Calculus Illustrated. Volume 5: Differential Equations [5]
his is the fifth volume of the series Calculus Illustrated, a textbook for undergraduate students. Mathematical thinkin
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English Pages [423] Year 2020
Preface
Ordinary differential equations
Incremental motion
Discrete models: how to set up ODEs
Discrete forms
Differential forms
Solution sets of ODEs
Separation of variables in ODEs
The method of integrating factors
Change of variables in ODEs
Euler's method: back to the discrete
How large is the difference between the discrete and the continuous?
Qualitative analysis of ODEs
Linearization of ODEs
Motion under forces: the acceleration
Discrete models: how to set up ODEs of second order
Discrete forms, continued
Vector variables
Where matrices come from
Transformations of the plane
Linear operators
Examining and building linear operators
The determinant of a matrix
It's a stretch: eigenvalues and eigenvectors
The significance of eigenvectors
Bases
Classification of linear operators according to their eigenvalues
Vector and complex variables
Algebra of linear operators and matrices
Compositions of linear operators
How complex numbers emerge
Classification of quadratic polynomials
The complex plane C is the Euclidean space R2
Multiplication of complex numbers: C isn't just R2
Complex functions
Complex linear operators
Linear operators with complex eigenvalues
Complex calculus
Series and power series
Solving ODEs with power series
Systems of ODEs
Parametric curves
The predator-prey model
Qualitative analysis of the predator-prey model
Solving the Lotka–Volterra equations
Vector fields and systems of ODEs
Discrete systems of ODEs
Qualitative analysis of systems of ODEs
The vector notation and linear systems
Classification of linear systems
Classification of linear systems, continued
Applications of ODEs
Vector-valued forms
The pursuit curves
ODEs of second order as systems
Vector ODEs of second order: a double spring
A pendulum
Planetary motion
The two- and three-body problems
A cannon is fired...
Boundary value problems
Partial differential equations
Heat transfer between adjacent objects
Heat transfer depends on permeability
Heat transfer is caused by temperature differences
Heat transfer depends on the geometry
The heat PDE
Cells and forms in higher dimensions
Heat transfer in dimension 2: a plate
The heat PDE for dimension 2
Wave propagation in dimension 1: springs and strings
The wave PDE
Wave propagation in dimension 2: a membrane
Exercises
Exercises: Basics
Exercises: Analytical methods
Exercises: Euler's method
Exercises: Generalities
Exercises: Models and setting up ODEs
Exercises: Qualitative analysis
Exercises: Systems
Exercises: Second order
Exercises: Advanced
Exercises: PDEs
Exercises: Computing
Index
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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❡
x✱
a
✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ x2 = a✳
a
✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r
s✉❝❤ t❤❛t
❚❤❡ 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❤ ♦❢ ❞✐❛❣♦♥❛❧ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛
1 × 1 sq✉❛r❡❄ ❚❤❡ sq✉❛r❡ ✐s ♠❛❞❡ ♦❢ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ a✳ ❚❤❡♥✱ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ t❤❡ sq✉❛r❡ ♦❢
❞✐❛❣♦♥❛❧ ✐s t❤❡✐r s❤❛r❡❞ ❤②♣♦t❡♥✉s❡✳ ▲❡t✬s ❝❛❧❧ ✐t 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 ❛♥❞
•
♠♦❞❡❧✐♥❣ ✐♥ 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 ✶✿ ❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ✶✳✶✶ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷ ✶✼ ✷✽ ✸✼ ✹✸ ✻✶ ✻✺ ✻✼ ✼✸ ✼✾ ✽✽ ✾✻ ✶✵✵ ✶✵✻ ✶✶✻
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷✻ ✶✸✸ ✶✹✶ ✶✹✾ ✶✻✷ ✶✻✽ ✶✼✾ ✶✽✹ ✶✾✶
❈❤❛♣t❡r ✸✿ ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✽ ✸✳✶ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✸✳✻ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✸✳✼ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤❛♣t❡r ✷✿ ❱❡❝t♦r ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✷✳✶ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❇❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✾✽ ✷✵✹ ✷✵✾ ✷✶✻ ✷✶✾ ✷✷✸ ✷✷✽ ✷✸✸ ✷✸✻ ✷✹✷ ✷✹✹ ✷✺✵
❈❤❛♣t❡r ✹✿ ❙②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹
✹✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹ ✹✳✷ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸
❈♦♥t❡♥ts
✶✶
✹✳✸ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✹✳✹ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✻ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✹✳✽ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✵ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✻✼ ✷✼✵ ✷✼✸ ✷✼✾ ✷✽✸ ✷✽✼ ✷✾✸ ✷✾✽
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✵✻
✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✻✳✹ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ❚❤❡ ❤❡❛t P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s ✻✳✶✵ ❚❤❡ ✇❛✈❡ P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✶ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✹✸
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✹✶✵
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✹✷✷
❈❤❛♣t❡r ✺✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s ✺✳✶ ✺✳✷ ✺✳✸ ✺✳✹ ✺✳✺ ✺✳✻ ✺✳✼ ✺✳✽ ✺✳✾
✳ ✳ ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s ✳ ✳ ✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ ❆ ♣❡♥❞✉❧✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❧❛♥❡t❛r② ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s ✳ ✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ s♣r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤❛♣t❡r ✻✿ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s
❊①❡r❝✐s❡s ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❊①❡r❝✐s❡s✿ ❇❛s✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊①❡r❝✐s❡s✿ ❆♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❊①❡r❝✐s❡s✿ ❊✉❧❡r✬s ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①❡r❝✐s❡s✿ ●❡♥❡r❛❧✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ❛♥❞ s❡tt✐♥❣ ✉♣ ❖❉❊s ✻ ❊①❡r❝✐s❡s✿ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❊①❡r❝✐s❡s✿ ❙②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❊①❡r❝✐s❡s✿ ❙❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❊①❡r❝✐s❡s✿ ❆❞✈❛♥❝❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❊①❡r❝✐s❡s✿ P❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
■♥❞❡①
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✵✻ ✸✵✼ ✸✶✷ ✸✶✺ ✸✷✶ ✸✷✹ ✸✸✵ ✸✸✻ ✸✸✾
✸✹✸ ✸✺✸ ✸✻✵ ✸✻✺ ✸✼✵ ✸✼✺ ✸✽✸ ✸✾✶ ✸✾✸ ✹✵✵ ✹✵✺ ✹✶✵ ✹✶✷ ✹✶✸ ✹✶✹ ✹✶✺ ✹✶✻ ✹✶✼ ✹✶✽ ✹✶✾ ✹✷✵ ✹✷✶
❈❤❛♣t❡r ✶✿ ❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s
❈♦♥t❡♥ts
✶✳✶ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ✶✳✶✶ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥
❲❡ ❝❛♥ ❡❛s✐❧② ❞❡r✐✈❡ t❤❡ s♣❡❡❞ ❢r♦♠ t❤❡ ❞✐st❛♥❝❡ t❤❛t ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞✿
❖♥ t❤❡ ✢✐♣ s✐❞❡✱ t❤❡ ❞❡r✐✈❡ t❤❡ ❞✐st❛♥❝❡ ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ❦♥♦✇♥ ✈❡❧♦❝✐t②✿
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✶✷ ✳ ✶✼ ✳ ✷✽ ✳ ✸✼ ✳ ✹✸ ✳ ✻✶ ✳ ✻✺ ✳ ✻✼ ✳ ✼✸ ✳ ✼✾ ✳ ✽✽ ✳ ✾✻ ✳ ✶✵✵ ✳ ✶✵✻ ✳ ✶✶✻
✶✳✶✳
■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥
✶✸
❚❤❡ t✇♦ ♣r♦❜❧❡♠s ❛r❡ s♦❧✈❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡s❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ s❛♠❡ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ❢♦r♠✉❧❛✿ s♣❡❡❞
=
❞✐st❛♥❝❡
/
t✐♠❡
❛♥❞
❞✐st❛♥❝❡
=
❲❤❛t t❛❦❡s t❤✐s ✐❞❡❛ ❜❡②♦♥❞ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ✐s t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t
s♣❡❡❞
×
t✐♠❡
✈❡❧♦❝✐t② ✈❛r✐❡s✳
. ❚❤❡ s✐♠♣❧❡st ❝❛s❡ ✐s
✇❤❡♥ ✐t ✈❛r✐❡s ✐♥❝r❡♠❡♥t❛❧❧②✳ ❲❡ ♥❡①t ❝♦♥s✐❞❡r ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡s ♦❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t②✳ ❋✐rst✱
t♦ ✈❡❧♦❝✐t②✳✳✳
❙✉♣♣♦s❡ t❤❛t t❤✐s t✐♠❡ ✇❡ ❤❛✈❡ ❛
s❡q✉❡♥❝❡
♦❢ ♠♦r❡ t❤❛♥
❢r♦♠ ❧♦❝❛t✐♦♥
30 ❞❛t❛ ♣♦✐♥ts ✭♠♦r❡ ✐s ✐♥❞✐❝❛t❡❞ ❜② ✏✳✳✳✑✮❀
t❤❡② ❛r❡
t❤❡ ❧♦❝❛t✐♦♥s ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡
♠✐♥✉t❡s
❧♦❝❛t✐♦♥
♠✐❧❡s
0 1 2 3 4 5 6 7 8 9 10 ... 0.00 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...
❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✭❧❡❢t✮✿
❊✈❡r② ♣❛✐r ♦❢ ♥✉♠❜❡rs ✐♥ t❤❡ t❛❜❧❡ ✐s t❤❡♥ ♣❧♦tt❡❞ ✭r✐❣❤t✮✳ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡
❞✐✛❡r❡♥❝❡s
❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ❢❛st ✇❡ ♠♦✈❡ ♦✈❡r t❤❡s❡
♦❢ ❧♦❝❛t✐♦♥s ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳
❋✐rst✱ t❤❡ t❛❜❧❡✳ ❲❡ ✉s❡ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ❧♦❝❛t✐♦♥s✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst ♦♥❡ ✐s ❝♦♠♣✉t❡❞✿ t✐♠❡
♠✐♥
❧♦❝❛t✐♦♥
♠✐❧❡s
❞✐✛❡r❡♥❝❡ ✈❡❧♦❝✐t②
♠✐❧❡s✴♠✐♥
0 0.00 ց
1 0.10 ↓ 0.10 − 0.00 || 0.10
... ... ... ...
❲❡ ❝♦♠♣✉t❡ t❤✐s ❞✐✛❡r❡♥❝❡ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡♥ ♣❧❛❝❡ ✐t ✐♥ ❛ r♦✇ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s
✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥
✶✹
t❤❛t ✇❡ ❝r❡❛t❡❞ ❛t t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡
♠✐♥
❧♦❝❛t✐♦♥
♠✐❧❡s
✈❡❧♦❝✐t②
♠✐❧❡s✴♠✐♥
0 0.00
1 2 3 4 5 6 7 8 9 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ 0.10 0.10 0.10 0.09 0.09 0.09 0.08 0.07 0.07
... ... ... ...
Pr❛❝t✐❝❛❧❧②✱ ✇❡✬❞ r❛t❤❡r ✉s❡ t❤❡ ❝♦♠♣✉t✐♥❣ ❝❛♣❛❜✐❧✐t✐❡s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❊①❛♠♣❧❡ ✶✳✶✳✶✿ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛s
❲❡ ✉s❡ ❢♦r♠✉❧❛s t♦ ♣✉❧❧ ❞❛t❛ ❢r♦♠ ♦t❤❡r ❝❡❧❧s✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s✳ ❋✐rst✱ t❤❡ ✏❛❜s♦❧✉t❡✧ r❡❢❡r❡♥❝❡✿
❂❘✷❈✸✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞ ❛t r♦✇
2
❛♥❞ ❝♦❧✉♠♥
3
❛♥❞
sq✉❛r❡ ✐t✿
❙❡❝♦♥❞✱ t❤❡ ✏r❡❧❛t✐✈❡✧ r❡❢❡r❡♥❝❡✿
❂❘❬✷❪❈❬✸❪✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞
2 r♦✇s ❞♦✇♥ ❛♥❞ 3 ❝♦❧✉♠♥s
r✐❣❤t ❢r♦♠ ✐t ❛♥❞ sq✉❛r❡ ✐t✿
❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❜② ♣✉❧❧✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ❧♦❝❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿
❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪ ❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ ❧❛st ❝♦❧✉♠♥✱
❈❬✲✶❪ ✱
s❛♠❡ r♦✇✱
❘✱
❛♥❞ ❧❛st r♦✇✱
❘❬✲✶❪ ✳
❇❡❧♦✇✱ ②♦✉
❝❛♥ s❡❡ t❤❡ t✇♦ r❡❢❡r❡♥❝❡s ✐♥ t❤❡ ❢♦r♠✉❧❛s ♠❛r❦❡❞ ✇✐t❤ r❡❞ ❛♥❞ ❜❧✉❡ ✭❧❡❢t✮ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ s❤♦✇♥ ✇✐t❤ t❤❡ ❛rr♦✇s ✭r✐❣❤t✮✿
✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥
✶✺
❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ✇❡ ❝r❡❛t❡❞ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s✿
❚❤✐s ♥❡✇ ❞❛t❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t✳ ❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛✱ ✉♥❧✐❦❡ t❤❡ ❧♦❝❛t✐♦♥✱ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s r❛t❤❡r t❤❛♥ t✐♠❡ ✐♥st❛♥❝❡s✱ ✇❡ ♣❧♦t ✐t ✇✐t❤ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts✳ ■♥ ❢❛❝t✱ t❤❡ ❞❛t❛ t❛❜❧❡ ❝❛♥ ❜❡ r❡❛rr❛♥❣❡❞ ❛s ❢♦❧❧♦✇s t♦ ♠❛❦❡ t❤✐s ♣♦✐♥t ❝❧❡❛r❡r✿ t✐♠❡ ❧♦❝❛t✐♦♥ ✈❡❧♦❝✐t②
0 1 2 3 4 ... 0.00 − 0.10 − 0.20 − 0.30 − .39 − ... · 0.10 · 0.10 · 0.10 · 0.09 · 0.09 ...
■♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣✱ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✳ ▲❡t✬s ❝❛❧❧ ✐t h✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ✐t ❛s ❞❡♣❡♥❞❡♥t ♦♥ t✐♠❡ ✐❢ ♥❡❡❞❡❞✳ ■❢ ❛ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥s ❛t t❤❡ t✐♠❡ ♠♦♠❡♥ts a, a + h, a + 2h, ...✱ t❤❡♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦✈❡r ❡❛❝❤ ♣❡r✐♦❞ ♦❢ t✐♠❡ [t, t + h] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ ∆f = f (t + h) − f (t)
❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ✈❡❧♦❝✐t② ♦✈❡r ❡❛❝❤ ♣❡r✐♦❞ ♦❢ t✐♠❡ [t, t + h] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ f (t + h) − f (t) ∆f = ∆t h
❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✳ ❆ ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t tr❛♥s✐t✐♦♥ ✐s ❢r♦♠ ✈❡❧♦❝✐t② t♦ ❧♦❝❛t✐♦♥✳✳✳ ❆❣❛✐♥✱ ✇❡ ❝♦♥s✐❞❡r 30 ❞❛t❛ ♣♦✐♥ts✳ ❚❤❡s❡ ♥✉♠❜❡rs ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛♥ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡ ✈❡❧♦❝✐t②
♠✐♥✉t❡s ♠✐❧❡s✴❤♦✉r
0 1 2 3 4 5 6 7 8 9 10 ... 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...
❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ♣❧♦tt❡❞ ♦♥❡ ❜❛r ❛t ❛ t✐♠❡✿
✶✳✶✳
✶✻
■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥
❚❤✐s ♣❛rt ❞❛t❛ ✐s ❢✉rt❤❡r♠♦r❡ ✐❧❧✉str❛t❡❞ ❛s ❛ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❆❣❛✐♥✱ ✇❡ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s ❜② ♣❧♦tt✐♥❣ ✐ts ✈❛❧✉❡s ✇✐t❤ ❤♦r✐③♦♥t❛❧ ❜❛rs✳ ❚❤✐s ♣❛rt✐❝✉❧❛r s❡t ♦❢ ❞❛t❛ ♠❛② ❜❡ ❞❡s❝r✐❜✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ♦❢ ❛ ❜❛❧❧ r♦❧❧✐♥❣ t❤r♦✉❣❤ ❛ tr♦✉❣❤✿
❚♦ ✜♥❞ ♦✉t ✇❤❡r❡ ✇❡ ❛r❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ s✉♠ ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ✈❡❧♦❝✐t✐❡s ❛s t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ✈❡❧♦❝✐t✐❡s ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst ♦♥❡ ✐s ❝♦♠♣✉t❡❞✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s 0✿ t✐♠❡ ♠✐♥ 0 1 ... ✈❡❧♦❝✐t② ♠✐❧❡s 0.10 ... s✉♠
↓ 0.00+ 0.10 ... ↑ || 0.00 0.10 ...
❧♦❝❛t✐♦♥ ♠✐❧❡s✴♠✐♥ ❲❡ ♣❧❛❝❡ t❤✐s ❞❛t❛ ✐♥ ❛ ♥❡✇ r♦✇ ❛❞❞❡❞ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡ ♠✐♥ 0 1 2 3 4 ✈❡❧♦❝✐t② ♠✐❧❡s 0.10 0.20 0.30 0.39 ❧♦❝❛t✐♦♥
♠✐❧❡s✴♠✐♥
5 0.48 ↓ ↓ ↓ ↓ ↓ 0.00 → 0.10 → 0.30 → 0.59 → 0.98 → 1.46 →
6 0.56 ↓ 2.03 →
7 0.64 ↓ 2.67 →
8 0.72 ↓ 3.39 →
Pr❛❝t✐❝❛❧❧②✱ ✇❡ ✉s❡ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s✉♠s ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ✈❡❧♦❝✐t✐❡s ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪
❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ s❛♠❡✱ ❈ ✱ ♦r ❧❛st✱ ❈❬✲✶❪ ✱ ❝♦❧✉♠♥ ❛♥❞ t❤❡ s❛♠❡✱ ❘ ✱ ❛♥❞ ❧❛st✱ ❘❬✲✶❪ ✱ r♦✇✱ ❛s ❢♦❧❧♦✇s✿
✶✳✷✳
✶✼
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ❢♦r t❤❡ ❧♦❝❛t✐♦♥s✿
❚❤❡ ❞❛t❛ ✐s ❛❧s♦ ✐❧❧✉str❛t❡❞ ❛s t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❲❡✱ ❛❣❛✐♥✱ r❡❛rr❛♥❣❡ t❤❡ ❞❛t❛ t❛❜❧❡ t♦ ♠❛❦❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ t②♣❡s ♦❢ ❞❛t❛ ❝❧❡❛r❡r✿ t✐♠❡ ✈❡❧♦❝✐t② ❧♦❝❛t✐♦♥
0 1 2 3 4 ... · 0.00 · 0.10 · 0.20 · 0.30 · 0.39 ... 0.00 − 0.10 − 0.30 − 0.59 − .98 − ...
■♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣✱ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✱ h✳ ■❢ ❛ ❢✉♥❝t✐♦♥ g r❡♣r❡s❡♥ts t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦✈❡r ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛❧ [t, t + h]✱ t❤❡♥ t❤❡ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ [a, a + nh] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ n X
t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t
g(a + kh) .
k=1
❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ s✉♠ ♦❢ g ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧② ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ v r❡♣r❡s❡♥ts t❤❡ ✈❡❧♦❝✐t✐❡s ♦✈❡r ❡❛❝❤ ✐♥t❡r✈❛❧ [t, t + h]✱ t❤❡♥ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ [a, a + nh] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ n X
v(a + kh)h .
k=1
❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❘✐❡♠❛♥♥
s✉♠
♦❢ v ✳
■♥ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ t❤❡ ✈❛❧✉❡s ♦❢ g ❛♥❞ v ❛r❡ ♣❧❛❝❡❞ ❛t t❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ♣r❡s❡♥t❡❞ ❜❡❧♦✇✱ ✇❡ tr❛♥s❝❡♥❞ t❤✐s ✐ss✉❡✳
❡♥❞s
♦❢ t❤❡ ✐♥t❡r✈❛❧s✳ ❲❤❡r❡ ❡❧s❡❄ ■♥ t❤❡
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
❇❡❧♦✇✱ ✇❡ ♣r♦❞✉❝❡ ❞✐s❝r❡t❡ ♠♦❞❡❧s ❛♥❞ t❤❡♥ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❢r♦♠ ✈❡r❜❛❧ ❞❡s❝r✐♣t✐♦♥s✳ ❲❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ♦r [a, +∞]✳ ■t ✐s✱ ✜rst✱ ❛ s❡q✉❡♥❝❡ ♦❢ n ♥♦❞❡s✱ ti ✿ a = t0 < t1 < t2 < ...
✶✳✷✳
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
❚❤❡
✐♥❝r❡♠❡♥ts
♦❢
t
✶✽
❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s✿
∆ti = ti − ti−1 , i = 1, 2, ...
■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♥♦❞❡s✱ t❤❡r❡ ❛r❡
❡❞❣❡s ✿
c1 = [t0 , t1 ], c2 = [t1 , t2 ], ... ❲❡ t❤❡♥ s♣❡❛❦ ♦❢ ❛
❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥
♦❢ t❤❡ ✐♥t❡r✈❛❧✳
■♥ ❛❧❧ ❡①❛♠♣❧❡s ✐♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡ ❡q✉❛❧✿
∆ti = ti − ti−1 = h, i = 1, 2, ... ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿
a = t0 , t1 = a + h, t2 = a + 2h, ... ◆❡①t✱ t❤❡
❞✐✛❡r❡♥❝❡
♦❢ ❛ ❢✉♥❝t✐♦♥
♣❛rt✐t✐♦♥✳ ❖♥ t❤❡ ❡❞❣❡
[tk−1 , tk ]✱
y
❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡
✇❡ ❤❛✈❡✿
∆y = y(tk ) − y(tk−1 ), k = 1, 2, ... ❙✉♣♣♦s❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡
y(a)
✐s s❡t ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢
y
✐s ❣✐✈❡♥ ♦♥❡ st❡♣ ❛t ❛ t✐♠❡✳ ❚❤❡♥ ✇❡ ✜♥❞
y
❜② ❛
r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿
y(tk+1 ) = y(tk ) + ∆y , k = 0, 1, 2, ... ▼❡❛♥✇❤✐❧❡✱ t❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r
❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✿ ∆y =f. ∆t
❊①❛♠♣❧❡ ✶✳✷✳✶✿ ✉♥✐❢♦r♠ ♠♦t✐♦♥
❯♥✐❢♦r♠ ♠♦t✐♦♥ ♠❡❛♥s t❤❛t ✇❡ ❝♦✈❡r t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ♦✈❡r ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✳ ■❢ t❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥
∆y
y ✐s ♦✉r ❧♦❝❛t✐♦♥✱ ∆t✳
✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡
❚❤❡ ❛❧❣❡❜r❛ ✐s ❛s ❢♦❧❧♦✇s✿
❢♦r s♦♠❡ ❝♦♥st❛♥t
∆y = k · ∆t ,
k✳
■❢ ✇❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❛ ❢❛♠✐❧✐❛r ✐❞❡❛ ❢r♦♠ ♣❤②s✐❝s✱ ✇❡ ❤❛✈❡ ❛
◮
❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❞②♥❛♠✐❝s ✿
✏❚❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✑✳
❚❤❡ ❛❧❣❡❜r❛ ✐s ❛s ❢♦❧❧♦✇s✿
∆y = k. ∆t
❙✉♣♣♦s❡ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
y(t0 ) = y0 . ❚❤❡♥ ♦✉r ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s❡q✉❡♥❝❡ ✈✐❛ t❤✐s
r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✿
y(tn+1 ) = y(tn ) + k · ∆t . ❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ ❝♦♥s✐❞❡r ❛ ✏❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✑✿
❲❡ ❝❤♦♦s❡ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥ts t♦ ❜❡
∆y = 2. ∆t ❝♦♥st❛♥t ∆t = h = 0.2 ✳
❚❤❡♥✿
y(tn+1 ) = y(tn ) + 2h . ❍❡r❡ ❛r❡ ❛ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r ✈❛r✐♦✉s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭♦♥ t❤❡
y ✲❛①✐s✮✿
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✶✾
❚❤❡② ❛r❡ ❥✉st ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✿ st❡♣ r✐❣❤t ❜②
h
❛♥❞ t❤❡♥ ✉♣ ❜②
2h✳
●❡♥❡r❛❧❧②✱ t❤❡② ❞♦♥✬t ❤❛✈❡
t♦ ❜❡ ✕ ✇❤❡♥ t❤❡ ♣❛rt✐t✐♦♥ ✐s ✉♥❡✈❡♥✿
❇✉t t❤❡ ♣♦✐♥ts st✐❧❧ ❧✐❡ ♦♥ t❤❡ s❛♠❡ str❛✐❣❤t ❧✐♥❡s✦
❊①❛♠♣❧❡ ✶✳✷✳✷✿ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t②
❚❤❡ ❢♦r♠✉❧❛ ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ ❱♦❧✉♠❡s ✷✱ ✸✱ ❛♥❞ ✹ t♦ ♠♦❞❡❧ ✉♥✐❢♦r♠ ♠♦t✐♦♥ ❛♥❞ ❛❝q✉✐r❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤✐s ✈❡❧♦❝✐t②✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡✿
◮
✏❚❤❡ ✈❡❧♦❝✐t② ❞❡♣❡♥❞s ♦♥ t✐♠❡✑✳
❲❡ t❛❦❡ ❛♥② ❢✉♥❝t✐♦♥
z = f (t)
❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ t❤✐♥❦ ♦❢
✈❛❧✉❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❞✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧
[ti−1 , ti ]✳
f (ci )
❛s t❤❡
❲❡ s❡t ✉♣ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿
∆y = f (cn ) . ∆t ❘❡❛rr❛♥❣❡❞ ✐t ♣r♦❞✉❝❡s ✐ts ♦✇♥ s♦❧✉t✐♦♥ ❛s ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿
y(tn+1 ) = y(tn ) + f (cn ) · ∆t , ✇❤❡r❡
tn+1 = tn + ∆t . f ❄ ■t ✐s ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ z = f (t) ✭♣❧♦tt❡❞ ❜❡❧♦✇ ❧❡❢t✮✳ ❍♦✇❡✈❡r✱ t❤❡ yn = y(tn ) ✐s t♦ ❜❡ ♣❧♦tt❡❞ ♦♥ t❤❡ ty ✲♣❧❛♥❡✳ ■t ✐s✱ t❤❡r❡❢♦r❡✱ ❜❡♥❡✜❝✐❛❧ t♦ t❤✐♥❦ ♦❢ f ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (t, y) t❤❛t ❥✉st ❤❛♣♣❡♥s t♦ ❤❛✈❡ t❤❡ s❛♠❡ ✈❛❧✉❡ ❢♦r ❡❛❝❤ t r❡❣❛r❞❧❡ss ♦❢ y ✳ ❲❡ s❛♠♣❧❡ f ♦✈❡r ✐ts ❞♦♠❛✐♥✱ ❛ r❡❝t❛♥❣❧❡ ♦❢ t✬s ❛♥❞ y ✬s✱ ❛♥❞ ✇r✐t❡ t❤❡ ♦✉t♣✉ts ✭♠✐❞❞❧❡✮✿
❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s ❣r❛♣❤ ♦❢ ❛ s♦❧✉t✐♦♥
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✷✵
❲❡ ❡♥❤❛♥❝❡ t❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ f ✐♥ t❡r♠s ♦❢ ❝♦❧♦r ✕ ❜❧✉❡ ❢♦r ♥❡❣❛t✐✈❡ ❛♥❞ r❡❞ ❢♦r ♣♦s✐t✐✈❡✳ ❲❡ ❛❧s♦ s❤♦✇ f ❛s ❛ s✉r❢❛❝❡ ✭r✐❣❤t✮✳ ❚❤❡ ♠✐❞❞❧❡ ✐s t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ♦❢ t❤❡s❡ ❛s ✐t ✐s s❡❡♥ ❛s ❛ ✜❡❧❞ ♦❢ ✈❡❧♦❝✐t✐❡s✳ ❙♦✱ ❛s y ✐s ♣❧♦tt❡❞ ♣♦✐♥t ❜② ♣♦✐♥t✱ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r ❝♦♥str✉❝t✐♥❣ ❛ s♦❧✉t✐♦♥✿ • ❲❡ ❣♦ ✉♣ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s r❡❞✳ • ❲❡ ❣♦ ❞♦✇♥ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s ❜❧✉❡✳ ❲❡ ❝❛♥ ❞r❛✇ ❛ ❝✉r✈❡ ✐♥ t❤✐s ♠❛♥♥❡r✿
▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ r❡❛❞ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ t❛❜❧❡ ❛s ✇❡ ♣r♦❣r❡ss ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✿ ✈❡❧♦❝✐t② t✐♠❡ ✐♥t❡r✈❛❧ h ❥✉♠♣ ∆y 0.2 · 0.1 = 0.02 0.7 · 0.1 = 0.07 1.2 · 0.1 = 0.12 .. .. .. ❖r ✐t ✐s ❞♦♥❡ ❜② t❤❡ ❝♦♠♣✉t❡r✿
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿
∆y =f. ∆t ❋✉rt❤❡r♠♦r❡✱ ✇❤❛t ✐❢ f ✐s♥✬t ❥✉st ❛ str❡❛♠ ♦❢ ♥✉♠❜❡rs ❜✉t ❛ s❛♠♣❧❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥❄ ❆♥❞ ✇❤❛t ✐❢ y ✐s ❛ s❛♠♣❧❡❞ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥❄ ❲❡ ❝❛♥ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛tt❡r ❡q✉❛t✐♦♥✱
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
∆t → 0✳
✷✶
❚❤❡ r❡s✉❧t ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿
dy =f, dt ♦r
y ′ = f (t) . ❊①❡r❝✐s❡ ✶✳✷✳✸
P❧♦t ♠♦r❡ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❛t ♦t❤❡r ♣♦✐♥ts✳ ❲❤❛t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠❄
❊①❛♠♣❧❡ ✶✳✷✳✹✿ ❞❛t❛ str❡❛♠
❚❤❡ q✉❛♥t✐t✐❡s ✉s❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥ ✭♦r ❛ st❛t❡✮ ✕ ❝✉rr❡♥t t✐♠❡ ❛♥❞ ✈❡❧♦❝✐t② ✕ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❝♦♠❡ ❢r♦♠ ❛ ❢♦r♠✉❧❛✦ ❚❤✐s ✐s ♥♦t❤✐♥❣ ❜✉t ❞❛t❛ ❛♥❞ ✐t ♠❛② ❝♦♠❡ ❛s str✐♥❣s ♦❢ ♥✉♠❜❡rs ✭♦r ✐t ❝❛♥ ❜❡ ❢❡❞ ✐♥t♦ t❤❡ ❝♦♠♣✉t❡r ❝♦♥t✐♥✉♦✉s❧②✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥
y0
❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t
❛♥❞✱ ❛s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿
tn , vn , yn , n = 1, 2, 3, ... ❚❤❡ s❛♠❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s ✉s❡❞ t♦ ✜♥❞ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✿
yn+1 = yn + vn+1 · ∆t . ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐t❡r❛t✐♦♥
n
t✐♠❡
0 1 ... 1000 ...
✐♥✐t✐❛❧✿
tn
✈❡❧♦❝✐t②
3.5 3.6 ... 103.5 ...
vn
❧♦❝❛t✐♦♥
−− 33 ... 4 ...
yn
22 25.3 ... 336 ...
❆❧❧ ❜✉t t❤❡ ❧❛st ❝♦❧✉♠♥ ♠❛② ❝♦♠❡ ❛s ❛ ❞❛t❛ ✜❧❡✳
❊①❛♠♣❧❡ ✶✳✷✳✺✿ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤
❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t ✐♥ t❤❡ ♥❡①t ❞❡s❝r✐♣t✐♦♥✿
◮
✏❚❤❡ r❛t❡ ♦❢ ❣r♦✇t❤✴❞❡❝❛② ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳✑
❆♥ ❡①❛♠♣❧❡ ✐s ❜❛❝t❡r✐❛ ♦r r❛❜❜✐ts ❣r♦✇✐♥❣ ✇✐t❤ ❛♥ ✉♥r❡str✐❝t❡❞ ❛♠♦✉♥t ♦❢ ❢♦♦❞ ❜❡❝❛✉s❡ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ♥❡✇❜♦r♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❛❞✉❧ts✳ ❲❡ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥✿
❢♦r s♦♠❡ ❝♦♥st❛♥t ❛s
k✳
∆y = ky , ∆t
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥
∆y
✐s ♣r♦♣♦rt✐♦♥❛❧ t♦
y
❛s ✇❡❧❧
∆t✿ ∆y = k · y · ∆t .
❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡✿
f (y) = ky . yn = y(tn ) ✐s ♣❧♦tt❡❞ ♦♥ t❤❡ ty ✲♣❧❛♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐t ✈❛r✐❛❜❧❡s z = f (t, y) t❤❛t ❥✉st ❤❛♣♣❡♥s t♦ ❤❛✈❡ t❤❡ s❛♠❡
❖♥❝❡ ❛❣❛✐♥✱ ❤♦✇❡✈❡r✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ s♦❧✉t✐♦♥ ✐s ❜❡tt❡r t♦ t❤✐♥❦ ♦❢ ✈❛❧✉❡ ❢♦r ❡❛❝❤
y
f
❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦
r❡❣❛r❞❧❡ss ♦❢
t✿
✶✳✷✳
✷✷
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ y(tn+1 ) = y(tn ) + ky(tn ) · ∆t .
❲❡ ❛ss✉♠❡ ❤❡r❡ t❤❛t ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧
❝♦♥❞✐t✐♦♥
✿
y(t0 ) = y0 .
❚❤❡ ❢♦r♠✉❧❛ ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t♦ st✉❞② ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧s ❛♥❞ ♥♦✇ ✇❡ ✇✐❧❧ ❛♣♣❧② ✐t t♦ ♠♦❞❡❧s ✇✐t❤ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ q✉❛♥t✐t② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ q✉❛♥t✐t②✬s ❝✉rr❡♥t ✈❛❧✉❡✿
❲❡ s❡❡ ❤♦✇ ✇❡ ♣r♦❣r❡ss ✐♥ t❤❡ ✉♣✇❛r❞ ❞✐r❡❝t✐♦♥ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ✈❛❧✉❡s ♦❢ f ❤✐❣❤❡r ❛♥❞✱ ❛s ❛ r❡s✉❧t✱ t❤❡ ❣r♦✇t❤ ♦❢ y ❢❛st❡r✳ ❲❡ ❝❛♥ ♣❧♦t ❛ ❧♦t ♠♦r❡ s♦❧✉t✐♦♥s ✐♥ t❤✐s ♠❛♥♥❡r✿
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ f ❛s s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❢♦r ❛❧❧ r❡❛❧ t ❛♥❞ y ✱ t❤❡ ❡q✉❛t✐♦♥ ✐s ❝♦♥✈❡rt❡❞ t♦ ❛ ♥❡✇ ❡q✉❛t✐♦♥✿ ∆y dy = ky =⇒ = ky ♦r y ′ = ky . ∆t→0 ∆t dt lim
❊①❡r❝✐s❡ ✶✳✷✳✻
P❧♦t ♠♦r❡ s♦❧✉t✐♦♥s✳ ❲❤❛t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠❄ ❈♦♠♣❛r❡ t♦ t❤❡ ❦♥♦✇♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s t❤❛t s❛t✐s❢② y ′ = y ✳
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✷✸
❊①❡r❝✐s❡ ✶✳✷✳✼
P❧♦t s♦❧✉t✐♦♥s ❢♦r
k < 0✳
❲❤❛t ❞♦❡s t❤❡ s②st❡♠ ♠♦❞❡❧❄
❊①❛♠♣❧❡ ✶✳✷✳✽✿ ❧♦❣✐st✐❝ ❣r♦✇t❤
❆ ❞✐✛❡r❡♥t✱ ❛♥❞ ♠♦r❡ ❝♦♠♣❧❡①✱ ♣♦♣✉❧❛t✐♦♥ ❞②♥❛♠✐❝s ✐s ❣✐✈❡ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡s❝r✐♣t✐♦♥✿
◮
✏❚❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ❛♥❞ t♦
t❤❡ r❡♠❛✐♥✐♥❣ ♣♦♣✉❧❛t✐♦♥ ♦❢ t❤❡ t♦t❛❧ t❤❛t ❝❛♥ ❜❡ s✉st❛✐♥❡❞✑✳ ❆♥ ❡①❛♠♣❧❡ ✐s ❜❛❝t❡r✐❛ ❣r♦✇✐♥❣ ✐♥ ❛ ❥❛r✳ ❲❡ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥✿
❢♦r s♦♠❡ ❝♦♥st❛♥t
T − y✱
✇❤❡r❡
T
k✳
∆y = ky(T − y) , ∆t
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥
✐s t❤❡ t♦t❛❧ ♣♦ss✐❜❧❡ ♣♦♣✉❧❛t✐♦♥ ✭❛s ✇❡❧❧ ❛s
∆t
∆y
✐s ♣r♦♣♦rt✐♦♥❛❧ t♦
y
✳✳✳ ❛♥❞
❛s ❜❡❢♦r❡✮✿
∆y = k · y · (T − y) · ∆t . ❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s s❧✐❣❤t❧② ♠♦r❡ ❝♦♠♣❧❡①✿
f (t, y) = ky(T − y) . ❚❤✐s ✐s ♦✉r ❢✉♥❝t✐♦♥ ✭k
= 5, T = 1✮✿
❲❡ s❡❡ ❧♦✇ ❧❡✈❡❧ ♦❢ ♣♦t❡♥t✐❛❧ ❣r♦✇t❤ ✐♥ t❤❡ ❛r❡❛s ✇❤❡r❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ❧♦✇ ♦r ✇❤❡r❡ ✐t ✐s t♦♦ ❝❧♦s❡ t♦ ❜❡ ❡①❤❛✉st✐♥❣ t❤❡ r❡s♦✉r❝❡s✳ ❲✐t❤ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
y(t0 ) = y0 , ✇❡ ♣r♦❞✉❝❡ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿
y(tn+1 ) = y(tn ) + ky(tn )(T − y(tn )) · ∆t, k > 0 . ❚❤❡ ♠♦❞❡❧ ❞❡s❝r✐❜❡s ❛ r❡str✐❝t❡❞ ❣r♦✇t❤ ✭k
= 5, ∆t = h = .05✮✿
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✷✹
❲❡ s❡❡ t❤❡ ❢❛st❡st ❣r♦✇t❤ ✇❤❡♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ❛❜♦✉t ❤❛❧❢ ♦❢ ✇❤❛t ❝❛♥ ❜❡ s✉st❛✐♥❡❞✳ ❆❧❧ s♦❧✉t✐♦♥s ❡①❤✐❜✐t t❤✐s ❜❡❤❛✈✐♦r✿
❈❤♦♦s✐♥❣
∆t
s♠❛❧❧ r❡❧❛t✐✈❡ t♦
k
❝♦✉❧❞ ❜r❡❛❦ ♦✉r ♠♦❞❡❧ ✭k
= 25, ∆t = .05✮✿
❚❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s s♦♠❡t✐♠❡s ❧❛r❣❡r t❤❛♥ t❤❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ❥❛r✦ ■❢
y
✐s ❛ s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ t❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥
∆y = y(T − y) . ∆t ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿
y ′ = ky(T − y) . ❊①❡r❝✐s❡ ✶✳✷✳✾
❲❤❛t ❞♦❡s t❤❡ ❝❛s❡ ♦❢
k 0✳
❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s s❧✐❣❤t❧②
s✐♠♣❧❡r t❤❛♥ ❧❛st✿
f (t, y) = k · (r − y) . ■t ✐s ♣❧♦tt❡❞ ❜❡❧♦✇ ✭k
= 10, ∆t = .05✮✿
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✷✺
❲❡ ❝❛♥ s❡❡ t❤❛t ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❤✐❣❤❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ❞❡❝r❡❛s❡s ❛♥❞ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❧♦✇❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ✐♥❝r❡❛s❡s✳ ❚❤✐s ✐s ✇❤② t❤❡ ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡ ✐s♥✬t ❡①♣❡❝t❡❞ t♦ ✏♦✈❡rs❤♦♦t✑ t❤❛t ♦❢ t❤❡ r♦♦♠✳
❚❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡
❢♦r♠✉❧❛✿
y(tn+1 ) = y(tn ) + k(r − y(tn )) · ∆t .
❚❤❡ ❢♦r♠✉❧❛ ✇❛s ✉s❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t♦ ♠♦❞❡❧ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✿
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ✐❢ ❛♥②✱ ✇♦✉❧❞ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿
y ′ = k(r − y) . ❊①❛♠♣❧❡ ✶✳✷✳✶✶✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣ ❝♦♥t✐♥✉❡❞
❈❤♦♦s✐♥❣
∆t
s♠❛❧❧ r❡❧❛t✐✈❡ t♦
k
❝♦✉❧❞ ❜r❡❛❦ ♦✉r ♠♦❞❡❧ ✭k
= 25, h = ∆t = .05✮✿
✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s
✷✻
❚❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❝♦✛❡❡ ✐s s♦♠❡t✐♠❡s ❝♦❧❞❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✦ ❈❛♥ ✇❡ ❣✉❛r❛♥t❡❡ t❤❛t t❤✐s ✇♦♥✬t ❤❛♣♣❡♥❄ ❙✉♣♣♦s❡ ❢♦r ❛❧❧
u0 > r ✱
✇❤❛t ❝♦♥❞✐t✐♦♥ ✇♦✉❧❞ ❡♥s✉r❡ t❤❛t
un > r
n❄
❇② ✐♥❞✉❝t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t
un > r un+1 ⇐= ⇐= ⇐= ⇐=
❚❤❛t✬s t❤❡ ❝♦♥❞✐t✐♦♥✦
❛♥❞ ♣r♦✈❡ t❤❛t
un+1 > r✿
= un + k(r − un )h un − kun h un (1 − kh) 1 − kh kh
>r > r − krh > r(1 − kh) >0 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r ∆t → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ y ′ (t) = f (y(t)) ,
♣r♦✈✐❞❡❞ y = y(t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y(t)✳ ❙✉❝❤ ❛♥ ❡q✉❛t✐♦♥ ♠❛② ❜❡ ♣♦ss✐❜❧❡ t♦ s♦❧✈❡✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥✿ ❡q✉❛t✐♦♥
❛ s♦❧✉t✐♦♥
1. f ′ (x)
= x2
−→ f (x) = x3 /3
2. f ′ (x)
= f (x) −→ f (x) = ex
❲❤❡♥ x r❡♣r❡s❡♥ts t✐♠❡ ❛♥❞ y = f (x) r❡♣r❡s❡♥ts ❧♦❝❛t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥s ❤❛✈❡ s✐♠♣❧❡ ✐♥t❡r♣r❡t❛t✐♦♥s✿ ✶✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✿ ♠✐ss✐❧❡✳ ✷✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✿ ❧✐q✉✐❞ ✢♦✇✳ ❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❥✉st✐✜❡s ✉s✐♥❣ t ❢♦r t❤❡ ♥❛♠❡ ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■♥ t❤❡ ✜❡❧❞ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✐t ✐s ❛❧s♦ ❝♦♠♠♦♥ ♥♦t t♦ ♥❛♠❡ t❤❡ ♥❛♠❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✉♥❞❡r t❤✐s ♠♦r❡ ❝♦♠♣❛❝t ♥♦t❛t✐♦♥✿
t❤❡ ❢✉♥❝t✐♦♥s ❛♥②♠♦r❡ ❜✉t ✉s❡ ✐♥st❡❛❞
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✹✹
❖❉❊s
1. y ′ = t2 −→ y = t3 /3 2. y ′ = y −→ y = et
✐s ❛ s♦❧✉t✐♦♥✳ ✐s ❛ s♦❧✉t✐♦♥✳
❚❤❡② ❛r❡ ❣❡♥❡r❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ ✇❡ ❤❛✈❡
g
✐♥❞❡♣❡♥❞❡♥t ♦❢
y✳
❲❡ ❛r❡ ❛❢t❡r ❢✉♥❝t✐♦♥s
y = y(t)
t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿
y ′ (t) = g(t) , ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥
(t, y(t))✳
g✳
■t ✐s
❧♦❝❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t✳
❚❤✐s s❧♦♣❡ ✐s t❤❡ s❛♠❡ ❛❧♦♥❣ ❛♥② ❣✐✈❡♥
❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥
g
❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜②✿
C
✈❡rt✐❝❛❧
❤❛♣♣❡♥s t♦ ❜❡ ❝♦♥st❛♥t✱ s❛②
♣❛r❛❜♦❧❛s✿
✇❤❡r❡
❍❡r❡✱ ❢♦r ❡✈❡r②
−1✱
t✱
✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t
❧✐♥❡✿
✐ts s♦❧✉t✐♦♥ s❡t ✐s t❤❡ ❢❛♠✐❧② ♦❢ t❤❡s❡ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞
1 y = − t2 + C , 2
✐s ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿
1 y = − t2 + C : C ∈ R 2
.
❆♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣✐❝t✉r❡ r❡✈❡❛❧s t❤❛t t❤✐s ❢❛♠✐❧② ♦❢ ❝✉r✈❡s s❛t✐s❢② t❤❡s❡ t✇♦ ♣r♦♣❡rt✐❡s✿ ✶✳ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✳ ✷✳ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳
❊①❛♠♣❧❡ ✶✳✺✳✶✿ q✉❛❞r❛t✐❝ ❝❛s❡ ▲❡t✬s ♣r♦✈❡ t❤❡s❡ st❛t❡♠❡♥ts✳
✶✳✺✳ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✹✺
❋✐rst✱ ✏❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✑ ✐s tr❛♥s❧❛t❡❞ ❛s ✏❚❤❡ ♣❧❛♥❡ ✐s t❤❡ ✉♥✐♦♥ t❤❡ ❝✉r✈❡s✑ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ✏❊✈❡r② ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❝✉r✈❡s✑✳ ❙✉♣♣♦s❡ ♣♦✐♥t✳ ▲❡t✬s ✜♥❞ ❛ ❝✉r✈❡ ❢♦r ✐t✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ s♣❡❝✐✜❝
(t0 , y0 )
❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡
1 y = − t2 + C 2
C✱
(t0 , y0 ) ✐s ♦♥❡ s✉❝❤
t❤❛t✬s ❛❧❧✳ ❚❤❡ ♣♦✐♥t
♠❡❛♥s t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ✐t✿
1 y0 = − t20 + C . 2 ❙♦❧✈❡ ❢♦r
C✿
1 C = y0 + t20 . 2
❉♦♥❡✦ ❙❡❝♦♥❞✱ ✏❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✑ ✐s tr❛♥s❧❛t❡❞ ❛s ✏❚✇♦ ❝✉r✈❡s t❤❛t ✐♥t❡rs❡❝t ❛r❡ t❤❡ s❛♠❡ ❝✉r✈❡✑✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ ❝✉r✈❡s✿
1 y = − t2 + C , 2
❛♥❞
1 y = − t2 + K , 2
❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs
C
❛♥❞
K✳
❙✉♣♣♦s❡ t❤❡② ✐♥t❡rs❡❝t✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ♣♦✐♥t
(t0 , y0 ) t❤❛t
❜❡❧♦♥❣s t♦ ❜♦t❤✳ ❚❤❡r❡❢♦r❡✱ ❜♦t❤ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ s❛t✐s✜❡❞ ❢♦r t❤✐s ♣♦✐♥t✿
1 y0 = − t20 + C , 2 ❛♥❞
1 y0 = − t20 + K . 2
❲❡ s♦❧✈❡ t❤✐s s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ ❝♦♥❝❧✉❞❡✿
C =K. ❚❤❛t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦
❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡
g
✐♥❞❡♣❡♥❞❡♥t ♦❢
t✳
❲❡ ❛r❡ ❛❢t❡r ❢✉♥❝t✐♦♥s
y = y(x)
t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿
y ′ (t) = h(y(t)) , ❢♦r s♦♠❡
h✳ ❲❛r♥✐♥❣✦ ❨♦✉ ❝❛♥✬t ✐♥t❡❣r❛t❡
■t ✐s t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✳ ❍❡r❡✱ ❢♦r ❡✈❡r② s❛♠❡ ❛❧♦♥❣ ❛♥② ❣✐✈❡♥ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✿
y✱
y′ = y✳
✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t
(t, y)✳
❚❤✐s s❧♦♣❡ ✐s t❤❡
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥
✹✻
h(x) = x
❤❛♣♣❡♥s t♦ ❜❡ t❤❡ ✐❞❡♥t✐t②✱ ✐ts s♦❧✉t✐♦♥ ✐s t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞
❝✉r✈❡s✿
❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜②✿
y = Cet , ✇❤❡r❡
C
✐s ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿
{y = Cet : C ∈ R} . ❆♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣✐❝t✉r❡ r❡✈❡❛❧s t❤❛t t❤✐s ❢❛♠✐❧② ♦❢ ❝✉r✈❡s s❛t✐s❢② t❤❡s❡ t✇♦ ♣r♦♣❡rt✐❡s✿ ✶✳ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✳ ✷✳ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❊①❛♠♣❧❡ ✶✳✺✳✷✿ ❡①♣♦♥❡♥t✐❛❧ ❝❛s❡
▲❡t✬s ♣r♦✈❡ t❤❡s❡ st❛t❡♠❡♥ts✳ ❋✐rst✱ ✏❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✑ ✐s ✉♥❞❡rst♦♦❞ ❛s ✏❊✈❡r② ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❝✉r✈❡s✑✳ ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ s♣❡❝✐✜❝
C✳
(t0 , y0 )
✐s ♦♥❡ s✉❝❤ ♣♦✐♥t✳ ▲❡t✬s ✜♥❞ ❛ ❝✉r✈❡ ❢♦r ✐t✳ ❚❤✐s ♠❡❛♥s t❤❛t
❚❤❡ ♣♦✐♥t
(t0 , y0 )
❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡
❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ✐t✿
1 y = − t2 + C 2
♠❡❛♥s t❤❛t t❤❡
y0 = Cet0 . ❙♦❧✈❡ ❢♦r
C✿ C = y0 /et0 .
