉♦t Pr♦❞✉❝t✱ ✶✵✺

▼❛❣♥✐t✉❞❡ ♦❢ ❱❡❝t♦r✱ ✼✽ ♠❛❣♥✐t✉❞❡ ♦❢ ✈❡❝t♦r✱ ✼✶✱ ✼✽ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✶✼✽ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❋♦r P❛rt✐t✐♦♥✱ ✶✽✵ ♠♦♠❡♥t✱ ✸✾✼✱ ✸✾✾ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✸✸✽ ▼✉❧t✐♣❧❡s ♦❢ ❱❡❝t♦rs✱ ✼✺ ♠✉❧t✐♣❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✶✾✼

r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡✱ ✷✶✼ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛t✐♦♥✱ ✷✵✺ ❘❡♣❛r❛♠❡tr✐③❛t✐♦♥✱ ✶✸✼ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✱ ✶✽✺✱ ✸✻✺✱ ✸✼✼✱ ✸✽✻✱ ✸✾✵ ❘✐❡♠❛♥♥ s✉♠✱ ✶✽✹✱ ✶✽✾✱ ✸✺✼✱ ✸✼✻✱ ✸✽✹✱ ✹✷✷ ❘✐❡♠❛♥♥ ❙✉♠ ♦❢ ❘✐❡♠❛♥♥ ❙✉♠s✱ ✸✻✹ ❘✐❣❤t ❆♥❣❧❡✱ ❩❡r♦ ❉♦t Pr♦❞✉❝t✱ ✾✹ ❘♦❧❧❡✬s ❚❤❡♦r❡♠✱ ✶✼✼ r♦t♦r✱ ✹✺✵✱ ✹✻✵

♥✲✈♦❧✉♠❡✱ ✸✾✶ ♥❛t✉r❛❧ ❞♦♠❛✐♥✱ ✷✹✺ ◆❡❣❛t✐✈❡ ■♥t❡❣r❛❧✱ ✶✽✻ ♥❡❣❛t✐✈❡ ✈❡❝t♦r✱ ✺✸ ◆❡❣❛t✐✈✐t② ♦❢ ■♥t❡❣r❛❧✱ ✹✷✾ ◆❡❣❛t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠✱ ✹✷✸ ◆❡❣❛t✐✈✐t② ♦❢ ❙✉♠✱ ✹✶✾ ♥♦r♠ ♦❢ ✈❡❝t♦r✱ ✼✽ ◆♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❱❡❝t♦rs✱ ✼✺ ❖r✐❡♥t❛t✐♦♥ ♦❢ ❉♦♠❛✐♥ ♦❢ ■♥t❡❣r❛t✐♦♥✱ ✸✻✻ ♦r✐❡♥t❡❞ ✵✲❝❡❧❧✱ ✶✷✽ ♦r✐❡♥t❡❞ ✶✲❝❡❧❧✱ ✶✷✽ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥✱ ✶✵✺ ❖rt❤♦❣♦♥❛❧✐t② ✐♥ ✷✲s♣❛❝❡✱ ✶✵✸ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♣♣r♦❛❝❤❡s ♣♦✐♥t✱ ✶✹✻ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❣♦❡s t♦ ✐♥✜♥✐t②✱ ✶✹✻ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥✱ ✽✹ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥✱ ✽✾ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❝✉r✈❡✱ ✶✸✼✱ ✷✵✹ P❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ●r❛♣❤✱ ✶✸✼ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡✱ ✸✵✾ P❛rt✐❛❧ ❉❡r✐✈❛t✐✈❡s✱ ✷✼✹✱ ✸✵✾ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✱ ✷✼✹ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡✱ ✷✻✽ ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✷✻✾✱ ✸✵✽

◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✷✾ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✷✾ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✷✽ ◗✉♦t✐❡♥t ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✸

s❝❛❧❛r ♣r♦❞✉❝t✱ ✺✸ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✼✶ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✼✶ ❙❝❛❧❛r Pr♦❞✉❝t ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✶✼✶ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡✱ ✷✾✸ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✶✾✻✱ ✷✾✶✱ ✷✾✸ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✱ ✸✷✽ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✶✵✾ ❙q✉❡❡③❡ ❚❤❡♦r❡♠✱ ✶✹✾ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❋♦r ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✹ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❡❧❧✐♣s❡✱ ✶✸✼ st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❣r❛♣❤✱ ✶✸✼ ❙✉❜s♣❛❝❡s✱ ✻✹ ❙✉❜st✐t✉t✐♦♥ ❘✉❧❡✱ ✶✹✽ s✉♠✱ ✶✸✵✱ ✶✽✹✱ ✸✺✻✱ ✸✼✺✱ ✹✶✽ ❙✉♠ ❆r♦✉♥❞ ❈❡❧❧✱ ✹✺✸ ❙✉♠ ❆r♦✉♥❞ ❘❡❣✐♦♥✱ ✹✺✺ s✉♠ ♦❢ ✈❡❝t♦rs✱ ✺✷ ❙✉♠ ❘✉❧❡✱ ✶✶✹✱ ✶✶✻✱ ✶✾✾ ❙✉♠ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s✱ ✶✻✾ ❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✶✻ ❙✉♠ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✶✻✾ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✶✺ ❙✉♠ ❘✉❧❡ ❋♦r ❉✐✛❡r❡♥❝❡s✱ ✶✻✾ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✶✺

✺✶✵

❙✉♠ ❘✉❧❡ ❋♦r ❉♦✉❜❧❡ ■♥t❡❣r❛❧s✱ ✸✻✾ ❙✉♠ ❘✉❧❡ ❋♦r ■♥t❡❣r❛❧s✱ ✶✾✹✱ ✸✾✺ ❙✉♠ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s✱ ✷✺✸ ❙✉♠ ❘✉❧❡ ❋♦r ▲✐♠✐ts ♦❢ P❛r❛♠❡tr✐❝ ❈✉r✈❡s✱ ✶✹✹ ❙✉♠ ❘✉❧❡ ❋♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✶✾✹✱ ✸✻✷ ❙✉♠ ❘✉❧❡ ❋♦r ❙✉♠s✱ ✶✾✹ ❙✉♠ ❘✉❧❡ ❋♦r ❚r✐♣❧❡ ■♥t❡❣r❛❧s✱ ✸✼✾ s✉♣ ♠❡tr✐❝✱ ✸✺

✉♥✐t ♥♦r♠❛❧ ✈❡❝t♦r✱ ✷✷✻ ✉♥✐t ✈❡❝t♦r✱ ✼✹

t❛①✐❝❛❜ ♠❡tr✐❝✱ ✸✺ ❚✇♦ P♦t❡♥t✐❛❧ ❋✉♥❝t✐♦♥s✱ ✸✹✻

❲❡✐❣❤t ♦❢ ❈✉r✈❡✱ ✷✷✸ ✇❡✐❣❤t ♦❢ ❝✉r✈❡✱ ✷✷✸ ✇♦r❦✱ ✹✷✵✱ ✹✷✸✱ ✹✷✼

❯♥✐q✉❡♥❡ss ♦❢ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥✱ ✸✶✷ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❙❡q✉❡♥❝❡✱ ✶✶✵

✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✷✹✷ ❱❛r✐❛❜❧❡ ▼✉❧t✐♣❧❡ ❘✉❧❡✱ ✶✹✺ ✈❡❝t♦r✱ ✸✾✱ ✹✾ ✈❡❝t♦r ✜❡❧❞✱ ✹✵✼ ✈❡❝t♦r s♣❛❝❡✱ ✻✹ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠✱ ✶✷✼

③❡r♦ ✈❡❝t♦r✱ ✺✸

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