❉♦♥❡✦ ❙❡❝♦♥❞✱ ✏❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✑ ✐s ✉♥❞❡rst♦♦❞ ❛s ✏❚✇♦ ❝✉r✈❡s t❤❛t ✐♥t❡rs❡❝t ❛r❡ t❤❡ s❛♠❡ ❝✉r✈❡✑✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ ❝✉r✈❡s✿
y = Cet , ❛♥❞
y = Ket , ❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs
C
❛♥❞
K✳
❙✉♣♣♦s❡ t❤❡② ✐♥t❡rs❡❝t✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ♣♦✐♥t
❜❡❧♦♥❣s t♦ ❜♦t❤✳ ❚❤❡r❡❢♦r❡✱ ❜♦t❤ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ s❛t✐s✜❡❞ ❢♦r t❤✐s ♣♦✐♥t✿
y0 = Cet0 , ❛♥❞
y0 = Ket0 .
❲❡ s♦❧✈❡ t❤✐s
s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ ❝♦♥❝❧✉❞❡✿
C =K. ❚❤❛t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦
(t0 , y0 ) t❤❛t
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✹✼
❲❡ ❝♦♥s✐❞❡r❡❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ♦❢ ✜rst ♦r❞❡r ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
∆y = f (t, y) . ∆t ❲❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳
❉❡✜♥✐t✐♦♥ ✶✳✺✳✸✿ ❖❉❊s ❙✉♣♣♦s❡
f
✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❆♥
♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥
♦❢ ✜rst ♦r❞❡r✱ ♦r s✐♠♣❧② ❛♥ ❖❉❊✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ dy = f (t, y) dt
♦r
y ′ = f (t, y)
❍❡r❡✱ ✏♦r❞✐♥❛r②✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛♥❞ ♦♥❧② ♦♥❡ ❞❡r✐✈❛t✐✈❡✳ ❚❤❡ ✏♦r❞❡r✑ r❡❢❡rs t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦
f
❛♥❞
h
❛s t❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑✳
●❡♥❡r❛❧❧②✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❛♥ ❖❉❊ ❤❛s ♥❡✐t❤❡r ♦❢ t❤❡ ❛❜♦✈❡ ♣❛tt❡r♥s✿
❈♦♥s✐❞❡r t❤❡ t✇♦ ❡①❛♠♣❧❡s ♦❢ ❖❉❊s ❢r♦♠ ❜❡❢♦r❡✿
y ′ = −gt + v0 =⇒ y(t) = −gt2 /2 + v0 t + C y′ = y =⇒ y(t) = Cet ❚❤❡r❡ ✐s ♦♥❡ s♦❧✉t✐♦♥ ❢♦r ❡❛❝❤
C✱
❛s ✇❡ ❦♥♦✇✳ ❊❛❝❤ ♦❢ t❤❡♠ ❤❛s ❞♦♠❛✐♥
(−∞, ∞)✳
❯♥❧✐❦❡ t❤❡s❡ t✇♦ ❖❉❊s✱ s♦♠❡ ♦t❤❡rs ❤❛✈❡ s♦❧✉t✐♦♥s t❤❛t ❝❛♥♥♦t ❜❡ ❞❡✜♥❡❞ t♦ t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡✳
❊①❛♠♣❧❡ ✶✳✺✳✹✿ ❛s②♠♣t♦t❡s ❙♦♠❡ ♦❢ t❤❡♠ s✐♠♣❧② r✉♥ ❛✇❛②✳ ❋♦r ❡①❛♠♣❧❡✱ ❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛♥ ❖❉❊✿
y ′ = 1/x . ❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥✱
f (x, y) = 1/x , ✐s ✉♥❞❡✜♥❡❞ ❛t
x = 0✳
❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ ✐ts ❞♦♠❛✐♥ ❝♦♥s✐sts ♥♦✇ ♦❢ t✇♦ ♦♣❡♥ ❤❛❧❢✲♣❧❛♥❡s✿
x 0. x = 0✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✹✽
❲❡ ❤❛✈❡✱ ✐♥ ❢❛❝t✱ t✇♦ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s✿ ❢♦r x > 0; ❢♦r x < 0.
y = ln(−x) + C y = ln(x) + K
❚❤❡② ❡①✐st s❡♣❛r❛t❡❧② ❢r♦♠ ❡❛❝❤ ♦t❤❡r ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♠❜✐♥❡❞ ❛r❜✐tr❛r✐❧②✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② ✇❡ t❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ s♦❧✉t✐♦♥s✳
❉❡✜♥✐t✐♦♥ ✶✳✺✳✺✿ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ ❆ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ y ′ = f (t, y) ✐s ❛ ❢✉♥❝t✐♦♥ y = y(t) ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ✐♥ I ✇❡ ❤❛✈❡✿ y ′ (t) = f (t, y(t))
■t ♠❛② ❛❧s♦ ❜❡ ❝❛❧❧❡❞ ❛ str♦♥❣
s♦❧✉t✐♦♥✳
◆♦t❡ t❤❛t ✇❡ r❡q✉✐r❡ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ s♦ t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✐s ♣r♦♣❡r❧② ❞❡✜♥❡❞✳ ▲❡t✬s ✐♥t❡❣r❛t❡ t❤❡ ❡q✉❛t✐♦♥ ♦✈❡r s♦♠❡ ✐♥t❡r✈❛❧ [a, x] ✇✐t❤ ✈❛r✐❛❜❧❡ r✐❣❤t ❡♥❞✿ Z
x ′
y (t) dt = a
Z
x
f (t, y(t)) dt . a
❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ s✐♠♣❧✐✜❡s✿ y(x) − y(a) =
❲❡ ❤❛✈❡ ❛♥ ✐♥t❡❣r❛❧
Z
x
f (t, y(t)) dt a
❡q✉❛t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✶✳✺✳✻✿ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ ❆ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ y ′ = f (t, y) ✐s ❛ ❢✉♥❝t✐♦♥ y ❝♦♥t✐♥✉♦✉s ♦♥ ❛♥ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ [a, b] s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ✐♥ (a, b]✱ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞✳
◆♦t❡ t❤❛t ✇❡ r❡q✉✐r❡ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡ ❛♥ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ s♦ t❤❛t ✇❡ ❝❛♥ ✐♥t❡❣r❛t❡✳
❊①❛♠♣❧❡ ✶✳✺✳✼✿ st❡♣✲❢✉♥❝t✐♦♥ ■♥ t❤❡ ❧❛t❡r ❝❛s❡ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❛ st❡♣✲❢✉♥❝t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ❡❛s② ✐t ✐s t♦ ♣r♦❞✉❝❡ t❤❡ s♦❧✉t✐♦♥s✿
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✹✾
❙✉❝❤ ❢✉♥❝t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ✏♣✐❡❝❡✲✇✐s❡ ❧✐♥❡❛r✑✳
❊①❡r❝✐s❡ ✶✳✺✳✽
❉♦ ❛♥② ♦❢ t❤❡ s♦❧✉t✐♦♥s ✐♥t❡rs❡❝t❄
❊①❡r❝✐s❡ ✶✳✺✳✾
P❧♦t ❛ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥✿
1 0 2 −1
1 −1 0 1 −1 −1 1 −2 0 1 0 2
❆❜♦✈❡✱ ✇❡ s❤♦✇❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✶✳✺✳✶✵✿ ✇❡❛❦ s♦❧✉t✐♦♥s
❊✈❡r② ✭str♦♥❣✮ s♦❧✉t✐♦♥ ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✶✿ ❝♦♥✈❡rs❡
❚❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ▲❡t✬s ❝❤♦♦s❡ t❤❡ r✐❣❤t✲❤❛♥❞ ❢✉♥❝t✐♦♥
f
t♦ ❜❡ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥✿
❚❤❡♥ ❛♥ ✏♦❜✈✐♦✉s✑ s♦❧✉t✐♦♥ ✐s t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✿
❚❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❢❛✐❧s ❛t
t=0
❤♦✇❡✈❡r✿
y ′ = sign(t), ❈♦♥t✐♥✉♦✉s ✈s✳ ❞✐✛❡r❡♥t✐❛❜❧❡ ✈s✳ ✐♥t❡❣r❛❜❧❡✿
y = |t| .
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✵
❊①❛♠♣❧❡ ✶✳✺✳✶✷✿ ✐♥❝r❡♠❡♥t❛❧ ❝❤❛♥❣❡
❲❡❛❦ s♦❧✉t✐♦♥ ♠❛② ❡♠❡r❣❡ ❢r♦♠ ♣r♦❝❡ss❡s ❣✉✐❞❡❞ ❜② ❢✉♥❝t✐♦♥s t❤❛t ❝❤❛♥❣❡ ✐♥❝r❡♠❡♥t❛❧❧②✳ ❖♥❡ ❡①❛♠♣❧❡ ✐s ❤♦✇ t❤❡ ✐♥❝♦♠❡ t❛① ❡♠❡r❣❡s ❢r♦♠ t❤❡ ✐♥❝♦♠❡ t❛① r❛t❡✿
❲❡ ❤❛✈❡ t❤❡ ❖❉❊ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ❛ s♦❧✉t✐♦♥ ♦♥ t❤❡ r✐❣❤t✳ ■t ✐s s♦❧✈❡❞ ❜② ❛ s✐♠♣❧❡ ✐♥t❡❣r❛t✐♦♥✳ ❲❡ ✐❣♥♦r❡ t❤❡ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ ❜r❛❝❦❡ts✱ ❜✉t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ ✐s st✐❧❧ r❡q✉✐r❡❞✦ ❆♥♦t❤❡r s♦✉r❝❡ ✐s s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳ ❲❡❛❦ s♦❧✉t✐♦♥s ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s❛♠♣❧✐♥❣ ♦❢ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ❖❉❊ ❛s ❛ ✇❛② t♦ s♦❧✈❡ ✐t ♥✉♠❡r✐❝❛❧❧②✿
❚❤❡ ❢✉♥❝t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ❛ st❡♣✲❢✉♥❝t✐♦♥✳ ❚❤❡ ✈❡r② ✜rst ♦❜s❡r✈❛t✐♦♥ ✇❡ s❤♦✉❧❞ ♠❛❦❡ ✐s t❤❛t✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ t❤❡r❡ ✐s ❛ s❡♣❛r❛t❡ s♦❧✉t✐♦♥ ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ✐♥s✐❞❡ I ✱ ✐✳❡✳✱ ✐❢ y ✐s ❛ s♦❧✉t✐♦♥ ♦♥ ✐♥t❡r✈❛❧ I t❤❡♥ s♦ ✐s ✐ts r❡str✐❝t✐♦♥ z = y t♦ ❛♥② ✭♦♣❡♥ J ♦r ❝❧♦s❡❞ r❡s♣❡❝t✐✈❡❧②✮ ✐♥t❡r✈❛❧ J ✐♥ I ✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✶
❚❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ✈❡r② s✐♠♣❧②✿
z(t) = y(t)
❢♦r ❡❛❝❤
t
✐♥
J.
❋♦r t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❡❛❝❤ s❡❣♠❡♥t ♦❢ ❛ ❝✉r✈❡ s❡❡♥ ❜❡❧♦✇ ✐s ❛ s♦❧✉t✐♦♥✿
❲❛r♥✐♥❣✦
❚❤❡ ✐♥t❡r✈❛❧ I ♠❛② ❜❡ ✐♥✜♥✐t❡✿ I = (−∞, +∞), (−∞, b), (a, +∞) . ❘❡❝❛❧❧ t❤❛t s♦❧✈✐♥❣ ❛♥ ❖❉❊ ♠❡❛♥s t♦ ❢❛❝❡ ❛ ✏✜❡❧❞ ♦❢ s❧♦♣❡s✑ ❛♥❞ t❤❡♥ s❡❛r❝❤ ❢♦r ❝✉r✈❡s ✇✐t❤ t❤❡s❡ t❛♥❣❡♥ts✿
❙♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❛❜♦✉t t❤❡ ❢❛❧❧ ♦❢ ❛ ❜❛❧❧ t❤r♦✇♥ ❢r♦♠ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ❛t ❛ ♣❛rt✐❝✉❧❛r t✐♠❡ r❡q✉✐r❡❞ ♣✐❝❦✐♥❣ ♦✉t ♦♥❡ ♦❢ t❤❡♠✿
❚❤❡r❡ ✐s ❛ r❡❧❛t❡❞ ❝♦♥❝❡♣t ✐♥ t❤❡ t❤❡♦r② ♦❢ ❖❉❊s✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✷
❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✸✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ ❖❉❊ y ′ = f (t, y) ❛♥❞ ❛ ❣✐✈❡♥ ♣❛✐r ♦❢ ✈❛❧✉❡s (t0 , y0 )✱ t❤❡ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠✱ ♦r ❛♥ ■❱P✱ ✐s y ′ = f (t, y),
❛♥❞ ✐ts
s♦❧✉t✐♦♥
y(t0 ) = y0 ✳
✈❛❧✉❡
y(t0 ) = y0
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡
✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱
❲❛r♥✐♥❣✦ ❆ ❞✐✛❡r❡♥t ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥❞✐t✐♦♥ ♠❛② ♣r♦❞✉❝❡ t❤❡ s❛♠❡ s♦❧✉t✐♦♥✳
■♥ t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❛❜❧❡ ❢♦r ❡❛❝❤ (t0 , y0 ) ✐♥ t❤❡ ❞♦♠❛✐♥✳ ❚❤❛t ✐s ✇❤② t❤❡ ❝✉r✈❡s ✜❧❧ t❤❡ ♣❧❛♥❡✳ ■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❡② ✜❧❧ ❛❧❧ ♣❧❛♥❡ ❡①❝❡♣t t❤❡ ❧✐♥❡ t = 0✳ ❚❤❡ ✜rst ♣r♦♣❡rt② ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s✿ ◮ ❚❤❡ ❝✉r✈❡s ❝♦✈❡r t❤❡ ♣❧❛♥❡✳
❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✹✿ ❡①✐st❡♥❝❡
❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , y0 ) ✇❤❡♥ t❤❡ ■❱P✿ y ′ = f (t, y),
y(t0 ) = y0 ,
❤❛s ❛ s♦❧✉t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❡✈❡r② ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ s♦❧✉t✐♦♥ t❤❛t st❛rts t❤❡r❡✳ ■t ❞♦❡s♥✬t ♠❛tt❡r ❤♦✇ s♠❛❧❧ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s s♦❧✉t✐♦♥✳ ❚❤❡r❡ ❛r❡ ♥♦ ✭str♦♥❣ ♦r ✇❡❛❦✮ s♦❧✉t✐♦♥s ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡✿
■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t ✐s ❛s ✐❢ t❤❡ ♣r♦❝❡ss st❛rts ❜✉t ♥❡✈❡r ❝♦♥t✐♥✉❡s✳ ❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✶✳✺✳✶✺✿ ❊①✐st❡♥❝❡ ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊ ❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,
s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✸
Pr♦♦❢✳ ■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❥✉st t❤❡ ❛♣♣r♦♣r✐❛t❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✳ ❚❤❡ s❡❝♦♥❞ ♣r♦♣❡rt② ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s✿ ◮ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳
❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✻✿ ✉♥✐q✉❡♥❡ss ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , y0 ) ✐❢ ❡✈❡r② ♣❛✐r ♦❢ s♦❧✉t✐♦♥s✱ y1 , y2 ✱ ♦❢ t❤❡ ■❱P✿ y ′ = f (t, y),
y(t0 ) = y0
❛r❡ ❡q✉❛❧✱ y1 (t) = y2 (t)
❢♦r ❡✈❡r② t ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s t0 ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❡✈❡r② ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ s♦❧✉t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡✳
✉♥✐q✉❡
s♦❧✉t✐♦♥✱ t❤❛t st❛rts
❚♦ ✈✐♦❧❛t❡ ✉♥✐q✉❡♥❡ss✱ t❤❡ t✇♦ str♦♥❣ s♦❧✉t✐♦♥s ✇✐❧❧ ❤❛✈❡ t♦ ❤❛✈❡ t❤❡ s❛♠❡ s❧♦♣❡ ✭✐✳❡✳✱ t❤❡② ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r✮ ❛t t❤❡ ❢♦r❦✿
❊①❛♠♣❧❡ ✶✳✺✳✶✼✿ ♥♦♥✲✉♥✐q✉❡♥❡ss ❲❡❛❦ s♦❧✉t✐♦♥s ❝❛♥ ❡❛s✐❧② ♣❛rt ✇❛②s✿
❚❤❡ ❞❡✜♥✐t✐♦♥ s❛②s t❤❛t t❤❡✐r r❡str✐❝t✐♦♥s t♦ s♦♠❡ ✐♥t❡r✈❛❧ J ❛r❡ ❡q✉❛❧✿ y1 = y2 . J
J
■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t✬s ❛s ✐❢ ②♦✉ ❤❛✈❡ ❛❧❧ t❤❡ ❞❛t❛ ❜✉t ❝❛♥✬t ♣r❡❞✐❝t ❡✈❡♥ t❤❡ ♥❡❛r❡st ❢✉t✉r❡✳ ❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✹
❚❤❡♦r❡♠ ✶✳✺✳✶✽✿ ❯♥✐q✉❡♥❡ss ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊
❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,
s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳ Pr♦♦❢✳
■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❛❧✇❛②s ♦♥❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡s❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛❧✇❛②s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡✐r ❣r❛♣❤s ♥❡✈❡r ✐♥t❡rs❡❝t✳ ❲❡ ✇✐❧❧ s❡❡ ❖❉❊s ✇✐t❤♦✉t t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❧❛t❡r✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✾✿ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥
❋♦r t❤❡ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆y = ky(1 − y)∆t ♠❛② ♣r♦❞✉❝❡ str❛♥❣❡ r❡s✉❧ts ✭❧❡❢t✮ ❜✉t ♥♦t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❖❉❊ y ′ = ky(1 − y) ✭r✐❣❤t✮✿
❖♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤✐s ✐s tr✉❡ ❜② ❛♣♣❡❛❧✐♥❣ t♦ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥s✿ t❤❡② ❝❛♥✬t ❝r♦ss t❤❡ tr✐✈✐❛❧ s♦❧✉t✐♦♥s y = 0 ❛♥❞ y = 1✳ ❙♦✱ ❢♦r ❛ ♠♦❞❡❧ t♦ ❜❡ ✈❛❧✐❞✱ ✐t ❤❛s t♦ s❛t✐s❢②✿ ✶✳ ❊①✐st❡♥❝❡✿ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ s♦❧✉t✐♦♥ ❝✉r✈❡s✳ ✷✳ ❯♥✐q✉❡♥❡ss✿ ❚❤❡ s♦❧✉t✐♦♥ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❚❤❡r❡ ✐s ❛♥♦t❤❡r ✐♠♣♦rt❛♥t ❝♦♥❞✐t✐♦♥✦ ❊①❛♠♣❧❡ ✶✳✺✳✷✵✿ ✇❡❛t❤❡r ❢♦r❡❝❛st
■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ❛ ✇❡❛t❤❡r ❢♦r❡❝❛st ❜❡②♦♥❞ 3 ❞❛②s ✐s ✈❡r② ✉♥r❡❧✐❛❜❧❡✳ ❆ s✐t✉❛t✐♦♥ ❧✐❦❡ t❤❛t ♦❝❝✉rs ✇❤❡♥ s♠❛❧❧ ❝❤❛♥❣❡ ✐♥ t❤❡ ✐♥♣✉t ♣r♦❞✉❝❡s ❛ ❧❛r❣❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ♦✉t♣✉t✿
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✺
❙✉♣♣♦s❡ t❤❡ ❢♦r❡❝❛st ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❡❛s✉r✐♥❣ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡✳ ■♥ ❛♥ ❛tt❡♠♣t t♦ ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❢♦r❡❝❛st✱ ✇❡ ♠❛❦❡ t❤✐s ♠❡❛s✉r❡♠❡♥t ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉❧ts ✇✐❧❧ ❡✈❡♥t✉❛❧❧② r✉♥ ❛✇❛② ❢r♦♠ t❤❡ ♥♦♠✐♥❛❧✱ y = 0✳ ❲❤❛t ✇❡ ❛r❡ ❛❢t❡r ✐s ♠♦❞❡❧s t❤❛t ♣r♦❞✉❝❡ s♦❧✉t✐♦♥s t❤❛t ❛♣♣r♦❛❝❤ t❤❡ ♥♦♠✐♥❛❧ ❛s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣r♦❛❝❤ t❤❡ ♥♦♠✐♥❛❧✿
❚❤✐s ✐s ✇❤❛t ✇❡ ♠❛② r❡q✉✐r❡ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ t❤✐s ❦✐♥❞ ♦❢ ✐♥st❛❜✐❧✐t②✳
❉❡✜♥✐t✐♦♥ ✶✳✺✳✷✶✿ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❛t (t0 , y0 ) ✐❢ ❢♦r ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ t❤❡ ■❱P✱ ❛♥② t1 > t0 ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ y ✱ ❛♥❞ ❛♥② s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s✱ yn ❞❡✜♥❡❞ ♦♥ [t0 , t1 ]✱ ♦❢ t❤❡ ❖❉❊ ✇❡ ❤❛✈❡✱ ❛s n → ∞✱ yn (t0 ) → y(t0 ) =⇒ yn (t1 ) → y(t1 ) .
❙✉❝❤ ❛♥ ❖❉❊ ❝❛♥ ❛❧s♦ ❜❡ ❝❛❧❧❡❞ st❛❜❧❡✳
❊①❛♠♣❧❡ ✶✳✺✳✷✷✿ ♣❛r❛❜♦❧❛s ▲❡t✬s t❛❦❡ t❤✐s ❢❛♠✐❧② ♦❢ ♣❛r❛❜♦❧❛s ❛❣❛✐♥✿ y = −t2 /2 + C .
❆ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✇✐❧❧ ❧♦♦❦ ❧✐❦❡ t❤✐s✿ yn (t) = −t2 /2 + Cn , n = 1, 2, 3...
❚❤❡♥✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ ❛t ♦♥❡ ❡♥❞✱ t = t0 ✿ yn (t0 ) = −t20 /2 + Cn , n = 1, 2, 3...
■❢ t❤✐s s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ❝♦♥✈❡r❣❡s✱ yn (t0 ) = −t20 /2 + Cn → y(t0 ) = −t20 /2 + C ,
t❤❡♥ Cn → C ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❝♦♥✈❡r❣❡♥❝❡ ♦♥ t❤❡ ♦t❤❡r ❡♥❞✱ t = t0 ✿ yn (t1 ) = −t21 /2 + Cn → y(t1 ) = −t21 /2 + C .
❊①❡r❝✐s❡ ✶✳✺✳✷✸ Pr♦✈❡ t❤❡ ♣r♦♣❡rt② ❢♦r t❤❡ ❢❛♠✐❧② ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s y = Cet ✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✻
❊①❡r❝✐s❡ ✶✳✺✳✷✹ ❲❤❛t ❛❜♦✉t ❝♦♥✈❡r❣❡♥❝❡ ♦❢
❢✉♥❝t✐♦♥s yn → y❄ ❲❤❛t ❛❜♦✉t t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡❄
❊①❛♠♣❧❡ ✶✳✺✳✷✺✿ ❝♦♥t✐♥✉✐t②❄ ■♥ ♦r❞❡r t♦ ❡①♣❧❛✐♥ t❤❡ t❡r♠ ✏❝♦♥t✐♥✉✐t②✑✱ ❧❡t✬s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥
Q(z) = y(t1 )
✇❤❡♥
Q
♦❢ t❤❡
y ✲❛①✐s
t♦ ✐ts❡❧❢ ❜②✿
y(t0 ) = z .
■t✬s ❛s ✐❢ t❤❡ ✢♦✇ r❡❛rr❛♥❣❡s t❤❡ ♠♦❧❡❝✉❧❡s✳ ❲❡ t❤❡♥ ❜✉✐❧❞ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿
■t ✐s ❝♦♥t✐♥✉♦✉s✳
❊①❡r❝✐s❡ ✶✳✺✳✷✻ Pr♦✈❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳
■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t✬s ❛s ✐❢ ❛♥② ❡rr♦r ✐♥ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ✕ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ✕ ♠❛② ❝❛✉s❡ ❛ s✐❣♥✐✜❝❛♥t ❡rr♦r ✐♥ ②♦✉r ♣r❡❞✐❝t✐♦♥✳
❊①❛♠♣❧❡ ✶✳✺✳✷✼✿ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ ■❱P ❲❡❛❦ s♦❧✉t✐♦♥ ❝❛♥ ❡❛s✐❧② ✈✐♦❧❛t❡ t❤✐s ♣r♦♣❡rt②✿
❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✶✳✺✳✷✽✿ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊
❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,
s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✼
Pr♦♦❢✳
■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❛❧✇❛②s ♦♥❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡s❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛❧✇❛②s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♥✈❡r❣❡ t♦ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ t❤✐s ❝♦♥st❛♥t ❣♦❡s t♦ ③❡r♦✳ ◆❡①t✱ ❖❉❊s ♣r♦❞✉❝❡ ❢❛♠✐❧✐❡s ♦❢ ❝✉r✈❡s ❛s t❤❡ s❡ts ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✳✳✳ ❛♥❞ ✈✐❝❡ ✈❡rs❛✿ ■❢ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s ✐s ❣✐✈❡♥ ❜② ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ s✐♥❣❧❡ ♣❛r❛♠❡t❡r C ✱ t❤❡♥ ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ✇❛② ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥ ❖❉❊✳ ❊①❛♠♣❧❡ ✶✳✺✳✷✾✿ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞
❋✐rst✱ t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ❣r❛♣❤s✱ y = g(t) + C ,
❝r❡❛t❡s✱ ✇❤❡♥ ❞✐✛❡r❡♥t✐❛t❡❞✱ ❛♥ ❖❉❊✿
y ′ = g ′ (t) .
❙❡❝♦♥❞✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ❣r❛♣❤s✱ y = Cet .
❲❤❡♥ ❞✐✛❡r❡♥t✐❛t❡❞✱ t❤❡ r❡s✉❧t✐♥❣ ❡q✉❛t✐♦♥ st✐❧❧ ❤❛s C ❤♦✇❡✈❡r✦ ❚♦ ❝r❡❛t❡s ❛♥ ❖❉❊ ✇✐t❤♦✉t C ✱ ❧❡t✬s ✜rst s♦❧✈❡ ❢♦r C ❛♥❞ t❤❡♥ ❞✐✛❡r❡♥t✐❛t❡✿ C = ye−t =⇒ 0 = y ′ e−t + y e−t
❙✐♥❝❡ e−t > 0✱ ✇❡ ❤❛✈❡✿
′
= y ′ e−t − ye−t = (y ′ − y)e−t .
y′ − y = 0 .
❊①❛♠♣❧❡ ✶✳✺✳✸✵✿ ❝✐r❝❧❡s
❲❤❛t ❛❜♦✉t t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s❄
❚❤❡② ❛r❡ ❣✐✈❡♥ ❜②
x2 + y 2 = C > 0 .
❲❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ x✿ 2x + 2yy ′ = 0 ,
♦r✱
x y′ = − . y
❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s y 6= 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❝❛♥♥♦t ❜❡ s❛t✐s✜❡❞ ♦♥ t❤❡ x✲❛①✐s✳ ❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✇❛s ❞❡✜♥❡❞ ♦♥ t❤❡ x✲❛①✐s✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ✇♦✉❧❞ st✐❧❧ ❜r❡❛❦ ❞♦✇♥✿
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✽
❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❧❡❢t ♦❢ t❤✐s ♣♦✐♥t✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ♣♦✐♥t ✐❢ ✐t ✐s ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ 0✳ ❊①❡r❝✐s❡ ✶✳✺✳✸✶
❲❤❛t ❛❜♦✉t ✉♥✐q✉❡♥❡ss❄ ❊①❛♠♣❧❡ ✶✳✺✳✸✷✿ ❤②♣❡r❜♦❧❛s
❚❤❡s❡ ❤②♣❡r❜♦❧❛s ❛r❡ ❣✐✈❡♥ ❜② t❤❡s❡ ❡q✉❛t✐♦♥s✿ xy = C .
✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ❛①❡s✳✮ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❖❉❊✿ y + xy ′ = 0 ,
♦r✱ y′ =
y . x
❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s x 6= 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❝❛♥♥♦t ❜❡ s❛t✐s✜❡❞ ♦♥ t❤❡ y ✲❛①✐s✳ ❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✇❛s ❞❡✜♥❡❞ ♦♥ t❤❡ y ✲❛①✐s✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ✇♦✉❧❞ st✐❧❧ ❜r❡❛❦ ❞♦✇♥✿
❲❡ ❝❛♥♥♦t ❤❛✈❡ ❛ s♦❧✉t✐♦♥ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ y ✲❛①✐s✳
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✺✾
■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♥t✐♥✉✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳
❚❤❡♦r❡♠ ✶✳✺✳✸✸✿ ❊①✐st❡♥❝❡
❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t (t0 , y0 )✳ ❆t ✐ts s✐♠♣❧❡st✱ t❤✐s ♦♣❡♥ s❡t
U
✐s ❛ r❡❝t❛♥❣❧❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t❡r✈❛❧s
j
❛♥❞
G✿
❲❛r♥✐♥❣✦ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ t❤❡ ✇❤♦❧❡
J
❜❡❝❛✉s❡ ✐t ♠❛② ❡①✐t t❤❡ r❡❝t❛♥❣❧❡ t❤r♦✉❣❤
✐ts t♦♣ ♦r ✐ts ❜♦tt♦♠✿
❊①❡r❝✐s❡ ✶✳✺✳✸✹ ❲❤❛t ✐❢
G = (−∞, ∞)❄
❊①❛♠♣❧❡ ✶✳✺✳✸✺✿ ♥♦♥✲✉♥✐q✉❡♥❡ss ❈♦♥s✐❞❡r t❤✐s ❖❉❊✿
◆♦t✐❝❡ t❤❛t t❤❡r❡ ✐s ♦♥❡ ♠♦r❡ s♦❧✉t✐♦♥
dy = y 2/3 . dx ♣❛ss✐♥❣ t❤r♦✉❣❤ (0, 0)
■t ✐s
y=
(
0 1 3 x 27
✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ tr✐✈✐❛❧ s♦❧✉t✐♦♥
❢♦r
x < 0,
❢♦r
x ≥ 0.
y = 0✿
✶✳✺✳
❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s
✻✵
❲❤❛t ♠❛❦❡s t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ y 2/3 s♣❡❝✐❛❧ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s❄ ■ts ❞❡r✐✈❛t✐✈❡ ✐s ✐♥✜♥✐t❡ ❛t y = 0✳ ■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡ s❡t✲✉♣ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠ ❡①❝❡♣t✿ ◮ ❝♦♥t✐♥✉✐t② ❢♦r y ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✳
❚❤❡♦r❡♠ ✶✳✺✳✸✻✿ ❯♥✐q✉❡♥❡ss ✭✰❊①✐st❡♥❝❡✮ ❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t (t0 , y0 )✳
❲❛r♥✐♥❣✦ ❚❤✐s ✐s
♥♦t
✇❤❛t ♥♦♥✲✉♥✐q✉❡♥❡ss ❧♦♦❦s ❧✐❦❡✿
■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡ s❡t✲✉♣ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✶✳✺✳✸✼✿ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P ✭✰❯♥✐q✉❡♥❡ss✰❊①✐st❡♥❝❡✮ ❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❛t (t0 , y0 )✳ ◆❡①t✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ y = y(t) ♦♥ ✐♥t❡r✈❛❧ I ✱ ❛♥② r❡str✐❝t✐♦♥ z = y ♦❢ ❢✉♥❝t✐♦♥ y t♦ ❛♥② ✐♥t❡r✈❛❧ J J ✐♥s✐❞❡ I ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✳ ❈♦♥✈❡rs❡❧②✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ z ♦♥ ✐♥t❡r✈❛❧ J ✱ t❤❡r❡ ♠❛② ❜❡ ❛♥ ❡①t❡♥s✐♦♥ y ♦❢ z ✱ ✐✳❡✳✱ y = y(t) ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ z = y ✳ ❙♦♠❡t✐♠❡s ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❢✉rt❤❡r ❡①t❡♥❞ ♦✉r s♦❧✉t✐♦♥✳ J
❉❡✜♥✐t✐♦♥ ✶✳✺✳✸✽✿ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ❆ s♦❧✉t✐♦♥ y = y(t) ✐s ❝❛❧❧❡❞ ❛ ♠❛①✐♠❛❧
s♦❧✉t✐♦♥ ✐❢ ✐t ❞♦❡s♥✬t ❤❛✈❡ ❡①t❡♥s✐♦♥s✳
❆s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡✱ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ✐s ♦❢t❡♥ ❞❡✜♥❡❞ ♦♥ (−∞, +∞) ❜✉t ❛❧s♦ ♠❛② ❜❡ ❞❡✜♥❡❞ ♦♥ ♦t❤❡r ♦♣❡♥ ✐♥t❡r✈❛❧s✳ ❇❡❧♦✇✱ ✇❡ ✇✐❧❧ r❡❢❡r t♦ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥s ❛s s✐♠♣❧② s♦❧✉t✐♦♥s✳
✶✳✻✳
❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✻✶
✶✳✻✳ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
❙♦♠❡ t②♣❡s ♦❢ ❖❉❊s ❝❛♥ ❜❡ s♦❧✈❡❞ ❡①♣❧✐❝✐t❧②✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❛❧❧♦✇ ✉s s♦♠❡t✐♠❡s t♦ s❡♣❛r❛t❡ x✱ ❛♥❞ dx✱ ❢r♦♠ y ✱ ❛♥❞ dy ✳ ❊①❛♠♣❧❡ ✶✳✻✳✶✿ ✐♥t❡❣r❛t✐♦♥
❲❡ r❡✲✇r✐t❡ t❤❡ ❖❉❊ ❛s ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❛♥❞ t❤❡♥ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s✿ dy = x2 =⇒ dy = x2 dx =⇒ dx
Z
dy =
Z
x2 dx =⇒ y + C = x3 /3 + K .
❚❤❡ t✇♦ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛t✐♦♥ ♣r♦❞✉❝❡ t✇♦ ❝♦♥st❛♥ts✳ ❙✐♥❝❡ ❜♦t❤ ❛r❡ ❛r❜✐tr❛r②✱ ❥✉st ♦♥❡ ✐s s✉✣❝✐❡♥t ✭Q = K − C ✮✿ y = x3 /3 + Q .
❚❤❡ ❖❉❊ ✐s s♦❧✈❡❞✳ ❊①❛♠♣❧❡ ✶✳✻✳✷✿ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧
❲❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊✿
dy = y. dx
❘❡❝❛❧❧ t❤❛t ♣❧♦tt✐♥❣ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ♣r♦❞✉❝❡s t❤✐s✿
❋♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✱ ✇❡ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❦❡❡♣✐♥❣ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✿ dy dy = y =⇒ = dx =⇒ ❲❤❡♥ y > 0 : dx y
Z
dy = y
Z
dx =⇒ ln y = x + Q .
❚❤✐s ✐s ❛ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❖❉❊ ❣✐✈❡♥ ✐♠♣❧✐❝✐t❧②✳ ■t ✐s ❛ ❢❛♠✐❧② ♦❢ r❡❧❛t✐♦♥s✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ♦❜t❛✐♥✿ ❲❤❡♥ y < 0 : ln(−y) = x + P . ❲❤❛t✬s ❧❡❢t ✐s t❤❡ ❝❛s❡ ♦❢ y = 0✳ ❚❤❛t✬s ❛ ✇❤♦❧❡ ✭❝♦♥st❛♥t✮ s♦❧✉t✐♦♥✿ ❲❤❡♥ y = 0 :
dy = 0 =⇒ y = ❝♦♥st❛♥t =⇒ y = 0 . dx
❲❡ ✈❡r✐❢② t❤✐s ❢❛❝t ❜② s✉❜st✐t✉t✐♦♥✳ ◆♦✇ ❡①♣❧✐❝✐t t❤❡ s♦❧✉t✐♦♥s✳ ❲❡ ❛♣♣❧② t❤❡ ❧♦❣❛r✐t❤♠ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥s✳ ❋✐rst✿ ❲❤❡♥ y > 0 : ln y = x + Q =⇒ y = ex+Q = eQ ex , Q r❡❛❧.
✶✳✻✳
❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✻✷
❚❤❡s❡ ❛r❡ ❛❧❧ ♣♦s✐t✐✈❡ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❙❡❝♦♥❞✿ ❲❤❡♥ y > 0 : ln(−y) = x + P =⇒ y = −ex+P = −eP ex , P r❡❛❧. ❚❤❡s❡ ❛r❡ ❛❧❧ ♥❡❣❛t✐✈❡ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❖❢ ❝♦✉rs❡✱ y = 0 ✐s t❤❡ ③❡r♦ ♠✉❧t✐♣❧❡✳ ❲❡ ❤❛✈❡ t❤❡♠ ❛❧❧ ✐♥ ♦♥❡ ❢♦r♠✉❧❛✿ y = Cex , C r❡❛❧. ❚❤❡ s♦❧✉t✐♦♥ s❡t t❤✐s r❡s✉❧t ♣r♦❞✉❝❡s ✐s ❢❛♠✐❧✐❛r✿
■♥ ❣❡♥❡r❛❧✱ t❤❡ ♠❡t❤♦❞ ❛♣♣❧✐❡s ✇❤❡♥❡✈❡r ♦✉r r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ s♣❧✐ts ✐♥t♦ ❛ ❢❛❝t♦r ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❡ ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ x ❛♥❞ t❤❡ ♦t❤❡r ♦♥❧② ♦♥ y ✿ dy dy = p(x)q(y) =⇒ = p(x) dx . dx q(y)
❙✉❝❤ ❖❉❊s ❛r❡ ❝❛❧❧❡❞ s❡♣❛r❛❜❧❡✳ ❊①❡r❝✐s❡ ✶✳✻✳✸
●✐✈❡ ❡①❛♠♣❧❡s ♦❢ s❡♣❛r❛❜❧❡ ❛♥❞ ♥♦♥✲s❡♣❛r❛❜❧❡ ❖❉❊s✳ ❚❤❡♦r❡♠ ✶✳✻✳✹✿ ❙♦❧✉t✐♦♥s ♦❢ ❙❡♣❛r❛❜❧❡ ❖❉❊s
❊✈❡r② s♦❧✉t✐♦♥ y = y(x) ♦❢ ❛ s❡♣❛r❛❜❧❡ ❖❉❊ y ′ = p(x)q(y) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿ Z
dy = q(y)
Z
p(x) dx
❲❛r♥✐♥❣✦ ❚❤❡r❡ ✐s ♥♦ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❤❡r❡ ❛s t❤❡s❡ ❛r❡ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡s✳
❚❤❡ q✉❡st✐♦♥ t❤❡♥ r❡♠❛✐♥s ✇❤❡t❤❡r t❤❡ t✇♦ ✐♥t❡❣r❛❧s ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✐♥ ❛♥ ❡❧❡♠❡♥t❛r② ❢❛s❤✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✻✳✺✿ ❛❧❣❡❜r❛
❈♦♥s✐❞❡r t❤✐s ❖❉❊✿
dy = y sin x . dx
❲❡ ❦♥♦✇ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✳
✶✳✻✳
❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✻✸
❲❡ ❝❛♥ ❛❧s♦ t❛❦❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ ∆y = y sin x . ∆x
❛♥❞ ♣❧♦t t❤❡ s♦❧✉t✐♦♥s✿
◆♦✇ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❖❉❊ ✐s s❡♣❛r❛❜❧❡✿ dy dx dy =⇒ Zy =⇒
= ysin x = sin x dx Z dy = sin x dx y
=⇒ ln y
= − cos x + Q ❢♦r y > 0, ln(−y) = − cos x + P ❢♦r y < 0
=⇒ y
= e− cos x+Q ❢♦r y > 0,
y = −e− cos x+P ❢♦r y < 0
=⇒ y
= e− cos x eQ ❢♦r y > 0,
y = −e− cos x eP ❢♦r y < 0
=⇒ y
= Ce− cos x
❊✈❡♥ t❤♦✉❣❤ s♦♠❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ t♦✉❝❤ t❤❡ x✲❛①✐s✱ t❤❡ ✉♥✐q✉❡♥❡ss ✐s s❛t✐s✜❡❞✳ ■♥❞❡❡❞✱ ❡✈❡r② s♦❧✉t✐♦♥ ✐s ❥✉st ❛ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ♦❢ t❤❡ s✐♠♣❧❡st ♦♥❡✿ y = e− cos x . ❊①❡r❝✐s❡ ✶✳✻✳✻
Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞ t♦ ❛♥ ❖❉❊ t❤❛t ✇❡ ✇✐❧❧ s❡❡ ❧❛t❡r✳ ❚❤❡♦r❡♠ ✶✳✻✳✼✿ ▲✐♥❡❛r ❖❉❊
❚❤❡ ❖❉❊
y ′ = a(x)y ,
✇✐t❤ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ a✱ s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss✳ ❚❤❡ s♦❧✉✲ t✐♦♥s ♦❢ t❤❡ ❖❉❊ ❛r❡ ❣✐✈❡♥ ❜②✿ y = CeA(x)
✶✳✻✳
❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✇❤❡r❡
A
✻✹
✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢
a✿
A(x) =
Z
a(x) dx
❚❤✉s t❤❡ s♦❧✉t✐♦♥ s❡t s♣❧✐ts ✐♥t♦ t❤r❡❡ ♣❛rts✿ (1) y = KeA(x) ✇✐t❤ K < 0, (2) y = 0, (3) y = CeA(x) ✇✐t❤ C > 0 .
❆❧❧ ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ y = eA(x) ✳ ❊①❡r❝✐s❡ ✶✳✻✳✽
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✶✳✻✳✾✿ ❣❡♥❡r✐❝
❈♦♥s✐❞❡r t❤✐s ❖❉❊✿
dy = y2x . dx
❲❡ ❦♥♦✇ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✳ ❲❡ ❝❛♥ ❛❧s♦ t❛❦❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ ∆y = y sin x . ∆x
❛♥❞ ♣❧♦t t❤❡ s♦❧✉t✐♦♥s✿
◆♦✇ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❖❉❊ ✐s s❡♣❛r❛❜❧❡✿ dy dx dy =⇒ 2 Zy
= y2x
= x dx Z dy = x dx =⇒ y2 1 1 =⇒ − = x2 + Q y 2 1 ♦♥ ❡❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦❢ x . =⇒ y = −1 2 x +Q 2
✶✳✼✳
❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs
✻✺
❊①❛♠♣❧❡ ✶✳✻✳✶✵✿ ❧♦❣✐st✐❝ ♠♦❞❡❧ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊✿
dy = y(1 − y) . dx
❘❡❝❛❧❧ t❤❛t ♣❧♦tt✐♥❣ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s t❤✐s✿
❋♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✱ ✇❡ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❦❡❡♣✐♥❣ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✿
dy dy = y(1 − y) =⇒ = dx =⇒ dx y(1 − y)
Z
dy = y(1 − y)
Z
dx =⇒
y =x+Q y−1
❢♦r
0 < y < 1.
❚❤✐s ✐s ❛ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❖❉❊ ❣✐✈❡♥ ✐♠♣❧✐❝✐t❧② ❛s ❛ ❢❛♠✐❧② ♦❢ r❡❧❛t✐♦♥s✳
❊①❡r❝✐s❡ ✶✳✻✳✶✶ ❋✐♥❞ t❤❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥s✳
✶✳✼✳ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs
❊①❛♠♣❧❡ ✶✳✼✳✶✿ ♥♦♥✲s❡♣❛r❛❜❧❡ ❡q✉❛t✐♦♥ ▲❡t✬s t❛❦❡ ❛ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭♥♦♥✲s❡♣❛r❛❜❧❡✮ ❡q✉❛t✐♦♥✿ 2
y ′ · sin x + y · cos x = xex . ❲❡ s❡❡ t✇♦ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ♣r❡s❡♥t✿ • y ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ y ′ ✱ ❛♥❞
• sin x
❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡
cos x✳
❚❤❡② ❛r❡ ✏❝r♦ss✲♠✉❧t✐♣❧✐❡❞✑✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ✐s ❢❛♠✐❧✐❛r❀ ✐t✬s t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✿
(y · sin x)′ = y ′ · sin x + y · cos x . ❚❤❡r❡❢♦r❡✱ ♦✉r ❖❉❊ t✉r♥s ✐♥t♦✿
❲❡ ✐♥t❡❣r❛t❡✿
(y · sin x)′ = xex . 2
Z
′
(y · sin x) dx =
Z
2
xex dx ,
✶✳✼✳
❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs
✻✻
❛♥❞ ❛♣♣❧② t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✿
1 2 y · sin x = ex + C . 2 ❲❡ ✇❡r❡ ❧✉❝❦②✳
❊①❛♠♣❧❡ ✶✳✼✳✷✿ ❢✉rt❤❡r st❡♣s ❈♦♥s✐❞❡r t❤❡ ❖❉❊✿
y′ + y = x . ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❛♥❞
y
′
♥♦t
t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳✳✳ ❈❛♥ ✇❡ ♠❛❦❡ ✐t s♦❄ ❆❢t❡r ❛❧❧✱ t❤❡ ♣❛✐r
y
✐s ❛❧r❡❛❞② t❤❡r❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❤❛t✬s ❧❡❢t ❞♦❡s♥✬t ✇♦r❦✿
y′ · 1 + y · 1 = x . ❲❡ ❝❛♥✬t ❥✉st r❡♣❧❛❝❡ t❤❡ t❤❡
s❛♠❡
1✬s
✇✐t❤ s♦♠❡ ❢✉♥❝t✐♦♥s
F
❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡
F ′✱
❝❛♥ ✇❡❄ ❲❡ ❝❛♥ ✐❢ t❤✐s ✐s
❢✉♥❝t✐♦♥✿
(ex )′ = ex .
❲❡ ♠✉❧t✐♣❧② ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❜② t❤✐s ❢❛❝t♦r
F (x) = ex ✿
y ′ · ex + y · ex = xex , ♦r
y ′ · ex + y · (ex )′ = xex . ❇② t❤❡ Pr♦❞✉❝t ❘✉❧❡ ✇❡ ❤❛✈❡ ✐♥st❡❛❞✿
(y · ex )′ = xex . ❆❢t❡r ✐♥t❡❣r❛t✐♦♥✱ ✇❡ ❤❛✈❡
y · ex = −ex + xex + C , ♦r ❡✈❡♥ s✐♠♣❧❡r✿
y = −1 + x + Ce−x . ❚❤❡♥✱ ✐t ✐s ♣r♦♠✐s✐♥❣ ❢♦r t❤✐s ❛♣♣r♦❛❝❤ t♦ ❤❛✈❡ ❜♦t❤ ❡q✉❛t✐♦♥ s❤♦✉❧❞ ❜❡
❧✐♥❡❛r
✇✐t❤ r❡s♣❡❝t t♦
y
❛♥❞
y
❛♥❞
y′✱
❜✉t ♥♦t
y2
♦r
sin y ′ ✳
■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥
′
y✳
❊①❛♠♣❧❡ ✶✳✼✳✸✿ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❲❡ ♠❛❦❡ ❛ s♠❛❧❧ ❛❞❥✉st♠❡♥t t♦ t❤❡ ❖❉❊✿
y ′ + 2y = x . ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ♥♦t t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳✳✳ ❡✈❡♥ ✐❢ ✇❡ ♠✉❧t✐♣❧② ❜②
y ′ · ex + 2y · ex = xex . ❲❤❛t ❝❤♦✐❝❡ ♦❢ ❛ ❢❛❝t♦r
F
✇♦✉❧❞ ✇♦r❦✿
y ′ · F (x) + 2y · F (x) = xF (x)? ❲❡✬❞ ♥❡❡❞ t❤✐s✿
F ′ = 2F . ❇✉t t❤❛t✬s ❥✉st ❛♥♦t❤❡r ❖❉❊✦ ❆♥❞ ❛ ❢❛♠✐❧✐❛r ♦♥❡ t♦♦✿
F (x) = e2x .
ex ✿
✶✳✽✳
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✻✼
❚❤✉s ✇❡ ❤❛✈❡✿
y ′ · e2x + 2y · e2x = xe2x ,
♦r
y ′ · e2x + y · e2x
❚❤❡♥✱
y · e2x
❆❢t❡r ✐♥t❡❣r❛t✐♦♥✱ ✇❡ ❤❛✈❡
′
′
= xe2x .
= xe2x .
y · e2x = −e2x /2 + xe2x + C .
■♥ ❣❡♥❡r❛❧✱ ✇❤❡♥ t❤❡ ❧✐♥❡❛r ❡q✉❛t✐♦♥
y ′ + a(x)y = b(x)
✐s ♠✉❧t✐♣❧✐❡❞ ❜② s♦♠❡ ❢❛❝t♦r F ✱ ✇❡ ❤❛✈❡ y ′ F (x) + a(x)yF (x) = b(x)F (x) .
❚❤❡♥✱ t❤❡ ❛♣♣r♦❛❝❤ ✈✐❛ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❛♣♣❧✐❡s ✇❤❡♥ t❤❛t ❡q✉❛t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤✐s ❡q✉❛t✐♦♥✿ y ′ F (x) + yF ′ (x) = b(x)F (x) ,
✐✳❡✳✱ ✇❤❡♥ F s❛t✐s✜❡s✿
F ′ = a(x)F .
❚❤✐s ❖❉❊ ❛❧✇❛②s ❤❛s ❛ s♦❧✉t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠✿ F =e
R
a(x) dx
,
❛s ❧♦♥❣ ❛s ❢✉♥❝t✐♦♥ a ✐s ✐♥t❡❣r❛❜❧❡✳ ❋✉♥❝t✐♦♥ F ❞❡✜♥❡❞ t❤✐s ✇❛② ✐s ❝❛❧❧❡❞ t❤❡ ✐♥t❡❣r❛t✐♥❣ ❡q✉❛t✐♦♥✳
❢❛❝t♦r ♦❢ t❤❡ ❧✐♥❡❛r
❚❤❡♦r❡♠ ✶✳✼✳✹✿ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ▲✐♥❡❛r ❖❉❊ ❊✈❡r② s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊
y ′ + a(x)y = b(x) ✐s ❣✐✈❡♥ ❜②✿
y=e ✇❤❡r❡
A
✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢
−A(x)
Z
b(x)eA(x) dx
a✿
A(x) =
Z
a(x) dx
✶✳✽✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❛ ❝♦♠♠♦♥ t♦♦❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ♠❛t❤❡♠❛t✐❝s✳ ❚❤❡ ✐❞❡❛ ♦❢ ✜♥❞✐♥❣ ❛ ♥❡✇ ✈❛r✐❛❜❧❡ t❤❛t ♠✐❣❤t ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ s✐♠♣❧❡r ✐s ❡s♣❡❝✐❛❧❧② ♣❡r✈❛s✐✈❡ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ❙♦♠❡ s✐♠♣❧❡ ❡①❛♠♣❧❡s ❛r❡✿ • ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✭s❡❝♦♥❞s t♦ ♠✐♥✉t❡s✱ ♠❡t❡rs t♦ ❢❡❡t✱ ❈❡❧s✐✉s t♦ ❋❛❤r❡♥❤❡✐t ❡t❝✳✮
✶✳✽✳
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✻✽
• ❝❤❛♥❣❡ ♦❢ ❢r❛♠❡ ♦❢ r❡❢❡r❡♥❝❡ ✭❢r♦♠ t❤❡ ❊❛rt❤ t♦ t❤❡ ❙✉♥ ❛s t❤❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠♦t✐♦♥
♦❢ t❤❡ ♣❧❛♥❡t✱ t❤❡♥ ❜❛❝❦ t♦ t❤❡ ❊❛rt❤ ❛s t❤❡ ❝❡♥t❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠♦t✐♦♥ ♦❢ ❛ s♣❛❝❡❝r❛❢t ❤❡❛❞✐♥❣ ❢♦r t❤❡ ▼♦♦♥✱ ❡t❝✳✮
• r❡♣r❡s❡♥t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥
❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✐s t♦ tr② t♦ ❝❤❛♥❣❡ t❤❡ ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❣r❛♣❤s ♦♥ ✐t✿
✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥
❊①❛♠♣❧❡ ✶✳✽✳✶✿ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥
▲❡t✬s r❡✈✐❡✇ ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ ✐♥t❡❣r❛t✐♦♥
❘❡❝♦❣♥✐③✐♥❣ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❝♦♠♣♦s✐t✐♦♥ ✿
❜② s✉❜st✐t✉t✐♦♥ ✿
Z
2
xex dx =?
2
y = ex ,
✇❡ s♣❧✐t ✐t✳ ❲❡ ✐♥tr♦❞✉❝❡ ✭✐♥s❡rt✦✮ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡✱ u✿ x 7→ u 7→ y ,
✇✐t❤
u = x2 .
❚❤❡♥✱ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦❢ u ✐s t❤✐s ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✿ du = 2xdx .
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧✿ Z
❆t t❤✐s ♣♦✐♥t✱
xe
x2
dx =
Z
xe x2
x2 =u
• ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✱ ❛♥❞ • t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ✐s s✐♠♣❧❡r✦
dx
= dx= du 2x
Z
u du
1 xe = 2x 2
Z
eu dx .
❊①❡r❝✐s❡ ✶✳✽✳✷
❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = x3 ✐♥ t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧✳ ❍✐♥t✿ ❨❡s✱ t❤✐r❞ ♣♦✇❡r✳
✶✳✽✳
✻✾
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
❊①❛♠♣❧❡ ✶✳✽✳✸✿ s✉❜st✐t✉t✐♦♥ ✐♥ ❖❉❊s
▲❡t✬s r❡❝❛st t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡ ✐♥t❡❣r❛❧ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ ❛♥s✇❡r✳✳✳ t♦ ✇❤❛t q✉❡st✐♦♥❄ ❆♥ ❖❉❊✳ ❚❤❡ ❖❉❊ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✐♥t❡❣r❛❧ ✐s✿ 2 y ′ = xex .
▲❡t✬s ✉s❡ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✿ ❲❡ ❤❛✈❡ t✇♦
✈❛r✐❛❜❧❡s
dy 2 = xex . dx
r❡❧❛t❡❞ ✈✐❛ ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(x)✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✿ u = x2 .
❲❡ ❤❛✈❡ t❤r❡❡ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✈✐❛✿ • ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(x) t❤❛t ✐s st✐❧❧ t♦ ❜❡ ❢♦✉♥❞✱ • t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❢✉♥❝t✐♦♥✿ u = x2 ✱ ❛♥❞ • ❛♥♦t❤❡r ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(u) t♦ ❜❡ ❢♦✉♥❞ ✜rst✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠ ✐s ❜❡❧♦✇✿ x → u ց ↓ y
❚❤❡ t❤r❡❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ ✈❛r✐❛❜❧❡s ❤❛✈❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✿ ♦r✐❣✐♥❛❧✿
x
❞❡❝♦♠♣♦s❡❞✿ x
−→
−→
−→
y
u
−→
y
dy 2 = xex dx
−→
du = 2x dx
dy =? du
❚✇♦ ❛r❡ ❦♥♦✇♥ ❛♥❞ ♦♥❡ ✐s ✉♥❦♥♦✇♥✳ ❋♦rt✉♥❛t❡❧②✱ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❝♦♥♥❡❝t❡❞ ✈✐❛ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✿ dy du dy · = . dx du dx
❚❤❡r❡❢♦r❡✱ ♦✉r ❖❉❊ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❢t❡r t❤✐s s✉❜st✐t✉t✐♦♥✿ dy du dy dy dy 1 2 = xex =⇒ · = xeu =⇒ 2x = xeu =⇒ = eu . dx dx du du du 2
❲❡ ❤❛✈❡ ❛❝❝♦♠♣❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✳ • ❚❤❡ ♥❡✇ ❖❉❊ ✐s s✐♠♣❧❡r✦ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿
dy dy dy 2 = xex −→ = . dx du du ❲❡ s♦❧✈❡ t❤❡ ♥❡✇ ❖❉❊ ✭✇✐t❤ r❡s♣❡❝t t♦ u✮ ❡❛s✐❧②✿ Z 1 1 u e du = eu + C . y= 2 2
❙✐♥❝❡ t❤❡ ✈❛r✐❛❜❧❡ u ✇❛s ♠❛❞❡ ✉♣✱ ✇❡ ♥❡❡❞ t❤❡ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✱ u = x2 ✱ ❣✐✈✐♥❣ ✉s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊ ✭✇✐t❤ r❡s♣❡❝t t♦ x✮✿ 1 2 y = ex + C . 2
✶✳✽✳
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✼✵
❊①❡r❝✐s❡ ✶✳✽✳✹
❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = x3 ✐♥ t❤❡ ❛❜♦✈❡ ❖❉❊✳ ❊①❡r❝✐s❡ ✶✳✽✳✺
❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = sin x ✐♥ t❤❡ ❖❉❊ y ′ = cos3 x✳ ❊①❡r❝✐s❡ ✶✳✽✳✻
❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥ ❚❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✐s t♦ tr② t♦ ❝❤❛♥❣❡ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ✈❡rt✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❣r❛♣❤s ♦♥ ✐t✿
❚❤❡ s✐♠♣❧❡st s✉❝❤ s✉❜st✐t✉t✐♦♥ ✐s t❤❡ s❤✐❢t ✿
y = z + a,
✇❤❡r❡ z ✐s t❤❡ ♥❡✇ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■❢ ✇❡ ✇❛♥t t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s✐♥❣❧❡ ♣♦✐♥t y = a ✭❛♥❞ ✐ts ✈✐❝✐♥✐t②✮ ❛t ❛ t✐♠❡✱ t❤❡ s❤✐❢t ❛❧❧♦✇s ✉s t♦ ♠♦✈❡ t❤❛t ♣♦✐♥t t♦ ③❡r♦✳ ❊①❛♠♣❧❡ ✶✳✽✳✼✿ ♠✐❣r❛t✐♦♥
❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r
❖❉❊ ✿ y ′ = my + b, m 6= 0 .
■t ❝❛♥ r❡♣r❡s❡♥t ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✴❞❡❝❛② ❛❝❝♦♠♣❛♥✐❡❞ ❜② ♠✐❣r❛t✐♦♥ ❛t ❛ ❝♦♥st❛♥t r❛t❡✳ ❚❤❡ ❖❉❊ ✐s ❝❧♦s❡ t♦ y ′ = my ✭♥♦ ♠✐❣r❛t✐♦♥✮ t❤❛t ✇❡ ❦♥♦✇ ❤♦✇ t♦ ❤❛♥❞❧❡✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ❣❡t t❤❡r❡❄ ▲❡t✬s ✐♥✈❡st✐❣❛t❡✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✿ ∆y ∆y = my ✈s✳ = my + b . ∆x ∆x
❲❡ ♣❧♦t ❛ ❝♦✉♣❧❡ ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✿
✶✳✽✳
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✼✶
■♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ st❛♥❞❛r❞ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✭❧❡❢t✮✱ t❤❡r❡ s❡❡♠s t♦ ❜❡ ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✭r✐❣❤t✮✳ ❇✉t ❤♦✇ ❢❛r❄ ▲❡t✬s ❝❤♦♦s❡ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✿ z = y − a.
❚❤❛t✬s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✭✇✐t❤ a st✐❧❧ t♦ ❜❡ ❢♦✉♥❞✮✳ ❚❤❡♥✱ ✜rst✱ y ′ = (z + a)′ = z ′ ,
❛♥❞✱ s❡❝♦♥❞✱
b my + b = m(z + a) + b = mz + ma + b = mz + m a + m
❋♦r t❤❡ ❧❛st t❡r♠ t♦ ❞✐s❛♣♣❡❛r ✇❡ ❥✉st s❡❧❡❝t✿ a=−
❲❡ ❤❛✈❡ ❛ ♥❡✇ ❖❉❊ ✇✐t❤ r❡s♣❡❝t t♦ z ✿
b . m
z ′ = mz .
❲❡ ❤❛✈❡ ❛❝❝♦♠♣❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✳ • ❚❤❡ ♥❡✇ ❖❉❊ ✐s s✐♠♣❧❡r✦ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ ❲❡ s♦❧✈❡ ✐t ❛s ❜❡❢♦r❡✿
dz dy = my + b −→ = mz . dx dx z = Cemt .
❆❢t❡r ❛ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊✿ ♦r ❚❤❡ ✇❤♦❧❡ ♣✐❝t✉r❡ ❥✉st s❤✐❢ts ✉♣✦
y − a = Cemt , y=−
b + Cemt . m
.
✶✳✽✳
❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s
✼✷
❊①❛♠♣❧❡ ✶✳✽✳✽✿ ❜❛♥❦✐♥❣
❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❞❡♣♦s✐ts ✇✐t❤ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st❄ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❡ s❛♠❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛s ✐♥ t❤❡ ❝❛s❡ ✇✐t❤♦✉t ❞❡♣♦s✐ts ❜✉t s❤✐❢t❡❞ ❞♦✇♥ ✭♥♦t ✉♣✦✮✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ✇✐t❤❞r❛✇❛❧s ✇✐t❤ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st❄ ❚❤❡ s❛♠❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛s ✐♥ t❤❡ ❝❛s❡ ✇✐t❤♦✉t ✇✐t❤❞r❛✇❛❧s ❜✉t s❤✐❢t❡❞ ✉♣✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❤❛t ✐❢ ♦✉r ✐♥✈❡st♠❡♥t ✐s ❧♦s✐♥❣ ✈❛❧✉❡ ❜✉t ✇❡ ❝♦♥t✐♥✉❡ t♦ ♣✉t ♠♦♥❡② ✐♥ ✐t❄ ❚❤❡ ❡①♣♦✲ ♥❡♥t✐❛❧ ❞❡❝❛② ♦r ❛ st❛❣♥❛t✐♥❣ ❣r♦✇t❤ t♦✇❛r❞ ❛ ♥♦♥✲③❡r♦ ✈❛❧✉❡✿
❊①❛♠♣❧❡ ✶✳✽✳✾✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣
▲❡t✬s ✇r✐t❡ t❤❡ ❖❉❊s ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ♠♦❞❡❧ ❛♥❞ t❤❡ ❝♦♦❧✐♥❣ ❧❛✇ ♠♦❞❡❧ ♥❡①t t♦ ❡❛❝❤ ♦t❤❡r✿ y ′ = ky ❛♥❞ y ′ = k · (r − y) .
❚❤❡② s❡❡♠ s✐♠✐❧❛r✦ ▲❡t✬s ♣❧♦t t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡s❡ ♠♦❞❡❧s✿
❚❤❡② s❡❡♠ s✐♠✐❧❛r✦ ❚❤❡ ❝♦♠♣❛r✐s♦♥ s✉❣❣❡sts ❛ s✉❜st✐t✉t✐♦♥✿ ❛ ✈❡rt✐❝❛❧ s❤✐❢t ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣✳ ❲❡ ❡①❡❝✉t❡ ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤❡ s❤✐❢t ✐s ❣✐✈❡♥ ❜②✿ ❚❤❡♥ t❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✐s✿
z = y −r. y = z +r.
❚❤❡♥ t❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ♦✉r ❖❉❊ ❛s ✇❡ s✉❜st✐t✉t❡✿ dy d(z + r) dz = k(r − y) =⇒ = k(r − (z + r)) =⇒ = −kz . dt dt dt
❊①❝❡♣t ❢♦r t❤❡ ♠✐♥✉s s✐❣♥ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣✱ t❤✐s ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊✦ ◆♦✇ t❤❡ ✢✐♣✿ s = −t .
✶✳✾✳
✼✸
❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
❚❤❡♥ t❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✐s✿ t = −s .
◆♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s✿
dz dz ds dz = =− . dt ds dt ds
❖✉r ❖❉❊ ❜❡❝♦♠❡s✿
dz = kz . ds
❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❝♦♦❧✐♥❣ ❖❉❊ ✐s t❤❛t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊ s✐❢t❡❞ ✈❡rt✐❝❛❧❧② ❜② r ❛♥❞ t❤❡♥ ✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧②✦ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥s t♦♦ ✈✐❛ ❛ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥s✿ z = Ceks =⇒ y − r = Cek(−t) =⇒ y = r + Ce−kt . ❊①❡r❝✐s❡ ✶✳✽✳✶✵
❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥
✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
❯♥❢♦rt✉♥❛t❡❧②✱ ❛♥ ❖❉❊ t②♣✐❝❛❧❧② ❤❛s ♥♦ ❛❧❣❡❜r❛✐❝ s♦❧✉t✐♦♥✦ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ♦❢ ❛ ❣❡♥❡r❛❧ ■❱P✿ y ′ = f (t, y), y(t0 ) = y0 .
❚❤❡ ■❱P t❡❧❧s ✉s✿ • ✇❤❡r❡ ✇❡ ❛r❡ ✭t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞ • t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ❛r❡ ❣♦✐♥❣ ✭t❤❡ ❖❉❊✮✳
■t ✐s t❤❡♥ ❡❛s② t♦ ♣r♦❞✉❝❡ ❛ s♦❧✉t✐♦♥ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝❤❛♥❣❡ ✐s ✐♥❝r❡♠❡♥t❛❧✿ ❛s ✇❡ ❛rr✐✈❡ t♦ ❛ ♥❡✇ ❧♦❝❛t✐♦♥✱ ✇❡✱ ♦♥❝❡ ❛❣❛✐♥✱ ❦♥♦✇ ✇❤❡r❡ ✇❡ ❛r❡ ❛♥❞ ✇❤❡r❡ ✇❡ ❛r❡ ❣♦✐♥❣✳
❚❤❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ❤❡r❡ ❢♦❧❧♦✇s t❤❡ ✐❞❡❛ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✿ t❤❡ ✉♥❦♥♦✇♥ s♦❧✉t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳
✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
✼✹
❊①❛♠♣❧❡ ✶✳✾✳✶✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✿
❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿
x y ′ = f (x, y) = − . y
❲❡ ✇✐❧❧ ❜❡ s♦❧✈✐♥❣ ♥✉♠❡r♦✉s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ✇❤✐❧❡ st❛②✐♥❣ ❛✇❛② ❢r♦♠ t❤❡
x✲❛①✐s
✇❤❡r❡ t❤❡r❡
❛r❡ ♥♦ s♦❧✉t✐♦♥s✳ ❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢
x✿ ∆x = 1 .
❲❡ st❛rt ✇✐t❤ t❤✐s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
x0 = 0,
y0 = 1 .
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿
0 y′ = − = 0 . 1 ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❍♦✇ ❢❛r❄ ❆s ❢❛r ❛s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥❝r❡♠❡♥t ♦❢
y
✐s
❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡
xy ✲♣❧❛♥❡
∆y = 0 · ∆x = 0 × 1 = 0 . ✐s t❤❡♥✿
x1 = x0 + ∆x = 0 + 1 = 1,
y1 = y0 + ∆y = 1 + 0 = 1 .
❚❤✐s ❝♦♠♣✉t❛t✐♦♥ ❣✐✈❡s ✉s ❛ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✿
x1 = 1,
y1 = 1 .
x
❛❧❧♦✇s✳ ❚❤❡r❡❢♦r❡✱
✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
✼✺
❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿
1 y ′ = − = −1 . 1 ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢
y
✐s
∆y = −1 · ∆x = −1 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡
xy ✲♣❧❛♥❡
✐s t❤❡♥✿
x2 = x1 + ∆x = 1 + 1 = 2, ❲❡ ❤❛✈❡ ❡♥❞❡❞ ✉♣ ♦♥ t❤❡
x✲❛①✐s
y2 = y1 + ∆y = 1 + (−1) = 0 .
❛♥❞ st♦♣✳ ❚❤❡s❡ t❤r❡❡ ♣♦✐♥ts ❢♦r♠ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✳
❲❡ st❛rt ❛❧❧ ♦✈❡r ❢r♦♠ ❛♥♦t❤❡r ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
x0 = 0,
y0 = 2 .
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿
0 y′ = − = 0 , 2 ♣r♦❞✉❝✐♥❣ t❤❡ s❧♦♣❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢
y
✐s
∆y = 0 · ∆x = 0 · 1 = 0 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡
xy ✲♣❧❛♥❡
✐s t❤❡♥✿
x1 = x0 + ∆x = 0 + 1 = 1, y1 = y0 + ∆y = 2 + 0 = 2 . ❆ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣❡❛rs✿
x0 = 1,
y0 = 2 .
❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿
1 y ′ = − = −1/2 , 2 ♣r♦❞✉❝✐♥❣ t❤❡ s❧♦♣❡ t♦ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢
y
✐s
∆y = −1/2 · ∆x = −1/2 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡
xy ✲♣❧❛♥❡
✐s t❤❡♥✿
x2 = x1 + ∆x = 1 + 1 = 2,
y2 = y1 + ∆y = 2 + (−1/2) = 3/2 .
❖♥❡ ♠♦r❡ ■❱P✿
x2 = 2, y2 = 3/2 . ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢
y
✐s
❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡
2 x · 1 = −4/3 . ∆y = − · ∆x = − y 3/2 xy ✲♣❧❛♥❡
✐s t❤❡♥✿
x3 = x2 + ∆x = 2 + 1 = 3, ❲❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ♣❛ss t❤❡
x✲❛①✐s
y3 = y2 + ∆y = 3/2 − 4/3 = 1/6 .
✇✐t❤ ♥❡①t st❡♣ ❛♥❞ ✇❡ st♦♣ ♥♦✇✳ ❚❤❡s❡ ❢♦✉r ♣♦✐♥ts✱ ✐♥ ❛❞❞✐t✐♦♥ t♦
t❤❡ ♣r❡✈✐♦✉s t❤r❡❡✱ ❢♦r♠ ❛ ✈❡r② ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♦✉r ❝✐r❝✉❧❛r s♦❧✉t✐♦♥s✿
✶✳✾✳
✼✻
❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
■♥ ❢❛❝t✱ ✇❡ s❡❡ ✇❤❛t ✐s♥✬t s✉♣♣♦s❡❞ t♦ ❤❛♣♣❡♥✿ t❤❡ s♦❧✉t✐♦♥ ❣♦❡s ❜❡❧♦✇ t❤❡ x✲❛①✐s✦ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♠♦t✐♦♥✱ t❤✐s ✐s t❤❡ ♠❡t❤♦❞✬s ✐♥t❡r♣r❡t❛t✐♦♥✿
◮ ❆t ♦✉r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t t✐♠❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❖❉❊ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ ♠♦✈❡ ✇✐t❤ t❤✐s ✈❡❧♦❝✐t② t♦ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✳
❉❡✜♥✐t✐♦♥ ✶✳✾✳✷✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡
❊✉❧❡r s♦❧✉t✐♦♥
✇✐t❤ ✐♥❝r❡♠❡♥t h > 0 ♦❢ t❤❡ ■❱P
y ′ = f (t, y), y(t0 ) = y0 , ✐s t❤❡ s❡q✉❡♥❝❡ {yn } ♦❢ r❡❛❧ ♥✉♠❜❡rs ❣✐✈❡♥ ❜②✿
yn+1 = yn + f (tn , yn ) · h ✇❤❡r❡ tn+1 = tn + h✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝t t♦ ❦♥♦✇ ❛❜♦✉t ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s t❤❛t ✐❢ ✇❡ ❞❡r✐✈❡❞ ♦✉r ❖❉❊ ❢r♦♠ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✕ ❛♥❞ ❢r♦♠ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ✈✐❛ ∆t → 0 ✕ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐❧❧ ❜r✐♥❣ ✉s r✐❣❤t ❜❛❝❦ t♦ ✐t✿ ❖❉❊ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥
ր
same!
←−−−−−−−
ց
❊✉❧❡r✬s ♠❡t❤♦❞
❊①❡r❝✐s❡ ✶✳✾✳✸ ❊①❡❝✉t❡ t❤❡ ♠❡t❤♦❞ ❢♦r t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ ❛♥❞ h = 1/2✳
❊①❛♠♣❧❡ ✶✳✾✳✹✿ ❝✐r❝❧❡s ▲❡t✬s ♥♦✇ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ❢♦r t❤❡ ❖❉❊✿
y y′ = − . x ❚❤❡ ❢♦r♠✉❧❛ ❢♦r yn ✐s✿ ❚❤❡ r❡s✉❧ts ❛r❡ ❧❡ss ❝r✉❞❡✿
❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✶❪✯❘✷❈✶
✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
✼✼
❍♦✇❡✈❡r✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛♣♣❡❛r t♦ ❜❡❤❛✈❡ ❡rr❛t✐❝❛❧❧② ✇❤❡♥ t❤❡② ❣❡t ❝❧♦s❡ t♦ t❤❡ r❡❛s♦♥ ✐s t❤❡ ❞✐✈✐s✐♦♥ ❜②
x✲❛①✐s✱
y
x✲❛①✐s✳
❚❤❡
✇❤✐❝❤ ♠❛② ❜❡ ✈❡r② s♠❛❧❧✳ ■♥ ❢❛❝t t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉♣♣♦s❡❞ t♦ st♦♣ ❛t t❤❡
❜✉t✱ ❜② ❞❡s✐❣♥ ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞✱ t❤❡② ❝❛♥✬t✳
❊❧s❡✇❤❡r❡✱ t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥s ❛r❡ ❝❧♦s❡ t♦ ♣❡r❢❡❝t✿
❚❤❡ s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s s♦❧✉t✐♦♥✿
dy x = − =⇒ ydy = −xdx =⇒ dx y
Z
ydy = −
Z
xdx =⇒
y2 x2 =− +C. 2 2
❊①❡r❝✐s❡ ✶✳✾✳✺ ❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳
❲❛r♥✐♥❣✦
❚❤❡ ❊✉❧❡r s♦❧✉t✐♦♥s ♠✐❣❤t ❜❡ ✉♥❛✛❡❝t❡❞ ❜② t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♣r♦♣❡rt② ♦❢ t❤❡ ❖❉❊✳ ❊①❛♠♣❧❡ ✶✳✾✳✻✿ ❤②♣❡r❜♦❧❛s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❤②♣❡r❜♦❧❛s✿
xy = C .
✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡
✼✽
❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿
y′ = ❋♦r ❧❛r❣❡r ✈❛❧✉❡s ♦❢
h✱
y . x
t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥ ♠✐❣❤t ❝r♦ss t❤❡
x✲❛①✐s
❞❡♠♦♥str❛t✐♥❣ ♥♦♥✲✉♥✐q✉❡♥❡ss✿
❙❡✈❡r❛❧ ❜❡tt❡r ❧♦♦❦✐♥❣ ❊✉❧❡r s♦❧✉t✐♦♥s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿
❊✈❡♥ t❤❡♥✱ ✐s t❤✐s ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡
x✲❛①✐s
♦r ❞♦ t❤❡② ♠❡r❣❡❄
❚❤❡ s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s s♦❧✉t✐♦♥✿
dy y dy dx = − =⇒ =− =⇒ dx x y x
Z
dy =− y
Z
dx =⇒ ln y = − ln x + K . x
❊①❡r❝✐s❡ ✶✳✾✳✼ ❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳
❲❛r♥✐♥❣✦ ❊✉❧❡r✬s ♠❡t❤♦❞ ♠✐❣❤t ✐♥tr♦❞✉❝❡ ♥♦♥✲✉♥✐q✉❡♥❡ss t♦ ❛♥ ❖❉❊✳
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✼✾
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✿ ❛♣♣r♦①✐♠❛t✐♦♥s❄
❍♦✇ ❢❛r ❛♣❛rt ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ♣❛✐r ♦❢ ♠❛t❝❤✐♥❣ ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❖❉❊s❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ❤♦✇ ❢❛r ✐s t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥ ❢r♦♠ t❤❡ ❛❝t✉❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s❄ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r✿ ∆y t t = − ❛♥❞ y ′ = − . ∆t
y
y
❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❝✐r❝❧❡s✳ ❙♦✱ ✐❢ ♦♥❡ s✉❝❤ s♦❧✉t✐♦♥ st❛rts ❛t (0, y0 )✱ ✐✳❡✳✱ t = 0 ❛♥❞ y = y0 ✱ t❤❡♥ t❤❡ ❝✉r✈❡ ✐s s✉♣♣♦s❡❞ t♦ ❡♥❞ ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ t✲❛①✐s✱ ✐✳❡✳✱ (y0 , 0)✳ ❚❤✐s ✐s ♥♦t ✇❤❛t ✇❡ s❡❡ ✇❤❡♥ ✇❡ ❧♦t ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r♠❡r ❡q✉❛t✐♦♥✿
❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠❡r ❞❡❝r❡❛s❡ s❧♦✇❡r t❤❛♥ t❤♦s❡ ♦❢ t❤❡ ❧❛tt❡r✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❞✐♠✐♥✐s❤❡s ✇✐t❤ t❤❡ ✈❛❧✉❡ ♦❢ h = ∆t✳ ❆♣♣r♦①✐♠❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐s ❧✐❦❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♥✉♠❜❡rs ✭s✉❝❤ ❛s π ♦r t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✮ ❜✉t ❤❛r❞❡r✳ ❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻ t❤❛t ❧✐♥❡❛r✐③❛t✐♦♥ ♠❡❛♥s r❡♣❧❛❝✐♥❣ ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ y = f (x) ✇✐t❤ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ y = L(x) t❤❛t ❜❡st ❛♣♣r♦①✐♠❛t❡s ✐t ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t✳ ■t ✐s ❝❛❧❧❡❞ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ✐ts ❤❛♣♣❡♥s t♦ ❜❡ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ♣♦✐♥t✳ ❚❤❡ r❡♣❧❛❝❡♠❡♥t ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿
✶✳✶✵✳
❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛ ♠♦r❡ ❜❛s✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✿ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱
✽✵
y = C(x)✳
❊①❛♠♣❧❡ ✶✳✶✵✳✷✿ sq✉❛r❡ r♦♦t
▲❡t✬s r❡✈✐❡✇ t❤✐s ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✶P❈✲✷❈✲✻✿
√ x❄ ❲❡ ❛♣♣r♦①✐♠❛t❡✳ f (x) = √ x ✏❛r♦✉♥❞✑ a = 4✳ ❢✉♥❝t✐♦♥ f (x) =
❡✈❛❧✉❛t✐♥❣ t❤❡
❲❡ ✜rst ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛
❆♥❞ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥✉♠❜❡r
❝♦♥st❛♥t
√ √4.1 ✇✐t❤♦✉t ❛❝t✉❛❧❧② ♦❢ 4.1✱ ✇❡ ❛♣♣r♦①✐♠❛t❡
❤♦✇ ❞♦ ✇❡ ❝♦♠♣✉t❡
❢✉♥❝t✐♦♥✿
C(x) = 2 . ❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡
f (a) =
√ √
4 = 2✳
❚❤❡♥ ✇❡ ❤❛✈❡✿
4.1 = f (4.1) ≈ C(4.1) = 2 .
■t ✐s ❛ ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥✿
❚❤❡ ♦t❤❡r✱ ❧✐♥❡❛r✱ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✈✐s✐❜❧② ❜❡tt❡r✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛
1 L(x) = 2 + (x − 4) . 4 ❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡
√
f (a) =
√
4=2
❛♥❞
f ′ (a) =
1 ✳ 4
❚❤❡♥ ✇❡ ❤❛✈❡✿
1 4.1 = f (4.1) ≈ L(4.1) = 2 + (4.1 − 4) = 2.025 . 4
❧✐♥❡❛r ❢✉♥❝t✐♦♥✿
✶✳✶✵✳
❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✶
❲❡ ❤❛✈❡ ❢♦r ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ x = a✿ ❆ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✿ ❆ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✿
C(x) = f (a) L(x) = f (a) +f ′ (a)(x − a)
❲❡ s❤♦✉❧❞ ♥♦t✐❝❡ ❡❛r❧② ♦♥ t❤❛t t❤❡ ❧❛tt❡r ❥✉st ❛❞❞s ❛ ♥❡✇ ✭❧✐♥❡❛r✮ t❡r♠ t♦ t❤❡ ❢♦r♠❡r✦ ❆❧s♦✱ t❤❡ ❧❛tt❡r ✐s ❜❡tt❡r t❤❛♥ t❤❡ ❢♦r♠❡r ✕ ❜✉t ♦♥❧② ✇❤❡♥ ✇❡ ♥❡❡❞ ♠♦r❡ ❛❝❝✉r❛❝②✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❢♦r♠❡r ✐s ✇♦rs❡ ❜❡❝❛✉s❡ ✐t r❡q✉✐r❡s ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ ✇❡ ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ ❛ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t (1, 1) ✇✐t❤ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ✜rst❀ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳ ❚❤✐s ❧✐♥❡ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ ♠❛♥② ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t❀ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳
◆♦✇✱ ✇❡ s❤❛❧❧ s❡❡ t❤❛t t❤❡s❡ ❛r❡ ❥✉st t❤❡ t✇♦ ✜rst st❡♣s ✐♥ ❛ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✦ ❚❤❡ t❛♥❣❡♥t ❧✐♥❡ ❜❡❝♦♠❡s ♦♥❡ ♦❢ ♠❛♥② q✉❛❞r❛t✐❝ ❝✉r✈❡s ✕ ♣❛r❛❜♦❧❛s ✕ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳✳✳ ❛♥❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡✳ ❲❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ❞♦ ✇❡ ❝❤♦♦s❡❄ ■♥ ♦r❞❡r t♦ ❛♥s✇❡r t❤❛t✱ ✇❡ ♥❡❡❞ t♦ r❡✈✐❡✇ ❛♥❞ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ✇❡r❡ ❝❤♦s❡♥✳ ■♥ ✇❤❛t ✇❛② ❛r❡ t❤❡② t❤❡ ❜❡st❄ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❛♥❞ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣r♦①✐♠❛t❡ ✐ts ❜❡❤❛✈✐♦r ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ♣♦✐♥t✱ x = a✱ ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ y = T (x)✳ ❚❤❡ ❧❛tt❡r ✐s t♦ ❜❡ t❛❦❡♥ ❢r♦♠ s♦♠❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ✇❡ ✜♥❞ s✉✐t❛❜❧❡✳ ❲❤❛t ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐s t❤❡
❡rr♦r✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥ T ✿ E(x) = |f (x) − T (x)|.
❲❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ✐♥ s♦♠❡ ✇❛②✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ y = E(x) ✐s ❧✐❦❡❧② t♦ ❣r♦✇ ✇✐t❤ ♥♦ ❧✐♠✐t ❛s ✇❡ ♠♦✈❡ ❛✇❛② ❢r♦♠ ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ x = a✳✳✳ ❜✉t ✇❡ ❞♦♥✬t ❝❛r❡✳ ❲❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ a ✇❤✐❝❤ ♠❡❛♥s ♠❛❦✐♥❣ s✉r❡ t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❡rr♦r ❛s x → a ❣♦❡s t♦ 0✦
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✷
❚❤❡♦r❡♠ ✶✳✶✵✳✸✿ ❇❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥
❙✉♣♣♦s❡ f ✐s ❝♦♥t✐♥✉♦✉s ❛t x = a ❛♥❞ C(x) = k
✐s ❛♥② ♦❢ ✐ts ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥s ✭✐✳❡✳✱ ❛r❜✐tr❛r② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✮✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥st❛♥t ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ lim (f (x) − C(x)) = 0 ⇐⇒ k = f (a)
x→a
❚❤❛t✬s t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✳ ❚❤❡♦r❡♠ ✶✳✶✵✳✹✿ ❇❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥
❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ L(x) = f (a) + m(x − a)
✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ x − a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❧✐♥❡❛r t❡r♠ ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ f (x) − L(x) = 0 ⇐⇒ m = f ′ (a) x→a x−a lim
❈♦♠♣❛r✐♥❣ t❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ f (x) − C(x) → 0 ❛♥❞
f (x) − L(x) → 0, x−a
r❡✈❡❛❧s t❤❡ s✐♠✐❧❛r✐t② ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❤♦✇ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐♥ t❤❡ ❞❡❣r❡❡✿ ❤♦✇ ❢❛st t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❣♦❡s t♦ ③❡r♦✳ ■♥❞❡❡❞✱ ✇❡ ❧❡❛r♥❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t❤❛t t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ♠❡❛♥s t❤❛t f (x) − L(x) ❝♦♥✈❡r❣❡s t♦ 0 ❢❛st❡r t❤❛♥ x − a✱ ✐✳❡✳✱ f (x) − L(x) = o(x − a),
t❤❡r❡ ✐s ♥♦ s✉❝❤ r❡str✐❝t✐♦♥ ❢♦r t❤❡ ❢♦r♠❡r✳ ❙♦ ❢❛r✱ t❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✿ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ ❧✐♥❡❛r t❡r♠✳ ❚❤❡ ❜❡st ♦♥❡ ♦❢ t❤♦s❡ ❤❛s t❤❡ s❧♦♣❡ ✭✐ts ♦✇♥ ❞❡r✐✈❛t✐✈❡✮ ❡q✉❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ❛t a✳ ❍♦✇ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐❧❧ ♣r♦❣r❡ss ✐s ♥♦✇ ❝❧❡❛r❡r✿ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ q✉❛❞r❛t✐❝ t❡r♠✳
✶✳✶✵✳
❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✸
❇✉t ✇❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ✐s t❤❡ ❜❡st❄
❚❤❡♦r❡♠ ✶✳✶✵✳✺✿ ❇❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥
❙✉♣♣♦s❡ f ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ Q(x) = f (a) + f ′ (a)(x − a) + p(x − a)2
✐s ❛♥② ♦❢ ✐ts q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ (x − a)2 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ t❡r♠ ✐s ❡q✉❛❧ t♦ ❤❛❧❢ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ f (x) − Q(x) 1 = 0 ⇐⇒ p = f ′′ (a) 2 x→a (x − a) 2 lim
❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ♠❡❛♥s t❤❛t
f (x) − Q(x)
❝♦♥✈❡r❣❡s t♦
0
❢❛st❡r t❤❛♥
(x − a)2 ✱
f (x) − Q(x) = o((x − a)2 ) . ❲❡ st❛rt t♦ s❡❡ ❛ ♣❛tt❡r♥✿
•
❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❣r♦✇✐♥❣✳
•
❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ❜❡✐♥❣ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛r❡ ❣r♦✇✐♥❣ t♦♦✳
❊①❛♠♣❧❡ ✶✳✶✵✳✻✿ s✐♥❡ ▲❡t✬s ❛♣♣r♦①✐♠❛t❡
f (x) = sin x
❛t
x = 0✳
❋✐rst✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿
f (x) = sin x =⇒ f (0) = 0 =⇒ C(x) = 0 ′ f (x) = cos x =⇒ f ′ (0) = 1 =⇒ L(x) = x f ′′ (x) = − sin x =⇒ f ′′ (0) = 0 =⇒ Q(x) = ?
♦r
✶✳✶✵✳
❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✹
❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿
0 Q(x) = 0 + 1(x − 0) − (x − 0)2 = x . 2 ❙❛♠❡ ❛s t❤❡ ❧✐♥❡❛r✦ ❲❤②❄ ❇❡❝❛✉s❡
♥✉♠❜❡rs
❲❡ ✉s❡❞ s❡q✉❡♥❝❡s ♦❢
❢✉♥❝t✐♦♥s
sin
✐s ♦❞❞✳
t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ♥✉♠❜❡rs ✭❈❤❛♣t❡r ✺✮❀ ♥♦✇ ✇❡ ✇✐❧❧ ✉s❡ s❡q✉❡♥❝❡s ♦❢
t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ❢✉♥❝t✐♦♥s✳ ■♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❣♦ ❜❡②♦♥❞ q✉❛❞r❛t✐❝ ✐♥ ♦✉r s❡q✉❡♥❝❡ ♦❢
♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ r❡♥❛♠❡ t❤❡♠ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r
❞❡❣r❡❡s ✿
T0 (x) = C(x) , T1 (x) = L(x) , T2 (x) = Q(x) , ... ❊①❛♠♣❧❡ ✶✳✶✵✳✼✿ sq✉❛r❡ r♦♦t
❇❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❡①❛♠♣❧❡ ♦❢
f (x) =
√
x
❛t
❝♦♥st❛♥t✿ ❧✐♥❡❛r✿
❝✉❜✐❝✿
1 (x − 4)2 2 · 32 1 (x − 4)2 2 · 32
−
q✉❛❞r❛t✐❝✿
(?)(x − 4)3 − ✳ ✳ ✳
✳ ✳ ✳
a = 4✳
❖♥❡ ❝❛♥ ❣✉❡ss ✇❤❡r❡ t❤✐s ✐s ❣♦✐♥❣✿
2 = T0 (x) 1 (x − 4) + 2 = T1 (x) 4 1 (x − 4) + 2 = T2 (x) + 4 1 + (x − 4) + 2 = T3 (x) 4 ✳ ✳ ✳
✳ ✳ ✳
❲❡ ❛❞❞ ❛ t❡r♠ ❡✈❡r② t✐♠❡ ✇❡ ♠♦✈❡ ❞♦✇♥ t♦ t❤❡ ♥❡①t ❞❡❣r❡❡❀ ✐t✬s ❛
✳ ✳ ✳
f − T0 = o(1)
f − T1 = o(x − a) f − T2 = o((x − a)2 ) f − T3 = o((x − a)3 ) ✳ ✳ ✳
r❡❝✉rs✐♦♥ ✦
❙✉❝❤ ❛ s❡q✉❡♥❝❡ ✭❛
s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✦✮ ✐s ❝❛❧❧❡❞ ❛ ✏s❡r✐❡s✑✳
❲❡ ❝♦♥s✐❞❡r ❛ s♦❧✉t✐♦♥ ♦❢ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ❛♥❞ t❤❛t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t✐♥✉♦✉s ❖❉❊ ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱
y(t0 ) = y0 ✳ t > t0 ✐s
✈❛❧✉❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❡✈❛❧✉❛t❡❞ ❛t s♦♠❡
❚❤❡✐r ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❛❜s♦❧✉t❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡
❡rr♦r✳
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✺
❚♦ ❡st✐♠❛t❡ t❤❡ ❡rr♦r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡rr♦r ❜♦✉♥❞ ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✳ ❙✉♣♣♦s❡ y ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = t0 ❛♥❞ L(t) = y(t0 ) + y ′ (t0 )(t − t0 ) ✐s ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t t0 ✳ ❚❤❡♥✱ t❤❡ ❡rr♦r s❛t✐s✜❡s✿ E(t) = |y(t) − L(t)| ≤ 12 K(t − t0 )2 ,
✇❤❡r❡ K ✐s ❛ ❜♦✉♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ✐♥t❡r✈❛❧ ❢r♦♠ t0 t♦ t✿ |y ′′ (c)| ≤ K ❢♦r ❛❧❧ c ✐♥ t❤✐s ✐♥t❡r✈❛❧ .
❙✉♣♣♦s❡ ♥♦✇ t❤❛t y ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ■❱P✿ y ′ = f (t, y), y(t0 ) = y0 .
Pr♦✈✐❞❡❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❜♦✉♥❞❡❞ ❛s ❛❜♦✈❡✱ t❤❡ ❡rr♦r ♦❢ ❛ s✐♥❣❧❡ st❡♣ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭✐✳❡✳✱ ❊✉❧❡r ♠❡t❤♦❞✮ ✐s ❜♦✉♥❞❡❞ ❜②✿ E(t0 + h) = |y(t0 + h) − y1 | ≤ 12 Kh2 ,
✇❤❡r❡ y1 = L1 (t0 + h) ✐s t❤❡ ✜rst st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ L1 (t) = y0 + f (t0 , y0 )(t − t0 ) ,
t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ y ❛t t0 ✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❡rr♦r ♦❢ ❛ s✐♥❣❧❡ st❡♣ ✐s ❞❡❝r❡❛s✐♥❣ q✉❛❞r❛t✐❝❛❧❧② ✇✐t❤ h✳ ◆♦✇✱ ❧❡t✬s ❝♦♥s✐❞❡r ♠✉❧t✐♣❧❡ st❡♣s✳ ❚❤✐s ✐s ✇❤❛t ✇✐❧❧ ❤❛♣♣❡♥✿
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✻
❲❤❛t ❝❛♥ ✇❡ ❞♦ ❛❜♦✉t t❤✐s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❡rr♦r ❄ ■❢ y ′′ ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ s❛♠❡ ❝♦♥st❛♥t K ✱ ✇❡ ❝❛♥ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛ t♦ t❤❡ s❡❝♦♥❞ st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✿ |y(t1 + h) − y2 | ≤ 12 Kh2 ,
✇❤❡r❡ y2 = L2 (t1 + h) ✐s t❤❡ s❡❝♦♥❞ st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ L2 (t) = y1 + f (t1 , y1 )(t − t1 ) ,
t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ y ❛t t1 ✳ ◆♦t✐❝❡ t❤❛t t❤❡ t✇♦ ❡rr♦r ❜♦✉♥❞s ❛♣♣❧② t♦ t✇♦ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s y ♦❢ t✇♦ ❞✐✛❡r❡♥t ■❱Ps✿
❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r ♦❢ t❤❡ t✇♦ st❡♣s ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ✭✐♥ ♦✉r ❝❛s❡✱ ✐❞❡♥t✐❝❛❧✮ ❜♦✉♥❞s✿ E(t0 + 2h) = |y(t0 + 2h) − y2 | ≤ 2 · 12 Kh2 .
❆♥❞ s♦ ♦♥✳✳✳ ❚❤❡ ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r ♦❢ t❤❡ n st❡♣s ✐s n t✐♠❡s t❤❡ ❜♦✉♥❞✿ E(t0 + nh) = |y(t0 + nh) − yn | ≤ n · 21 Kh2 .
❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✇❡ ❡①t❡♥❞ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥ ✇✐t❤ n st❡♣s t♦ t0 + H ✳ ❚❤❡♥ n=
H . h
❚❤❡r❡❢♦r❡✱ E(t0 + nh) = |y(t0 + nh) − yn | ≤
1 H 1 · Kh2 = KHh . h 2 2
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❡rr♦r ✐s ❞❡❝r❡❛s✐♥❣ ❧✐♥❡❛r❧② ✇✐t❤ h✳ ❚❤❡♦r❡♠ ✶✳✶✵✳✽✿ ❊rr♦r ❇♦✉♥❞
❙✉♣♣♦s❡ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ z = f (t, y) s❛t✐s✜❡s ∂f ∂f (t, y) + (t, y) f (t, y) ≤ K , ∂t ∂y
❢♦r ❡✈❡r② (t, y) ✐♥ t❤❡ r❡❝t❛♥❣❧❡ t0 ≤ t ≤ t0 + H, A ≤ y ≤ B ✳ ❙✉♣♣♦s❡ y ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ■❱P y ′ = f (t, y), y(t0 ) = y0 ,
✇✐t❤ t❤❡ ❞♦♠❛✐♥ [t0 , t0 + H] ❛♥❞ t❤❡ r❛♥❣❡ ✇✐t❤✐♥ [A, B]✳ ❙✉♣♣♦s❡ yn ✐s t❤❡ n st❡♣s ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❝♦✉♥t❡r♣❛rt ♦❢ t❤✐s ■❱P ✇✐t❤ h = H/n✳ ❚❤❡♥
✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄
✽✼
t❤❡✐r ❞✐✛❡r❡♥❝❡ s❛t✐s✜❡s✿ |y(t0 + nh) − yn | ≤ 12 KHh . Pr♦♦❢✳
❚❤❡ ♣r♦♦❢ ✐s ❛❜♦✈❡❀ ✇❡ ❥✉st ✉s❡ t❤❡ ❈❤❛✐♥ ❘✉❧❡ t♦ ❞❡r✐✈❡ t❤✐s✿ y ′′ (t) =
∂f ∂f (t, y(t)) + (t, y(t)) f (t, y(t)) . ∂t ∂y
❋♦r ❡❛❝❤ h✱ t❤❡ ❡rr♦r ❜♦✉♥❞ ♣r♦✈✐❞❡s ❛ t✉♥♥❡❧✱ ♦❢ ❝♦♥st❛♥t ✇✐❞t❤✱ ❛r♦✉♥❞ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✭✉♥❦♥♦✇♥✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ❖❉❊✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✿
❊①❡r❝✐s❡ ✶✳✶✵✳✾
Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ❍✐♥t✿ ②♦✉ ♥❡❡❞ ♠♦r❡ t❤❛♥ ❥✉st n ♣♦✐♥ts ❛s ✐♥ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✵
❆♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✶✿ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❝✐r❝❧❡s
■♥ ♦✉r ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✱ ✇❡ ❛❧s♦ s❛✇ t❤❛t t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥s ♦✈❡rs❤♦♦t ✇❤❡♥ t❤❡② r❡❛❝❤ t❤❡ t✲❛①✐s✳ ❚❤❡ r❡❛s♦♥ ✐s ❝♦♥❝❛✈✐t②✳ ❙✉♣♣♦s❡ y ✐s ❛ s♦❧✉t✐♦♥ ❛❜♦✈❡ t❤❡ t✲❛①✐s✱ t❤❡♥ y > 0 ❛♥❞ y ′ < 0✳ ❚❤❡♥✱ ✇❡ ❝❛♥ ✜♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ y ✿ d d y = f (t, y) = dt dt ′′
t − y
=−
1 · y − t · y′ t · y′ − y = < 0. y2 y2
■t✬s ♥❡❣❛t✐✈❡✦ ■t✬s ♣♦s✐t✐✈❡ ❜❡❧♦✇ t❤❡ t✲❛①✐s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ ❝♦♥❝❛✈❡ ❞♦✇♥ ✐♥ t❤❡ ❢♦r♠❡r ❛♥❞ ❝♦♥❝❛✈❡ ✉♣ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡✳ ❚❤❡♥✱ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s t❤❛t ✇❡ ✉s❡ ❢♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ♦✈❡r❡st✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞ ✉♥❞❡r❡st✐♠❛t❡ ✐♥ t❤❡ ❧❛tt❡r✳
❚❤❡ ❜❡♥❡✜t ✐s t❤❛t ✇❡ ❛r❡ ❝✉tt✐♥❣ t❤❡ t✉♥♥❡❧ ✐♥ ❤❛❧❢✳
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✽✽
❈♦r♦❧❧❛r② ✶✳✶✵✳✶✷✿ ❖♥❡✲s✐❞❡❞ ❊rr♦r ❇♦✉♥❞ ❯♥❞❡r t❤❡ ❝♦♥❞✐t♦♥s ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
•
❲❤❡♥
f ′ > 0✱
✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤
t
✐♥
[t0 , t0 + H]✿
yn ≤ y(t0 + nh) ≤ yn + 12 KHh . •
❲❤❡♥
f ′ < 0✱
✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤
t
✐♥
[t0 , t0 + H]✿
yn − 21 KHh ≤ y(t0 + nh) ≤ yn . ❊①❡r❝✐s❡ ✶✳✶✵✳✶✸
❆♣♣❧② t❤❡ ❝♦r♦❧❧❛r② t♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳
✶✳✶✶✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
❯♥❢♦rt✉♥❛t❡❧②✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ h ❛♥❞✱ ❢r♦♠ s✐♠♣❧❡ ❡①♣❡r✐♠❡♥t❛t✐♦♥✱ ♦♥❡ ❝❛♥♥♦t ❦♥♦✇ ✇❤❡t❤❡r h ✐s s♠❛❧❧ ❡♥♦✉❣❤✳ ■t✬s ❥✉st ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥✦ ❚❤❡ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❢❛✐❧ ♥❡❛r t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❲✐t❤ t❤❡s❡ ❧✐♠✐t❛t✐♦♥ ♦❢ t❤❡ q✉❛♥t✐t❛t✐✈❡ ♠❡t❤♦❞s✱ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♣r♦✈✐❞❡s ❢✉❧❧② ❛❝❝✉r❛t❡ ✐❢ ❜r♦❛❞ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s✳ ❚❤❡ ♠❡t❤♦❞ ❛♠♦✉♥ts t♦ ❣❛t❤❡r✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤♦✉t s♦❧✈✐♥❣ t❤❡ ❖❉❊ ✕ ❡✐t❤❡r ❛♥❛❧②t✐❝❛❧❧② ♦r ♥✉♠❡r✐❝❛❧❧②✳ ❆s ✇❡ s❛✇ ❡❛r❧② ✐♥ t❤✐s ❝❤❛♣t❡r✱ s♦♠❡ ❖❉❊s y ′ = f (y) ❤❛✈❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t ❛s ✐♥ f (y) = y 2 ✱ ❡t❝✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✜❡❧❞ ♦❢ s❧♦♣❡s ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✉♥❞❡r ❤♦r✐③♦♥t❛❧ s❤✐❢t ✇✐❧❧ ❧❛♥❞ ♦♥ ✐ts❡❧❢✿
❚❤❡♥ s♦ ✇✐❧❧ ✐ts s♦❧✉t✐♦♥ s❡t✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ✈❡r② ✐♠♣♦rt❛♥t✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✶✿ ❚✐♠❡ ■♥❞❡♣❡♥❞❡♥t ❖❉❊s
y ′ = f (y) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✳ ❚❤❡♥✱ s♦ ✐s y = y(t + s) ❢♦r ❛♥② r❡❛❧ s✳
❙✉♣♣♦s❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❛♥ ❖❉❊
y = y(t) ❊①❡r❝✐s❡ ✶✳✶✶✳✷
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❡♥
✐❢
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✽✾
❊①❛♠♣❧❡ ✶✳✶✶✳✸✿ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥
❈♦♥s✐❞❡r t❤❡ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥✿
❈♦♥s✐❞❡r ❛❧s♦ t❤❡ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧✿
◆♦t ♦♥❧② ✐t ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❤♦r✐③♦♥t❛❧ s❤✐❢ts✱ ❜✉t ❛❧s♦ ✉♥❞❡r ✈❡rt✐❝❛❧ str❡t❝❤❡s✦ ❚❤❡ ♠❛✐♥ t♦♦❧s ❝♦♠❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ❈❤❛♣t❡r ✷❉❈✲✺✮✳ ❋✐rst✱ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ y ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ ✐t✬s ♥❡❣❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ y ′ > 0 =⇒ y ր y ′ < 0 =⇒ y ց ❚❤❡♦r❡♠ ✶✳✶✶✳✹✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙♦❧✉t✐♦♥s
■❢
y
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊
❤❛✈❡ ♦♥ t❤✐s s❡t✿
y ′ = f (t, y)
✇✐t❤✐♥ ❛♥ ♦♣❡♥ s❡t ✐♥ t❤❡
f (t, y) > 0 =⇒ y f (t, y) < 0 =⇒ y
ty ✲♣❧❛♥❡✱
✇❡
✐s ✐♥❝r❡❛s✐♥❣. ✐s ❞❡❝r❡❛s✐♥❣.
❚❤❡ s❡❝♦♥❞ t♦♦❧ ✐s ❛❧s♦ ❡❧❡♠❡♥t❛r②✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✺✿ ❙t❛t✐♦♥❛r② ❙♦❧✉t✐♦♥s
■❢
f (t, y0 ) = 0
❢♦r ❛❧❧
t
✇✐t❤✐♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❛♥❞ s♦♠❡
✭❝♦♥st❛♥t✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊
y ′ = f (t, y)✳
y0 ✱
t❤❡♥
y(t) = y0
✐s ❛
❊①❛♠♣❧❡ ✶✳✶✶✳✻✿ tr✐❣♦♥♦♠❡tr✐❝
❈♦♥s✐❞❡r✿
y ′ = − tan(y) .
❋✐rst✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ t❤❡ t❛♥❣❡♥t✱ ✐s ✉♥❞❡✜♥❡❞ ❛t y = π/2 + kπ ❢♦r ❛❧❧ ✐♥t❡❣❡r k✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
❤♦r✐③♦♥t❛❧ ❧✐♥❡s ♦♥ t❤❡ ty ✲♣❧❛♥❡ ✇✐❧❧ ❝✉t t❤❡ ❞♦♠❛✐♥ ✐♥t♦ str✐♣s s♦ t❤❛t ❡✈❡r② s♦❧✉t✐♦♥ ✇✐❧❧ st❛② ✐♥s✐❞❡ ♦♥❡ ♦❢ t❤❡♠✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s✐❣♥ ♦❢ t❤❡ t❛♥❣❡♥t ❝❤❛♥❣❡s ❛t t❤♦s❡ ✈❛❧✉❡s ♦❢ y ❛s ✇❡❧❧ ❛s ❛t y = kπ ✱ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ❡❛❝❤ str✐♣✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t t❤❡ str✐♣ (−π/2, π/2)✿ • ❋♦r −π/2 < y < 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y > 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ր ✳ • ❋♦r y = 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y = 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ • ❋♦r 0 < y < π/2✱ ✇❡ ❤❛✈❡ y ′ = − tan y < 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ց ✳
❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ❛♥❛❧②s✐s ❛r❡ s✉♠♠❛r✐③❡❞ ❜❡❧♦✇ ✇✐t❤ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥ ♠❛t❝❤❡❞ ✇✐t❤ t❤❡ s✐❣♥ ♦❢ t❤❡ t❛♥❣❡♥t✿
❙♦✱ ❡✈❡r② s♦❧✉t✐♦♥ y ✐s ❞❡❝r❡❛s✐♥❣ ✭♦r ✐♥❝r❡❛s✐♥❣ r❡s♣❡❝t✐✈❡❧②✮ t❤r♦✉❣❤♦✉t ✐ts ❞♦♠❛✐♥✳ ❚❤❡r❡ ❛r❡ ♠♦r❡ s✉❜t❧❡ ❝♦♥❝❧✉s✐♦♥s✳ ❆❝❝♦r❞✐♥❣ t♦ ❛ t❤❡♦r❡♠✱ t❤❡ ❡①✐st❡♥❝❡ ✐s s❛t✐s✜❡❞ ✇✐t❤✐♥ t❤❡ str✐♣✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r✳ ❆❝❝♦r❞✐♥❣ t♦ ❛♥♦t❤❡r t❤❡♦r❡♠✱ t❤❡ ✉♥✐q✉❡♥❡ss ✐s s❛t✐s✜❡❞ ✇✐t❤✐♥ t❤❡ str✐♣✳ ❚❤❡r❡❢♦r❡✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥s ❝❛♥✬t ❝r♦ss t❤❡ ❧✐♥❡ y = 0✳ ❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❊✈❡r② s♦❧✉t✐♦♥ ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ♠✐❞❞❧❡ ❧✐♥❡ ♦❢ t❤❡ str✐♣✱ y = 0✱ ❛s t → +∞✱ ❛♥❞ t❤✐s ❧✐♥❡ ✐s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✱ ✐✳❡✳✱ y → 0+ ✳
❇✉t ❤♦✇ ❞♦❡s ✐t ❛♣♣r♦❛❝❤ t❤❡ ♥♦✲s♦❧✉t✐♦♥ ❧✐♥❡s y = π/2 + kπ ❄ ❆s t❤❡ t❛♥❣❡♥t ✐s ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t②✱ s♦ ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❊✈❡r② s♦❧✉t✐♦♥ ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ❡❞❣❡ ❧✐♥❡s ♦❢ t❤❡ str✐♣✱ y = ±π/2✱ ❛♥❞ ✐t ❜❡❝♦♠❡s ♠♦r❡ ❛♥❞ ♠♦r❡ ✈❡rt✐❝❛❧✱ ✐✳❡✳✱ y ′ → ±∞✳
❚❤✐s ✐s ✇❤❛t t❤❡ s♦❧✉t✐♦♥ s❡t ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ ✇❤❡♥ ❞r❛✇♥ ❜② ❤❛♥❞✿
◆♦✇ ✇❡ ✉t✐❧✐③❡ s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ f ✳ ✶✳ ❲❡ ✉s❡ t❤❛t ❢❛❝t t❤❛t f ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t t♦ ♣r♦❞✉❝❡ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✮ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢ts✳ ✷✳ ❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t f ✐s ♦❞❞ t♦ ♣r♦❞✉❝❡ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ❛ ✈❡rt✐❝❛❧ ✢✐♣✳ ✸✳ ❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t f ✐s ♣❡r✐♦❞✐❝ t♦ ♣r♦❞✉❝❡ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ✈❡rt✐❝❛❧ s❤✐❢ts✳ ❚❤✐s ✐s t❤❡ s♦❧✉t✐♦♥ s❡t✿
✾✵
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✾✶
❚❤❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿
❲❡ ♥♦t✐❝❡ ♥♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❛r❡ r✉♥♥✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❡❞❣❡ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t♦✇❛r❞ t❤❡ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❛r❡ r❡♣❡❛t❡❞ ✉♥❞❡r ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t✳
❲❛r♥✐♥❣✦
❲❡ ❝♦♥✜r♠ ♦✉r q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✳✳✳ ❛♥❞ ✈✐❝❡ ✈❡rs❛✦ ❊①❡r❝✐s❡ ✶✳✶✶✳✼ ■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✐s y = π/2 ❛❧s♦ ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡❄ ❈♦♥✜r♠ ②♦✉r ❝♦♥❝❧✉s✐♦♥ ✇✐t❤ ❊✉❧❡r✬s y ′ ♥❡❛r t❤✐s ❧✐♥❡❄
♠❡t❤♦❞✳ ❲❤❛t ✐s
◆♦t✐❝❡ t❤❛t ❜♦t❤ ❛❜♦✈❡ ❛♥❞ ❜❡❧♦✇ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ❢♦r ✈❛r✐♦✉s ✈❛❧✉❡s ♦❢
t0 = 0, 1, 2, 3...
✐♥ ♦r❞❡r t♦
❛✈♦✐❞ ❡♠♣t② s♣❛❝❡s ❝r❡❛t❡❞ ❜② t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s✳
❊①❛♠♣❧❡ ✶✳✶✶✳✽✿ ♠♦r❡ ❝♦♠♣❧❡① ❈♦♥s✐❞❡r ♥❡①t✿
y ′ = cos y ·
p
❲❡ st❛rt ✇✐t❤ t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱
|y| . f (y) = cos y ·
p |y|✱
✇❤✐❝❤ ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞
✶✳✶✶✳
✾✷
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
❜② cos y ✉♥❧❡ss y = 0✿ y f (y) y = y(t) 3π/2 0 y(t) = 3π/2, ❝♦♥st❛♥t − yց 0 y(t) = π/2, ❝♦♥st❛♥t π/2 + yր 0 y(t) = 0, ❝♦♥st❛♥t 0 + yր 0 y(t) = −π/2, ❝♦♥st❛♥t −π/2 − yց 0 y(t) = −3π/2, ❝♦♥st❛♥t −3π/2
→→→→ ցցցց →→→→ րրրր →→→→ րրրր →→→→ ցցցց →→→→
❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿
❚❤❡ ♣❧♦tt❡❞ ❊✉❧❡r s♦❧✉t✐♦♥s ❝r♦ss t❤❡ ♣❛t❤ ♦❢ y = 0 ✐♥❞✐❝❛t✐♥❣ ♥♦♥✲✉♥✐q✉❡♥❡ss✳ ❈♦♥s✐❞❡r✐♥❣ ♥♦♥✲ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ f (y) ❛t y = 0✱ t❤✐s ✐s ❛ ♣♦ss✐❜✐❧✐t②✳ ❊❧s❡✇❤❡r❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❧❜❧❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❤♦❧❞s✳ ❚❤❡ ❜❡❤❛✈✐♦r ✐s ❛s②♠♣t♦t✐❝ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❊①❛♠♣❧❡ ✶✳✶✶✳✾✿ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s
❚❤❡ ♥❡①t ♦♥❡ ✐s ❞❡♣❡♥❞❡♥t ♦♥ t✿ y′ =
1 . t−y
❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ✉♥❞❡✜♥❡❞ ✇❤❡♥❡✈❡r t − y = 0✳ ❚❤❡r❡❢♦r❡✱ ♥♦ s♦❧✉t✐♦♥ ❝r♦ss❡s t❤❡ ❞✐❛❣♦♥❛❧ ❧✐♥❡ y = t✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❛❜♦✈❡ ✐t ❛r❡ ❞❡❝r❡❛s✐♥❣ ❛♥❞ t❤❡ ♦♥❡s ❜❡❧♦✇ ✐t ❛r❡ ✐♥❝r❡❛s✐♥❣ ❛s ❢♦❧❧♦✇s✿
❈❤❡❝❦✿
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✾✸
❊①❛♠♣❧❡ ✶✳✶✶✳✶✵✿ tr✐❣ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s
❚❤❡ ♥❡①t ♦♥❡ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿ y ′ = cos(t + y) .
❚❤❡ s✐❣♥ ✐s ❝❤❛♥❣✐♥❣ ✇❤❡♥❡✈❡r t + y = π/2 + kπ ❢♦r ❛❧❧ ✐♥t❡❣❡r k ✳ ❚❤✉s t❤❡ s♦❧✉t✐♦♥ ❝❤❛♥❣❡ t❤❡✐r ♠♦♥♦t♦♥✐❝✐t② ✇❤❡♥❡✈❡r t❤❡② r❡❛❝❤ ♦♥❡ ♦❢ t❤❡ ❧✐♥❡s y = −t + π/2 + kπ
♦❢ s❧♦♣❡ −1✳ ❖♥❡ ♦❢ t❤❡s❡ ❧✐♥❡s ❤♦✇❡✈❡r ✐s s♣❡❝✐❛❧ ❛♥❞ ✇❡ ✇♦✉❧❞♥✬t t❤✐♥❦ ♦❢ ✐t ✇✐t❤♦✉t ❞♦✐♥❣ ❊✉❧❡r✬s ♠❡t❤♦❞ ✜rst✿
■♥❞❡❡❞✱ y = π − t ✐s ❛ s♦❧✉t✐♦♥✦ ❚❤❡ r❡st ♦❢ t❤❡♠ ❛r❡ ✇❛✈❡s ♠♦✈✐♥❣ ❞✐❛❣♦♥❛❧❧②✳ ❊①❛♠♣❧❡ ✶✳✶✶✳✶✶✿ ❞✐s❝♦♥t✐♥✉♦✉s ❘❍❙
❚❤❡ ♥❡①t ♦♥❡ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✿
y ′ = [t + y] .
❙✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ❝❤❛♥❣❡s ❛❜r✉♣t❧②✱ s♦ ❞♦❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ✭t❤❡ s❧♦♣❡✮ ♦❢ ❡❛❝❤ s♦❧✉t✐♦♥✳ ❋♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ✇❡ ✉s❡ t❤❡ ❋▲❖❖❘ ❢✉♥❝t✐♦♥✿
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✾✹
❚❤❡ ❝♦r♥❡rs ✐♥❞✐❝❛t❡ t❤❛t t❤❡s❡ ❛r❡ ✇❡❛❦ s♦❧✉t✐♦♥s✳
❙✉♣♣♦s❡ t❤❡ ❖❉❊ ✐s
t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✱
y ′ = f (y) .
❚❤❡♥ ✐t ✐s t❤♦✉❣❤t ♦❢ ❛ ❧✐q✉✐❞ ✢♦✇✿ ✐♥ ❛ ❝❛♥❛❧ ♦r ❛ ♣✐♣❡✳
❊s♣❡❝✐❛❧❧② ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛
♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞✳
❲✐t❤ s✉❝❤ ❛♥ ❡q✉❛t✐♦♥✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✐s ♠✉❝❤ s✐♠♣❧❡r✳ ■♥ ❢❛❝t✱ ♦♥❡ ♦❢ t❤❡ ❖❉❊s ❛❜♦✈❡✱
p cos y · |y|,
❡①❤✐❜✐ts ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ♦❢
❧♦❝❛❧
f (y) =
❜❡❤❛✈✐♦r✳
❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❣✐✈❡♥ ❧♦❝❛t✐♦♥
y = a✳
f (a) 6= 0✳ ❚❤❡♥✱ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t f (y) > 0 ♦r f (y) < 0 ❝♦♥t❛✐♥s a✳ ❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥s ❧♦❝❛t❡❞ ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × I ❛r❡ ❡✐t❤❡r
❚❤❡ ✜rst ♠❛✐♥ ♣♦ss✐❜✐❧✐t② ✐s ✐♥ s♦♠❡ ✐♥t❡r✈❛❧
I
t❤❛t
❛❧❧ ✐♥❝r❡❛s✐♥❣ ♦r ❛❧❧ ❞❡❝r❡❛s✐♥❣ r❡s♣❡❝t✐✈❡❧②✿
❚❤❡ ❜❡❤❛✈✐♦r ✐s ✏❣❡♥❡r✐❝✑✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❜❡❤❛✈✐♦rs ❛r❡ s❡❡♥ ❛r♦✉♥❞ ❛ ③❡r♦ ♦❢
◮ f (a) = 0 =⇒ y = a
f✿
✐s ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ✭❛♥ ❡q✉✐❧✐❜r✐✉♠✮✳
❚❤❡♥ t❤❡ ♣❛tt❡r♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✱ ✐✳❡✳✱ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦❢
f✱
I ✱ ❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r t❤✐s ✐s ❛ ♠❛①✐♠✉♠
❛ ♠✐♥✐♠✉♠✱ ♦r ♥❡✐t❤❡r✳
❋✐rst✱ ✐❢
f
❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♣♦s✐t✐✈❡ t♦ ♥❡❣❛t✐✈❡✱ ✇❡ ❤❛✈❡ ❛
st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏s✐♥❦✑✿
✶✳✶✶✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s
✾✺
■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❛❧❧ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❜❛♥❞
(−∞, +∞)×I ✱ ✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ ❛s②♠♣t♦t✐❝❛❧❧②
y = a✿
❛♣♣r♦❛❝❤
y→a ❙❡❝♦♥❞✱ ✐❢
f
❛s
❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♥❡❣❛t✐✈❡ t♦ ♣♦s✐t✐✈❡✱ ✇❡ ❤❛✈❡ ❛♥
■t ♠❡❛♥s t❤❛t t❤❡r❡ ❛r❡ ♥♦ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤
y=a
❛s
x → +∞✳
f
❞♦❡s ♥♦t ❝❤❛♥❣❡ ✐ts s✐❣♥ ❛t
y = a✱
✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏s♦✉r❝❡✑✿
(−∞, +∞) × I ✱
✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ t❤❛t
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❤❛✈❡
y→a ❚❤✐r❞✱ ✐❢
x → +∞ .
❛s
x → −∞ .
✇❡ ❤❛✈❡ ❛♥
s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏♣❛ss✑✿
■t ♠❡❛♥s t❤❛t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ❧✐♥❡
y=a
✇✐t❤✐♥ t❤❡ ❜❛♥❞
(−∞, +∞) × I ✱
✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤ t❤✐s ❧✐♥❡ ❛♥❞ t❤❡ ♦♥❡s ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ❞♦ ♥♦t✳ ❚❤❡ ❧❛st ♣♦ss✐❜✐❧✐t② ✐s
f (y) = 0
❞❡❣❡♥❡r❛t❡ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤✐s
❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❡q✉✐❧✐❜r✐❛
♦♥ t❤❡ ✇❤♦❧❡
I✳
❚❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛r❡ st❛t✐♦♥❛r②✳ ❚❤✐s ✐s ❛
✐s ❡❛s② t♦ ✉♥❞❡rst❛♥❞ ✐♥ t❡r♠s ♦❢ t❤❡ ✢♦✇ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❖❉❊✿
•
s✐♥❦✿ ✢♦✇ ✐♥ ♦♥❧② ✭st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮❀
•
s♦✉r❝❡✿ ✢♦✇ ♦✉t ♦♥❧② ✭✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮❀
•
♣❛ss✿ ✢♦✇✱ st♦♣✱ ✢♦✇ ✭s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮✳
✶✳✶✷✳
▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s
✾✻ ❲❛r♥✐♥❣✦ ■♥ s♣✐t❡ ♦❢ t❤❡ ♥❛♠❡s✱ ❛ ♣❛rt✐❝❧❡ ♣❧❛❝❡❞ ❛t ❛ s♦✉r❝❡ ✇✐❧❧ ♥❡✈❡r ❧❡❛✈❡ ✐t ❛♥❞ ❛ ♣❛rt✐❝❧❡ ♣❧❛❝❡❞ ♥❡❛r ❛ s✐♥❦ ✇✐❧❧ ♥❡✈❡r r❡❛❝❤ ✐t✱ ✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✳
❚❤❡r❡ ❛r❡ ♦♥❧② t❤❡s❡ t❤r❡❡ ♠❛✐♥ ♣♦ss✐❜✐❧✐t✐❡s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❲❡ ✇✐❧❧ s❡❡ ✐♥ ❈❤❛♣t❡r ✸ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ❛r♦✉♥❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡♥ t❤❡ s♣❛❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✐s 2✲❞✐♠❡♥s✐♦♥❛❧✱ ❛ ♣❧❛♥❡✳
✶✳✶✷✳ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s
❊✉❧❡r✬s ♠❡t❤♦❞ ❛♣♣r♦①✐♠❛t❡s s♦❧✉t✐♦♥s ♦❢ ❛♥ ❖❉❊ ❜② ❧✐♥❡❛r✐③✐♥❣ t❤❡♠✱ ♦♥❡ ❧♦❝❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❲❡ ❝❛♥ ❛❧s♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊s ❜② ❧✐♥❡❛r✐③✐♥❣✱ ❧♦❝❛❧❧②✱ ✐ts r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❆s ✇❡ ❛r❡ ♦❢t❡♥ ✉♥❛❜❧❡ t♦ s♦❧✈❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r✱ ❧❡t✬s tr② t♦ r❡♣❧❛❝❡ ✐t ✇✐t❤ ❛ s✐♠♣❧❡r ♦♥❡ t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞✳ ❍♦✇ ❛❝❝✉r❛t❡ ❞♦ ✇❡ ♥❡❡❞ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❜❡❄ ❆t t❤❡ ✈❡r② ❧❡❛st✱ t❤❡ ♥❡✇ ❖❉❊ s❤♦✉❧❞ ❤❛✈❡ t❤❡ s❛♠❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✕ ✐♥❝r❡❛s✐♥❣✱ ❞❡❝r❡❛s✐♥❣✱ ❛♥❞ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ✕ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❝❤♦s❡♥ ♣♦✐♥t✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❖❉❊ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s r✐❣❤t ❤❛♥❞✲s✐❞❡✱
y ′ = f (y) , ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❝❤♦s❡♥ ♣♦✐♥t y = a✳ ❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ r❡s✉❧ts st❛t❡❞ ♣r❡✈✐♦✉s❧②✳ ❇❡❢♦r❡ ✇❡ ❣❡t t♦ t❤❡ ❧✐♥❡❛r✱ ❧❡t✬s tr② t❤❡
❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳
❲❤❛t ✐s t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❛t x = a ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ y = f (x)❄ ■t✬s y = f (a)✦ ❊①❛♠♣❧❡ ✶✳✶✷✳✶✿ s✐♠♣❧❡st ❛♣♣r♦①✐♠❛t✐♦♥
❙♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧②✿
y ′ = y 2 ❛r♦✉♥❞ y = 1 .
❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿
y 2
= 1. y=1
❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ g(y) = 1✳ ❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱ ❖❉❊✿ y′ = 1 . ■ts s♦❧✉t✐♦♥ ✐s✿
y =x+C. ■t✬s ❛ ♣♦♦r s✉❜st✐t✉t❡✳ ❍♦✇❡✈❡r✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡✦ ❍♦✇❡✈❡r✱ ❧❡t✬s s♦❧✈❡ t❤❡ s❛♠❡ ❖❉❊ ❛r♦✉♥❞ ❛♥♦t❤❡r ♣♦✐♥t✿
y ′ = y 2 ❛r♦✉♥❞ y = 0 . ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿
y 2
= 0. y=0
❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ g(y) = 0✳ ❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱ ❖❉❊✿ y′ = 0 .
✶✳✶✷✳
▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s
✾✼
■ts s♦❧✉t✐♦♥ ✐s ❝♦♥st❛♥t✿ y =C.
■t✬s ❛♥ ✉♥❛❝❝❡♣t❛❜❧② ❜❛❞ s✉❜st✐t✉t❡ ❜❡❝❛✉s❡ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ ❞✐✛❡r❡♥t✦ ■♥ ❣❡♥❡r❛❧✱ ❢♦r ❛ ❝❤♦s❡♥ ♣♦✐♥t y = a✱ ✇❡ r❡♣❧❛❝❡ z = f (y) ✇✐t❤ z = f (a)✿ ◮ ■♥st❡❛❞ ♦❢ y ′ = f (y)✱ ✇❡ s♦❧✈❡ y ′ = f (a)✳
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✈❛❧✉❡ f (a) ♦❢ f ❛t a ✐s ✉s❡❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ♥❡✇ ❖❉❊✿ y ′ = f (a) ,
❢♦r ❛❧❧ y ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ I t❤❛t ❝♦♥t❛✐♥s a✳ ■ts s♦❧✉t✐♦♥s ❛r❡ t❤❡s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ t✿ y = f (a)(t + C) ,
❢♦r ❡❛❝❤ r❡❛❧ C ✳ ❉♦❡s t❤❡✐r ❜❡❤❛✈✐♦r ♠❛t❝❤ t❤❡ ♦r✐❣✐♥❛❧❄ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❝❛s❡s✿ ✶✳ f (a) 6= 0
✷✳ f (a) = 0
❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ s✐♠♣❧✐✜❡❞ ❖❉❊✳ ■♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇❡ ❤❛✈❡✿ f (a) > 0 =⇒ y ր ❛♥❞ f (a) < 0 =⇒ y ց ,
❢♦r ❛❧❧ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × I ✳ ❚❤❡② ❛r❡ ❝✉r✈❡❞ ♦♥ t❤❡ ❧❡❢t ❛♥❞ str❛✐❣❤t ♦♥ t❤❡ r✐❣❤t✿
■♥❞❡❡❞✱ t♦ ❝♦♠♣❛r❡ t♦ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊✱ ✇❡ ❝❛♥ ❥✉st ❛♣♣❧② t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❢r♦♠ ❝❛❧❝✉❧✉s✳ ■t✬s ❛ ♠❛t❝❤✦ ❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ❝❛s❡✳ ❚❤❡♦r❡♠ ✶✳✶✷✳✷✿ ❈♦♥st❛♥t ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s
❙✉♣♣♦s❡ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y = a✳ ❙✉♣♣♦s❡ f (a) 6= 0 .
❚❤❡♥ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊ y ′ = f (y) ♠❛t❝❤ t❤♦s❡ ♦❢ ✐ts ❝♦♥st❛♥t s✉❜st✐t✉t❡ y ′ = f (a)✱ ❛s ❢♦❧❧♦✇s✿ t❤❡r❡ ✐s s✉❝❤ ❛♥ ε > 0 t❤❛t ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ y ′ = f (y) t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × (a − ε, a + ε) s❛t✐s✜❡s✿ f (a) > 0 =⇒ y ✐s ✐♥❝r❡❛s✐♥❣. f (a) < 0 =⇒ y ✐s ❞❡❝r❡❛s✐♥❣. Pr♦♦❢✳
❚❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ε > 0 s♠❛❧❧ ❡♥♦✉❣❤ t❤❛t ❢♦r ❡❛❝❤ y ✇✐t❤✐♥ (a − ε, a + ε)✱ ✇❡
✶✳✶✷✳
▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s
✾✽
❤❛✈❡
f (a) > 0 =⇒ f (y) > 0 f (a) < 0 =⇒ f (y) < 0
■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ ✇❡ ❤❛✈❡✿
f (a) = 0 =⇒ y
.
✐s ❝♦♥st❛♥t
❚❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛r❡ st❛t✐♦♥❛r②✳ ❇✉t t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊ ♠✐❣❤t ❤❛✈❡ ❛ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ❛ ♠✐s♠❛t❝❤✦ ❈♦♥❝❧✉s✐♦♥✿
◮
❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢
❚♦ ✐♠♣r♦✈❡ ♦✉r ❝❤❛♥❝❡s✱ ✇❡ tr② ✐s t❤❡
y = f (y)
❛t
x=a
❢❛✐❧s ✇❤❡♥
❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥
♦❢
f (a) = 0✳
f✳
❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❝❛s❡✱ ♦❢ ❝♦✉rs❡✱ t❤❛t ❤❛s ♥♦t ❜❡❡♥✱ q✉❛❧✐t❛t✐✈❡❧②✱ s♦❧✈❡❞ ❜② t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✲ ✐♠❛t✐♦♥✱ ✐✳❡✳✱
f (a) = 0 . ❊①❛♠♣❧❡ ✶✳✶✷✳✸✿ ❛r♦✉♥❞ st❛t✐♦♥❛r② ♣♦✐♥t ▲❡t✬s s♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧②✿
y′ = y2 + y
❛r♦✉♥❞
y = 1.
❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿
y + y 2
= 2. y=1
g(y) = 2✳
❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥
❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱
❖❉❊✿
y′ = 2 . ■ts s♦❧✉t✐♦♥ ✐s✿
y = 2x + C . ❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦
▲❡t✬s s♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧② ❡❧s❡✇❤❡r❡✿
y′ = y2 + y
❛r♦✉♥❞
y = 0.
❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿
y + y 2
= 0. y=0
❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❖❉❊✿
y′ = 0 . ■ts s♦❧✉t✐♦♥ ✐s✿
y =C.
g(y) = 0✳
❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱
✶✳✶✷✳
▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s
✾✾
❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s ♥♦t t❤❡ s❛♠❡✳ ❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦
◆♦✇ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳ ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡✿
(y + y) 2
′
y=0
= 2y + 1
❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❧✐♥❡❛r ❢✉♥❝t✐♦♥
= 1. y=0
g(y) = y ✳
❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱
❖❉❊✿
y′ = y . ■ts s♦❧✉t✐♦♥ ✐s✿
y = Cex . ❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡✳ ❚❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦
◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ♣r♦❝❡ss ♦❢ ❧✐♥❡❛r✐③❛t✐♦♥ st❛rts ✇✐t❤ ♠♦✈✐♥❣ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ ❝✉rr❡♥t❧②
y = a✱
t♦
0✳
❙✉❝❤ ❛ s✉❜st✐t✉t✐♦♥ ✐s ❞❡♠♦♥str❛t❡❞ ❡❛r❧✐❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳ ❚❤✐s ✐s t❤❡ ♥❡✇ ❖❉❊✿
y ′ = f (y)
✇✐t❤
f (0) = 0 .
❚❤❡ ❧✐♥❡❛r✐③❡❞ ❖❉❊ ✐s
y ′ = f ′ (0)y , ❢♦r ❛❧❧
y
✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧
I
t❤❛t ❝♦♥t❛✐♥s
0✳
■ts s♦❧✉t✐♦♥s ❛r❡ t❤❡s❡
❡①♣♦♥❡♥t✐❛❧
❢✉♥❝t✐♦♥s ♦❢
t✿
′
y = Cef (0)t , ❢♦r ❡❛❝❤ r❡❛❧
C✳
❉♦❡s t❤❡✐r ❜❡❤❛✈✐♦r ♠❛t❝❤ t❤❡ ♦r✐❣✐♥❛❧❄ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❝❛s❡s✿ ✶✳ ✷✳
f ′ (0) 6= 0 f ′ (0) = 0
❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t
❛❧❧
s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r✐③❡❞ ❖❉❊✳ ❋✐rst ✇❡ ❤❛✈❡✿
f ′ (0) < 0 =⇒ f ց =⇒ f (y) > 0 ❢♦r ❛❧❧ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛♥ ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✿
(−∞, +∞) × I ✳
❚❤❡♥✱
f
❢♦r
y 0,
❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♣♦s✐t✐✈❡ t♦ ♥❡❣❛t✐✈❡ ❛♥❞ ✇❡ ❤❛✈❡
❯♥❧✐❦❡ t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❤❛s ♣r♦❞✉❝❡❞ ❛ ♠❛t❝❤✦ ❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡ ✇✐t❤✐♥ t❤❡ ❜❛♥❞✿
f ′ (0) > 0 =⇒ f ր =⇒ f (y) < 0 ❚❤❡♥✱
f
❢♦r
y 0
❢♦r
y > 0.
❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♥❡❣❛t✐✈❡ t♦ ♣♦s✐t✐✈❡ ❛♥❞ ✇❡ ❤❛✈❡ ❛ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✳ ❖♥❝❡ ❛❣❛✐♥✱ ❛ ♠❛t❝❤✦
❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ❝❛s❡✳
✶✳✶✸✳
✶✵✵
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
❚❤❡♦r❡♠ ✶✳✶✷✳✹✿ ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s
❙✉♣♣♦s❡ z = f (y) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t y = 0✳ ❙✉♣♣♦s❡ f (0) = 0 ❛♥❞ f ′ (0) 6= 0 .
❚❤❡♥ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊ y ′ = f (y) ♠❛t❝❤ t❤♦s❡ ♦❢ ✐ts ❧✐♥❡❛r s✉❜st✐t✉t❡ y ′ = f ′ (0)y ✱ ❛s ❢♦❧❧♦✇s✿ t❤❡r❡ ✐s s✉❝❤ ❛♥ ε > 0 t❤❛t ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ y ′ = f (y) t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞)×(ε, ε) s❛t✐s✜❡s✿ f ′ (0) > 0 =⇒ y ✐s ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ y < 0 ❛♥❞ y ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ y > 0 . f ′ (0) < 0 =⇒ y ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ y < 0 ❛♥❞ y ✐s ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ y > 0 . ❊①❡r❝✐s❡ ✶✳✶✷✳✺
▲✐♥❡❛r✐③❡ t❤❡ ❖❉❊ y ′ = yey ❛t y = 0✳ ❲❤❛t ❛❜♦✉t t❤❡ ❝❛s❡ f ′ (0) = 0❄ ❲❡ ❛r❡ ✐♥ ❛ s✐♠✐❧❛r ♣❧❛❝❡ t♦ t❤❡ ♦♥❡ ❢♦r t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❍❡r❡✱ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❣✐✈❡s ✉s t❤❡ ❖❉❊ y ′ = 0 ✇✐t❤ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ♦♥❧②✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ f ♠✐❣❤t ❤❛✈❡ ❛ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✿
❆ ♠✐s♠❛t❝❤✦ ❈♦♥❝❧✉s✐♦♥✿ ◮ ❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ y = f (y) ❛t x = 0 ❢❛✐❧s ✇❤❡♥ f (0) = 0 ❛♥❞ f ′ (0) = 0✳
❚❤❡ ❛♥s✇❡r ✐s t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ s❡❝♦♥❞ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ T2 ♦❢ f ❛r♦✉♥❞ y = 0✱ ♣r♦✈✐❞❡❞ f ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ♠❛② st✐❧❧ ❜❡ ❡①❝❡♣t✐♦♥s t❤❛t ✇✐❧❧ ❝❛❧❧ ❢♦r ✉s✐♥❣ t❤❡ ❝✉❜✐❝ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ T3 ♦❢ f ✳ ❆♥❞ s♦ ♦♥✳ ❲❡ ✇✐❧❧ ♥❡❡❞ ❛❧❧ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✱ ✐✳❡✳✱ t❤❡ ❚❛②❧♦r s❡r✐❡s✳ ❚❤❡ ✐❞❡❛ ✐s ❞❡✈❡❧♦♣❡❞ ✐♥ ❈❤❛♣t❡r ✷✳
✶✳✶✸✳ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
❊①❛♠♣❧❡ ✶✳✶✸✳✶✿ ♠♦✈✐♥❣ ❜❛❧❧
❲❡ ❦♥♦✇ t❤❛t ❛ ❜❛❧❧
r♦❧❧✐♥❣
♦♥ ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✇✐❧❧ ❤❛✈❡ ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t②✿
✶✳✶✸✳
✶✵✶
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
❲❤❛t ✐❢ t❤❡ ❜❛❧❧ ✐s ♥♦✇ t❤r♦✇♥ ✉♣ ✐♥ t❤❡ ❛✐r ❄ ❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t✳ ■♥ t❤❡ ❢♦r♠❡r✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ■♥ t❤❡ ❧❛tt❡r✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ■♠❛❣✐♥❡ t❤❛t ✇❡ ❤❛✈❡ t❤✐s ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ♦❢ t❤❡ ❤❡✐❣❤ts ♦❢ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ❢❛❧❧✐♥❣ ❞♦✇♥ r❡❝♦r❞❡❞ ❛❜♦✉t ❡✈❡r② .1 s❡❝♦♥❞ ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s✿
❲❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ♣❧♦t t❤❡ ✐✳❡✳✱ t❤❡ ✈❡❧♦❝✐t②✱ vn ✭❣r❡❡♥✮✿
❧♦❝❛t✐♦♥
s❡q✉❡♥❝❡✱ pn ✭r❡❞✮✳ ❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ pn ✱
■t ❧♦♦❦s ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❇✉t t❤✐s t✐♠❡✱ ✇❡ t❛❦❡ ♦♥❡ ♠♦r❡ st❡♣✿ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② s❡q✉❡♥❝❡✳ ■t ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ an ✭❜❧✉❡✮✳ ■t ❛♣♣❡❛rs ❝♦♥st❛♥t✦ ❚❤❡r❡ ♠✐❣❤t ❜❡ ❛ ❧❛✇ ♦❢ ♥❛t✉r❡ ❤❡r❡✳ ▲❡t✬s ❛❝❝❡♣t t❤❡ ♣r❡♠✐s❡ ✇❡✬✈❡ ♣✉t ❢♦r✇❛r❞✿
◮
✳
❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❢r❡❡ ❢❛❧❧ ✐s ❝♦♥st❛♥t
❚❤❡♥ ✇❡ ❝❛♥ tr② t♦ ♣r❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛♥ ♦❜❥❡❝t t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ✕ ❢r♦♠ ❛♥② ✐♥✐t✐❛❧ ❤❡✐❣❤t ❛♥❞ ✇✐t❤ ❛♥② ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦✉r ❝♦♠♣✉t❛t✐♦♥ ✐s ♦♣♣♦s✐t❡ t♦ t❤❛t ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✿
✶✳✶✸✳
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ◮
✶✵✷
❲❡ ✉s❡ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ t♦ ❞❡r✐✈❡ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ ❞❡r✐✈❡ t❤❡ ♣♦s✐t✐♦♥
♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t✐♠❡✳ ❲❤✐❧❡ ✇❡ ✉s❡❞
❞✐✛❡r❡♥❝❡s
✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ✉s❡
s✉♠s
✭❈❤❛♣t❡r ✶P❈✲✶✮ ♥♦✇✳
❲❡ ♣❧♦t t❤❡s❡ ♣♦s✐t✐♦♥s ❛❣❛✐♥st t✐♠❡✿
❚❤❡ ❣r❛♣❤ ✏❧♦♦❦s ❧✐❦❡✑ ❛ ♣❛r❛❜♦❧❛✿
❲❡ ✉s❡ t❤❡s❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r♠✉❧❛s t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿
vn = ✇❤❡r❡
h
∆p pn+1 − pn = ∆t h
❛♥❞
an =
∆v vn+1 − vn = , ∆t h
✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦♥
t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳ ❊①❛♠♣❧❡ ✶✳✶✸✳✷✿ s♣r❡❛❞s❤❡❡t
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✐s ✉s❡❞ ❛❣❛✐♥✿
❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴❘✷❈✶ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿
✶✳✶✸✳
✶✵✸
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
◆♦✇ ✐♥ r❡✈❡rs❡✦ ❋♦r s✐♠✉❧❛t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ❣♦❡s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿ • t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡♥ • t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳
❚❤❡ ❢♦r♠✉❧❛s ❛r❡ s♦❧✈❡❞ ❢♦r pn+1 ❛♥❞ vn+1 r❡s♣❡❝t✐✈❡❧②✿ pn+1 − pn =⇒ vn+1 = vn + han h vn+1 − vn an = =⇒ pn+1 = pn + hvn h vn =
❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦♥ t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳ ❙✉♠♠❛r②✿ ♣♦s✐t✐♦♥ ✈❡❧♦❝✐t② ❛❝❝❡❧❡r❛t✐♦♥
✈❡rt✐❝❛❧ pn ❣✐✈❡♥
pn+1 − pn h vn+1 − vn an = h vn =
=⇒
❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥
✈❡rt✐❝❛❧ an ❣✐✈❡♥ vn+1 = vn + han pn+1 = pn + hvn
❊①❛♠♣❧❡ ✶✳✶✸✳✸✿ ♠♦✈✐♥❣ ❜❛❧❧
▲❡t✬s ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ♣r♦❜❧❡♠✳ ◮ P❘❖❇▲❊▼✿ ❋r♦♠ ❛ 100 ❢❡❡t ❜✉✐❧❞✐♥❣✱ ❛ ❜❛❧❧ ✐s t❤r♦✇♥ ✉♣ ❛t 50 ❢❡❡t ♣❡r s❡❝♦♥❞ s♦ t❤❛t ✐t ❢❛❧❧s ♦♥ t❤❡ ❣r♦✉♥❞✳ ❍♦✇ ❤✐❣❤ ✇✐❧❧ t❤❡ ❜❛❧❧ ❣♦❄ ❲❡ ✉s❡ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥✿ ❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✲✶❪✯❘✷❈✶
◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡r❡ ✐s ❥✉st ♦♥❡ ❢♦r❝❡✱ t❤❡ ❣r❛✈✐t②✱ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❦♥♦✇♥ t♦ ❜❡ a = −g ✱ ✇❤❡r❡ g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✿ g = 32 ❢t/s❡❝2 .
◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ p0 = 100✳ • ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ✐s ❣✐✈❡♥ ❜②✿ v0 = 50✳ ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❡✈❡r② h = .20 s❡❝♦♥❞✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡✿
✶✳✶✸✳
✶✵✹
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
❲✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ❛s❦ ❛♥❞ ❛♥s✇❡r ❛ ✈❛r✐❡t② ♦❢ ♦t❤❡r q✉❡st✐♦♥s ❛❜♦✉t s✉❝❤ ♠♦t✐♦♥ ✭❤♦✇ ❤❛r❞ ✐t ❤✐ts t❤❡ ❣r♦✉♥❞✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦♥❧② ❞♦ ♦♥❡ ❡①❛♠♣❧❡ ❛t ❛ t✐♠❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞r❛✇ ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ t♦ t❤❡s❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡
❝♦♥t✐♥✉♦✉s
❝❛s❡✱ ✐✳❡✳✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✈❡r②t❤✐♥❣ ❛❜♦✈❡✿ h = ∆x → 0 .
■♥st❡❛❞ ♦❢ s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤✐s t✐♠❡ t❤❡s❡ ❢✉♥❝t✐♦♥s
♦❢ t✐♠❡
✿
• p ✐s t❤❡ ❤❡✐❣❤t✱ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✳ • v = p′ ✐s t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②✳
• a = v ′ ✐s t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✳
◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✿
a = −g .
❲❡ ❦♥♦✇ t❤❛t✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✳ ❈♦♥✈❡rs❡❧②✿ ✶✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❧✐♥❡❛r ✐s ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ✷✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ◮ p = p(t) ✐s q✉❛❞r❛t✐❝✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❲❤❛t ♠❛❦❡s t❤❡s❡ s♣❡❝✐✜❝ ❛r❡ t❤❡ ✐♥✐t✐❛❧
p(t) = ax2 + bx + c . ❝♦♥❞✐t✐♦♥s
✿
✶✳✶✸✳
✶✵✺
▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥
• p0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ p0 = p(0)✳ • v0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ v(0) = ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿
dp ✳ dt t=0
1 p(t) = p0 +v0 t − gt2 2 ❊①❛♠♣❧❡ ✶✳✶✸✳✹✿ ♠♦✈✐♥❣ ❜❛❧❧
■♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♦✉rs✱ ✇❡ ❤❛✈❡✿
p0 = 100, v0 = 50 . ❖✉r ❡q✉❛t✐♦♥s ❜❡❝♦♠❡✿
p = 100 +50t −16t2 .
■♥ ❝♦♥tr❛st t♦ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥ ✐s♥✬t r❡❝✉rs✐✈❡ ❜✉t ❞✐r❡❝t ❛♥❞ ❡①♣❧✐❝✐t✦ ❇❡❢♦r❡ ✇❡ ✉t✐❧✐③❡ t❤❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r❡s✉❧ts ❜② ♣❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♥❡①t t♦ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ r❡❝✉rs✐✈❡❧②✿
❊①❡r❝✐s❡ ✶✳✶✸✳✺
❲❤❛t ❤❛♣♣❡♥❡❞ t♦ ±❄ ❊①❡r❝✐s❡ ✶✳✶✸✳✻
❍♦✇ ❤✐❣❤ ❞♦❡s t❤❡ ♣r♦❥❡❝t✐❧❡ ❣♦ ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡❄ ❊①❡r❝✐s❡ ✶✳✶✸✳✼
❯s✐♥❣ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣r♦❥❡❝t✐❧❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✜r❡❞
❞♦✇♥
❄
❊①❡r❝✐s❡ ✶✳✶✸✳✽
❯s❡ t❤❡ ❛❜♦✈❡ ♠♦❞❡❧ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣ ✐t ✇✐❧❧ t❛❦❡ ❢♦r ❛♥ ♦❜❥❡❝t t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✐t ✐s ❞r♦♣♣❡❞✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ q✉❡st✐♦♥s ❛❜♦✉t t❤❡ s✐t✉❛t✐♦♥ ❛♥❞ ❛♥s✇❡r t❤❡♠✳ ❘❡♣❡❛t✳ ❊①❛♠♣❧❡ ✶✳✶✸✳✾✿ ❛❝❝❡❧❡r❛t✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥ t✐♠❡
❲❤❛t ②♦✉r ❛❧❣❡❜r❛✐❝ ❛♥❛❧②s✐s ✐s ✐♥❝❛♣❛❜❧❡ ♦❢ ❞♦✐♥❣ ✭❢♦r ♥♦✇✮ ✐s t♦ ❤❛♥❞❧❡ t❤❡ ❝❛s❡ ♦❢ t✐♠❡✲❞❡♣❡♥❞❡♥t ❛❝❝❡❧❡r❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❊❛rt❤ ♠♦✈❡s ✐♥ ❛♥❞ ♦✉t ✕ ♣❡r✐♦❞✐❝❛❧❧② ✕ s♦ ❝❧♦s❡ t♦ t❤❡ ❙✉♥ t❤❛t t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ❧❛tt❡r ✐s str♦♥❣❡r t❤❛♥ t❤❛t ♦❢ t❤❡ ❢♦r♠❡r✳ ❚❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐❧❧ ❜❡
✶✳✶✹✳
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
✶✵✻
❛❧s♦ ❝❤❛♥❣✐♥❣ ♣❡r✐♦❞✐❝❛❧❧②✳ ❲❤❛t ✇✐❧❧ ❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❜❛❧❧❄ ❚❤❡ s♣r❡❛❞s❤❡❡t ✇♦r❦s ✇✐t❤ ♥♦ ❝❤❛♥❣❡✦ ❲❡ ❥✉st ✐♥s❡rt ❛ ♣❡r✐♦❞✐❝ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✿ ❂✶✵✵✯❙■◆✭❘❈❬✲✶❪✮
❚❤❡♥ t❤❡ r❡st ✐s t❛❦❡♥ ❝❛s❡ ♦❢✿
❊①❡r❝✐s❡ ✶✳✶✸✳✶✵
❉❡✈✐s❡ s✉❝❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❧❛st ❡①❛♠♣❧❡ t❤❛t t❤❡ ♦❜❥❡❝t ♦s❝✐❧❧❛t❡s ❜❡t✇❡❡♥ t❤❡ ❊❛rt❤ ❛♥❞ t❤❡ ❙✉♥✳
✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❡ s✐♠♣❧❡st ✭✜rst ♦r❞❡r✮ ❖❉❊s ❢♦r ♠♦t✐♦♥✿ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t✱ ✇❡ ❝❛♥ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s✳ ❙✉♣♣♦s❡ ✐♥st❡❛❞ ✇❤❛t ✇❡ ❦♥♦✇ ✐s t❤❡ ✇❡ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s❄
❢♦r❝❡s
❛✛❡❝t✐♥❣ t❤❡ ♦❜❥❡❝t ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ❛❝❝❡❧❡r❛t✐♦♥✳ ❍♦✇ ❝❛♥
▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❲❡ st❛rt ✇✐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
• t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ a1
✶✳✶✹✳
✶✵✼
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❲❡ ❝❤♦♦s❡✿ h = ∆t .
❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿ tn , an , vn , pn , n = 1, 2, 3, ...
✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿ • tn+1 = tn + h • vn+1 = vn + an · h • pn+1 = pn + vn · h
❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐♥✐t✐❛❧✿
✐t❡r❛t✐♦♥ n t✐♠❡ tn 0 1 ... 1000 ...
3.5 3.6 ... 103.5 ...
❛❝❝❡❧❡r❛t✐♦♥ an ✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn −− 66 ... 666 ...
33 38.5 ... 4 ...
22 25.3 ... 336 ...
❲❤❡r❡ ❞♦❡s t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ ❝♦♠❡ ❢r♦♠❄ ❖✉r ♠❛✐♥ ✐♥t❡r❡st ✐s t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❧❛st ❧♦❝❛t✐♦♥✱ ❡✳❣✳✱ an+1 = 1/p2n ✱ s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❣r❛✈✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♣❧❛♥❡t ♦r t❤❡ ❢♦r❝❡ ♦❢ t❤❡ s♣r✐♥❣ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ ♦❢ ✐ts ❡♥❞ ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✶✿ ♥♦ ❢♦r❝❡s
❘❡❝❛❧❧ s♦♠❡ ❡①❛♠♣❧❡s✳ ❆ r♦❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ s✐♠♣❧✐❢② ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② vn+1 = vn + an · h = vn = v0 ✐s ❝♦♥st❛♥t✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · h = pn + v0 · h ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s ❧✐♥❡❛r❧② ♦♥ t❤❡ t✐♠❡✳ ❆s ∆x → 0✱ ✇❡ ❤❛✈❡ ❛♥ ❡q✉❛t✐♦♥✿ y ′′ = 0 .
❚❤❡ s♦❧✉t✐♦♥ s❡t ❧♦♦❦s ✐s✿ y = p0 + v0 t ,
t❛❦❡♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ■t ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ❛s ❢♦❧❧♦✇s✿
❆❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛r❡ ❤❡r❡✦ ❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✿ t❤❡ y ✲✐♥t❡r❝❡♣ts ❛♥❞ t❤❡ s❧♦♣❡s✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✷✿ ❢r❡❡ ❢❛❧❧
❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐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 · h = vn + a · h ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · h ❣r♦✇s ❛t ❧✐♥❡❛r❧② ✐♥❝r❡❛s✐♥❣ ✐♥❝r❡♠❡♥ts✳
✶✳✶✹✳
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛s ❛ s❛♠♣❧❡❞ t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿ y ′′ = −g .
❚❤❡ s♦❧✉t✐♦♥ s❡t ❧♦♦❦s ✐s✿
1 y = p0 + v0 t − gt2 , 2
t❛❦❡♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ✐s ❛ s❦❡t❝❤ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿
❚❤✐s ✐s ❛❧s♦ ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✿ t❤❡ y ✲✐♥t❡r❝❡♣ts ❛♥❞ t❤❡ s❧♦♣❡s ❛t x = 0✳ P❘❖❇▲❊▼✿
●✐✈❡♥
• t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ z = f (y)✱
r❡♣r❡s❡♥t
• t❤❡ ✈❡❧♦❝✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ v = v(t) ❛♥❞
• t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ y = y(t)✳
❚❤❡♥ ♦✉r t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢②✿
an = f (pn ), vn = v(tn ), ❛♥❞ pn = y(tn ) .
❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ♦❢ ♦✉r ♣❛✐r r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥s✱ vn+1 = vn +an · h pn+1 = pn +vn · h
❢♦r ❡✈❡r② h > 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦✱ ❛s ✇❡❧❧ ❛s t = tn ✱ ✐♥t♦ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿
❚❤❡♥✱
v(t + h) = v(t) +f (y(t + h)) · h y(t + h) = y(t) +v(t + h) · h v(t + h) − v(t) = f (y(t + h)) h y(t + h) − y(t) = v(t + h) h
❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r h → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ♦✉r ❢✉♥❝t✐♦♥s✿ v ′ (t) = f (y(t)), y ′ (t) = v(t),
✶✵✽
✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
✶✵✾
♣r♦✈✐❞❡❞ y = y(t) ❛♥❞ v = v(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y(t)✳ ❚❤❡ ♦✉t❝♦♠❡ ✐s tr❡❛t❡❞ ❡✐t❤❡r ❛s ❛ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ♦r❞❡r ❞✐s❝✉ss❡❞ ❧❛t❡r✿
y ′ = v, v ′ = f (y),
✇❤✐❝❤ ✐s ❛ ✈❡❝t♦r ✜❡❧❞✱ ♦r ❛s ❛ s✐♥❣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r y ′′ = f (y) . ❊①❛♠♣❧❡ ✶✳✶✹✳✸✿ r♦❧❧✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡
❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♥♦t ❦♥♦✇♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✦ ❲❡ ❥✉st ❦♥♦✇ ❤♦✇ ✐t ❞❡♣❡♥❞s ♦♥ ❧♦❝❛t✐♦♥✳ ❆s t❤❡ ❜♦❧❧ ✐s r♦❧❧✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡✱ t❤❡ ♣♦rt✐♦♥ ♦❢ t❤❡ ❣r❛✈✐t② ❛❝❝❡❧❡r❛t✐♦♥ ✈❛r✐❡s✿
❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ∆v ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥ y ❛♥❞ t♦ ∆t ❛s ✐♥ t❤✐s ❡q✉❛t✐♦♥✿ ∆v = −f (y) · h, f (y) > 0 .
❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ v(tn+1 ) = v(tn ) − f (y(tn )) · h .
❋♦r t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛♥② ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✿ ∆y = v · h ,
✇❤❡r❡ v ✐s ❦♥♦✇♥ ❢r♦♠ ❛❜♦✈❡✳ ❲❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ y(tn+1 ) = y(tn ) + v(tn ) · h .
▲❡t✬s ❝♦♥s✐❞❡r
f (y) = y 2 .
❋♦r t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❧♦❝❛t✐♦♥✿ ❂✲✭❘❬✲✶❪❈❬✷❪✮✂ ✷
❚❤❡ r❡s✉❧t ✐s ❛❝❝❡❧❡r❛t❡❞ r♦❧❧ ✿
✶✳✶✹✳
✶✶✵
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ v ❛s s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s✱ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥✈❡rt❡❞ t♦ ❖❉❊s ❛s ❢♦❧❧♦✇s✿ y ′ = v, ❛♥❞ v ′ = −f (y) .
❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿
y ′′ = −f (y) .
❲❤❡♥ f ✐s ❝♦♥st❛♥t✱ ✇❡ ❛r❡ ❜❛❝❦ t♦ ❢r❡❡ ❢❛❧❧✦ ❆ ♠♦r❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡ ✐s f ❧✐♥❡❛r✳ ❋♦r ❡①❛♠♣❧❡✿ y ′′ = y .
❲❤❛t ❛r❡ t❤❡ s♦❧✉t✐♦♥s❄ ❋✐rst✱ ✐❢ y ′ = y ✱ t❤✐s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✳ ❖♥❡ ❝❧❛ss ✐s✱ t❤❡r❡❢♦r❡✿ y = Aet .
▼♦r❡❄ ■❢ y ′ = −y ✱ t❤✐s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✿ y ′′ = (y ′ )′ = (−y)′ = −y ′ = −(−y) = y .
❆♥♦t❤❡r ❝❧❛ss ✐s✱ t❤❡r❡❢♦r❡✿
y = Be−t .
▼♦r❡❄ ❆❢t❡r s♦♠❡ ❣✉❡ss✐♥❣✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ s✉♠ ♦❢ t❤❡ ❛❜♦✈❡✿ y = Aet + Be−t .
❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✦ ❊①❡r❝✐s❡ ✶✳✶✹✳✹
❲❤❛t s❤❛♣❡ ♦❢ ❛ s❧✐❞❡ ✇✐❧❧ ♣r♦✈✐❞❡ t❤❡ ❢❛st❡st tr✐♣ ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B ❄ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✺✿ ●❡♥❡r❛❧ ❙♦❧✉t✐♦♥ ♦❢ ❘♦❧❧ ❖❉❊ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿
y ′′ = ky, k > 0 , ❛r❡ ❣✐✈❡♥ ❜②✿
y(t) = Ae ✇❤❡r❡
A, B
√
kt
+ Be−
√
kt
,
❛r❡ t✇♦ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳
✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
✶✶✶
❊①❡r❝✐s❡ ✶✳✶✹✳✻
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✼✿ s♣r✐♥❣
❚❤❡ ❢♦r❝❡ ♦❢ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳
❉❡s❝r✐♣t✐♦♥ ✭❍♦♦❦❡✬s ▲❛✇✮✿
◮
✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❝✉rr❡♥t
❧♦❝❛t✐♦♥✑✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t②
∆v
✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥
y
❛♥❞ t♦
∆t
❛s ✐♥ t❤✐s ❡q✉❛t✐♦♥✿
∆v = −k · y · h, k > 0 . ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ✈❡❧♦❝✐t②✿
v(t0 ) = v0 . ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿
v(tn+1 ) = v(tn ) − ky(tn ) · h . ❋♦r t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛♥② ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✿
∆y = v · h , ✇❤❡r❡
v
✐s ❦♥♦✇♥ ❢r♦♠ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❧♦❝❛t✐♦♥✿
y(t0 ) = y0 . ❚❤✐s s❡ts ✉♣ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿
y(tn+1 ) = y(tn ) + v(tn ) · h . ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❣✐✈❡s ✉s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❲❡ ♠♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t ❜② ❣✐✈✐♥❣ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❧♦❝❛t✐♦♥✿
❂✲❘❬✲✶❪❈❬✷❪ ❚❤❡ r❡s✉❧t ✐s ♦s❝✐❧❧❛t✐♦♥ ✿
✶✳✶✹✳
✶✶✷
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ v ❛s s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s✱ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥✈❡rt❡❞ t♦ ❖❉❊s ❛s ❢♦❧❧♦✇s✿ y ′ = v, ❛♥❞ v ′ = −ky . ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿
❚❤✐s ✐s ❛ s❦❡t❝❤ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿
y ′′ = −ky, k > 0 .
❲❤❡♥ k = 1✱ ❛ ❝♦✉♣❧❡ ♦❢ s♦❧✉t✐♦♥s ♦❢ y ′′ = −y ❛r❡ ✇♦rt❤ r❡♠❡♠❜❡r✐♥❣✿ y1 (t) = cos t ❛♥❞ y2 (t) = sin t .
▼♦r❡ s♦❧✉t✐♦♥s❄ ❆❢t❡r s♦♠❡ ❣✉❡ss✐♥❣✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ ❧✐♥❡❛r
❝♦♠❜✐♥❛t✐♦♥
♦❢ t❤❡ ❛❜♦✈❡✿
y = A cos t + B sin t .
❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✽✿ ●❡♥❡r❛❧ ❙♦❧✉t✐♦♥ ♦❢ ❙♣r✐♥❣ ❖❉❊ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿
y ′′ = −ky, k > 0 ,
✶✳✶✹✳
✶✶✸
❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❛r❡ ❣✐✈❡♥ ❜②✿
y(t) = A cos ✇❤❡r❡
A, B
√
kt + B sin
√
kt
❛r❡ t✇♦ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳
❊①❡r❝✐s❡ ✶✳✶✹✳✾
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ♠♦t✐♦♥ ✐s t❤❡ ♣❡♥❞✉❧✉♠ ✭❈❤❛♣t❡r ✹✮✳ ❋♦r ♠♦r❡ ❡①❛♠♣❧❡s ♦❢ ✇❛✈❡ ❢✉♥❝t✐♦♥s✱ s❡❡ ❈❤❛♣t❡r ✶P❈✲✺✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✶✵✿ s♣r✐♥❣ ✇✐t❤ ❞❛♠♣❡♥✐♥❣
❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❝❛♥ ❛❧s♦ ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❡❧♦❝✐t②✦ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠✐❣❤t ✇❛♥t t♦ st♦♣ t❤❡ ✉♣ ❛♥❞ ❞♦✇♥ ♠♦t✐♦♥ ♦❢ t❤❡ s✉s♣❡♥s✐♦♥ ♦❢ ❛ ❝❛r✳ ❍♦✇❄ ❇② ✐♥tr♦❞✉❝✐♥❣ s♦♠❡ ❢r✐❝t✐♦♥✳ ❆ ♠♦❞❡❧ ♦❢ ❢r✐❝t✐♦♥ ♠❛② ❜❡ t❤❛t ✐ts ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♠♦t✐♦♥ ❜✉t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ ✐✳❡✳✱ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ✈❡❧♦❝✐t②✿ −my ′ .
❲❡ ❥✉st ♠♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t ❜② ❣✐✈✐♥❣ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ ❜♦t❤ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t②✿ ❂✲❘❬✲✶❪❈❬✷❪✲❘❬✲✶❪❈❬✶❪
❚❤❡ r❡s✉❧t ✐s ❛ q✉✐❝❦ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡ ♦s❝✐❧❧❛t✐♦♥✿
❚❤❡ ❖❉❊ ✐s ❛s ❢♦❧❧♦✇s✿ ■t ♥❡❡❞s ❢✉rt❤❡r ❛♥❛❧②s✐s✳
y ′′ = −my ′ − ky, m, k > 0 .
❊①❛♠♣❧❡ ✶✳✶✹✳✶✶✿ ❢r❡❡ ❢❛❧❧ ✇✐t❤ ❛✐r r❡s✐st❛♥❝❡
❚❤❡ ❡✛❡❝t ♦❢ ❛✐r r❡s✐st❛♥❝❡ ♦♥ ❛ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s t❤❡ s❛♠❡✿ y ′′ = −my ′ − g .
❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts ♦❢ s✐♠✉❧❛t✐♦♥s✿
✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
✶✶✹
❚❤❡ t❡r♠✐♥❛❧ ✈❡❧♦❝✐t② ✐s ❜❡✐♥❣ ❛♣♣r♦❛❝❤❡❞✦
❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✶✷✿ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❙✉♣♣♦s❡ h ✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✳ ❚❤❡♥ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆2 y ∆y = h t, y, ∆t2 ∆t
❛♥❞ ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣✿ dy d2 y ♦r y ′′ = h(t, y, y ′ ) = h t, y, dt2 dt
■t ✐s ♠♦r❡ ♣r♦❞✉❝t✐✈❡ t♦ ❛❞❞r❡ss t❤❡s❡ ❡q✉❛t✐♦♥s ❛s s②st❡♠s ♦❢ ❖❉❊s ♦❢ ✜rst ♦r❞❡r ✭❈❤❛♣t❡r ✸✮✿ (
y′ = v v ′ = h(t, y, v)
❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✶✸✿ ❧✐♥❡❛r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❚❤❡s❡ ❖❉❊s ❛r❡ ❝❛❧❧❡❞ ❧✐♥❡❛r ✇❤❡♥ h ✐s ❧✐♥❡❛r✿ ay ′′ + by ′ + cy = 0 , a 6= 0 .
❆❧♠♦st ❛❧❧ ❡①❛♠♣❧❡s ❤❛✈❡ ❜❡❡♥ ❧✐♥❡❛r✳ ❚❤❡② ❤❛✈❡ s♣❡❝✐❛❧ ♣r♦♣❡rt✐❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛tt❛❝❤✐♥❣ t♦ ♦✉r s♣r✐♥❣ ❛♥♦t❤❡r✱ ✐❞❡♥t✐❝❛❧ s♣r✐♥❣ ✇✐❧❧ ❝r❡❛t❡ ❛♥ ♦s❝✐❧❧❛t✐♦♥ ❡q✉✐✈❛❧❡♥t t♦ t❤❛t ♦❢ ❛ s✐♥❣❧❡ s♣r✐♥❣✿
✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
✶✶✺
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s✉♠ ♦❢ t✇♦ s♦❧✉t✐♦♥s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✦ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✶✹✿ ▲✐♥❡❛r✐t② ♦❢ ✷♥❞ ❖r❞❡r ▲✐♥❡❛r ❖❉❊ ■❢
y1
❛♥❞
y2
❛r❡ s♦❧✉t✐♦♥s ♦❢ ❛ ✷♥❞ ❖r❞❡r ▲✐♥❡❛r ❖❉❊✱ t❤❡♥ s♦ ✐s ❛♥② ♦❢ t❤❡✐r
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❀ ✐✳❡✳✱ ✐❢
y1
❛♥❞
y2
s❛t✐s❢②
y ′′ + by ′ + cy = 0 , t❤❡♥ s♦ ❞♦❡s
y = Ay1 + By2 , ✇❤❡r❡
A, B
❛r❡ ❛♥② r❡❛❧ ♥✉♠❜❡rs✳
Pr♦♦❢✳
■t ❢♦❧❧♦✇s ❢r♦♠ ❧✐♥❡❛r✐t② ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳
❚❤❡ r❡s✉❧t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ✏❙✉♣❡r♣♦s✐t✐♦♥ Pr✐♥❝✐♣❧❡✑✳ ■♥ t✇♦ t❤❡♦r❡♠s✱ ✇❡ s❤♦✇❡❞ t❤❛t ❢♦r♠✉❧❛s ❢♦r t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❘♦❧❧ ❖❉❊✱
y ′′ = ky (k > 0), t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❙♣r✐♥❣ ❖❉❊✱
y ′′ = −ky (k > 0), s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ❜② ❞✐r❡❝t s✉❜st✐t✉t✐♦♥s✳ ❇✉t ❤♦✇ ✇♦✉❧❞ ♦♥❡ ❞✐s❝♦✈❡r t❤❡♠❄ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❢♦r♠❡r ✐s t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿
y = ert . ❲❡ s✉❜st✐t✉t❡✿
(ert )′′ = kert . ❈♦♠♣✉t❡✿
r2 ert = kert . ❲❡ ❞✐s❝♦✈❡r t❤❛t
√ r = ± k.
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✶✻
❚❤❡ r❡s✉❧t ♠❛t❝❤❡s t❤❡ t❤❡♦r❡♠✳ ■❢ ✇❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ t❤❡ ❧❛tt❡r✱ ✇❡ ❛rr✐✈❡ t♦✿
√ r = ± −k . ❚❤❡s❡ ❛r❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✦ ❍♦✇ ✐♠❛❣✐♥❛r② ✭❛♥❞ ❝♦♠♣❧❡①✮ ♥✉♠❜❡rs ♣r♦❞✉❝❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ✐s s❤♦✇♥ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
▲❡t✬s r❡✈✐❡✇✳ ❲❡ ❞✐✈✐❞❡ t❤❡
x✲❛①✐s ✭✐✳❡✳✱ t❤❡ r❡❛❧ ❧✐♥❡ R✮ ✐♥t♦ ❞✐s❝r❡t❡ ♣✐❡❝❡s Q✳ ❚❤❡s❡ ❛r❡ t❤❡ t✇♦ t②♣❡s ♦❢ ♣✐❡❝❡s✿
✐♥t♦ ✐♥t❡r✈❛❧s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤
❢r♦♠ s♦♠❡ ❧♦❝❛t✐♦♥
•
❚❤❡ ♥♦❞❡s ❛r❡
0✲❝❡❧❧s✱ x = ..., Q − 2h , Q − h, Q, Q + h, Q + 2h, ...✳
•
❚❤❡ ❡❞❣❡s ❛r❡ t❤❡
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡
1✲❝❡❧❧s✱ [x, x + h] = ... [Q − h, Q], [Q, Q + h], [Q + h, Q + 2h], ...✳
y ✲❛①✐s
✐s ❥✉st t❤❡ r❡❛❧s✿
◆♦✇✱ t❤❡ ♠❛✐♥ ♦❜❥❡❝ts ✐♥ ♦✉r st✉❞② ✇✐❧❧ ❜❡ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❞✐s❝r❡t❡ ❢✉♥❝t✐♦♥s✿
•
❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ ♥♦❞❡s ❛s ✐♥♣✉ts ❛r❡ ❝❛❧❧❡❞
0✲❢♦r♠s✳
•
❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ ❡❞❣❡s ❛s ✐♥♣✉ts ❛r❡ ❝❛❧❧❡❞
1✲❢♦r♠s✳
❚❤❡ ♦✉t♣✉ts ❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❲❡ ✉s❡ ❛rr♦✇s t♦ ♣✐❝t✉r❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s ❝♦rr❡s♣♦♥❞❡♥❝❡s✿
h>0
st❛rt✐♥❣
✶✳✶✺✳
✶✶✼
❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
❲❡ ❝❛♥
❧✐st t❤❡ ✈❛❧✉❡s
♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿
• ❛ 0✲❢♦r♠ ✭♥♦❞❡ ❢✉♥❝t✐♦♥✮ f ✇✐t❤ f (0) = 2, f (1) = 4, f (2) = 3, ... • ❛ 1✲❢♦r♠ ✭❡❞❣❡ ❢✉♥❝t✐♦♥✮ s ✇✐t❤ s [0, 1] = 3, s [1, 2] = .5, s [2, 3] = 1, ...
●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f ✱ ✐ts ❣r❛♣❤ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ xy ✲♣❧❛♥❡ s♦ t❤❛t ✇❡ ❛❧✇❛②s ❤❛✈❡ y = f (x)✿
❆❜♦✈❡ ✇❡ s❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❋♦r ❛ ♥♦❞❡ ❢✉♥❝t✐♦♥✱ x ✐s ❛ ♥♦❞❡✱ ❛ ♥✉♠❜❡r✱ ❛♥❞ y = f (x) ✐s ❛❧s♦ ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ t❤❡② ♣r♦❞✉❝❡ (x, y)✱ ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✭✇✐t❤ t❤❡ x✲❛①✐s s♣❧✐t ✐♥t♦ ❝❡❧❧s ❛s s❤♦✇♥ ❛❜♦✈❡✮✳ ✷✳ ❋♦r ❛♥ ❡❞❣❡ ❢✉♥❝t✐♦♥✱ [A, B] ✐s ❛♥ ❡❞❣❡✱ ❛♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡ x✲❛①✐s✱ ❛♥❞ y = g([A, B]) ✐s ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ t❤❡② ♣r♦❞✉❝❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ xy ✲♣❧❛♥❡ s✉❝❤ ❛s (x, y) ❢♦r ❡✈❡r② x ✐♥ [A, B]✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t✳ ❲❡ ♠✐❣❤t ❞✐s❝♦✈❡r s✉❝❤ ❢✉♥❝t✐♦♥s ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛ ❝♦♥t✐♥✉♦✉s
◆♦✇ ❛ ♥❡✇ q✉❡st✐♦♥✿ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡
❝✉r✈❡
✿
❝❤❛♥❣❡ ♦❢ t❤❡ ❝❤❛♥❣❡
❄
❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❧✐♥❡ t♦ t❤❡ ❝❤❛♥❣❡s ♦❢ ❧♦❝❛t✐♦♥s ✭❞✐s♣❧❛❝❡♠❡♥ts✮ ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s t♦ t❤❡ ❝❤❛♥❣❡s ♦❢ ❞✐s♣❧❛❝❡♠❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ❛❣❛✐♥✿ f: − 2 −−− 5 −−− 10 − ∆f : − − • − 5 − 2 = 3 −•− 10 − 5 = 5 − • − − ∆∆f : − −?− −−− 5−3=2 −−− −?− − t: 1 2 3 4 5
●❡♥❡r❛❧❧②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧②
t❤r❡❡
✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst ❧✐♥❡✮ ❛t t❤❡ ❡♥❞s ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❝♦♠♣✉t❡ t❤❡
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✶✽
❞✐✛❡r❡♥❝❡s ❛❧♦♥❣ t❤❡ t✇♦ ✐♥t❡r✈❛❧s ✭s❡❝♦♥❞ ❧✐♥❡✮ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿ − f (x1 )
− − − − − • − ∆ f [x1 , x2 ]
− −•− x1
−−−
f (x2 ) −•− ∆f [x2 , x3 ] − ∆ f [x1 , x2 ] x2
− − − f (x3 ) − ∆f [x2 , x3 ] − • − − −−−
−•− − x3
❚♦ ✜♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ♠✐❞❞❧❡ ✭t❤✐r❞ ❧✐♥❡✮✳ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❄ ❆❞❞ ❞✐✈✐s✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❧✐♥❡ t♦ t❤❡ ✈❡❧♦❝✐t✐❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s t♦ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ❛❣❛✐♥✿ ❧♦❝❛t✐♦♥✿ − 2 −−− 5−2 = 3/2 ✈❡❧♦❝✐t②✿ − − • − ❛❝❝❡❧❡r❛t✐♦♥✿ − −?− t✐♠❡✿ 1
5
−−− 10 − 10 − 5 −•− = 5/2 − • − − 2 5/2 − 3/2 = 1/2 −−− −?− − 2 3 4 5
2 −−− 2
●❡♥❡r❛❧❧②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧② t❤r❡❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst ❧✐♥❡✮ ❛t t❤❡ ❡♥❞s ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛❧♦♥❣ t❤❡ t✇♦ ✐♥t❡r✈❛❧s ✭s❡❝♦♥❞ ❧✐♥❡✮ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿ − f (x1 ) − − − ∆f − −•− h − −•− −−− x1
f (x2 ) −•−
∆f h
− ∆f h h x2
− − − f (x3 ) − ∆f −•− − h −−− −•− − x3
❚♦ ✜♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ♠✐❞❞❧❡ ✭t❤✐r❞ ❧✐♥❡✮ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥ ❢✉❧❧ ❣❡♥❡r❛❧✐t②✳ ❋✐rst✱ ✇❡ ❤❛✈❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ♣❛rt✐t✐♦♥ ✐t ✐♥t♦ n ❡❞❣❡s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b .
❚❤❡s❡ ❛r❡ t❤❡ ❡❞❣❡s✿ c1 = [x0 , x1 ], c2 = [x1 , x2 ], ..., cn = [xn−1 , xn ] .
■❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ❜②✿ ∆f (ck ) = f (xk+1 ) − f (xk )
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✶✾
❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳ ❆❧s♦✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s✿ f (xk+1 ) − f (xk ) ∆f (ck ) = ∆x xk+1 − xk
❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳ ❚❤❡ ❢♦r♠❡r ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛❧✉❡ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡✱ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✿
■♥ ♦r❞❡r t♦ r❡♣❡❛t t❤❡ ❞✐✛❡r❡♥❝❡ ♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝♦♥str✉❝t✐♦♥ ♦♥ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ♥♦✇ ❛ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿
❲❡ ♥❡❡❞ ❝♦♥✈❡rt ❡❞❣❡ ❢✉♥❝t✐♦♥s t♦ ♥♦❞❡ ❢✉♥❝t✐♦♥s ❛♥❞ ✈✐❝❡ ✈❡rs❛ ❜② s✇✐t❝❤✐♥❣ ❜❡t✇❡❡♥ ♥♦❞❡s ❛♥❞ ❡❞❣❡s✳ ❚♦ s❡t t❤✐s ✉♣ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❝❡❧❧s✿
❚❤✉s t❤❡r❡ ❛r❡ ❛ ♥❡✇ ♥♦❞❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ❛♥❞ ❛ ♥❡✇ ❡❞❣❡ ❢♦r ❡❛❝❤ ♥♦❞❡✳ ❚♦❣❡t❤❡r✱ t❤❡s❡ ♥❡✇ ♥♦❞❡s ❛♥❞ ❡❞❣❡s ❢♦r♠ ❛ ♥❡✇ ❝♦♣② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ❝❡❧❧s ❛❜♦✈❡ ❛r❡ ♠❛t❝❤❡❞ ❛❜♦✈❡✿ • ❆♥ ❡❞❣❡ [x, x + h] ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♥♦❞❡ x + h/2✳
• ❆ ♥♦❞❡ x ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❡❞❣❡ [x − h/2, x + h/2]✳
❋♦r t❤❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✶✿ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ❞♦♠❛✐♥s ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦♣② ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♠❛❧ ❞♦♠❛✐♥✳ ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥♦t❤❡r ❝♦♣② ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R ✐s ❝❛❧❧❡❞ t❤❡ ❞✉❛❧ ❞♦♠❛✐♥✳ ❆ ❢✉♥❝t✐♦♥ ❢r♦♠ 0✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡ 1✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r ❛♥❞ ❢r♦♠ 1✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡ 0✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r t❤❛t ✐s ♠♦♥♦t♦♥❡ ✐s ❝❛❧❧❡❞ t❤❡ st❛r ♦♣❡r❛t♦r ♦r t❤❡ ⋆✲♦♣❡r❛t♦r✳
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✷✵
❚❤✐s ✐s ❤♦✇ t❤❡② ❛r❡ ❞❡♥♦t❡❞✿
❉✉❛❧✐t② ♦❢ ❝❡❧❧s ❆ ♣r✐♠❛❧ ❝❡❧❧ ❛ ❞✉❛❧ ❝❡❧❧
a
⋆
a ❝♦rr❡s♣♦♥❞s t♦ ✳
❚❤✐s ✐s ❤♦✇ t❤❡② ♠❛t❝❤ ✉♣✿
❚❤❡ r❡❧❛t✐♦♥ ❝❛♥ ❜❡ r❡✈❡rs❡❞✿
•
❆ ❞✉❛❧ ❡❞❣❡
a = [x − h/2, x + h/2]
•
❆ ❞✉❛❧ ♥♦❞❡
x + h/2
❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣r✐♠❛❧ ♥♦❞❡
❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣r✐♠❛❧ ❡❞❣❡
a⋆ = x✳
x⋆ = [x, x + h]✳
❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✷✿ ❞✉❛❧ ❝❡❧❧s ❋♦r ❜♦t❤ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ❛♥❞ ❢♦r ❜♦t❤ ♥♦❞❡s ❛♥❞ ❡❞❣❡s✱
a ❛♥❞ a⋆
❛r❡ ❝❛❧❧❡❞ ❞✉❛❧
♦❢ ❡❛❝❤ ♦t❤❡r✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿
◮
❆ ♣r✐♠❛❧
k ✲❝❡❧❧
❝♦rr❡s♣♦♥❞s t♦ ❛ ❞✉❛❧
(1 − k)✲❝❡❧❧✳
◆♦✇ t❤❡ ❞✉❛❧✐t② ♦❢ ❢♦r♠s✳ ❚❤❡ ✈❛❧✉❡s ❛r❡ t❤❡ s❛♠❡✱ ♦♥❧② t❤❡ ✐♥♣✉ts ❝❤❛♥❣❡✦ ❚✇♦ ❞✉❛❧ ♣❛✐rs ♦❢ ❢♦r♠s ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿
❚❤❡② ❝♦♠❡ ✐♥ ♣❛✐rs✱ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧✿
•
■❢
•
■❢
❛
f ✐s ❛ ♣r✐♠❛❧ 1✲❢♦r♠✳ g
♥♦❞❡ ❢✉♥❝t✐♦♥✱ ❛
0✲❢♦r♠✱
t❤❡♥
✐s ❛ ♣r✐♠❛❧ ❡❞❣❡ ❢✉♥❝t✐♦♥✱ ❛
1✲❢♦r♠✱
t❤❡♥
0✲❢♦r♠✳
❚❤❡s❡ ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❞✉❛❧ ❢♦r♠s✿
f ⋆ [x − h/2, x + h/2] = f (x) g ⋆ (x) = g [x − h/2, x + h/2]
✐s ❛ ❞✉❛❧ ❡❞❣❡ ❢✉♥❝t✐♦♥✱
✐s ❛ ❞✉❛❧ ♥♦❞❡ ❢✉♥❝t✐♦♥✱
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✷✶
■❢ ✇❡ ③♦♦♠ ♦✉t✱ t❤❡ ❢✉♥❝t✐♦♥s ❞✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r ✇✐❧❧ ❧♦♦❦ ✐❞❡♥t✐❝❛❧✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤✳
❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✸✿ ❞✉❛❧ ❢♦r♠s ●✐✈❡♥ ❛ ♥♦❞❡✴❡❞❣❡ ❢✉♥❝t✐♦♥ s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❡❞❣❡✴♥♦❞❡ ❢✉♥❝t✐♦♥✿ s⋆ (a) = s(a⋆ )
❚❤❡♥✱ s⋆ ✐s ❝❛❧❧❡❞ t❤❡ ❞✉❛❧ ❢✉♥❝t✐♦♥ ♦❢ s✳ ❚❤✐s s❡t✉♣ ❛❧❧♦✇s ✉s t♦ ❝♦♥str✉❝t t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ 0✲❢♦r♠✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ s❡❣♠❡♥t ✐♥t♦ n − 1 ✐♥t❡r✈❛❧s ❜② ❣✐✈✐♥❣ ♥♦❞❡s t♦ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ ♥❛♠❡s✿ p = c1 , c2 , c3 , ..., cn−1 , cn = q .
❚❤❡♥ t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡✿ ∆ck = ck+1 − ck .
◆♦✇✱ ✇❤❛t ❛r❡ t❤❡ ♥♦❞❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡❞❣❡s ♦❢ t❤✐s ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥❄ ❚❤❡ ♥♦❞❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❝♦✉rs❡✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿ x1 ✐♥ [c1 , c2 ], x2 ✐♥ [c2 , c3 ], ..., xn−1 ✐♥ [cn−1 , cn ] . ∆f
✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❲❡ ❛♣♣❧② t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥s t♦ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ g = ∆x g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿ ∆g(xk ) = g(ck+1 ) − g(ck )
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ♦❢ g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿ g(ck+1 ) − g(ck ) ∆g (xk ) = ∆x ck+1 − ck
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳
❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✹✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡
❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐❢✲ ❢❡r❡♥❝❡✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆f (ck+1 ) − ∆f (ck )
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✷✷
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳
❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✺✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆x2
− ∆f (c ) ∆x k ck+1 − ck
∆f (c ) ∆x k+1
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡✿ • n + 1 ✈❛❧✉❡s ♦❢ f ✭❛t t❤❡ ♥♦❞❡s✮✱
∆f ✭❛t t❤❡ ❡❞❣❡s✮✱ ❛♥❞ ∆x ∆2 f ✭❛t t❤❡ ♥♦❞❡s ❡①❝❡♣t a ❛♥❞ b✮✳ • n − 1 ✈❛❧✉❡s ♦❢ ∆x2
• n ✈❛❧✉❡s ♦❢
❲❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ s✉❜s❝r✐♣ts ❢♦r t❤❡ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✿
❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ∆2 f (x) = ∆f (c+∆c)−∆f (c) .
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✷✸
❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆2 f (x) = ∆x2
∆f (c ∆x
+ ∆c) − ∆c
∆f (c) ∆x
.
❊①❛♠♣❧❡ ✶✳✶✺✳✻✿ ❝✉r✈❛t✉r❡ ❆s ✇❡ ❦♥♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s✱ t❤❡r❡❢♦r❡✱ ③❡r♦✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ♥♦♥✲③❡r♦ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥❞✐❝❛t❡s ❛ ♥♦♥✲❧✐♥❡❛r ❣r❛♣❤✿
❆❜♦✈❡✱ t❤❡ s❧♦♣❡s r❡♠❛✐♥ t❤❡ s❛♠❡✱
2✱
❛t ✜rst❀ t❤❡r❡❢♦r❡✱ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ③❡r♦✿
∆2 f = 0. ∆x2 ❚❤❡♥✱ t❤❡ s❧♦♣❡ ❝❤❛♥❣❡s t♦
1
❛♥❞ t❤✐s ❝❤❛♥❣❡ ✐s t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✭❛ss✉♠✐♥❣
∆x = 1✮✿
∆2 f = −1 . ∆x2 ❆s ❛♥♦t❤❡r ✇❛② t♦ s❡❡ t❤✐s ✐❞❡❛✱ ✐♠❛❣✐♥❡ ②♦✉rs❡❧❢ ❞r✐✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ♣❛rt ♦❢ t❤❡ r♦❛❞ ❛♥❞ s❡❡✐♥❣ ❛ ♣❛rt✐❝✉❧❛r tr❡❡ ❛❤❡❛❞ ✭♥♦ ❝✉r✈❛t✉r❡✮✱ t❤❡♥✱ ❛s ②♦✉ st❛rt t♦ t✉r♥✱ t❤❡ tr❡❡s st❛rt t♦ ♣❛ss ②♦✉r ✜❡❧❞ ♦❢ ✈✐s✐♦♥ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ✭❝✉r✈❛t✉r❡✮✿
❋✉rt❤❡r♠♦r❡✱ ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢
y = f (x)✳
❚❤❡ ❤✐❣❤❡r ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞
❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♠❡❛♥s ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢
y = f (x)✳
❊①❛♠♣❧❡ ✶✳✶✺✳✼✿ ❢r❡❡ ❢❛❧❧ ❉❡s❝r✐♣t✐♦♥✿ ✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✑✳ ❲❡ ❤❛✈❡ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r✿
∆2 p = a(∆t)2 . ❚❤✐s ✐s ❥✉st ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇❡ ❤❛✈❡ ✉s❡❞ ❜❡❢♦r❡✳ ❚♦ ✐❧❧✉str❛t❡✱ ❧❡t✬s tr② t❤✐s s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢
a=1
❛♥❞
∆t = 1✿
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
✶✷✹
❲❡ st❛rt ✇✐t❤ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ ❞♦ s✉♠♠❛t✐♦♥ t✇✐❝❡✱ ❛♥❞ ❛rr✐✈❡ t♦ ❛ ❢✉♥❝t✐♦♥ t❤❛t s❡❡♠s q✉❛❞r❛t✐❝✳
❊①❛♠♣❧❡ ✶✳✶✺✳✽✿ ♦s❝✐❧❧❛t✐♥❣ s♣r✐♥❣
❉❡s❝r✐♣t✐♦♥✿ ✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥✑✳ ❲❡ ❤❛✈❡ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❢♦r t❤❡ ♣♦s✐t✐♦♥
p
♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s♣r✐♥❣✿
∆2 p = −kp(∆t)2 . ❚❤✐s ✐s ❥✉st ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇❡ ❤❛✈❡ ✉s❡❞ ❜❡❢♦r❡✳ ❚♦ ✐❧❧✉str❛t❡✱ ❧❡t✬s tr② t❤✐s s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢ ❛ ♣❡r✐♦❞✐❝
p
✇✐t❤
k=1
❛♥❞
∆t = 1✿
❆❢t❡r t❛❦✐♥❣ t✇♦ ❞✐✛❡r❡♥❝❡s ✭t❤❡ st❛r ♦♣❡r❛t♦rs ❛r❡ ♦♠✐tt❡❞✮✱ ✇❡ ❛rr✐✈❡ t♦
−p✳
❆ ♠❛t❝❤✦
✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞
❊①❡r❝✐s❡ ✶✳✶✺✳✾ ■❧❧✉str❛t❡ s✐♠✐❧❛r❧② ♦t❤❡r ♠♦❞❡❧s ❢r♦♠ t❤✐s ❝❤❛♣t❡r✳
✶✷✺
❈❤❛♣t❡r ✷✿ ❱❡❝t♦r ✈❛r✐❛❜❧❡s
❈♦♥t❡♥ts ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ✷✳✾
❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷✻ ✶✸✸ ✶✹✶ ✶✹✾ ✶✻✷ ✶✻✽ ✶✼✾ ✶✽✹ ✶✾✶
✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ s❡t ❛s✐❞❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ✕ ♠❡❛s✉r✐♥❣ ❞✐st❛♥❝❡s ❛♥❞ ❛♥❣❧❡s ✕ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ ♣✉r❡ ❛❧❣❡❜r❛✳ ▲❡t✬s r❡❝❛❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ❛❜♦✉t Pr♦❜❧❡♠✱ ❞✐♠❡♥s✐♦♥
$60❄
✳
♠✐①t✉r❡s
1✿ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ t②♣❡ ♦❢ ❝♦✛❡❡ t❤❛t ❝♦sts $3 ♣❡r ♣♦✉♥❞✳ ❍♦✇ ♠✉❝❤ ❞♦ ✇❡ ❣❡t ❢♦r
▲❡t x ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❝♦✛❡❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❙❡t✉♣✿ 3x = 60 . ❙♦❧✉t✐♦♥✿ x =
60 . 3
2✿ ❙✉♣♣♦s❡ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❝♦sts $2 ♣❡r ♣♦✉♥❞ ❛♥❞ t❤❡ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡ ❝♦sts $3 ♣❡r ♣♦✉♥❞✳ ❍♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡ 6 ♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ ❛ t♦t❛❧ ♣r✐❝❡ ♦❢ $14❄
Pr♦❜❧❡♠✱ ❞✐♠❡♥s✐♦♥
▲❡t x ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❛♥❞ ❧❡t y ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❙❡t✉♣✿
x + y =6 2x + 3y = 14
❙♦❧✉t✐♦♥✿ ❋r♦♠ t❤❡ ✜rst ❡q✉❛t✐♦♥✱ ✇❡ ❞❡r✐✈❡✿ y = 6 − x✳ ❚❤❡♥ s✉❜st✐t✉t❡ ✐t ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✿ 2x + 3(6 − x) = 14✳ ❙♦❧✈❡ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥✿ −x = −4✱ ♦r x = 4✳ ❙✉❜st✐t✉t❡ t❤✐s ❜❛❝❦ ✐♥t♦ t❤❡ ✜rst ❡q✉❛t✐♦♥✿ (4) + y = 6✱ t❤❡♥ y = 2✳ ❇✉t ✐t ✇❛s s♦ ♠✉❝❤ s✐♠♣❧❡r ❢♦r t❤❡ ❢♦r♠❡r ♣r♦❜❧❡♠✦ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♠✐♠✐❝ t❤❡ s❡t✉♣✱ ✐✳❡✳✱ t❤❡ ❡q✉❛t✐♦♥✱ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ❢♦r t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡❄ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ s✉❣❣❡sts t❤❛t ✐t ♠✐❣❤t ❜❡ ♣♦ss✐❜❧❡✳
✷✳✶✳
❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
✶✷✼
▲❡t✬s r❡❝❛❧❧ t❤❡ ✇❛②s ✇❡ ❤❛✈❡ ✐♥t❡r♣r❡t❡❞ t❤❡ ♣r♦❜❧❡♠✳ ❋✐rst✿ ♣♦✐♥ts ❛♥❞ ❧✐♥❡s✳ ❲❡ t❤✐♥❦ ♦❢ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛s ❡q✉❛t✐♦♥s ❛❜♦✉t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts✱
(
(x, y)✱
✐♥ t❤❡ ♣❧❛♥❡✿
x + y = 6, 2x + 3y = 14.
❊✐t❤❡r ❡q✉❛t✐♦♥ ✐s ❛ ❧✐♥❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡ s♦❧✉t✐♦♥
(x, y) = (4, 2)
✐s t❤❡ ♣♦✐♥t ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿
❙❡❝♦♥❞✿ ✈❡❝t♦rs ❛♥❞ t❤❡✐r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✳ ▲❡t✬s ♣✉t t❤❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡s❡ t❛❜❧❡s✿
1·x + 1·y = 6 2 · x + 3 · y = 14
❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs
❚❤❡ t❛❜❧❡ ✐s s♣❧✐t ❤♦r✐③♦♥t❛❧❧② t♦ r❡✈❡❛❧ t❤❡ ❡q✉❛t✐♦♥s✳ ◆❡①t✱ ✇❡ st❛rt t♦ s♣❧✐t ✈❡rt✐❝❛❧❧② ❛♥❞ r❡❛❧✐③❡ t❤❛t ✇❡ s❡❡ ❛ ❝♦♠♣♦♥❡♥t✇✐s❡
✿
1 · x + 1 · y = 6 2 · x + 3 · y = 14 ❲❡ ❤❛✈❡✿
1 · x
❇✉t
x✬s
❛♥❞
y ✬s
+
2 · x
1 · y
6 =
3 · y
14
❛r❡ r❡♣❡❛t❡❞✦ ❲❡ r❡❛❧✐③❡ t❤❛t ✇❡ s❡❡ ❛ ❝♦♠♣♦♥❡♥t✇✐s❡
1 2
1 · x +
3
s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs
6 · y =
14
❱❡❝t♦rs st❛rt t♦ ❛♣♣❡❛r✳ ■♥❞❡❡❞✱ ♦✉r s②st❡♠ ❤❛s ❜❡❡♥ r❡❞✉❝❡❞ t♦ ❛ s✐♥❣❧❡
✈❡❝t♦r
❡q✉❛t✐♦♥✿
6 1 1 y= x+ 14 3 2 2✳ ❚❤✐s ✐s♥✬t ❛ ❝♦✐♥❝✐❞❡♥❝❡✳ ❚❤❡② ✇❡✐❣❤t ✭✐♥ ♣♦✉♥❞s✮ 6 1 1 . ❛r❡ , , ❝♦st ✭✐♥ ❞♦❧❧❛rs✮ 14 3 2
❲❡ s❡❡ t❤r❡❡ ✈❡❝t♦rs ♦❢ t❤❡ s❛♠❡ ❞✐♠❡♥s✐♦♥
❚❤❡② ❧✐✈❡ ✐♥ t❤❡ s❛♠❡ s♣❛❝❡
R2
❛r❡ ♦❢ t❤❡ s❛♠❡ ♥❛t✉r❡✿
❛♥❞✱ t❤❡r❡❢♦r❡✱ s✉❜❥❡❝t t♦ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛✳
✿
✷✳✶✳
❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
✶✷✽
❚❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ s②st❡♠ ✐♥ t❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ ♠❡❛♥✐♥❣✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡
x ❛♥❞ y ✱ 6 1 1 . = +y x 14 3 2
t✇♦ ❡q✉❛t✐♦♥s ❛s ❛ s✐♥❣❧❡ ❡q✉❛t✐♦♥ ❛❜♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts✱
♦❢ t❤❡s❡ ✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡✿
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s
●❡♦♠❡tr✐❝❛❧❧②✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ✇❛② t♦ str❡t❝❤ t❤❡s❡ t✇♦ ✈❡❝t♦rs s♦ t❤❛t ❛❢t❡r ❛❞❞✐♥❣ t❤❡♠ t❤❡ r❡s✉❧t ✐s t❤❡ ✈❡❝t♦r ♦♥ t❤❡ r✐❣❤t✳ ❲❡ s♣❡❛❦ ♦❢
✳
❚❤❡ s❡t✉♣ ✐s ♦♥ t❤❡ ❧❡❢t ❢♦❧❧♦✇❡❞ ❜② ❛ tr✐❛❧✲❛♥❞✲❡rr♦r ♦♥ t❤❡ r✐❣❤t✿
❙♦✱ t❤❡ ♥❡✇ ♣♦✐♥t ♦❢ ✈✐❡✇ ❤❛s ❝❤❛♥❣❡❞✿ ■♥st❡❛❞ ♦❢ t❤❡
❧♦❝❛t✐♦♥s
✱ ✇❡ ❛r❡ ❛❢t❡r t❤❡
❞✐r❡❝t✐♦♥s
✳
❊①❡r❝✐s❡ ✷✳✶✳✶
❆r❡ t❤❡r❡ ♦t❤❡r ✈❡❝t♦rs ❤❡r❡❄
❚❤✐r❞✿ tr❛♥s❢♦r♠❛t✐♦♥s✳ ■♥✐t✐❛❧❧②✱ ✇❡ ✉s❡ ♣♦✐♥ts✳ ❉✐♠❡♥s✐♦♥
•
1
♣r♦❜❧❡♠✿
❆ tr❛♥s❢♦r♠❛t✐♦♥
f :R→R
✐s ❣✐✈❡♥ ❜②
f (x) = 30x . •
❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿
f (x) = 60 .
❉✐♠❡♥s✐♦♥
•
2
♣r♦❜❧❡♠✿
❆ tr❛♥s❢♦r♠❛t✐♦♥
f : R2 → R2
✐s ❣✐✈❡♥ ❜②
F (x, y) = (x + y, 2x + 3y) . •
❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿
F (x, y) = (6, 14) .
◆♦✇✱ ✇❡ ♣r❡❢❡r t♦ ✉s❡ ✈❡❝t♦rs✳ ▲❡t✬s ✜♥❞ ❛❧❧
2✲❞✐♠❡♥s✐♦♥❛❧
✈❡❝t♦rs ✐♥ t❤❡ ❡q✉❛t✐♦♥s✿
1·x + 1·y = 6 2 · x + 3 · y = 14 ❚❤❡ ✜rst ✐s ♦♥ t❤❡ r✐❣❤t❀ ✐t ❝♦♥s✐sts ♦❢ t❤❡ t✇♦ ✏❢r❡❡✑ t❡r♠s ✭❢r❡❡ ♦❢
6 B= . 14
x✬s
❛♥❞
y ✬s✦✮
♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿
✷✳✶✳
✶✷✾
❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
❆♥♦t❤❡r ♦♥❡ ✐s ❧❡ss ✈✐s✐❜❧❡❀ ✐t ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ t✇♦ ✉♥❦♥♦✇♥s✿ x . X= y
❊✈❡♥ t❤♦✉❣❤ ✐ts ❞✐♠❡♥s✐♦♥ ✐s ❛❧s♦ 2✱ ✐t✬s ♥♦t ♦❢ t❤❡ s❛♠❡ ♥❛t✉r❡ ❛s t❤❡ ♦t❤❡rs✿ # ♦❢ ♣♦✉♥❞s x . ✐s # ♦❢ ♣♦✉♥❞s y
■t ❧✐✈❡s ✐♥ ❛ ❞✐✛❡r❡♥t R2 ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ s♣❛❝❡s✿ F : R2 → R2 ,
Y = F (X) .
■ts ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ ✈❡❝t♦rs✿ x+y x . = F 2x + 3y y
❖✉r ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ❛ ♣r♦❜❧❡♠ ♦❢ s♦❧✈✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ❢♦r X ✿ F (X) = B . ❲❛r♥✐♥❣✦ ❙✐♥❝❡ ✇❡ ❛r❡♥✬t ❞♦✐♥❣ ❛♥② ❣❡♦♠❡tr② ❜✉t ✇❡ ❛r❡ ❞♦✲ ✐♥❣ ✈❡❝t♦r ❛❧❣❡❜r❛✱ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ✇✐❧❧ ❜❡ ♣r❡✲ ❢❡rr❡❞ t❤r♦✉❣❤♦✉t t❤❡ ❝❤❛♣t❡r✳
❚❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♥❡❡❞❡❞ t♦ ❝♦♠♣✉t❡ F ❛r❡ s♦ s✐♠♣❧❡ t❤❛t t❤❡② ✇✐❧❧ ❜❡ ❡❛s② t♦ ❛❜❜r❡✈✐❛t❡✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣✳ ❚❤❡ ♣r♦❜❧❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ n ❤❛s n ✐♥❣r❡❞✐❡♥ts✿ t❤❡ ✉♥❦♥♦✇♥ ♠✉❧t✐♣❧✐❡❞ ❜② ✐s ❡q✉❛❧ t♦
dim 1 dim 2 x X =< x, y > 3 ? 60 B =< 6, 14 >
❲❡ ❤❛✈❡ tr❛♥s✐t✐♦♥❡❞ ❢r♦♠ ♥✉♠❜❡rs t♦ ✈❡❝t♦rs✳ ❇✉t ✇❤❛t ✐s t❤❡ ♦♣❡r❛t✐♦♥ t❤❛t ♠❛❦❡s B ❢r♦♠ X ❄ ◆♦♥❡ ♦❢ t❤❡ ❢❛♠✐❧✐❛r ♦♥❡s✳ ❚❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts ♦❢ x, y ❢♦r♠ ❛ t❛❜❧❡✿
1 1 F = . 2 3
■t ❤❛s t✇♦ r♦✇s ❛♥❞ t✇♦ ❝♦❧✉♠♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛ 2 × 2 ♠❛tr✐①✳
❇♦t❤ X ❛♥❞ B ❛r❡ ❝♦❧✉♠♥✲✈❡❝t♦rs ✐♥ ❞✐♠❡♥s✐♦♥ 2✱ ❛♥❞ ♠❛tr✐① F t✉r♥s X ✐♥t♦ B ✳ ❚❤✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♥✉♠❜❡rs❀ ❛❢t❡r ❛❧❧✱ t❤❡② ❛r❡ ✈❡❝t♦rs ♦❢ ❞✐♠❡♥s✐♦♥ 1✳ ▲❡t✬s ♠❛t❝❤ t❤❡ s❡t✉♣s ♦❢ t❤❡ t✇♦ ♣r♦❜❧❡♠s✿ dim 1 : m · x = b dim 2 : F · X = B
■❢ ✇❡ ❝❛♥ ❥✉st ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ ♥❡✇ ❛❧❣❡❜r❛✦
❍❡r❡ F X = B ✐s ❛ ♠❛tr✐① ❡q✉❛t✐♦♥✱ ❛♥❞ ✐t✬s s✉♣♣♦s❡❞ t♦ ❝❛♣t✉r❡ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ F X = B ✿ x +y = 6 ✱ r❡✇r✐tt❡♥ ❛s 2x +3y = 14
1 1 x 6 · = . 2 3 y 14
✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
✶✸✵
❲❡ ❝❛♥ s❡❡ t❤❡s❡ ❡q✉❛t✐♦♥s ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❛s t❤❡s❡ t✇♦ ❞♦t ♣r♦❞✉❝ts✳ ❋✐rst✿
1·x+1·y =6
✱ r❡✇r✐tt❡♥ ❛s
❙❡❝♦♥❞✿
2x + 3y = 14 ❚❤✐s s✉❣❣❡sts ✇❤❛t t❤❡ ♠❡❛♥✐♥❣ ♦❢
X
FX
✱ r❡✇r✐tt❡♥ ❛s
x 1 1 · = 6. y
x 2 3 · = 14 . y
s❤♦✉❧❞ ❜❡✳ ❲❡ ✏♠✉❧t✐♣❧②✑ ❡✐t❤❡r r♦✇ ✐♥
A✱
❛s ❛ ✈❡❝t♦r✱ ❜② t❤❡ ✈❡❝t♦r
✕ ✈✐❛ t❤❡ ❞♦t ♣r♦❞✉❝t✿
❉❡✜♥✐t✐♦♥ ✷✳✶✳✷✿ ♣r♦❞✉❝t ♦❢ ♠❛tr✐① ❛♥❞ ✈❡❝t♦r ❚❤❡ ♣r♦❞✉❝t
FX
♦❢ ❛
2×2
♠❛tr✐①
F
❛♥❞ ❛
2✲✈❡❝t♦r X ✱
x a b , X= F = y c d
✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣
2✲✈❡❝t♦r✿
ax + by x a b = · FX = cx + dy y c d
❲❡ ❝❛♥ st✐❧❧ s❡❡ t❤❡s❡ ❞♦t ♣r♦❞✉❝ts ✐♥ t❤❡ r❡s✉❧t✿
a b x ax + by = · y cx + dy c d
❛♥❞
x a b ax + by · = . y c d cx + dy
❲❛r♥✐♥❣✦ ❆ ♠❛tr✐① ✐s ♥♦t❤✐♥❣ ❜✉t ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ ❛ tr❛♥s✲ ❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✿
F X = F (X) . ❍♦✇❡✈❡r✱
♥♦t
❛❧❧
tr❛♥s❢♦r♠❛t✐♦♥s
❝❛♥
❜❡
r❡♣r❡✲
s❡♥t❡❞ ❜② ♠❛tr✐❝❡s✳
❊①❛♠♣❧❡ ✷✳✶✳✸✿ 3 ✈❛r✐❛❜❧❡s ❲❤❛t ✐❢ t❤❡ ❜❧❡♥❞ ✐s t♦ ❝♦♥t❛✐♥ ❛♥♦t❤❡r✱ t❤✐r❞✱ t②♣❡ ♦❢ ❝♦✛❡❡❄ ●✐✈❡♥ t❤r❡❡ ♣r✐❝❡s ♣❡r ♣♦✉♥❞✱ ❤♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡
x✱ y ✱ ❛♥❞ z ❜❡ t❤❡ ❜❧❡♥❞ ✐s 14✳ ❚❤❡r❡❢♦r❡✱ ▲❡t
6
♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ ❛ t♦t❛❧ ♣r✐❝❡ ♦❢
2, 3, 5✱
14❄
✇❡✐❣❤ts ♦❢ t❤❡ t❤r❡❡ t②♣❡s ♦❢ ❝♦✛❡❡✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ t❤❡ t♦t❛❧ ♣r✐❝❡ ♦❢ t❤❡ ✇❡ ❤❛✈❡ ❛ s②st❡♠✿
x + y + z =6 2x + 3y + 5z = 14
❊✐t❤❡r ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s r❡♣r❡s❡♥ts ❛ ♣❧❛♥❡ ✐♥
R3 ✳
❚❤❡ s♦❧✉t✐♦♥ s❡t t❤❡♥ ❝♦♠❡s ❢r♦♠ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿
✷✳✶✳
❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠
✶✸✶
❚❤❡r❡ ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s✳ ❆♥ ❛❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛♥♦t❤❡r ❧✐♥❡❛r ❡q✉❛t✐♦♥ ♠❛② r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r t♦ ♦♥❡✳✳✳ ♦r ♥♦♥❡✳ ❚❤❡ ✈❛r✐❡t② ♦❢ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✐s✱ ❜② ❢❛r✱ ❤✐❣❤❡r t❤❛♥ ✐♥ t❤❡
2✲❞✐♠❡♥s✐♦♥❛❧
❝❛s❡❀ t❤❡② ❛r❡ ♥♦t ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳
❚❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛✱ ❤♦✇❡✈❡r✱ ✐s t❤❡ s❛♠❡✦ ❚❤❡ t❤r❡❡ ✇❡✐❣❤ts ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❛ ✈❡❝t♦r✱
< 1, 1, 1 >✱
❛♥❞ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s t❤❡ ❞♦t ♣r♦❞✉❝t✿
< 1, 1, 1 > · < x, y, z >= 6 . ❚❤❡ t❤r❡❡ ♣r✐❝❡s ♣❡r ♣♦✉♥❞ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❛ ✈❡❝t♦r✱
< 2, 3, 5 >✱
❛♥❞ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s t❤❡
❞♦t ♣r♦❞✉❝t✿
❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❛
♠❛tr✐① ❡q✉❛t✐♦♥
< 2, 3, 5 > · < x, y, z >= 14 . ✿
x 6 1 1 1 . · y = 14 2 3 5 z
❲✐t❤♦✉t ❤❛r♠✱ ✇❡ ❝❛♥ ♠❛❦❡ t❤❡ ♠❛tr✐① sq✉❛r❡✿
6 x 1 1 1 2 3 5 · y = 14 . 0 z 9 0 0 ❊①❛♠♣❧❡ ✷✳✶✳✹✿ s♣r❡❛❞s❤❡❡t
❖♥❡ ❝❛♥ ✉t✐❧✐③❡ ❛ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦t❤❡r s♦❢t✇❛r❡ t♦ t❤✐s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥s✳ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ✇♦r❦✱ t❤❡ ✈❡❝t♦r
❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r t❤❡ tr❛♥s♣♦s❡ ♦❢
X
❤❛s t♦ ❜❡ ✏tr❛♥s♣♦s❡❞✑ ✭❜♦tt♦♠✮✿
X✿
❂❚❘❆◆❙P❖❙❊✭❘❬✲✺❪❈✿❘❬✲✸❪❈✮
✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r Y ✿
✶✸✷ ❂❙❯▼P❘❖❉❯❈❚✭❘❈✷✿❘❈✹✱❘✽❈❬✲✷❪✿❘✽❈✮
❚❤❡ ✇❤♦❧❡ s②st❡♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❡①❛❝t❧② t❤❡ s❛♠❡ ♠❛tr✐① ❡q✉❛t✐♦♥✿ FX = B .
❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❡①❡❝✉t❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② t♦♦✿ ❝ ♦ ❧ r r ♦ ♦ ✇ ✇ · ✉ = rc + ro + ol + ou + wm + wn ♠ ♥
●❡♥❡r❛❧❧②✱ ✇❡ ❢❛❝❡ ❛ s②st❡♠ ✇✐t❤✿ ✶✳ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s m✱ ❛♥❞ ✷✳ t❤❡ ♥✉♠❜❡r ♦❢ ❡q✉❛t✐♦♥s n✳ ❲❡ ✇✐❧❧ ❤❛✈❡ ❛♥ n × m ♠❛tr✐①✿
1 1 2 2 0 n 3
2 3 ... m 0 3 ... 2 6 2 ... 0 ... 1 0 ... 12
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
✳✳✳
❍❡r❡✱ t❤❡ ♥✉♠❜❡r ❛t t❤❡ ij ✲♣♦s✐t✐♦♥ ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ j t❤ ✈❛r✐❛❜❧❡ ✐♥ t❤❡ it❤ ❡q✉❛t✐♦♥✳ ❘❡❝❛❧❧ ❤♦✇ ❛♥ ✐♥❞❡① ♣♦✐♥ts ❛t ❛ ❧♦❝❛t✐♦♥ ✇✐t❤✐♥ ❛ s❡q✉❡♥❝❡✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ✉s❡ ❞♦✉❜❧❡ ✐♥❞❡① t♦ ♣♦✐♥t ❛t t❤❡ ❝♦rr❡❝t ❧♦❝❛t✐♦♥ ✇✐t❤✐♥ ❛ t❛❜❧❡✳ ❉❡✜♥✐t✐♦♥ ✷✳✶✳✺✿ ❡♥tr✐❡s ♦❢ ♠❛tr✐①
❚❤❡ ij ✲❡♥tr② ✐♥ ❛♥ n × m ♠❛tr✐① A ✐s t❤❡ ♥✉♠❜❡r ❛t t❤❡ it❤ r♦✇ ❛♥❞ j t❤ ❝♦❧✉♠♥✱ ❞❡♥♦t❡❞ ❜② Aij
❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ ❛❜♦✈❡ ♠❛tr✐①✿ 1 2 3 i\j A1,1 = 2 A1,2 = 0 A1,3 = 3 1 A 2 2,1 = 0 A2,2 = 6 A2,3 = 2 m Am,1 = 3 Am,2 = 1 Am,3 = 0
✳✳✳
✳✳✳
✳✳✳
✳✳✳
... ... ...
n A1,n = 2 A2,n = 0 ... ... Am,n = 12
✳✳✳
❲❛r♥✐♥❣✦ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❛❜❧❡s ♦❢ ♥✉♠❜❡rs ❛♥❞ ♠❛tr✐❝❡s❄ ❚❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❞✐s❝✉ss❡❞ ❤❡r❡✳
✷✳✷✳
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✸
✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ♠❛❞❡ ✉♣ ♦❢ t✇♦ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳
❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❆
tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✐s ❛ ❢✉♥❝t✐♦♥ F : ❘2 → ❘2 ,
❣✐✈❡♥ ❜② ❛♥② ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s f, g ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿
F (x, y) = (u, v) =
f (x, y), g(x, y)
❲❤❡♥ ❛♣♣r♦♣r✐❛t❡✱ ✇❡ ❝❛♥ ❛❧s♦ ❧♦♦❦ ❛t t❤❡ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❛s ✈❡❝t♦rs ✿ u x , 7→ < u, v >= < x, y >= v y
✐♥st❡❛❞ ♦❢ ♣♦✐♥ts✳
▲❡t✬s r❡✈✐❡✇ ❛ ❢❡✇ ❡①❛♠♣❧❡s ♦❢ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s✳
❊①❛♠♣❧❡ ✷✳✷✳✷✿ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❲❡ ♥❡❡❞ t✇♦ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❈♦♥s✐❞❡r u = f (x, y) = 2x − 3y ✿
f : R2 → R, ♠❡❛♥✐♥❣ f : (x, y) → u = 2x − 3y ❈♦♥s✐❞❡r ❛❧s♦ v = g(x, y) = x + 5y ✿
g : R2 → R, ♠❡❛♥✐♥❣ g : (x, y) → v = x + 5y ▲❡t✬s ❜✉✐❧❞ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡s❡ t✇♦✳ ❲❡ t❛❦❡ t❤❡ ✐♥♣✉t t♦ ❜❡ t❤❡ s❛♠❡ ✕ ❛ ♣♦✐♥t ✐♥ t❤❡ ♣❧❛♥❡ ✕ ❛♥❞ ✇❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ♦✉t♣✉ts ✐♥t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t (u, v) ✕ ✐♥ ❛♥♦t❤❡r ♣❧❛♥❡✳ ❚❤❡♥ ✇❤❛t ✇❡ ❤❛✈❡ ✐s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✿ F : R2 → R2 , ♠❡❛♥✐♥❣ F : (x, y) → (u, v) = (2x − 3y, x + 5y)
■♥ s❤♦rt✱ t❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿
F (x, y) = (2x − 3y, x + 5y) . ■♥ t❡r♠s ♦❢ ✈❡❝t♦rs✿
2x − 3y x 7→ F : x + 5y y
❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♠❛tr✐① ♦❢ F ❛r❡ r❡❛❞ ❢r♦♠ t❤❛t r❡♣r❡s❡♥t❛t✐♦♥✿ 2 −3 F = 1 5 ❲❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❞♦❡s t♦ t❤❡ ♣❧❛♥❡ r❡♠❛✐♥s t♦ ❜❡ ❞❡t❡r♠✐♥❡❞✳
✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✹
❊①❛♠♣❧❡ ✷✳✷✳✸✿ ✈❡rt✐❝❛❧ s❤✐❢t ❚❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② F (x, y) = (x, y + 3)
✐s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✿
✉♣ k
(x, y) −−−−−−→ (x, y + k)
❲❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ❜② ❞r❛✇✐♥❣ s♦♠❡t❤✐♥❣ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ✭t❤❡ ❞♦♠❛✐♥✮ ❛♥❞ t❤❡♥ s❡❡ ✇❤❛t t❤❛t ❧♦♦❦s ❧✐❦❡ ✐♥ t❤❡ ♥❡✇ ♣❧❛♥❡ ✭t❤❡ ❝♦✲❞♦♠❛✐♥✮✿
Pr❡❞✐❝t❛❜❧②✱ t❤❡ ❢♦r♠✉❧❛✿ F (x, y) = (x + a, y + b) = (x, y)+ < a, b > ,
❣✐✈❡s t❤❡ s❤✐❢t ❜② ✈❡❝t♦r < a, b >✳
❊①❡r❝✐s❡ ✷✳✷✳✹ ❊①♣❧❛✐♥ ✇❤② t❤❡r❡ ✐s ♥♦ ♠❛tr✐①✳
❊①❛♠♣❧❡ ✷✳✷✳✺✿ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ✢✐♣ ◆♦✇ t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣✳ ❲❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ y ✲❛①❡s ❛❧✐❣♥✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② F (x, y) = (−x, y) ,
t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❛r❛❜♦❧❛✬s ❧❡❢t ❜r❛♥❝❤ ✐s ❛ ♠✐rr♦r ✐♠❛❣❡ ♦❢ ✐ts r✐❣❤t ❜r❛♥❝❤✿
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ✈❡❝t♦rs✿ (−1)x + 0y −x x = 7→ F : 0x + 1y y y
✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✺
❚❤❡♥✱ ✇❡ ❤❛✈❡ ✐ts ♠❛tr✐①✿ ■♥❞❡❡❞✱
−1 0 . F = 0 1
−x −1 0 x x x . = = =F F y 0 1 y y y
◆❡①t✱ ❝♦♥s✐❞❡r ✈❡rt✐❝❛❧ ✢✐♣✳ ❲❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ x✲❛①❡s ❛❧✐❣♥✳ ■❢ G(x, y) = (x, −y) ,
t❤❡♥ ✇❡ ❤❛✈❡✿
✈❡rt✐❝❛❧ ✢✐♣
(x, y) −−−−−−−−−−→ (x, −y).
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ❛ ♠❛tr✐①✿ 1 0 . G= 0 −1
■♥❞❡❡❞✱
x −1 0 x x . = = G −y 0 1 y y
❊①❛♠♣❧❡ ✷✳✷✳✻✿ ❝❡♥tr❛❧ s②♠♠❡tr②
❍♦✇ ❛❜♦✉t t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ♦r✐❣✐♥ ❄ ❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛✱ F (x, y) = (−x, −y) ,
♦❢ ✇❤❛t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❝❡♥tr❛❧ s②♠♠❡tr②✿
❚❤✐s ✐s ✇❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❞♦❡s t♦ ❛ st❛r✿
✷✳✷✳
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✻
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ✈❡❝t♦rs✿
(−1)x + 0y −x x = 7→ F : 0x + (−1)y y y ❚❤❡♥✱ ✇❡ ❤❛✈❡ ✐ts ♠❛tr✐①✿
−1 0 . F = 0 −1
■♥❞❡❡❞✱
−x x −1 0 x x . = = =F F −y 0 −1 y y y ❊①❛♠♣❧❡ ✷✳✷✳✼✿ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ str❡t❝❤ ◆♦✇ t❤❡
❤♦r✐③♦♥t❛❧ str❡t❝❤
✳ ❲❡ ❣r❛❜ ❛ r✉❜❜❡r s❤❡❡t ❜② t❤❡ t♦♣ ❛♥❞ t❤❡ ❜♦tt♦♠ ❛♥❞ ♣✉❧❧ t❤❡♠ ❛♣❛rt
✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡
y ✲❛①✐s
❞♦❡s♥✬t ♠♦✈❡✳ ❍❡r❡✱
F (x, y) = (kx, y) .
❙✐♠✐❧❛r❧②✱ t❤❡
❤♦r✐③♦♥t❛❧ str❡t❝❤
✐s ❣✐✈❡♥ ❜②✿
G(x, y) = (x, ky) .
✷✳✷✳
✶✸✼
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ♠❛tr✐❝❡s✿ 1 0 k 0 . ❛♥❞ G = F = 0 k 0 1 ❊①❛♠♣❧❡ ✷✳✷✳✽✿ r❡✲s❝❛❧✐♥❣
❍♦✇ ❛❜♦✉t t❤❡
✉♥✐❢♦r♠ str❡t❝❤
✭s❛♠❡ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s✮❄ ❚❤✐s ✐s ✐ts ❢♦r♠✉❧❛✿
F (x, y) = (kx, ky) .
❚❤❡ r❡s✉❧t ✐s r❡✲s❝❛❧✐♥❣✳ ❚❤❡ r❡❛s♦♥ ❝♦♠❡s ❢r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❣❡♦♠❡tr②✿ ❙✐♠✐❧❛r tr✐❛♥❣❧❡s ❤❛✈❡ t❤❡ s❛♠❡ ❛♥❣❧❡s✳ ❍❡r❡ ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ✇❤②✿
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ❛ ♠❛tr✐①✿ k 0 . F = 0 k ❊①❡r❝✐s❡ ✷✳✷✳✾
❋✐♥❞ t❤❡ ♠❛tr✐① ❢♦r ❛ ❞✐s♣r♦♣♦rt✐♦♥❛❧ str❡t❝❤✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ♣✉t ❛❧❧ ♥❡✇❧② ✐♥tr♦❞✉❝❡❞ ❢✉♥❝t✐♦♥s ✐♥ ❜r♦❛❞ ❝❛t❡❣♦r✐❡s✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❝❛t❡❣♦r✐❡s ✕ s✉❝❤ ❛s ♠♦♥♦t♦♥✐❝✐t② ✕ ❤❛✈❡ ❜❡❝♦♠❡ ✐rr❡❧❡✈❛♥t✳ ❖t❤❡rs ✕ s✉❝❤ ❛s s②♠♠❡tr② ✕ ❤❛✈❡ ❜❡❝♦♠❡ ❜② ❢❛r ♠♦r❡ ❝♦♠♣❧❡①✳ ❚✇♦ t❤❛t ✇✐❧❧ ❜❡ ♣✉rs✉❡❞ ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❘❡❝❛❧❧✿
• ❲❡ ❝❛❧❧ ❛ ❢✉♥❝t✐♦♥
• ❲❡ ❝❛❧❧ ❛ ❢✉♥❝t✐♦♥
♦♥❡✲t♦✲♦♥❡ ♦♥t♦
✐❢ t❤❡r❡ ✐s ♥♦ ♠♦r❡ t❤❛t ♦♥❡ ✐♥♣✉t ❢♦r ❡❛❝❤ ♦✉t♣✉t✳
✐❢ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ❢♦r ❡❛❝❤ ♦✉t♣✉t✳
❚❤❡ ❢✉♥❝t✐♦♥s ❛❜♦✈❡ ❛r❡ ❛❧❧ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❡r❝✐s❡ ✷✳✷✳✶✵
Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳
✷✳✷✳
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✽
❊①❛♠♣❧❡ ✷✳✷✳✶✶✿ ♣r♦❥❡❝t✐♦♥s ❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦ ❛r❡ t❤❡
♣r♦❥❡❝t✐♦♥s
✳ ❚❤❡r❡ ❛r❡ ❛t ❧❡❛st t✇♦ t②♣❡s✿
❚❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ ♦♥❡✿
F (x, y) = (x, 0) . ■t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡
x✲❛①✐s✳
■t✬s ❛s ✐❢ t❤❡ s❤❡❡t ♦❢ t❤❡
xy ✲♣❧❛♥❡
✐s r♦❧❧❡❞ ✐♥t♦ ❛ t❤✐♥ s❝r♦❧❧✿
❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ❛ ♠❛tr✐①✿
1 0 . F = 0 0
❊①❡r❝✐s❡ ✷✳✷✳✶✷ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤❡ ♣r♦ ❥❡❝t✐♦♥ ♦♥ t❤❡
y ✲❛①✐s✳
❊①❡r❝✐s❡ ✷✳✷✳✶✸ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤❡ ♣r♦ ❥❡❝t✐♦♥ ♦♥ t❤❡ ❧✐♥❡
❊①❛♠♣❧❡ ✷✳✷✳✶✹✿ ❝♦❧❧❛♣s❡ ❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡
❝♦❧❧❛♣s❡
✿
y = x✳
✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✸✾
■t ✐s ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✿ F (x, y) = (x0 , y0 ) .
■t✬s ❛s ✐❢ t❤❡ s❤❡❡t ✐s ❝r✉s❤❡❞ ✐♥t♦ ❛ t✐♥② ❜❛❧❧✿
❚❤❡r❡ ✐s ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ✇❤❡♥ t❤✐s ♣♦✐♥t ✐s t❤❡ ♦r✐❣✐♥✿ 0 0 . F = 0 0
❊①❡r❝✐s❡ ✷✳✷✳✶✺ ❙❤♦✇ t❤❛t t❤❡r❡ ✐s ♥♦ ♠❛tr✐① ✉♥❧❡ss x0 = 0, y0 = 0✳ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✐♥ t❤❡ ♠❛tr✐① ❞❡♣❡♥❞s ♦♥ ✐ts ❧♦❝❛t✐♦♥✿ x y x a b y c d
❚❤✐s ✐s ❛ s♣❡❝✐❛❧ ❝❛s❡ t❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t s♦ ❢❛r✿
▼❛tr✐① ❉❡❝♦♥str✉❝t✐♦♥ str❡t❝❤❡❞ ♦r ✢✐♣♣❡❞ x → a, 0 ← t❤❡r❡ ✐s ♥♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ x, y t❤❡r❡ ✐s ♥♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ x, y → 0, d ← str❡t❝❤❡❞ ♦r ✢✐♣♣❡❞ y ❚❤❡r❡ ❛r❡ ♠❛♥② ♠♦r❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❤♦✇❡✈❡r✱ t❤❛t ❛r❡♥✬t ❝♦✈❡r❡❞ s♦ ❢❛r ❜❡❝❛✉s❡ t❤❡② ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥s✳
❊①❛♠♣❧❡ ✷✳✷✳✶✻✿ ✢✐♣ ❛❜♦✉t ❞✐❛❣♦♥❛❧ ❆ ✢✐♣ ❛❜♦✉t t❤❡ ❧✐♥❡ x = y t❤❛t ❛♣♣❡❛r❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✜♥❞✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✿
❆s ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✈❡rs❡ ❜② ✐♥t❡r❝❤❛♥❣✐♥❣ x ❛♥❞ y ✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ❤❡r❡✿ (x, y) 7→ (y, x) .
❚❤❡ ♠❛tr✐① ✐s✿
0 1 . F = 1 0
✷✳✷✳
❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
✶✹✵
❲❡ s❡❡ ❤❡r❡ s♦♠❡ ♥❡✇ ❢❡❛t✉r❡s✿
▼❛tr✐① ❉❡❝♦♥str✉❝t✐♦♥
x
❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥
y
❞❡♣❡♥❞s ♦♥
x → 0, 1 ← x ❞❡♣❡♥❞s ♦♥ y x → 1, 0 ← y ❞♦❡s♥✬t ❞❡♣❡♥❞s
♦♥
y
❊①❡r❝✐s❡ ✷✳✷✳✶✼ ❋✐♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤✐s r♦t❛t✐♦♥✿
❊①❛♠♣❧❡ ✷✳✷✳✶✽✿ ❝♦♠♣♦s✐t✐♦♥s ▼♦r❡ ❝♦♠♣❧❡① tr❛♥s❢♦r♠❛t✐♦♥s✱ ❤♦✇❡✈❡r✱ ✇✐❧❧ r❡q✉✐r❡ ❢✉rt❤❡r st✉❞②✳ ❇❡❧♦✇✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r♦t❛t✐♦♥ ✇✐t❤ ❛ str❡t❝❤ ✇✐t❤ ❛ ❢❛❝t♦r ♦❢
2✿
❊①❛♠♣❧❡ ✷✳✷✳✶✾✿ ❢♦❧❞✐♥❣ ❆♠♦♥❣ ♦t❤❡rs✱ ✇❡ ♠❛② ❝♦♥s✐❞❡r
❢♦❧❞✐♥❣
t❤❡ ♣❧❛♥❡✿
F (x, y) = (|x|, y) .
90✲❞❡❣r❡❡
✷✳✸✳
▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✶
❊①❡r❝✐s❡ ✷✳✷✳✷✵
❈♦♥✜r♠ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♠❛tr✐①✳ ❖❢ ❝♦✉rs❡✱ ❛♥② ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn ❝❛♥ ❜❡ ✕ ✐♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r ✕ r♦t❛t❡❞ ✭❛r♦✉♥❞ ✈❛r✐♦✉s ❛①❡s✮✱ str❡t❝❤❡❞ ✭✐♥ ✈❛r✐♦✉s ❞✐r❡❝t✐♦♥s✮✱ ♣r♦❥❡❝t❡❞ ✭♦♥t♦ ✈❛r✐♦✉s ❧✐♥❡s ♦r ♣❧❛♥❡s✮✱ ♦r ❝♦❧❧❛♣s❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡
F (x, y, z) = (x, y, 0) ✇♦r❦s✿
❚❤✐s ✐s ❤♦✇ ♠❛♣s ❛r❡ ♠❛❞❡ ✭st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥✮✿
✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs
❲❡ ❝♦♥s✐❞❡r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛s ❜❡❢♦r❡ ❜✉t ✇❡ t❤✐♥❦ ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❛s t❤❡ ❡♥❞s ♦❢ 2✲✈❡❝t♦rs✳ ■t ♠❛❦❡s ♥♦ ❞✐✛❡r❡♥❝❡ ✇❤❡♥ t❤❡② ❛r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♠♣♦♥❡♥ts✿
< x, y > 7→ < u, v > ❚❤❡♥✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❢✉♥❝t✐♦♥s❄ ■t ✉s❡❞ t♦ ❜❡ t❤❡ ❝❛s❡ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♦✉t♣✉t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥♣✉t✿ F (x, y) = (u, v) = 2x − 3y, x + 5y) .
✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✷
◆♦✇ ♦✉r ❢✉♥❝t✐♦♥ s❤♦✇s ❤♦✇ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ♦✉t♣✉t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✐♥♣✉t✿ F < x, y >
=< u, v >=< 2x − 3y, x + 5y > .
❍❡r❡ < x, y > ✐s t❤❡ ✐♥♣✉t ❛♥❞ < u, v > ✐s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t
❢✉♥❝t✐♦♥
< x, y > →
❇♦t❤ ❛r❡ ✈❡❝t♦rs❀ ✐t✬s ❛ ✈❡❝t♦r ❢✉♥❝t✐♦♥ ✿
♦✉t♣✉t → < u, v >
F
F : R2 → R2 . ❲❛r♥✐♥❣✦
❲❡ ❝♦✉❧❞ ✐♥t❡r♣r❡t F ❛s ❛ ✈❡❝t♦r ✜❡❧❞✱ ❜✉t ✇❡ ✇♦♥✬t✳ ❊①❛♠♣❧❡ ✷✳✸✳✶✿ r❡✲s❝❛❧✐♥❣
❙♦♠❡ ♦❢ t❤❡ ❜❡♥❡✜ts ♦❢ t❤✐s ♣♦✐♥t ♦❢ ✈✐❡✇ ❛r❡ ✐♠♠❡❞✐❛t❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❡✈❡♥ ✐❢ ✇❡ ❞✐❞♥✬t ❦♥♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣✐✈❡♥ ❜② F (x, y) = (2x, 2y)
✐s ❛ ✉♥✐❢♦r♠ str❡t❝❤✱ ✇❡ ❝❛♥ ❞✐s❝♦✈❡r t❤❛t ❢❛❝t ✇✐t❤ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛✳ ❲❡ ✇r✐t❡ t❤✐s ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ✈❡❝t♦rs ❛♥❞ ❞✐s❝♦✈❡r s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ F (< x, y >) =< 2x, 2y >= 2 < x, y > .
■♥ ❢❛❝t✱ t❤✐s ✐❞❡❛ ✇✐❧❧ ✇♦r❦ ✐♥ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❚❤✐s ✐s ❤♦✇ ②♦✉ str❡t❝❤ t❤❡ s♣❛❝❡ ❜② ❛ ❢❛❝t♦r ♦❢ 2✱ ✉s✐♥❣ t❤❡ ❝♦♠♣♦♥❡♥t✲❢r❡❡ ❛♣♣r♦❛❝❤✿ F : Rn → Rn ❣✐✈❡♥ ❜② F (X) = 2X .
❚❤❡ ❛♣♣r♦❛❝❤ ❣✐✈❡s ✉s ❛ ❜❡tt❡r r❡♣r❡s❡♥t❛t✐♦♥ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ♠❛❦❡ ✉♣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❤❛♣♣❡♥ t♦ ❜❡ ❧✐♥❡❛r✳ ▼❛tr✐❝❡s r❡♣❧② ♦♥ ✈❡❝t♦r ❛❧❣❡❜r❛✿ x a b =F ·X. · F (X) = f (x, y), g(x, y) = y c d
❊①❡r❝✐s❡ ✷✳✸✳✷
❉♦❡s ✐t ✇♦r❦ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ❧✐♥❡❛r❄ ❚r② F (x, y) =
1 e , ✳ y x
❈♦♥✈❡rs❡❧②✱ ✐❢ ✇❡ ❞♦ ❤❛✈❡ ❛ ♠❛tr✐①✱ ✇❡ ❝❛♥ ❛❧✇❛②s ✉♥❞❡rst❛♥❞ ✐t ❛s ❛ ❢✉♥❝t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿
a b c d
ax + by x =< ax + by, cx + dy > , = cx + dy y
❢♦r s♦♠❡ a, b, c, d ✜①❡❞✳ ❙♦✱ ♠❛tr✐① F ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❢✉♥❝t✐♦♥ F ✳ ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ F ✭❛ t❛❜❧❡✮ ❛s ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ F X ✭❛ ❢♦r♠✉❧❛✮✳ ❲❛r♥✐♥❣✦
❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❧❡tt❡r ❢♦r t❤❡ ❢✉♥❝✲ t✐♦♥ ❛♥❞ t❤❡ ♠❛tr✐①✳
❈❧❡❛r❧②✱ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ❛ ♠❛tr✐① ✐s ❛ s♣❡❝✐❛❧ ♦♥❡✳ ❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t ✐t❄
✷✳✸✳
▲✐♥❡❛r ♦♣❡r❛t♦rs
◆♦✇✱ t❤❡ ❞♦♠❛✐♥
R2
✶✹✸
♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❛
✈❡❝t♦r s♣❛❝❡✱
❛♥❞ s♦ ✐s ✐ts ❝♦❞♦♠❛✐♥✳ ❍♦✇ ❞♦❡s s✉❝❤ ❛ ❢✉♥❝t✐♦♥
✐♥t❡r❛❝t ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡s❡ t✇♦ s♣❛❝❡s❄ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡
✈❡❝t♦r ♦♣❡r❛t✐♦♥s
✉♥❞❡r
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❝❛rr✐❡❞ ♦✉t ✐♥ t❤❡ ❞♦♠❛✐♥ s♣❛❝❡ ♦❢
F❄
F✳
❖♥❝❡
F
❤❛s
tr❛♥s❢♦r♠❡❞ t❤❡ ♣❧❛♥❡✱ ✇❤❛t ❞♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s ❧♦♦❦ ❧✐❦❡ ♥♦✇❄ ❊①❛♠♣❧❡ ✷✳✸✳✸✿ ❞✐♠❡♥s✐♦♥
❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②
1
2✱ f (x) = 2x
✏♣r❡s❡r✈❡s ❛❞❞✐t✐♦♥✑✿
f (x + x′ ) = 2(x + x′ ) = 2x + 2x′ = f (x) + f (x′ ) . ❆❢t❡r ❛❧❧✱ t❤✐s ✐s ❥✉st ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢
2✳
❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠✿
x,x′ ·2 y
+
′ x+ x ·2 y
−−−−→ +
2x, 2x′ −−−−→ 2x + 2x′ = 2(x + x′ ) ■♥ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿
• •
❘✐❣❤t✿ ❆❞❞ t❤❡♠✳ ❚❤❡♥ ❞♦✇♥✿ ❆♣♣❧② t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡ r❡s✉❧t✳ ❉♦✇♥✿ ❆♣♣❧② t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡♠✳ ❚❤❡♥ r✐❣❤t✿ ❆❞❞ t❤❡ r❡s✉❧ts✳
❆ s❤✐❢t ❜②
1✱ f (x) = x + 1✱
❞♦❡s♥✬t ♣r❡s❡r✈❡ ❛❞❞✐t✐♦♥✿
f (x + x′ ) = (x + x′ ) + 1 = x + x′ + 1 6= x + x′ + 2 = (x + 1) + (x′ + 1) = f (x) + f (x′ ) . ❊①❡r❝✐s❡ ✷✳✸✳✹
❲❤❛t ❡✛❡❝t ❞♦❡s t❤❡ ❢✉♥❝t✐♦♥
f :R→R
❣✐✈❡♥ ❜②
f (x) = 2x
❤❛✈❡ ♦♥ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ✐ts ❞♦♠❛✐♥❄
❊①❛♠♣❧❡ ✷✳✸✳✺✿ ❛❞❞✐t✐♦♥ ✉♥❞❡r r♦t❛t✐♦♥
♣❧❛♥❡ ✐s tr❛♥s❢♦r♠❡❞❄ r♦t❛t❡❞✱ ❛s ✐❢ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐t ✇✐❧❧ r❡♠❛✐♥ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✿
❲❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✭t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❝♦♥str✉❝t✐♦♥✮ ✇❤❡♥ t❤❡ ■❢ s✉❝❤ ❛ ❞✐❛❣r❛♠ ✐s
❲❡ s❡❡ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ r✉❧❡ ♦❢ ❛❞❞✐t✐♦♥ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✿
A, B r♦t❛t❡❞ y
+
−−−−→ +
A+ B r♦t❛t❡❞ y
F (A), F (B) −−−−→ F (A) + F (B) = F (A + B) ❲❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✇❤❡♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ ✐s tr❛♥s❢♦r♠❡❞❄ ❲❤❡♥ ✐t ✐s st✐❧❧ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✱ t❤✐s ✐s t❤❡ ❧❛♥❣✉❛❣❡ ✇❡ ✇✐❧❧ ✉s❡✿
✷✳✸✳
▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✹
❉❡✜♥✐t✐♦♥ ✷✳✸✳✻✿ ❛❞❞✐t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ❲❡ s❛② t❤❛t
❛❞❞✐t✐♦♥ ✐s ♣r❡s❡r✈❡❞
✉♥❞❡r ❛ ❢✉♥❝t✐♦♥
F : Rn → Rm
✐❢
F (A + B) = F (A) + F (B) ❢♦r ❛♥② ✈❡❝t♦rs
A
❛♥❞
B✳
❊①❛♠♣❧❡ ✷✳✸✳✼✿ str❡t❝❤ ❞✐♠❡♥s✐♦♥ ✷ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ str❡t❝❤ ❜❡❧♦✇ s❤♦✇s t❤❛t t❤❡ tr✐❛♥❣❧❡s ♦❢ t❤❡ ❞✐❛❣r❛♠✱ ✐❢ ♥♦t ✐❞❡♥t✐❝❛❧✱ ❛r❡
s✐♠✐❧❛r
t♦ t❤❡ ♦r✐❣✐♥❛❧✿
■t✬s ❛s ✐❢ ✇❡ ❥✉st st❡♣♣❡❞ ❛✇❛② ❢r♦♠ t❤❡ ♣✐❡❝❡ ♦❢ ♣❛♣❡r t❤❛t ❤❛s t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ♦♥ ✐t ♦r ♣✉t t❤❡ ❞✐❛❣r❛♠ ✉♥❞❡r ❛ ♠❛❣♥✐❢②✐♥❣ ❣❧❛ss✳
❊①❛♠♣❧❡ ✷✳✸✳✽✿ r❡✢❡❝t✐♦♥ ❞✐♠❡♥s✐♦♥ ✷ ❲❡ ❝❛♥ ❛❧s♦ s❡❡ t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✐♥ t❤❡ ♠✐rr♦r✱ ❛♥❞ ✐t✬s st✐❧❧ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✿
❊①❡r❝✐s❡ ✷✳✸✳✾ ❙❤♦✇ t❤❛t ❛ ❢♦❧❞ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ ❙✉❣❣❡st ♦t❤❡r ❡①❛♠♣❧❡s✳
❲❤❛t ❛❜♦✉t t❤❡ ❣❡♥❡r❛❧ ❝❛s❡❄
❚❤❡♦r❡♠ ✷✳✸✳✶✵✿ Pr❡s❡r✈✐♥❣ ❆❞❞✐t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥ ❛❞❞✐t✐♦♥✳
F : Rn → Rm
✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✱
F (X) = F X ✱
✐t ♣r❡s❡r✈❡s
✷✳✸✳
▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✺
Pr♦♦❢✳
❈♦♥s✐❞❡r
F : R 2 → R2
❛♥❞ t✇♦ ✐♥♣✉t ✈❡❝t♦rs✿
x A= y
❛♥❞
′ x B= ′ . y
▲❡t✬s ❝♦♥✜r♠ t❤❡ ❢♦r♠✉❧❛✿
F (A + B) = F (A) + F (B) . ▲❡t✬s ❝♦♠♣❛r❡✿
▲❡❢t✲❤❛♥❞ s✐❞❡✿
′ x x F + y y′ ′ a b x x = + c d y y′ ′ a b x + x = ′ c d y+y ′ ′ a(x + x ) + b(y + y ) = ′ ′ c(x + x ) + d(y + y )
❘✐❣❤t✲❤❛♥❞ s✐❞❡✿
′ x x F +F y y′ ′ a b x a b x = + c d y c d y′ ′ ′ ax + by ax + by = + ′ ′ cx + dy cx + dy ′ ′ ax + by + ax + by = ′ ′ cx + dy + cx + dy
❚❤❡s❡ ❛r❡ t❤❡ s❛♠❡✱ ❛❢t❡r ❢❛❝t♦r✐♥❣✳
❊①❛♠♣❧❡ ✷✳✸✳✶✶✿ str❡t❝❤ ❞✐♠❡♥s✐♦♥ ✷
❍❡r❡ ✐s ❛ ❝♦♥✜r♠❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧t ❢♦r ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤✿
❲❡ s✐♠♣❧② ❝♦♠♣❛r❡ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❞♦♠❛✐♥ ✇✐t❤ ✐ts ✏r❡✢❡❝t✐♦♥✑ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥✳
❚❤❡ ❞✐❛❣r❛♠ ♦❢ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♠✉❝❤ s✐♠♣❧❡r✿
✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✻
■t ✐s ❛ str❡t❝❤ ♦❢ t❤❡ ✈❡❝t♦r❀ r♦t❛t❡❞ ♦r str❡t❝❤❡❞✱ ✐t r❡♠❛✐♥s ❛ str❡t❝❤✳ ❚❤✐s ✐s t❤❡ ❧❛♥❣✉❛❣❡ ✇❡ ✇✐❧❧ ✉s❡✿
❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✷✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ❲❡ s❛② t❤❛t s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❛ ❢✉♥❝t✐♦♥ F : Rn → Rm ✐❢ F (kA) = kF (A)
❢♦r ❛♥② ✈❡❝t♦r X ❛♥❞ r❡❛❧ k✳ ❚❤❡ ❢♦r♠✉❧❛s ✐♥ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❛r❡ ❥✉st ❛❜❜r❡✈✐❛t✐♦♥s ♦❢ t❤❡s❡ ❞✐❛❣r❛♠s✿ A,B F y
+
−−−−→
A+ B F y
+
F (A), F (B) −−−−→ F (A) + F (B) = F (A + B)
A F y
·k
−−−−→ ·k
kA F y
F (A) −−−−→ kF (A) = F (kA)
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦r❞❡r ♦❢ t❤❡s❡ ♦♣❡r❛t✐♦♥s ♠❛❦❡s ♥♦ ❞✐✛❡r❡♥❝❡✳
❊①❛♠♣❧❡ ✷✳✸✳✶✸✿ ♠♦t✐♦♥s ❚❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ✐s q✉✐❝❦❧② r❡❛❝❤❡❞ ❢♦r t❤❡ ✢✐♣ ❛♥❞ ♦t❤❡r ♠♦t✐♦♥s✿ ❚❤❡ tr✐❛♥❣❧❡s ♦❢ t❤❡ ♥❡✇ ❞✐❛❣r❛♠ ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧✳ ❲❡ ❝❛♥ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✐s ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ✇✐t❤ ♥♦ ❣r✐❞✱ ✇❤✐❝❤ t❤❡♥ ❤❛s ❜❡❡♥ r♦t❛t❡❞✿
❚❤❡♦r❡♠ ✷✳✸✳✶✹✿ Pr❡s❡r✈✐♥❣ ❙❝❛❧❛r ▼✉❧t✐♣❧✐❝❛t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥
F : Rn → Rn
✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✱
F (A) = F A✱
✐t ♣r❡s❡r✈❡s s❝❛❧❛r
♠✉❧t✐♣❧✐❝❛t✐♦♥✳
Pr♦♦❢✳ ❈♦♥s✐❞❡r ❛♥ ✐♥♣✉t ✈❡❝t♦r X =< x, y > ❛♥❞ ❛ s❝❛❧❛r k✳ ❚❤❡♥✱
a b c d
a b x ax + by akx + bky kx a b x . =k =k = = k c d y cx + dy ckx + dky ky c d y
◆♦✇✱ t❤✐s ✐s t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✿
✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✼
❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✺✿ ❧✐♥❡❛r ♦♣❡r❛t♦r ❆ ❢✉♥❝t✐♦♥ F : Rn → Rm t❤❛t ♣r❡s❡r✈❡s ❜♦t❤ ❛❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✭♦r ❛ ❧✐♥❡❛r ♠❛♣✮❀ ✐✳❡✳✱✿ F (U + V ) = F (U ) + F (V ) F (kV ) = kF (V )
❲❛r♥✐♥❣✦ y = ax + b ❤❛s ❜❡❡♥ ❝❛❧❧❡❞ ❛ ✏❧✐♥❡❛r ◆♦✇✱ y = ax ✐s ❝❛❧❧❡❞ ❛ ✏❧✐♥❡❛r ♦♣❡r❛t♦r✑✳
Pr❡✈✐♦✉s❧②✱ ❢✉♥❝t✐♦♥✑✳
❲❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s t♦❣❡t❤❡r✿
❚❤❡♦r❡♠ ✷✳✸✳✶✻✿ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❛♥❞ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r F : Rn → Rm ♣r❡s❡r✈❡s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❀ ✐✳❡✳✱ X = xA + yB =⇒ F (X) = xF (A) + yF (B) ,
❢♦r ❛♥② ✈❡❝t♦rs A ❛♥❞ B ❛♥❞ ❛♥② r❡❛❧ ❝♦❡✣❝✐❡♥ts x ❛♥❞ y ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐❛❣r❛♠ ♦❢ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✇✐❧❧ r❡♠❛✐♥ s✉❝❤ ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿
❊①❡r❝✐s❡ ✷✳✸✳✶✼ ❉❡s❝r✐❜❡ ✇❤❛t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❛♥❞ ✜♥❞ ✐ts ♠❛tr✐①✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ♦✉r ❛♥❛❧②s✐s✳
❚❤❡♦r❡♠ ✷✳✸✳✶✽✿ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✈s✳ ▼❛tr✐❝❡s • ❚❤❡ ❢✉♥❝t✐♦♥ F : R2 → R2 ❞❡✜♥❡❞ ✈✐❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ 2 × 2 ♠❛tr✐① F ✱ F (X) = F X ,
✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳
✷✳✸✳
▲✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✽
• ❈♦♥✈❡rs❡❧②✱ ❡✈❡r② ❧✐♥❡❛r ♦♣❡r❛t♦r F : R2 → R2 ✐s ❞❡✜♥❡❞ ✈✐❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② s♦♠❡ 2 × 2 ♠❛tr✐①✳
Pr♦♦❢✳ ❚❤❡ ✜rst ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t✇♦ t❤❡♦r❡♠s ❛❜♦✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♣r♦✈❡♥ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳
❲❛r♥✐♥❣✦ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ❛r❡♥✬t ✐♥t❡r❝❤❛♥❣❡✲ ❛❜❧❡✱ ❜❡❝❛✉s❡ ♠❛tr✐❝❡s ❡♠❡r❣❡ ♦♥❧② ✇❤❡♥ ❛ ❈❛rt❡✲ s✐❛♥ s②st❡♠ ❤❛s ❜❡❡♥ s♣❡❝✐✜❡❞✳
❈♦r♦❧❧❛r② ✷✳✸✳✶✾✿ ▲✐♥❡❛r ❖♣❡r❛t♦r ❛t 0 ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r t❛❦❡s t❤❡ ③❡r♦ ✈❡❝t♦r t♦ t❤❡ ③❡r♦ ✈❡❝t♦r✿ F (0) = 0 .
❊①❡r❝✐s❡ ✷✳✸✳✷✵ Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r② ✭❛✮ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✭❜✮ ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ♠❛tr✐① ♠✉❧t✐♣❧✐✲ ❝❛t✐♦♥✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✇✐❧❧ ❜❡ ✈✐s✐❜❧❡ ✐♥ t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ ❧❛tt❡r ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ♠❛t❡r✐❛❧✐③❡❞ ✇❤❡♥ ❛ ♠❛tr✐① ✐s ❢♦✉♥❞ ❢♦r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞❡s❝r✐❜❡❞ ❜② ✇❤❛t ✐t ❞♦❡s✳ ❲❡ st❛rt ✇✐t❤ ❛ ❝♦✉♣❧❡ ♦❢ s✐♠♣❧❡ ❡①❛♠♣❧❡s✳ ▲❡t✬s ♥♦t ❢♦r❣❡t✿
◮ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❛r❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡ t✇♦ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s ✕ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛r❡ ✕ ❛r❡ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✐❞❡♥t✐t② ❢✉♥❝t✐♦♥✳ ❍♦✇❡✈❡r✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st ❝♦r♦❧❧❛r②✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✱ 0✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❚❤✐s ✐s t❤❡ s✐♠♣❧❡st ❧✐♥❡❛r ♦♣❡r❛t♦r✿
❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✶✿ ③❡r♦ ♦♣❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥ F : Rn → Rm t❤❛t t❛❦❡s ❡✈❡r② ✈❡❝t♦r t♦ t❤❡ ③❡r♦ ✈❡❝t♦r✱
F (0) = 0 , ✐s ❝❛❧❧❡❞ t❤❡
③❡r♦ ♦♣❡r❛t♦r✳
❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿
0 : Rn → Rm , ❚❤❡ ♠❛tr✐① ♦❢ t❤❡ ③❡r♦ ♦♣❡r❛t♦r ❝♦♥s✐sts✱ ♦❢ ❝♦✉rs❡✱ ♦❢ ❛❧❧ ③❡r♦s✿
0 0 0= ... 0
0 0 ... 0
0 0 ... 0
... ... ... ...
0 0 ... 0
0(X) = 0 .
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✹✾
❲❡ ❝❛♥ ✇r✐t❡✿
0ij = 0 . ❲❤❛t ❛❜♦✉t t❤❡ ✐❞❡♥t✐t② ❢✉♥❝t✐♦♥❄ ❚❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ ♠✉st ❜❡ t❤❡ s❛♠❡✿
❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✷✿ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥
F : Rn → Rn
t❤❛t t❛❦❡s ❡✈❡r② ✈❡❝t♦r t♦ ✐ts❡❧❢✱
F (X) = X , ✐s ❝❛❧❧❡❞ t❤❡
✐❞❡♥t✐t② ♦♣❡r❛t♦r✳
❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿
I : Rn → R n ,
I(X) = X .
❚❤❡ ♠❛tr✐① ♦❢ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿
1 0 I= ... 0 ■t ❤❛s
1✬s
♦♥ t❤❡ ♠❛✐♥ ❞✐❛❣♦♥❛❧ ❛♥❞
0✬s
0 1 ... 0
0 0 ... 0
... ... ... ...
0 0 ... 1
❡❧s❡✇❤❡r❡✳ ❲❡ ❝❛♥ ✇r✐t❡✿
Iij =
(
1 0
✐❢ ✐❢
i=j, i 6= j .
✷✳✹✳ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ✐❧❧✉str❛t❡❞ ♣r❡✈✐♦✉s❧② ✇✐t❤ ❝✉r✈❡s ♣❧♦tt❡❞ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ❛♥❞ t❤❡♥ s❡❡♥ tr❛♥s❢♦r♠❡❞✳
❇❡②♦♥❞ ❥✉st s❛②✐♥❣ t❤❛t t❤❡ ❝✉r✈❡ ✇❛s ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ♦r ❛ s❤❡❡t ♦❢
r✉❜❜❡r✱ ✇❤❛t ❡①❛❝t❧② ❤❛♣♣❡♥s t♦ t❤♦s❡ ❝✉r✈❡s❄ ❯s✐♥❣ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s t♦ r❡♣r❡s❡♥t t❤❡s❡ ❝✉r✈❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢❛✐❧s✳ ❡①❛♠♣❧❡✱ r♦t❛t✐♥❣ s✉❝❤ ❛ ❝✉r✈❡✱
u = g(v)✱
✐♥ t❤❡ ❝♦❞♦♠❛✐♥✿
y = f (x)✱
❋♦r
✐s ❧✐❦❡❧② t♦ ♣r♦❞✉❝❡ ❛ ❝✉r✈❡ t❤❛t ✐s♥✬t t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✱
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
❖✉r ❝❤♦✐❝❡ ✐s✱ t❤❡♥✱ ♣❛r❛♠❡tr✐❝ ❣✐✈❡♥ ❜②✿
❝✉r✈❡s ✿
✶✺✵
P : R → R2 ,
X = P (t) ♦r x = x(t), y = y(t) .
❊①❛♠♣❧❡ ✷✳✹✳✶✿ ♥♦♥✲✉♥✐❢♦r♠ r❡✲s❝❛❧❡
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥✿ ❍❡r❡✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ♠❛tr✐①✿
u = 2x v = 4y
2 0 F = 0 4
❲❡ ❝❛♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❧✐♥❡s t❤r♦✉❣❤ 0✿
❚❤❡② ❛❧❧ r❡♠❛✐♥ str❛✐❣❤t ❡✈❡♥ t❤♦✉❣❤ s♦♠❡ ♦❢ t❤❡♠ r♦t❛t❡✳ ▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ str❛✐❣❤t ❧✐♥❡s t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ s✉❝❤ ❛ ❧✐♥❡ ✐s ✈❡r② s✐♠♣❧❡✿ P (t) = tV ,
✇❤❡r❡ V 6= 0 ✐s s♦♠❡ ✜①❡❞ ✈❡❝t♦r✳ ❆s V ✐s tr❛♥s❢♦r♠❡❞ ❜② F ✱ t❤❡♥ s♦ ❛r❡ ❛❧❧ ♦❢ ✐ts ♠✉❧t✐♣❧❡s✿
■t✬s ❛♥♦t❤❡r str❛✐❣❤t ❧✐♥❡✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❞♦♥✬t ❜❡♥❞✦ ❲❡ ❝♦♥✜r♠ t❤❡ ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✹✳✷✿ ■♠❛❣❡s ♦❢ ▲✐♥❡s ❚❤❡ ✐♠❛❣❡ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥♦t❤❡r str❛✐❣❤t ❧✐♥❡ t❤♦r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✶
Pr♦♦❢✳
❙✉♣♣♦s❡
F
✐s s✉❝❤ ❛♥ ♦♣❡r❛t♦r ❛♥❞
V
P (t) = tV ❚❤✐s ✐s ❛ ❧✐♥❡ ✇✐t❤
F (V )
✐s t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ♦❢ t❤❡ ❧✐♥❡✳ ❚❤❡♥✿
F
−−−−→ (F ◦ P )(t) = F (tV ) = tF (V )
❛s t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✳
■♥ ❣❡♥❡r❛❧✱ ✇❡ ✇✐t♥❡ss t❤❡ ❢♦❧❧♦✇✐♥❣✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡
F
−−−−→
♣❛r❛♠❡tr✐❝ ❝✉r✈❡
▲❡t✬s ❝♦♥s✐❞❡r t✇♦ ♠♦r❡ ❜❛s✐❝ t②♣❡s ♦❢ ❝✉r✈❡s✳ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❝❛❧❧❡❞ ❛ ✏♣❛r❛❜♦❧❛✑✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❝❛r❡ ❛❜♦✉t✿
t❤❡ ♣❛r❛❜♦❧❛ ♦❢ f (x) = x2 ✈✐❛ ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ♦r s❤r✐♥❦✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❡✈❡r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❤❛s ✐ts ✈❡rt❡① ❢♦r♠ ✿ ◮
❆♥② ♣❛r❛❜♦❧❛ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠
f (x) = a(x − h)2 + k .
❚❤❡r❡ ❛r❡ ♠❛♥② ❝✐r❝❧❡s ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❝❛r❡ ❛❜♦✉t✿
◮
❆♥② ❝✐r❝❧❡ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠
t❤❡
❝✐r❝❧❡ ♦❢
■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❡✈❡r② ❝✐r❝❧❡ ❤❛s ✐ts
x2 + y 2 = 1
✈✐❛ ❛ ✉♥✐❢♦r♠ str❡t❝❤ ♦r s❤r✐♥❦✳
❝❡♥t❡r❡❞ ❢♦r♠ ✿
(x − h)2 + (y − k)2 = r2 .
❊①❡r❝✐s❡ ✷✳✹✳✸
❊①♣❧❛✐♥ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✳ ❙❛♠❡ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠s✳
ax
❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ ♥❛t✉r❛❧ ♦♥❡
ex
✈✐❛
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✷
❲❡ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ t♦ s✉♠♠❛r✐③❡ t❤❡s❡ t❤r❡❡ ❝❛s❡s✿ ✏t❡♠♣❧❛t❡✑
/
❧✐♥❡✿
y=x ⌣
♣❛r❛❜♦❧❛✿
y = x2 ◦
❝✐r❝❧❡✿
x2 + y 2 = 1
r❡❧❛t✐♦♥
♣❛r❛♠❡tr✐❝
y − 3 = 2(x − 1)
(x, y) = (1, 3) + t < 1, 2 >
✈❡rt✐❝❛❧ str❡t❝❤ ❜②
2,
s❤✐❢t ✉♣ ❜②
< 1, 3 >
y − 3 = 2(x − 1)2
(x, y) = (1, 3)+ < t, 2t2 >
✈❡rt✐❝❛❧ str❡t❝❤ ❜②
2,
s❤✐❢t ✉♣ ❜②
< 1, 3 >
(y − 3)2 + (x − 1)2 = 22 ✉♥✐❢♦r♠ str❡t❝❤ ❜②
2,
(x, y) = (1, 3) + 2 < cos t, sin t >
s❤✐❢t ✉♣ ❜②
< 1, 3 >
❲❡ ✇✐❧❧ ♥❡❡❞ t♦ ✈✐s✉❛❧✐③❡ ❡①❛♠♣❧❡s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ♦♥ t❤❡ ♣❧❛♥❡✿
F : R2 → R2 . ❲❡ ✐❧❧✉str❛t❡ t❤❡♠ ❜② ❝r❡❛t✐♥❣ ♠❛r❦s ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ❛♥❞ t❤❡♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡♠ ❛s t❤❡② ❛♣♣❡❛r ♦♥ t❤❡ ♥❡✇ ♣❧❛♥❡✳ ❊❛❝❤ ♦❢ t❤❡s❡ ♠❛r❦s ✇✐❧❧ ❜❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ t❤❡ ❞♦♠❛✐♥✿
P : R → R2 , X = P (t) . ❲❡ t❤❡♥ ♣❧♦t ✐ts ✐♠❛❣❡ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥
❚♦ ❜❡ ♣r❡❝✐s❡✱ ✇❡ ♣❧♦t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡
❝♦♠♣♦s✐t✐♦♥
Y = F (X)✿
♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿
F ◦ P : R → R2 , Y = F (P (t)) . ■t ✐s ❛❧s♦ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿
F
R2
−−−−→
↑P
րF ◦P
R2
R ❲❡ ✇✐❧❧ ♣❧♦t ♠❛♥② ♣❛✐rs ♦❢ ❝✉r✈❡s ❡❛❝❤ t✐♠❡✿
♦❧❞ ❝✉r✈❡✿
R2 ↑P R
R2 −→
♥❡✇ ❝✉r✈❡✿
R
րF ◦P
❲❡ ❝❛♥ ✉s❡ t❤✐s s❡t✉♣ ✐♥ t✇♦ ✇❛②s✿ ✶✳ ❲❡ ❝❛♥ st✉❞② ❛ ❝✉r✈❡ ❜② ❛♣♣❧②✐♥❣ ✈❛r✐♦✉s tr❛♥s❢♦r♠❛t✐♦♥s t♦ t❤❡ ♣❧❛♥❡✳ ✷✳ ❲❡ ❝❛♥ st✉❞② ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❜② ❛♣♣❧②✐♥❣ ✐t t♦ ✈❛r✐♦✉s ❝✉r✈❡s✳ ❲❡ ❞✐❞ t❤❡ ❢♦r♠❡r ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ◆♦✇ t❤❡ ❧❛tt❡r✳
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✸
❊①❛♠♣❧❡ ✷✳✹✳✹✿ str❡t❝❤
❙♦✱ ❛♣♣❧②✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s t♦ ❝✉r✈❡s ✇✐❧❧ ❣✐✈❡ ✉s ♥❡✇ ❝✉r✈❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ st❛rt ✇✐t❤ t❤❡ ❝✐r❝❧❡s✿ P (t) = r < cos t, sin t >, r > 0 .
❚❤❡♥✱ ✉s✐♥❣ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② 2 ♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜②✿
str❡t❝❤✐♥❣ r❛❞✐❛❧❧② t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❲❡
Q(t) = 2P (t) = 2r < cos t, sin t > .
■t ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 2r✿
❲❤❛t ❝✉r✈❡s ❞♦ ✇❡ ❝❤♦♦s❡❄ ❋❛♠✐❧✐❛r ♦♥❡s✿ str❛✐❣❤t ❧✐♥❡s ❛♥❞ ❝✐r❝❧❡s✳ ❍♦✇ ♠❛♥②❄ ❚❤❡ ✇❤♦❧❡ ❣r✐❞✳ ❲❡ ❤❛✈❡ t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ❈❛rt❡s✐❛♥ ❣r✐❞✿ P♦❧❛r ❣r✐❞✿
❛ r❡❝t❛♥❣✉❧❛r ❣r✐❞ ♦❢ ❧✐♥❡s
❛ ❣r✐❞ ♦❢ ❝✐r❝❧❡s ❛♥❞ r❛❞✐✐
❚❤❡ ❈❛rt❡s✐❛♥ ❣r✐❞ ✐s ❝r❡❛t❡❞ ✇✐t❤ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❧✐♥❡s✿
❚❤❡s❡ ❧✐♥❡s ❛r❡ ❞❡✜♥❡❞ ♣❛r❛♠❡tr✐❝❛❧❧②✿ ✶✳ ❤♦r✐③♦♥t❛❧✿ x = t, y = k, k = ... − 3, −2, −1, 0, 1, 2, 3, ... ✷✳ ✈❡rt✐❝❛❧✿ x = k, y = t, k = ... − 3, −2, −1, 0, 1, 2, 3, ...
❚❤❡ ♣♦❧❛r ❣r✐❞ ✐s ❝r❡❛t❡❞ ✇✐t❤ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❧✐♥❡s✿
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✹
❍❡r❡ t❤❡② ❛r❡ ❞❡✜♥❡❞ ♣❛r❛♠❡tr✐❝❛❧❧②✿ ✶✳ r❛②s✿ x = at, y = bt, a, b r❡❛❧❀ ❛♥❞ ✷✳ ❝✐r❝❧❡s✿ x = r cos t, y = r sin t, r r❡❛❧✳ ❲❡ ✇✐❧❧ ❛♣♣❧② t❤❡ ❈❛rt❡s✐❛♥ ♦r t❤❡ ♣♦❧❛r ❛s ♥❡❡❞❡❞✳ ❊①❛♠♣❧❡ ✷✳✹✳✺✿ ❝♦❧❧❛♣s❡ ♦♥ ❛①✐s
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ✈❡r② s✐♠♣❧❡ ❢✉♥❝t✐♦♥✿
u = 2x v =0
x 2 0 u . · = ✱ r❡✲✇r✐tt❡♥✿ y 0 0 v
❇❡❧♦✇✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤✐s ❢✉♥❝t✐♦♥ ❝♦❧❧❛♣s❡s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡ x✲❛①✐s✿
❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡✈❡r② ❝✐r❝❧❡✿
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ x✲❛①✐s ✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❲❡ ❝❛♥ s❡❡ ❜♦t❤ ✐♥ t❤❡ ♠❛tr✐①✿ str❡t❝❤ ♦❢ x → 2, 0 ← y ❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥ x → 0, 0 ←
❝♦❧❧❛♣s❡
❇❡❝❛✉s❡ ♦❢ t❤❡ ❝♦❧❧❛♣s❡ ♦❢ t❤❡ y ✲❛①✐s✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✻✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s
▲❡t✬s r❡✈✐s✐t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ❲❡ ❧♦♦❦ ❛t t❤❡ ❝✐r❝❧❡s ✜rst✿
2 0 . F = 0 4
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✺
❚❤❡ ❝✐r❝❧❡s ❤❛✈❡ ❜❡❝♦♠❡ ❡❧❧✐♣s❡s✦ ❲❡ ❝❛♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ♠❛tr✐①✿ str❡t❝❤ ♦❢
y
❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥
x → 2, 0 ← x ❞♦❡s♥✬t ❞❡♣❡♥❞s x → 0, 4 ← str❡t❝❤ ♦❢ y
♦♥
y
❚❤❡ ❛①❡s st❛② ♣✉t✳ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ r❡st ♦❢ t❤❡ ♣❧❛♥❡❄ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❧✐♥❡s ♥♦✇✿
❙✐♥❝❡ t❤❡ str❡t❝❤✐♥❣ ✐s ♥♦♥✲✉♥✐❢♦r♠✱ t❤❡ ✈❡❝t♦rs t✉r♥✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ t❤❡ ❜❛s✐s ✈❡❝t♦rs
e1
❛♥❞
e2
❞♦♥✬t
t✉r♥✱ t❤✐s ✐s ♥♦t ❛ r♦t❛t✐♦♥ ❜✉t r❛t❤❡r ❛ ✏❢❛♥♥✐♥❣ ♦✉t✑ ♦❢ t❤❡ ✈❡❝t♦rs✳ ❚❤❡✐r s❧♦♣❡s ❤❛✈❡ ✐♥❝r❡❛s❡❞✳ ❲❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳
❊①❛♠♣❧❡ ✷✳✹✳✼✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s ❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥ ✐s✿
u = −x v = 4y
■t ✐s s✐♠♣❧❡ ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❏✉st t❤❡ ❝✐r❝❧❡s✿
❚❤❡ s❧✐❣❤t ❝❤❛♥❣❡ t♦ t❤❡ ❢✉♥❝t✐♦♥ ♣r♦❞✉❝❡s ❛ s✐♠✐❧❛r ❜✉t ❞✐✛❡r❡♥t ♣❛tt❡r♥✿ ❲❡ s❡❡ t❤❡ r❡✈❡rs❛❧ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳ ❲❡ ✇❛② t❤❛t
t❤❡ ♦r✐❡♥t❛t✐♦♥ ❤❛s ❝❤❛♥❣❡❞✳
❚❤❡ ♠❛tr✐① ♦❢
F
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✐s st✐❧❧ ❞✐❛❣♦♥❛❧✿ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳
✶✺✻
−1 0 . F = 0 4
❊①❛♠♣❧❡ ✷✳✹✳✽✿ ❡①♣❡r✐♠❡♥t
▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥✿
u = x +2y v = 2x +4y
1 2 . =⇒ F = 2 4
■t ✐s ❤❛r❞ t♦ t❡❧❧ ✇❤❛t ✐t ❞♦❡s✱ ❥✉❞❣✐♥❣ ❜② ✐ts ♠❛tr✐①✳ ❲❡ ❡①♣❡r✐♠❡♥t ✿
■t ❛♣♣❡❛rs t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s str❡t❝❤✐♥❣ t❤❡ ♣❧❛♥❡ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦❧❧❛♣s✐♥❣ ✐♥ ❛♥♦t❤❡r✳ ❚❤❛t✬s ✇❤② t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚♦ ✜♥❞ ✐t✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿
x +2y = 0 2x +4y = 0
=⇒ x = −2y .
❚❤❡ ✈❡❝t♦r < 1, 2 > ✐s✱ ✐♥ ❢❛❝t✱ ✈✐s✐❜❧❡ ✐♥ t❤❡ ♠❛tr✐①✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❝♦❧❧❛♣s❡ ♦❢ t❤❡ ❣r❡❡♥ ❧✐♥❡ t♦ t❤❡ ♦r✐❣✐♥✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♦♥t♦✳ ❊①❡r❝✐s❡ ✷✳✹✳✾
❙❤♦✇ ✇❤❡r❡ ❡❛❝❤ ♣♦✐♥t ❣♦❡s✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✵✿ ❡①♣❡r✐♠❡♥t
❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① F ✿
−1 −2 . F = 1 −4
❆❣❛✐♥✱ ✐t✬s t♦♦ ❝♦♠♣❧❡① t♦ r❡✈❡❛❧ ✇❤❛t ✐t ❞♦❡s✱ ❛♥❞ ✇❡ ❤❛✈❡ t♦ ❡①♣❡r✐♠❡♥t✿
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✼
■t ❧♦♦❦s ❧✐❦❡ ❛ ♥♦♥✲✉♥✐❢♦r♠ str❡t❝❤ ❛❧♦♥❣ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥s✳
❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞
♦♥t♦✳
❊①❡r❝✐s❡ ✷✳✹✳✶✶
F✿ 1 2 . F = 3 2
❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r
❊①❛♠♣❧❡ ✷✳✹✳✶✷✿ s❦❡✇✐♥❣✲s❤❡❛r✐♥❣ ❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿
❇❡❧♦✇✱ ✇❡ r❡♣❧❛❝❡ ❝✐r❝❧❡s ✇✐t❤
❡❧❧✐♣s❡s
1 1 . F = 0 1
❛♥❞ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡♠ ✉♥❞❡r s✉❝❤ ❛ ❢✉♥❝t✐♦♥✿
❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ str❡t❝❤ ❛❧♦♥❣ t❤❡
x✲❛①✐s✳
❚❤❡r❡ ✐s st✐❧❧ ❛♥❣✉❧❛r str❡t❝❤✲s❤r✐♥❦ ❜✉t t❤✐s t✐♠❡ ✐t ✐s
❜❡t✇❡❡♥ t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ s❛♠❡ ❧✐♥❡✳
❚♦ s❡❡ ♠♦r❡ ❝❧❡❛r❧②✱ ❝♦♥s✐❞❡r t❤❡ ❈❛rt❡s✐❛♥ ❣r✐❞✳ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❛ sq✉❛r❡✿
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✽
❚❤❡ ♣❧❛♥❡ ✐s s❦❡✇❡❞✱ ❧✐❦❡ ❛ ❞❡❝❦ ♦❢ ❝❛r❞s✿
❙✉❝❤ ❛ s❦❡✇✐♥❣ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ❛♥② ✐♠❛❣❡✲❡❞✐t✐♥❣ s♦❢t✇❛r❡✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✸✿ r♦t❛t✐♦♥
π/2
❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 90 ❞❡❣r❡❡s✿ (x, y) ❜❡❝♦♠❡s (−y, x)✳ ❲❡ ❤❛✈❡✿
x 0 1 u . · = ✱ r❡✲✇r✐tt❡♥✿ y −1 0 v
u = −y v =x
❚❤❡ ❡①♣❡r✐♠❡♥t ❝♦♥✜r♠s ✇❤❛t ✇❡ ❦♥♦✇✿
❲❡✬✈❡ ❤❛❞ ♠❛♥② ❡①❛♠♣❧❡s✱ ❜✉t ❤♦✇ ❞♦ ✇❡ ❜✉✐❧❞ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❢r♦♠ ❛ ❞❡s❝r✐♣t✐♦♥❄ ❚❤❡ s♦❧✉t✐♦♥ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✹✳✶✹✿ ❈♦❧✉♠♥s ❛r❡ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs ❚❤❡ t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ ♠❛tr✐① ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r✿
a b F = c d
a 1 = =⇒ F c 0
❛♥❞
b 0 . = F d 1
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✺✾
❊①❡r❝✐s❡ ✷✳✹✳✶✺
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ❥✉st ❛s ✐♠♣♦rt❛♥t✿ ❚❤❡♦r❡♠ ✷✳✹✳✶✻✿ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs ❆r❡ ❈♦❧✉♠♥s ❚❤❡ ♠❛tr✐① ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♠❡r❣❡ t❤❡ t✇♦ ❝♦❧✉♠♥✲✈❡❝t♦rs ✐♥t♦ ❛ ♠❛tr✐①✿ 1 a 7→ F : , 0 c
0 b a b 7→ F : ▼❡r❣❡✿ F = . 1 d c d
❊①❛♠♣❧❡ ✷✳✹✳✶✼✿ ♠❛tr✐❝❡s ❢r♦♠ ✈❛❧✉❡s
❚❤✐s ✐s t❤❡ ③❡r♦ ♦♣❡r❛t♦r✿
❚❤✐s ✐s t❤❡ ✐❞❡♥t✐t②✿
0 1 , 7→ 0 0
0 0 7→ 0 1
0 0 . ▼❡r❣❡✿ 0 = 0 0
1 1 , 7→ 0 0
0 0 7→ 1 1
1 0 . ▼❡r❣❡✿ I = 0 1
2 1 , 7→ 0 0
0 0 7→ 1 1
2 0 . ▼❡r❣❡✿ Sx = 0 1
0 0 7→ 3 1
1 0 . ▼❡r❣❡✿ Sy = 0 3
0 0 7→ 1 1
−1 0 . ▼❡r❣❡✿ Fx = 0 1
0 0 7→ −1 1
1 0 . ▼❡r❣❡✿ Fy = 0 −1
❚❤✐s ✐s t❤❡ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜② 2✿
❆♥❞ t❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤ ❜② 3✿ 1 1 , 7→ 0 0
❚❤✐s ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣✿ −1 1 , 7→ 0 0
❆♥❞ t❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ ✢✐♣✿ 1 1 , 7→ 0 0
❚❤✐s ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✿ 1 0 7→ , 0 1
0 1 7→ 1 0
0 1 . ▼❡r❣❡✿ Fd = 1 0
❲❛r♥✐♥❣✦ ■ts ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✶✻✵
❊①❡r❝✐s❡ ✷✳✹✳✶✽
❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • ❧❡❛✈❡s t❤❡ x✲❛①✐s ✐♥t❛❝t✱ ❛♥❞ • str❡t❝❤❡s t❤❡ y ✲❛①✐s ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✶✾
❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • r♦t❛t❡s t❤❡ x✲❛①✐s 45 ❞❡❣r❡❡s ❝❧♦❝❦✇✐s❡✱ ❛♥❞ • ✢✐♣s t❤❡ y ✲❛①✐s✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✵
❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • ❧❡❛✈❡s t❤❡ x✲❛①✐s ✐♥t❛❝t✱ ❛♥❞ • str❡t❝❤❡s t❤❡ ❞✐❛❣♦♥❛❧ y = x ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✶
▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✜♥❞ ✐ts ♠❛tr✐①✳ ❘❡♣❡❛t✳ ▲❡t✬s ❛♣♣❧② t❤✐s r❡s✉❧t t♦ s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✇❡ ❤❛✈❡ ❜❡❡♥ ✐♥t❡r❡st❡❞ ✐♥✳ ❚❤❡♦r❡♠ ✷✳✹✳✷✷✿ ▼❛tr✐① ♦❢ ❘♦t❛t✐♦♥ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦❢ r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛♥❣❧❡ ♠❛tr✐①✿
cos α − sin α R= sin α cos α
Pr♦♦❢✳
❲❡ ♦♥❧② ♥❡❡❞ t♦ s❡❡ ✇❤❡r❡ t❤❡ ❜❛s✐s ✈❡❝t♦rs < 1, 0 > ❛♥❞ < 0, 1 > ❣♦✳
❚❤❡ ✜rst ♦♥❡ ✐s s✐♠♣❧❡✿
cos α 1 . = R sin α 0
❚❤❡ s❡❝♦♥❞ ♦♥❡ ✢✐♣s t❤❡ s✐❣♥ ♦❢ t❤❡ x✲❝♦♠♣♦♥❡♥t✿
− sin α 0 . = R cos α 1
α
✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣
✷✳✹✳
❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs
❊①❛♠♣❧❡ ✷✳✹✳✷✸✿ r♦t❛t✐♦♥
✶✻✶
π/4
❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 45 ❞❡❣r❡❡s✿ (
u = cos π4 x − sin π4 y
✱ r❡✲✇r✐tt❡♥✿
v = sin π4 x + cos π4 y
" # u v
=
"
cos π4 − sin π4 sin π4
cos π4
# " # x · . y
❲❡ ♣❧♦t ❡❧❧✐♣s❡s ✐♥st❡❛❞ ♦❢ ❝✐r❝❧❡s t♦ ♠❛❦❡ ✐t ❡❛s✐❡r t♦ s❡❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✿
❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✷✹✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦
▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✿
❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ♦✉r ❡❧❧✐♣s❡s✿
u = 3x −13y . v = 5x +y 3 −13 . F = 5 1
❲❡ s❡❡♠ t♦ ❤❛✈❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ str❡t❝❤✐♥❣✱ ✢✐♣♣✐♥❣✱ ❛♥❞ r♦t❛t✐♥❣✳✳✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❋♦r ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ t❤❡r❡ ✐s ❛♥ ❡❛s② ✇❛② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ✇❡ ❤❛✈❡ ❜❡❡♥ ❛s❦✐♥❣✿ ❲❤❡♥ ✐s ✐t ♦♥❡✲t♦✲♦♥❡❄
✷✳✺✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
✶✻✷
❚❤❡♦r❡♠ ✷✳✹✳✷✺✿ ❖♥❡✲t♦✲♦♥❡ ▲✐♥❡❛r ❖♣❡r❛t♦r ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r
F
✐s ♦♥❡✲t♦✲♦♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❡q✉❛t✐♦♥
F (X) = 0
❤❛s ♦♥❧②
t❤❡ ③❡r♦ s♦❧✉t✐♦♥✳
Pr♦♦❢✳
❙✉♣♣♦s❡ t❤❡r❡ ❛r❡ t✇♦ ❞✐st✐♥❝t s♦❧✉t✐♦♥s X 6= Y ✳ ❚❤❡♥✱ ✇❡ ❝♦♥❝❧✉❞❡✿ F (X) = F (Y ) =⇒ F (X) − F (Y ) = 0 =⇒ F (X − Y ) = 0 .
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦✉♥❞ s✉❝❤ ❛ Z = X − Y 6= 0 t❤❛t F (Z) = 0 = F (0)✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✻
Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ F ✐s ♦♥❡✲t♦✲♦♥❡ ✇❤❡♥ F (X) = 0 =⇒ X = 0 .
❙♦✱ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛ ♠✐①t✉r❡ ♣r♦❜❧❡♠ ❤❛s ❛ s✐♥❣❧❡ s♦❧✉t✐♦♥✱ ✇❡ ❝❤♦♦s❡✱ ✐♥ ❛ t✇✐st✱ t♦ r❡♣❧❛❝❡ ✐t ✇✐t❤ ❛ ♠✐①t✉r❡ ♣r♦❜❧❡♠ t❤❛t r❡q✉✐r❡s t♦ ♣r♦❞✉❝❡ ③❡r♦s ✐♥ ❛❧❧ ❡q✉❛t✐♦♥s✳ ❚❤❡♥ ✇❡ ❛s❦ ✐❢ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥✳
✷✳✺✳ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
❊①❛♠♣❧❡ ✷✳✺✳✶✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♥♦t
❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿
1 1 . A= 0 1
■s t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦♥❡✲t♦✲♦♥❡❄ ❊✈❡r② ♦♥❡ ♦❢ ♦✉r ❡❧❧✐♣s❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ❤❛s ❜❡❡♥ str❡t❝❤❡❞ ❛♥❞ ♠❛②❜❡ r♦t❛t❡❞ ❜✉t t❤❡② st✐❧❧ ❝♦✈❡r t❤❡ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥✿
■t ✐s ♦♥❡✲t♦✲♦♥❡✳ ❚❤✐s ♦♥❡ ✐s ❞✐✛❡r❡♥t✿
1 2 . A= 2 4
❲❡ ❥✉st ✇❛t❝❤ ✇❤❡r❡ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ❣♦✿
✷✳✺✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
✶✻✸
❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ ❢♦r♠❡r ❝❛s❡❄ ❚❤❡② ❣♦ t♦ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿ ❚❤❡✐r ✈❛❧✉❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❝t♦r < 1, 2 >✳ ❲❡ ❝❛♥ s❡❡ t❤❛t ✐♥ t❤❡ ♠❛tr✐① t❤❛t t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ✐s t✇✐❝❡ t❤❡ ✜rst✿ 2
2 1 . = 4 2
■♥ ❢❛❝t✱ ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ✈❡❝t♦rs ❣♦❡s t♦ 0❀ ✐t✬s ♥♦t ♦♥❡✲t♦✲♦♥❡✦
❉❡✜♥✐t✐♦♥ ✷✳✺✳✷✿ s✐♥❣✉❧❛r ♠❛tr✐① ❆ 2 × 2 ♠❛tr✐① A ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r ✇❤❡♥ ✐ts t✇♦ ❝♦❧✉♠♥s ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❙♦✱ ❢♦r ❛ s✐♥❣✉❧❛r ♠❛tr✐①
a b , A= c d
t❤❡r❡ ✐s s✉❝❤ ❛♥ x t❤❛t✿
b a . =x d c
▲❡t✬s ❡①❛♠✐♥❡ t❤✐s ✐❞❡❛✿ ❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s ✐s ❛ ♠❛tr✐① s✐♥❣✉❧❛r❄ ❲❡ ❜r❡❛❦ t❤❡ ✈❡❝t♦r ❡q✉❛t✐♦♥ ❛❜♦✈❡ ✐♥t♦ t✇♦ s❝❛❧❛r ❡q✉❛t✐♦♥s✿ a = xb, c = xd .
■♥st❡❛❞ ♦❢ s♦❧✈✐♥❣ t❤❡♠ ❢♦r x✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s s✉❝❤ ❛♥ x✳ ❲❡ ♠✉❧t✐♣❧② t❤❡ ✜rst ❜② d✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ❜② b✱ ❛♥❞ t❤❡♥ s✉❜tr❛❝t t♦ ❡❧✐♠✐♥❛t❡ x✿ ad = xbd cb = xbd ad − bc = 0
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐❢ s✉❝❤ ❛♥ x ❡①✐sts✱ t❤❡♥
ad − bc = 0 .
■♥ t❤✐s ❡①♣r❡ss✐♦♥✱ t❤❡ t❡r♠s ♦❢ t❤❡ ♠❛tr✐① ❛r❡ ❝r♦ss✲♠✉❧t✐♣❧✐❡❞ ❛♥❞ s✉❜tr❛❝t❡❞✿
a b c d
→ ad − bc .
❚❤✐s ♥✉♠❜❡r ✐s ❛♥ ✐♠♣♦rt❛♥t ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ♠❛tr✐①✿
❉❡✜♥✐t✐♦♥ ✷✳✺✳✸✿ ❞❡t❡r♠✐♥❛♥t
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ 2 × 2 ♠❛tr✐① A ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ a b = ad − bc det A = det c d
❲❤❛t ❞♦❡s t❤❡ ❞❡t❡r♠✐♥❛♥t ❞❡t❡r♠✐♥❡❄
✷✳✺✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
✶✻✹
❚❤❡♦r❡♠ ✷✳✺✳✹✿ ❙✐♥❣✉❧❛r ▼❛tr✐① ❛♥❞ ❉❡t❡r♠✐♥❛♥t ❆
2×2
♠❛tr✐①
A
✐s s✐♥❣✉❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢
det A = 0✳
Pr♦♦❢✳
✭⇒✮ ❙✉♣♣♦s❡ A ✐s s✐♥❣✉❧❛r✱ t❤❡♥ a = xb b a =⇒ =x =⇒ det A = ad − bc = (xb)d − b(xd) = 0 . c = xd d c
✭⇐✮ ❙✉♣♣♦s❡ ad − bc = 0✱ t❤❡♥ ❧❡t✬s ✜♥❞ x✱ t❤❡ ♠✉❧t✐♣❧❡✳ a • ❈❛s❡ ✶✿ ❆ss✉♠❡ b 6= 0✱ t❤❡♥ ❝❤♦♦s❡ x = ✳ ❚❤❡♥ b a = a, xb = b b a ad bc xd = d = = = c. b b b ❙♦ a b . = x c d • ❈❛s❡ ✷✿ ❆ss✉♠❡ a 6= 0 ✳✳✳ ❊①❡r❝✐s❡ ✷✳✺✳✺
❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ❲❡ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t t❤❡ ❞❡t❡r♠✐♥❛♥t✱ ✇❤✐❝❤ ✇✐❧❧ ❛❧s♦ r❡❛♣♣❡❛r ✐♥ t❤❡ ❝❛s❡ ♦❢ n × n ♠❛tr✐❝❡s✿ ◮ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ❛♥ ❛❧t❡r♥❛t✐♥❣ s✉♠ ♦❢ t❡r♠s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ n ♦❢ t❤❡
♠❛tr✐①✬s ❡♥tr✐❡s✱ ❡①❛❝t❧② ♦♥❡ ❢r♦♠ ❡❛❝❤ r♦✇ ❛♥❞ ❡①❛❝t❧② ♦♥❡ ❢r♦♠ ❡❛❝❤ ❝♦❧✉♠♥✳
▲❡t✬s ❝♦♥s✐❞❡r ❛ s♣❡❝✐❛❧ ♠❛tr✐① ❡q✉❛t✐♦♥✱ ✇✐t❤ t❤❡ ③❡r♦ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ F (X) = 0 .
■t ✐s ❝❛❧❧❡❞ ❛ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡r❡ ✐s ❛❧✇❛②s ❛t ❧❡❛st ♦♥❡ s♦❧✉t✐♦♥✱ t❤❡ ③❡r♦ ✈❡❝t♦r✦ ❚❤❡ q✉❡st✐♦♥ ✐s t❤❡♥ ❜❡❝♦♠❡s ◮ ❆r❡ t❤❡r❡ ❛♥② ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥s❄
❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛♥s✇❡r ♠❛② ❜❡ ♣r♦✈✐❞❡❞ ❜② ❛ ♠♦r❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ◮ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡❄
▲❡t✬s st❛rt ✇✐t❤ ❞✐♠❡♥s✐♦♥ 1✿ ❚❤✐s ✐s s✐♠♣❧❡✿
f (x) = mx, s♦❧✈❡ f (x) = 0 . mx = 0 =⇒ x = 0 ✳✳✳ ✉♥❧❡ss m = 0.
❆❧❧ ❢✉♥❝t✐♦♥s ❡①❝❡♣t t❤❡ ❝♦♥st❛♥t ③❡r♦ ❢✉♥❝t✐♦♥ ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝❛♥ ♦♥❧② ♣r♦❞✉❝❡ ❛ s✐♥❣❧❡ s♦❧✉t✐♦♥ ❢♦r ♦✉r ❡q✉❛t✐♦♥✿
✷✳✺✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
✶✻✺
■s t❤❡r❡ ❛ s✐♠✐❧❛r ❞❡❝✐s✐✈❡ ❝♦♥❞✐t✐♦♥ ✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡❄ ❲❡ ❞♦ ❤❛✈❡ t❤❡ ③❡r♦ ♦♣❡r❛t♦r ✭♠❛tr✐①✮✿
F (X) = 0
❢♦r ❛❧❧
X.
■t ❝♦❧❧❛♣s❡s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡ ♦r✐❣✐♥✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♦t❤❡r✱ ❧❡ss ❡①tr❡♠❡ ❝♦❧❧❛♣s❡s✱ ♣r♦❥❡❝t✐♦♥s ♦♥t♦ ❧✐♥❡s✿
❚❤❡♥ ❛ ✇❤♦❧❡ ❧✐♥❡ ✐s t❛❦❡♥ t♦
0✳
❚❤❡② ❛r❡ ❛❧s♦ ♥♦t ♦♥❡✲t♦✲♦♥❡✦
❲❡ ❞❡♣❧♦② s✐♠♣❧❡ ❛❧❣❡❜r❛ ✐♥ ♦r❞❡r t♦ r❡s♦❧✈❡ t❤✐s ✐ss✉❡✳
❚❤❡♦r❡♠ ✷✳✺✳✻✿ ◆♦♥✲③❡r♦ ❙♦❧✉t✐♦♥s ❙✉♣♣♦s❡
A
t❤❡ ♠❛tr✐①
2 × 2 ♠❛tr✐①✳ ❚❤❡♥✱ det A 6= 0 ✐❢ ❡q✉❛t✐♦♥ AX = 0 ❝♦♥s✐sts ♦❢ ♦♥❧② 0✳ ✐s ❛
❛♥❞ ♦♥❧② ✐❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢
Pr♦♦❢✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❩❡r♦ ❋❛❝t♦r Pr♦♣❡rt②✿ ❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♥✉♠❜❡rs ✐s ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡♠ ✭♦r ❜♦t❤✮ ✐s ③❡r♦❀ ✐✳❡✳✱
a = 0 ❖❘ b = 0 ⇐⇒ ab = 0 . ▲❡t✬s s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
ax + by = 0, cx + dy = 0.
(1) (2)
❋r♦♠ ✭✶✮✱ ✇❡ ❞❡r✐✈❡✿
y = −ax/b,
♣r♦✈✐❞❡❞
b 6= 0. (3)
❙✉❜st✐t✉t❡ t❤✐s ✐♥t♦ ✭✷✮✿
cx + d(−ax/b) = 0 . ❚❤❡♥
x(c − da/b) = 0 ,
✷✳✺✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
✶✻✻
♦r✱ ❛❧t❡r♥❛t✐✈❡❧②✱
x(cb − da) = 0, ✇❤❡♥ b 6= 0 .
❖♥❡ ♣♦ss✐❜✐❧✐t② ✐s x = 0❀ ✐t ❢♦❧❧♦✇s ❢r♦♠ ✭✸✮ t❤❛t y = 0 t♦♦✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t✇♦ ❝❛s❡s ❢♦r b 6= 0✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❈❛s❡ ✶ ❞♦❡s♥✬t ✐♥t❡r❡st ✉s✳ ■♥ ❝❛s❡ ✷✱ x ✐s ❛r❜✐tr❛r② ❛♥❞ t❤❡r❡ ♠❛② ❜❡ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥s✳ ◆♦✇✱ ✇❡ ❛♣♣❧② t❤✐s ❛♥❛❧②s✐s t♦ y ✐♥ ✭✶✮ ✐♥st❡❛❞ ♦❢ x❀ ✇❡ ❤❛✈❡ ❢♦r a 6= 0✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ✇❡ ❛♣♣❧② t❤✐s ❛♥❛❧②s✐s ❢♦r x ❛♥❞ y ✐♥ ✭✷✮ ✐♥st❡❛❞ ♦❢ ✭✶✮✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ t✇♦ ❝❛s❡s✳ ❚❤✉s✱ ✇❤❡♥❡✈❡r ♦♥❡ ♦❢ t❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts✱ a, b, c, d✱ ✐s ♥♦♥✲③❡r♦✱ ✇❡ ❤❛✈❡ t❤❡s❡ ❝❛s❡s✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❇✉t ✇❤❡♥ a = b = c = d = 0✱ ❈❛s❡ ✷ ✐s s❛t✐s✜❡❞✳✳✳ ❛♥❞ ✇❡ ❝❛♥ ❤❛✈❡ ❛♥② ✈❛❧✉❡s ❢♦r x ❛♥❞ y ✦ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❛♥❛❧②s✐s ❛❜♦✈❡✿ det A 6= 0 =⇒ x = y = 0 .
❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✳ ■♥❞❡❡❞✱ ❧❡t✬s ❝♦♥s✐❞❡r ♦✉r s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❛❣❛✐♥✿
ax + by = 0, cx + dy = 0.
(1) (2)
❲❡ ♠✉❧t✐♣❧② ✭✶✮ ❜② c ❛♥❞ ✭✷✮ ❜② a✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ cax acx (ca − ac)x 0·x
+ + + −
cby ady (cb − ad)y det A · y
=0 =0 =0 =0
c · (1) a · (2)
❚❤❡ t❤✐r❞ ❡q✉❛t✐♦♥ ✐s t❤❡ r❡s✉❧t ♦❢ s✉❜tr❛❝t✐♦♥ ♦❢ t❤❡ ✜rst t✇♦✳ ❚❤❡ ✇❤♦❧❡ ❡q✉❛t✐♦♥ ✐s ③❡r♦ ✇❤❡♥ det A = 0✦ ❚❤✐s ♠❡❛♥s t❤❛t ❡q✉❛t✐♦♥s ✭✶✮ ❛♥❞ ✭✷✮ r❡♣r❡s❡♥t t✇♦ ✐❞❡♥t✐❝❛❧ ❧✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s✳ ❊①❡r❝✐s❡ ✷✳✺✳✼
❲❤❛t ✐❢ a = 0 ♦r c = 0❄ ❊①❛♠♣❧❡ ✷✳✺✳✽✿ ❝♦♠♣✉t✐♥❣ ❞❡t❡r♠✐♥❛♥ts
❚❤❡ ✢✐♣ ♦✈❡r t❤❡ y ✲❛①✐s✿
❚❤❡ str❡t❝❤✿
−1 0 = (−1) · 1 − 0 · 0 = −1 . det 0 1
λ 0 = λ · µ − 0 · 0 = λ · µ. det 0 µ
❚❤❡ r♦t❛t✐♦♥✿ cos α − sin α = cos α · cos α − (− sin α) · sin α = cos2 α + sin2 α = 1 , det sin α cos α
❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳
✷✳✺✳
✶✻✼
❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①
❚❤❡s❡ ❤❛✈❡ ♥♦♥✲③❡r♦ ❞❡t❡r♠✐♥❛♥ts✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ❤❛s ❛ ③❡r♦ ❞❡t❡r♠✐♥❛♥t✿ λ 0 = λ · 0 − 0 · 0 = 0. det 0 0
❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ s♦♠❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t t❤❛♥ ❥✉st t❤❛t ✐t✬s ♦♥❡✲t♦✲♦♥❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❝♦♥tr❛✲♣♦s✐t✐✈❡ ❢♦r♠ ♦❢ t❤❡ t❤❡♦r❡♠✿ ❈♦r♦❧❧❛r② ✷✳✺✳✾✿ ◆♦♥✲③❡r♦ ❙♦❧✉t✐♦♥s
A ✐s ❛ 2 × 2 det A = 0✳
❙✉♣♣♦s❡ ♦♥❧② ✐❢
X 6= 0
♠❛tr✐①✳ ❚❤❡♥✱ t❤❡r❡ ✐s s✉❝❤ ❛♥
t❤❛t
AX = 0
✐❢ ❛♥❞
❙✐♥❝❡ A(0) = 0✱ t❤✐s ✐♥❞✐❝❛t❡s t❤❛t A ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✳ ❚❤❡r❡ ✐s ♠♦r❡✿ ❈♦r♦❧❧❛r② ✷✳✺✳✶✵✿ ❇✐❥❡❝t✐♦♥s ❛♥❞ ❉❡t❡r♠✐♥❛♥ts ❙✉♣♣♦s❡
A
✐s ❛
2×2
♠❛tr✐①✳ ■t ✐s ❛ ❜✐❥❡❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢
det A 6= 0✳
❊①❡r❝✐s❡ ✷✳✺✳✶✶
Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳ ❙♦✱ ❛ ③❡r♦ ❞❡t❡r♠✐♥❛♥t ✐♥❞✐❝❛t❡s t❤❛t s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r X ✐s t❛❦❡♥ t♦ 0 ❜② A✳ ■t ❢♦❧❧♦✇s t❤❛t ❛❧❧ t❤❡ ♠✉❧t✐♣❧❡s✱ kX ✱ ♦❢ X ❛r❡ ❛❧s♦ t❛❦❡♥ t♦ 0✿ A(kX) = kA(X) = k0 = 0 .
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐s ❝♦❧❧❛♣s❡❞ t♦ 0✳ ❚❤❡♦r❡♠ ✷✳✺✳✶✷✿ ▲✐♥❡ ❈♦❧❧❛♣s❡s ■❢ ❛ ✈❡❝t♦r ✐s t❛❦❡♥ t♦ ③❡r♦ ❜② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ t❤❡♥ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ✈❡❝t♦r ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✐s t❛❦❡♥ t♦ ③❡r♦✿
A(X) = 0 =⇒ A {Y : Y = kX, k
r❡❛❧
}
= 0.
❲❡ ❝❛♥ ♣❧❛❝❡ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s ♦♥ t❤❡ s❛♠❡ ♣❧❛♥❡✳ ❚❤❡ ♦r✐❣✐♥ ✐s t❤❡ s❛♠❡✱ ❜✉t t❤❡ ✉♥✐ts ❛♥❞ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ ❛①❡s ♠❛② ❜❡ ❞✐✛❡r❡♥t✿
❲❡ ❦♥♦✇ t❤❛t t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ r❡♠❛✐♥s t❤❡ s❛♠❡✳ ❍♦✇❡✈❡r✱ ❛ ❝♦♠♣♦♥❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❞♦❡s ❞❡♣❡♥❞ ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞❡♣❡♥❞s ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ t♦♦✳ ❘❡♠❛r❦❛❜❧②✱ t❤✐s ✐s♥✬t tr✉❡ ❢♦r t❤❡ ❞❡t❡r♠✐♥❛♥t✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❢❛❝t ✐s ❛❝❝❡♣t❡❞ ✇✐t❤♦✉t ♣r♦♦❢✿
✷✳✻✳
✶✻✽
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❈♦r♦❧❧❛r② ✷✳✺✳✶✸✿ ❉❡t❡r♠✐♥❛♥t ■s ■♥tr✐♥s✐❝ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♠❛✐♥s t❤❡ s❛♠❡ ✐♥ ❛♥② ❈❛rt❡s✐❛♥ ❝♦♦r❞✐✲ ♥❛t❡ s②st❡♠✳
❚❤❡ ❞❡t❡r♠✐♥❛♥t ✇✐❧❧ t❡❧❧ ✉s ❛ ❧♦t ❛❜♦✉t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ • det A < 0 ✐♥❞✐❝❛t❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ✢✐♣✳
• | det A| = 1 ✐♥❞✐❝❛t❡s t❤❛t t❤✐s ✐s ❛ ♠♦t✐♦♥✳
• det A = 0 ✐♥❞✐❝❛t❡s t❤❡ ❝♦❧❧❛♣s❡ ♦r t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♣r♦❥❡❝t✐♦♥✳
❇✉t ❤♦✇ ❞♦ ✇❡ ❞❡t❡❝t str❡t❝❤❡s ♦r r♦t❛t✐♦♥s❄
✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❆♥ ❡❛s② ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t r♦t❛t✐♦♥ ✐s t❤❛t ✐❢ ❥✉st ♦♥❡ ✈❡❝t♦r ✐s♥✬t r♦t❛t❡❞✱ t❤❡r❡ ✐s ♥♦ r♦t❛t✐♦♥✦ ■❢ ❛ ✈❡❝t♦r ✐s♥✬t r♦t❛t❡❞✱ ✇❤❛t ❝❛♥ ♣♦ss✐❜❧② ❤❛♣♣❡♥ t♦ ✐t ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r❄ ❆ str❡t❝❤ ✭✇✐t❤ ❛ ♣♦ss✐❜❧❡ ✢✐♣✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t✬s s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ V 7→ λV ,
❢♦r s♦♠❡ r❡❛❧ λ✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✿ r❡✲s❝❛❧✐♥❣
❈♦♥s✐❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❣✐✈❡♥ ❜② t❤❡ ♠❛tr✐①✿ 2 0 . A= 0 3
❲❤❛t ❡①❛❝t❧② ❞♦❡s t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❞♦ t♦ ✐t❄ ❚♦ ❛♥s✇❡r✱ ❥✉st ❝♦♥s✐❞❡r ✇❤❡r❡ A t❛❦❡s t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✿ " # " # A : e1 =
A : e2 =
1 0
" # 0 1
7→
7→
2 0
" # 0 3
= 2e1
= 3e2
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❤❛♣♣❡♥s t♦ ❡✐t❤❡r ✐s ❛ ✭❞✐✛❡r❡♥t✮ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ A(e1 ) = 2e1 ❛♥❞ A(e2 ) = 3e2
❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡♥t✐r❡t② ♦❢ ❡❛❝❤ ♦❢ t❤❡ ❛①❡s ✐s str❡t❝❤❡❞ t❤✐s ✇❛②✳ ❙♦✱ ✇❡ ❝❛♥ s❛② t❤❛t A str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❤♦r✐③♦♥t❛❧❧② ❜② ❛ ❢❛❝t♦r ♦❢ 2 ❛♥❞ ✈❡rt✐❝❛❧❧② ❜② 3✱ ✐♥ ❡✐t❤❡r ♦r❞❡r✿
✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
✶✻✾
❊✈❡♥ t❤♦✉❣❤ ✇❡ s♣❡❛❦ ♦❢ str❡t❝❤✐♥❣ t❤❡ ♣❧❛♥❡✱ t❤✐s ✐s ♥♦t t♦ s❛② t❤❛t ❛❧❧ ✈❡❝t♦rs ❛r❡ str❡t❝❤❡❞✳ ■♥❞❡❡❞✱
< 1, 1 > 1 2 1 , 6= λ = A 1 3 1
♦t❤❡r ✈❡❝t♦rs ♠❛② ❜❡ r♦t❛t❡❞❀ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ✈❛❧✉❡s ♦❢
❢♦r ❛♥② r❡❛❧
✐s♥✬t ✐ts ♠✉❧t✐♣❧❡✿
λ✳
❊①❡r❝✐s❡ ✷✳✻✳✷ ❆♥❛❧②③❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿
h 0 . A= 0 v
❊①❛♠♣❧❡ ✷✳✻✳✸✿ str❡t❝❤ ❛❧♦♥❣ ♦t❤❡r ❛①❡s ❲❤❛t ✐❢ t❤❡ ♦♣❡r❛t♦r str❡t❝❤❡s ❛❧♦♥❣ ♦t❤❡r ❧✐♥❡s❄ ❍❡r❡ ✇❡ s✐♠♣❧② r♦t❛t❡ t❤❡ ♣✐❝t✉r❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ t❤r♦✉❣❤
45
❞❡❣r❡❡s t♦ ♠❛❦❡ t❤✐s ♣♦✐♥t✿
❚❤❡ ❝✐r❝❧❡ ✐s str❡t❝❤❡❞✱ ❜✉t ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ♦r ❞✐r❡❝t✐♦♥s❄ ■s t❤❡r❡ ❛ r♦t❛t✐♦♥ t♦♦❄ ■t ✐s ❤❛r❞ t♦ t❡❧❧ ✇✐t❤♦✉t ♣r✐♦r ❦♥♦✇❧❡❞❣❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ s❡❡♥ t❤✐s✿
❚❤❡ ♣❧❛♥❡ ✐s ✈✐s✐❜❧② str❡t❝❤❡❞✱ ❜✉t ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ♦r ❞✐r❡❝t✐♦♥s❄ ■t ✐s ❤❛r❞ t♦ t❡❧❧ ❜❡❝❛✉s❡ t❤❡ r❡s✉❧t s✐♠♣❧② ❧♦♦❦s s❦❡✇❡❞✳
■t ♠✐❣❤t ❜❡ t②♣✐❝❛❧ t❤❡♥ t❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❖♥ s✉❝❤ ❛ ✈❡❝t♦r
V✱ A
A
r♦t❛t❡s s♦♠❡ ✈❡❝t♦rs✱ ❜✉t
A
❛❧s♦ str❡t❝❤❡s ♦t❤❡r ✈❡❝t♦rs✳
❛❝ts ❛s ❛ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿
A(V ) = λV , ❢♦r s♦♠❡ ♥✉♠❜❡r
λ✳
❋♦r ❡①❛♠♣❧❡✱ ✇❡ s❡❡ ❞✐s♣r♦♣♦rt✐♦♥❛❧ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ str❡t❝❤✐♥❣✿
✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
✶✼✵
❚❤✐s ✐❞❡❛ ❜r✐♥❣ ✉s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t✿
❉❡✜♥✐t✐♦♥ ✷✳✻✳✹✿ ❡✐❣❡♥✈❛❧✉❡ ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦❢
A
✐❢ ✐t s❛t✐s✜❡s✿
A : R2 → R 2 ✱
❛ ✭r❡❛❧✮ ♥✉♠❜❡r
λ
✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❛❧✉❡
A(V ) = λV ❢♦r s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r s♣♦♥❞✐♥❣ t♦
V
✐♥
R2 ✳
❚❤❡♥✱
V
✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢
A
❝♦rr❡✲
λ✳
❲❛r♥✐♥❣✦ V =0 A(0) = 0✳ ❱❡❝t♦r
✐s ❡①❝❧✉❞❡❞ ❜❡❝❛✉s❡ ✇❡ ❛❧✇❛②s ❤❛✈❡
◆♦t❡ t❤❛t ✏❡✐❣❡♥✑ ♠❡❛♥s ✏❝❤❛r❛❝t❡r✐st✐❝✑ ✐♥ ●❡r♠❛♥✳ ◆♦✇✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡s❡❄
❊①❛♠♣❧❡ ✷✳✻✳✺✿ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ■❢ t❤✐s ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✱
A = I✱
t❤❡ ❡q✉❛t✐♦♥ ✐s ❡❛s② t♦ s♦❧✈❡✿
λV = AV = IV = V. ❙♦✱
λ = 1✳
❚❤✐s ✐s t❤❡ ♦♥❧② ❡✐❣❡♥✈❛❧✉❡✳ ❲❤❛t ❛r❡ ✐ts ❡✐❣❡♥✈❡❝t♦rs❄ ❆❧❧ ✈❡❝t♦rs ❜✉t
0✳
■♥❞❡❡❞✱ ♥♦ ✈❡❝t♦r
✐s r♦t❛t❡❞✦
❊①❡r❝✐s❡ ✷✳✻✳✻ ❲❤❛t ❛❜♦✉t ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢
k❄
❊①❛♠♣❧❡ ✷✳✻✳✼✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ▲❡t✬s r❡✈✐s✐t t❤✐s ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿
❚❤❡♥ ♦✉r ✈❡❝t♦r ❡q✉❛t✐♦♥
AV = λV
2 0 . A= 0 3
❜❡❝♦♠❡s✿
2 0 0 3
x x . =λ y y
▲❡t✬s r❡✇r✐t❡✿
2x λx 2x = λx ❆◆❉ x(2 − λ) = 0, = =⇒ =⇒ 3y λy 3y = λy y(3 − λ) = 0.
(1) (2)
❚❤❡ t✇♦ ❡q✉❛t✐♦♥s ♠✉st ❜❡ s❛t✐s✜❡❞ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❩❡r♦ ❋❛❝t♦r ❘✉❧❡ ❛❣❛✐♥✳ ◆♦✇✱ ✇❡ ❤❛✈❡
V =< x, y >6= 0✱
▲❡t✬s ✉s❡ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s t♦ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❝❛s❡s✿
s♦ ❡✐t❤❡r
x 6= 0
♦r
y 6= 0✳
✷✳✻✳
✶✼✶
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
• ❈❛s❡ ✶✿ x 6= 0✱ t❤❡♥ ❢r♦♠ ✭✶✮✱ ✇❡ ❤❛✈❡✿ 2 − λ = 0 =⇒ λ = 2✳ • ❈❛s❡ ✷✿ y = 6 0✱ t❤❡♥ ❢r♦♠ ✭✷✮✱ ✇❡ ❤❛✈❡✿ 3 − λ = 0 =⇒ λ = 3✳
❚❤❡s❡ ❛r❡ t❤❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✳ ❲❡ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s✦ ❚❤❡ s❡❝♦♥❞ ♣❛rt ✐s t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ■❢ λ = 2✱ t❤❡♥ y = 0 ❢r♦♠ ✭✷✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs ❛r❡✿ x , x 6= 0 , 0
A
x x . =2 0 0
■❢ λ = 3✱ t❤❡♥ x = 0 ❢r♦♠ ✭✶✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs ❛r❡✿ 0 0 . =3 A y y
0 , y 6= 0 , y
❚❤❡s❡ t✇♦ s❡ts ❛r❡ ❛❧♠♦st ❡q✉❛❧ t♦ t❤❡ t✇♦ ❛①❡s✦ ■❢ ✇❡ ❛♣♣❡♥❞ 0 t♦ t❤❡s❡ s❡ts ♦❢ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❋♦r λ = 2✱ t❤❡ s❡t ✐s t❤❡ x✲❛①✐s✿ x : x r❡❛❧ 0
❆♥❞ ❢♦r λ = 3✱ t❤❡ s❡t ✐s t❤❡ y ✲❛①✐s✿
0 : y r❡❛❧ y
❚❤❡ s♦❧✉t✐♦♥ ✐s t✐♠❡✲❝♦♥s✉♠✐♥❣✱ ❜✉t t❤❡r❡ ✇✐❧❧ ❜❡ ❛ s❤♦rt✲❝✉t ❧❛t❡r✳ ❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❡ ❢❛❝t t❤❛t✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♥♦♥✲✉♥✐❢♦r♠ r❡✲s❝❛❧✐♥❣✱ ❛❧❧ ✈❡❝t♦rs ❛r❡ r♦t❛t❡❞ ❡①❝❡♣t ❢♦r t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ♦♥❡s✳ ❊①❡r❝✐s❡ ✷✳✻✳✽
❆♥❛❧②③❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿ h 0 . A= 0 v
❋r♦♠ t❤❡ ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❣✉❡ss ❛ ♣❛tt❡r♥✳ ❚❤❡♦r❡♠ ✷✳✻✳✾✿ ▼✉❧t✐♣❧❡s ♦❢ ❊✐❣❡♥✈❡❝t♦rs
❆♥② ♥♦♥✲③❡r♦ ♠✉❧t✐♣❧❡ ♦❢ ❛♥ ❡✐❣❡♥✈❡❝t♦r ✐s ❛❧s♦ ❛♥ ❡✐❣❡♥✈❡❝t♦r ✕ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ❡✐❣❡♥✈❛❧✉❡✳ Pr♦♦❢✳
❙✉♣♣♦s❡ V ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡ λ✿ AV = λV .
■❢ W = kV ✱ t❤❡♥ AW = A(kV ) = kAV = kλV = λ(kV ) = λW
❙✉❜st✐t✉t❡✳ ❯s❡ t❤❡ ❢❛❝t t❤❛t ✐t ♣r❡s❡r✈❡s s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❯s❡ t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ λ. ❘❡❛rr❛♥❣❡✳ ❙✉❜st✐t✉t❡ ❜❛❝❦✳
❚❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐s ♠❛❞❡ ✉♣ ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ■t✬s ❛ ❝♦♣② ♦❢ R✦ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿
✷✳✻✳
✶✼✷
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❉❡✜♥✐t✐♦♥ ✷✳✻✳✶✵✿ ❡✐❣❡♥s♣❛❝❡ ❋♦r ❛♥ ❡✐❣❡♥✈❛❧✉❡ λ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✱ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ E(λ) = {V : A(V ) = λV } .
■t✬s ❛❧❧ ❡✐❣❡♥✈❡❝t♦rs ♦❢ λ ♣❧✉s 0✳ ❲❡ ✐♥❝❧✉❞❡ ✐t ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s s❡t ✐♥t♦ ❛ s♣❛❝❡✱ ❛ ✈❡❝t♦r s♣❛❝❡✳ ❋r♦♠ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❊①❛♠♣❧❡ ✷✳✻✳✶✶✿ ✐❞❡♥t✐t② ♠❛tr✐① ❋♦r t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✱ ✇❡ ❤❛✈❡✿
E(1) = R2 .
❊①❛♠♣❧❡ ✷✳✻✳✶✷✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ❋♦r
2 0 , A= 0 3
✇❡ ❤❛✈❡✿
• E(2) ✐s t❤❡ x✲❛①✐s✳ • E(3) ✐s t❤❡ y ✲❛①✐s✳ ❚✇♦ ❝♦♣✐❡s ♦❢ R✦
❊①❛♠♣❧❡ ✷✳✻✳✶✸✿ r♦t❛t✐♦♥ ❆ r♦t❛t✐♦♥ ❞♦❡s♥✬t str❡t❝❤ ❛♥② ✈❡❝t♦rs✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ✭r❡❛❧✮ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❡❝t♦rs ❛♥❞ ♥♦ ❡✐❣❡♥s♣❛❝❡s✳
❊①❛♠♣❧❡ ✷✳✻✳✶✹✿ ③❡r♦ ♠❛tr✐① ❋♦r t❤❡ ③❡r♦ ♠❛tr✐①✱ A = 0✱ ✇❡ ❤❛✈❡✿ AV = λV, ♦r 0 = λV .
❚❤❡r❡❢♦r❡✱ λ = 0 s✐♥❝❡ V 6= 0✳ ❋✉rt❤❡r♠♦r❡✿ E(0) = R2 .
❊①❛♠♣❧❡ ✷✳✻✳✶✺✿ ♣r♦❥❡❝t✐♦♥ ❈♦♥s✐❞❡r ♥♦✇ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s✱ 1 0 . P = 0 0
❚❤❡♥ ♦✉r ♠❛tr✐① ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ❛s ❢♦❧❧♦✇s✿
1 0 0 0
x = λx ❆◆❉ λx x x x =⇒ = =⇒ =λ 0 = λy λy 0 y y
❙♦✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ❝❛s❡s ❛r❡✿
λ = 0 ❛♥❞ λ = 1 .
■t ❛♣♣❡❛rs t❤❛t t❤❡ ♦♣❡r❛t♦r ✐s ♣r♦❥❡❝t✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❞♦✐♥❣ ♥♦t❤✐♥❣ ✐♥ ❛♥♦t❤❡r✳
✷✳✻✳
✶✼✸
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
◆❡①t✱ ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ♥♦✇ ❣♦ ❜❛❝❦ t♦ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❢♦r x ❛♥❞ y ✳ ❲❡ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❝❛s❡s✳ ❋✐rst✿ ❈❛s❡ ✶✿ λ = 0 =⇒
(
x = 0 · x ❆◆❉ 0=0·y
=⇒
(
=⇒
(
x = 0 ❆◆❉ y ❛♥②
0 =⇒ E(0) = : y r❡❛❧ y
x ❛♥② ❆◆❉ y=0
x =⇒ E(1) = : x r❡❛❧ 0
❚❤✐s ✐s t❤❡ y ✲❛①✐s✳ ❙❡❝♦♥❞✿ ❈❛s❡ ✷✿ λ = 1 =⇒
(
x = 1 · x ❆◆❉ 0=1·y
❚❤✐s ✐s t❤❡ x✲❛①✐s✳ ❚②♣✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs t❤❛t ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿ • A(V1 ) = λ1 V1 ❛♥❞ • A(V2 ) = λ2 V2 ✱
❢♦r s♦♠❡ ♥✉♠❜❡rs λ1 6= λ2 ✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✻✿ str❡t❝❤ ❛❧♦♥❣ ♦t❤❡r ❛①❡s
▲❡t✬s r❡✈✐s✐t t❤❡ str❡t❝❤ ❛❧♦♥❣ s♣❡❝✐❛❧ ❧✐♥❡s✳ ■t ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿
❍♦✇❡✈❡r✱ ❤♦✇ ✇♦✉❧❞ ✇❡ ❡✈❡♥ ✜♥❞ t❤❡s❡ s♣❡❝✐❛❧ ❞✐r❡❝t✐♦♥s❄ ❇❡❧♦✇✱ ✇❡ tr② t❤❡ ❜❛s✐s ✈❡❝t♦rs✱ ❧♦♦❦ ❛t ✇❤❡r❡ t❤❡② ❣♦ ✭❢r♦♠ t❤❡ ♠❛tr✐① ✐ts❡❧❢✮✱ ❛♥❞ s❡❡ t❤❛t t❤❡② ❤❛✈❡ r♦t❛t❡❞✿
❚❤❡② ❛r❡ ♥♦t ❡✐❣❡♥✈❡❝t♦rs✦ ❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ❜❡❧♦✇ ♠❛② ❜❡ ❢♦✉♥❞ ❜② tr✐❛❧ ❛♥❞ ❡rr♦r ♦r ❜② t❤❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ❜❡❧♦✇❀ t❤❡②✱ ✐♥❞❡❡❞✱ ❞♦♥✬t r♦t❛t❡✿
✷✳✻✳
✶✼✹
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❊①❛♠♣❧❡ ✷✳✻✳✶✼✿ ♥♦ ❡✐❣❡♥✈❡❝t♦rs
❈❛♥ ✇❡ ❞❡r✐✈❡ ✇❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❢r♦♠ ✐ts ♠❛tr✐① ♦♥❧②✱ ✇✐t❤♦✉t ✈✐s✉❛❧✐③❛t✐♦♥❄ ❙✉♣♣♦s❡ A ✐s ❣✐✈❡♥✿ 0 −1 . 1 0
A=
❚❤❡♥✱ t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s✱ ✇❡ ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ AV = λV =⇒
0 −1 1 0
x x . =λ y y
❲❡ s♦❧✈❡ ✐t ❛s ❢♦❧❧♦✇s✿ =⇒
−y = λx ❆◆❉ =⇒ x = λy
−xy = λx2 ❆◆❉ xy = λy 2
=⇒ λx2 = −λy 2 =⇒ x2 = −y 2 ❖❘ λ = 0 .
❆ ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥ r❡✈❡❛❧s t❤❛t λ = 0 ✐s ♥♦t ❛♥ ❡✐❣❡♥✈❛❧✉❡✿ AV = 0 · V =⇒
0 −1 1 0
x = 0 =⇒ x = 0, y = 0 . y
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❡q✉❛t✐♦♥ x2 = −y 2 ✐s ✐♠♣♦ss✐❜❧❡ ✉♥❧❡ss ❜♦t❤ x ❛♥❞ y ❛r❡ ③❡r♦s✱ ✇❤✐❝❤ ✐s ♥♦t ❛❧❧♦✇❡❞✳ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ ❡✐❣❡♥✈❛❧✉❡s✱ ❝❡rt❛✐♥❧② ♥♦t r❡❛❧ ♦♥❡s✳✳✳ ❚❤✐s ♠❡❛♥s t❤❛t ❡✈❡r② ✈❡❝t♦r ✐s r♦t❛t❡❞✳ ▼❛②❜❡ t❤✐s ✐s ❛ r♦t❛t✐♦♥❄ ❨❡s✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ ♠❛tr✐① ♦❢ t❤❡ 90✲❞❡❣r❡❡ r♦t❛t✐♦♥✳ ❊①❡r❝✐s❡ ✷✳✻✳✶✽
❍♦✇ ❞♦❡s t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ A t❡❧❧ ②♦✉ ✇❤❡t❤❡r 0 ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡❄ ❊①❡r❝✐s❡ ✷✳✻✳✶✾
❙❤♦✇ t❤❛t ❛ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ ✐♠♣❧✐❡s ❛ ❝♦❧❧❛♣s❡✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✵✿ ♥❡❡❞ ❢♦r ❤♦♠♦❣❡♥❡♦✉s s②st❡♠
▲❡t✬s r❡✈✐s✐t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿
−1 −2 . A= 1 −4
✷✳✻✳
✶✼✺
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❖✉r ✈❡❝t♦r ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿
−1 −2 1 −4
x x . =λ y y
❲❡ r❡✇r✐t❡✱ ❛❣❛✐♥✱ ❛s ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
(−1 − λ)x − 2y = 0 ❆◆❉ λx −x − 2y =⇒ = x + (−4 − λ)y = 0 λy x − 4y
❚❤✐s ✐s ❛♥♦t❤❡r s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t♦ ❜❡ s♦❧✈❡❞✱ ❛❣❛✐♥✳ ■t ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ t❤❡ ♦♥❡s ✇❡ s❛✇ ❛❜♦✈❡ ❛♥❞ ♥♦♥❡ ♦❢ t❤❡ s❤♦rt❝✉ts ❛r❡ ❛✈❛✐❧❛❜❧❡✳✳✳ ❚❤❡ s②st❡♠ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❤♦♠♦❣❡♥❡♦✉s ✈❡❝t♦r ❡q✉❛t✐♦♥✿
❲❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t t❤♦s❡❄
−1 − λ −2 1 −4 − λ
0 x = . y 0
▲❡t✬s r❡✈✐❡✇✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs✳ ▲❡t✬s✱ ❢♦r ♥♦✇✱ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❢♦r♠❡r✳ ❙✉♣♣♦s❡ λ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ A✳ ❚❤✐s ♠❡❛♥s t❤❛t λ ✐s ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞ t❤❡r❡ ✐s s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r V t❤❛t s❛t✐s✜❡s✿ AV = λV
▲❡t✬s ❞♦ s♦♠❡ ✈❡❝t♦r ❛❧❣❡❜r❛✿ AV = λV =⇒ AV − λV = 0 .
❲❡ ✇❛♥t t♦ t✉r♥ t❤✐s ❡q✉❛t✐♦♥ ♦❢ ✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s ✐♥t♦ ♦♥❡ ❡♥t✐r❡❧② ♦❢ ♠❛tr✐❝❡s✳ ❲❡ ❝❛♥ t❛❦❡ t❤✐s ❡q✉❛t✐♦♥ ♦♥❡ st❡♣ ❢✉rt❤❡r ❜② ♦❜s❡r✈✐♥❣ t❤❛t λV = λIV ,
✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❚❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ❛❧❧♦✇s t♦ ❢❛❝t♦r V ♦✉t ♦❢ ♦✉r ❡q✉❛t✐♦♥✳ ■t t❛❦❡s ❛ ♥❡✇ ❢♦r♠✿ (A − λI)V = 0
❚❤❡ ❡q✉❛t✐♦♥ ❝❤❛r❛❝t❡r✐③❡s ❛♥ ❡✐❣❡♥✈❡❝t♦r ❛♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s ✐♥ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✶✿ ❞✐♠❡♥s✐♦♥
2
■♥ t❤❡ R2 ❝❛s❡✱ ✇❡ ♠❛❦❡ t❤✐s s♣❡❝✐✜❝ ✇❤❡♥ ♦✉r ❧✐♥❡❛r ♦♣❡r❛t♦r A ✐s s♣❡❝✐✜❝✳ ❙✉♣♣♦s❡ ✐t ✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✿ A=
a b . c d
❲❡ ❝❛rr② ♦✉t t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✿
a b AV = c d
λx x ax + by x . = ❛♥❞ λ = λy y cx + dy y
❚❤❡s❡ t✇♦ ✈❡❝t♦rs ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❡q✉❛❧✱ s♦ ✇❡ ❤❛✈❡ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤ ✐s t❤❡♥ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛ ❤♦♠♦❣❡♥❡♦✉s ❢♦r♠✿
ax + by = λx ❆◆❉ cx + dy = λy.
❚❤❡ ♠❛tr✐① ♦❢ t❤✐s s②st❡♠ ✐s✿
⇐⇒
(a − λ)x + by = 0 ❆◆❉ cx + (d − λ)y = 0.
a−λ b . G= c d−λ
✷✳✻✳
✶✼✻
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❋♦❧❧♦✇✐♥❣ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✱ ✇❡ r❡❝♦❣♥✐③❡ s♦♠❡ ♠❛tr✐① ❛❧❣❡❜r❛✿ 1 0 a b . −λ G= 0 1 c d ❲❡ ❣♦ ❜❛❝❦ t♦ ♦✉r ❝♦♠♣❛❝t r❡♣r❡s❡♥t❛t✐♦♥✿
❚❤❡♦r❡♠ ✷✳✻✳✷✷✿ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ ❊✐❣❡♥✈❡❝t♦rs ❙✉♣♣♦s❡
A
❡✐❣❡♥✈❡❝t♦r
✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳
V
♦❢
A
❚❤❡♥ ❡✈❡r② ♣❛✐r ♦❢ ❛♥ ❡✐❣❡♥✈❛❧✉❡
λ
❛♥❞ ✐ts
s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ❡q✉❛t✐♦♥✿
GV = 0,
✇❤❡r❡
G = A − λI
◆♦✇✱ t❤❡ q✉❡st✐♦♥ ❛❜♦✉t t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ♠❛tr✐① A ❜❡❝♦♠❡s ♦♥❡ ❛❜♦✉t t❤❡ ♠❛tr✐① G✿
◮ ❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s ❞♦❡s t❤❡ s②st❡♠ GV = 0 ❤❛✈❡ ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥❄ ❲❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✿
◮ ❚❤❡ s②st❡♠ GV = 0 ❤❛s ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ det G = 0✳ ❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿
❚❤❡♦r❡♠ ✷✳✻✳✷✸✿ ❊✐❣❡♥✈❛❧✉❡s ❛s ❘♦♦ts ❙✉♣♣♦s❡
A
✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r
R2 ✳
❚❤❡♥ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡
λ
♦❢
A
✐s ❛ s♦❧✉t✐♦♥
t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿
det(A − λI) = 0 ■♥ ❝♦♥tr❛st t♦ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❧❛st t❤❡♦r❡♠✱ t❤✐s s✐♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡♥✱ ♣♦ss✐❜❧②✱ t❤❡✐r ❡✐❣❡♥✈❡❝t♦rs✳
❛❧❣❡❜r❛✐❝
❡q✉❛t✐♦♥ ❛❧❧♦✇s t♦ ❞✐s❝♦✈❡r
❲❡ ❝♦❞✐❢② t❤✐s ✐❞❡❛ ❜❡❧♦✇✿
❉❡✜♥✐t✐♦♥ ✷✳✻✳✷✹✿ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❚❤❡
❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧
♦❢ ❛ 2 × 2 ♠❛tr✐① A ✐s ❞❡✜♥❡❞ t♦ ❜❡✿
χA (λ) = det(A − λI) ▼❡❛♥✇❤✐❧❡✱ t❤❡ ❡q✉❛t✐♦♥ χA (λ) = 0 ✐s ❝❛❧❧❡❞ t❤❡
✳
❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥
❚❤✐s ✐s t❤❡ ❝♦♥✈❡♥✐❡♥t ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❛r❡ t♦ s♦❧✈❡ ❢♦r ❞✐♠❡♥s✐♦♥ n = 2✿ a−λ b = (a − λ)(d − λ) − bc = 0 . χA (λ) = det c d−λ ■t✬s
q✉❛❞r❛t✐❝
✦
❲❡ ❞♦♥✬t ❦♥♦✇ ✇❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❜✉t ✕ ❡✈❡♥ ✇✐t❤♦✉t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ✕ ✇❡ ❝❛♥ t❡❧❧ ❛ ❧♦t ❢r♦♠ ✐ts ❡✐❣❡♥✈❛❧✉❡s✳ ❲❡ r❡❞✐s❝♦✈❡r s♦♠❡ ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❢❛♠✐❧✐❛r ♦♣❡r❛t♦rs ❜❡❧♦✇✳
✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
✶✼✼
❊①❛♠♣❧❡ ✷✳✻✳✷✺✿ r❡✲s❝❛❧✐♥❣❄
❈♦♥s✐❞❡r ❛❣❛✐♥✿ ❚❤❡♥ ✇❡ s♦❧✈❡✿ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿
2 0 . A= 0 3
2−λ 0 = (2 − λ)(3 − λ) = 0 . χA (λ) = det 0 3−λ
λ = 2, 3 .
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❜② t❤❡s❡ ❢❛❝t♦rs ✐♥ t✇♦ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ❲❡ ❝❛♥✬t t❡❧❧ ✇✐t❤♦✉t ✜♥❞✐♥❣ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✻✿ ♣r♦❥❡❝t✐♦♥❄
■❢
1 0 , A= 0 0
t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥ ✐s✿
1−λ 0 = (1 − λ)(−λ) = 0 . χA (λ) = det 0 −λ
❚❤❡♥✱ λ = 1, 0 .
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ♥♦t❤✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦❧❧❛♣s❡s ✐♥ ❛♥♦t❤❡r✳ ❚❤❛t✬s ❛ ♣r♦❥❡❝t✐♦♥✦ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ❲❡ ❞♦♥✬t ❦♥♦✇ ✇✐t❤♦✉t t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✼✿ r♦t❛t✐♦♥❄
❈♦♥s✐❞❡r ❚❤❡♥✱
0 −1 . A= 1 0
−λ −1 = λ2 + 1 = 0 . χA (λ) = det 1 −λ
◆♦ r❡❛❧ s♦❧✉t✐♦♥s✦ ❙♦✱ ♥♦ ♥♦♥✲③❡r♦ ✈❡❝t♦r ✐s t❛❦❡♥ ❜② A t♦ ✐ts ♦✇♥ ♠✉❧t✐♣❧❡✳ ▼❛②❜❡ t❤✐s ✐s ❛ r♦t❛t✐♦♥✳✳✳ ❚❤❡s❡ t❤r❡❡ ❡①❛♠♣❧❡s s✉❣❣❡st ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ♦❢ t❤❡ ♣❧❛♥❡✳ ❇✉t ✜rst ❛ q✉✐❝❦ r❡✈✐❡✇ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❈♦♥s✐❞❡r ♦♥❡✿
f (x) = x2 + px + q .
❚❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ t❤❡♥ ♣r♦✈✐❞❡s t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ p x=− ± 2
p p2 − 4q . 2
❖❢ ❝♦✉rs❡✱ t❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ t❤❡ r❡❛❧ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❛t ✐s ✇❤② t❤❡ r❡s✉❧t ♦♥❧② ♠❛❦❡s s❡♥s❡ ✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ D = p2 − 4q ,
✐s ♥♦♥✲♥❡❣❛t✐✈❡✳
✷✳✻✳
✶✼✽
■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
■♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢r❡❡ t❡r♠ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ♣❛ss t❤❡ x✲❛①✐s ❡♥t✐r❡❧②✳ ❲❡ ❝❛♥ ♦❜s❡r✈❡ ❤♦✇ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿
❚❤✐s ♣r♦❝❡ss ✐s ❡①♣❧❛✐♥❡❞ ❜② ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✱ ✇✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ q ✱ t♦ t❤❡ r♦♦ts ❣✐✈❡♥ ❜② t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ √ D p x1,2 = − ± . 2 2 ❚❤❡r❡ ❛r❡ t❤r❡❡ st❛t❡s✿ √ D ✶✳ ❙t❛rt✐♥❣ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡✱ D ❞❡❝r❡❛s❡s✱ ❛♥❞ ❞❡❝r❡❛s❡s✳ 2 √ D = 0✳ ✷✳ ❚❤❡♥ D ❜❡❝♦♠❡s 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ 2 ✸✳ ❚❤❡♥ D ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡✱ ❛♥❞ t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ r♦♦ts ✭❝♦♠♣❧❡① r♦♦ts ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✮✳ ❙♦✱ ✇❡ ❤❛✈❡✿
• ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ t❤❡ r❡❛❧ r♦♦ts ♦❢ t❤❡ ✭q✉❛❞r❛t✐❝✮ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ χA ✳
• ❚❤❡r❡❢♦r❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡✐❣❡♥✈❛❧✉❡s ✐s ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ 2✱ ❝♦✉♥t✐♥❣ t❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s✳
▲❡t✬s tr② t♦ ❡①♣❛♥❞ t❤❡ ❝❤❛r❛❝t❡rst✐❝ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s❡❡ ✐❢ ♣❛tt❡r♥s ❡♠❡r❣❡✿ b a − λ χ(λ) = det c d−λ
= (a − λ)(d − λ) − bc
= ad − aλ − λd + λ2 − bc = λ2 − (a + d)λ + (ad − bc) = λ2 − tr A λ + det A . ❚❤❡ t❡r♠ ✐♥ t❤❡ ♠✐❞❞❧❡ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿
❉❡✜♥✐t✐♦♥ ✷✳✻✳✷✽✿ tr❛❝❡ ♦❢ ♠❛tr✐① ❚❤❡
tr❛❝❡
♦❢ ❛ ♠❛tr✐① A ✐s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts✳ ■t ✐s ❞❡♥♦t❡❞ ❜②✿
a b =a+d tr A = tr c d
❙♦✱ t❤❡ tr❛❝❡ ❛♣♣❡❛rs ✕ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❞❡t❡r♠✐♥❛♥t ✕ ✐♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿
✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs
✶✼✾
❚❤❡♦r❡♠ ✷✳✻✳✷✾✿ ❈❤❛r❛❝t❡r✐st✐❝ P♦❧②♥♦♠✐❛❧ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ♠❛tr✐①
A
t❛❦❡s t❤✐s ❢♦r♠✿
χA (λ) = λ2 − tr A · λ + det A
■t ✐s ❦♥♦✇♥ t❤❛t ♥♦t ♦♥❧② t❤❡ ❞❡t❡r♠✐♥❛♥t ❜✉t ❛❧s♦ t❤❡ tr❛❝❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦✉r ❝❤♦✐❝❡ ♦❢ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ s♦ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ♥♦✇ ❜❡ ✉s❡❞ t♦ t❡❧❧ ✇❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❞♦❡s✿ D = (tr A)2 − 4 det A .
❚❤❡♦r❡♠ ✷✳✻✳✸✵✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❙✉♣♣♦s❡
A ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❣✐✈❡♥ ❜② ❛ 2 × 2 ♠❛tr✐① ❛♥❞ D
✐s t❤❡ ❞✐s❝r✐♠✐♥❛♥t
♦❢ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s✿ ✶✳
D > 0✳
❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❞✐st✐♥❝t✿ ❖♣❡r❛t♦r
A
♥♦♥✲✉♥✐❢♦r♠❧② r❡✲s❝❛❧❡s
t❤❡ ♣❧❛♥❡ ✐♥ t❤❡ ❞✐st✐♥❝t ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs✳ ✷✳
D = 0✳
❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❡q✉❛❧✿ ❖♣❡r❛t♦r
A
✉♥✐❢♦r♠❧② r❡✲s❝❛❧❡s t❤❡
♣❧❛♥❡ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s ✉♥❧❡ss t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❛❧❧ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸✳
D < 0✳
❚❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❛❧✉❡s✿ ❖♣❡r❛t♦r
A
r♦t❛t❡s t❤❡ ♣❧❛♥❡ ✭✇✐t❤ ❛
♣♦ss✐❜❧❡ r❡✲s❝❛❧✐♥❣✮✳
Pr♦♦❢✳ ❋♦r P❛rt ✶✱ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ ❞✐st✐♥❝t ❜❡❝❛✉s❡ ✐❢ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❡✐❣❡♥✈❛❧✉❡✳ ■♥❞❡❡❞✿ A(V ) = λV =⇒ A(kV ) = kA(V ) = kλV = λ(kV ) .
P❛rts ✷ ❛♥❞ ✸ ❛r❡ ❛❞❞r❡ss❡❞ ❧❛t❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳
✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ❲❡ ❤❛✈❡ s❤♦✇♥ ❤♦✇ ♦♥❡ ❝❛♥ ✈✐s✉❛❧✐③❡ t❤❡ ✇❛② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡✿ ❜② ❡①❛♠✐♥✐♥❣ ✇❤❛t ❤❛♣♣❡♥s t♦ ✈❛r✐♦✉s ❝✉r✈❡s ✐♥ t❤❡ ❞♦♠❛✐♥✳ ❇② ♠❛♣♣✐♥❣ t❤❡s❡ ❝✉r✈❡s✱ ♦♥❡ ❝❛♥ ❞✐s❝♦✈❡r str❡t❝❤✐♥❣✱ s❤r✐♥❦✐♥❣ ✐♥ ✈❛r✐♦✉s ❞✐r❡❝t✐♦♥s✱ r♦t❛t✐♦♥s✱ ❡t❝✳ ■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❛❧s♦ s❛✇ ❤♦✇ ♦♥❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❡ ✇❛② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡✿ ❜② ❡①❛♠✐♥✐♥❣ ✐ts ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ♠❡t❤♦❞ ✐s ❡♥t✐r❡❧② ❛❧❣❡❜r❛✐❝ r❛t❤❡r t❤❛♥ ❡①♣❡r✐♠❡♥t❛❧✳ ❲❡ s✐♠♣❧② ✜♥❞ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ♣✉r❡ str❡t❝❤ ❢♦r F ✿ F V = λV
❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥s ❛r❡ ♣r♦❞✉❝❡❞ ❜② ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ❛❧s♦ ❝♦♠♣✉t❡s t❤❡ ❡✐❣❡♥✈❡❝t♦r ❛♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s❀ t❤❡② ❛r❡ s❤♦✇♥ ❛❜♦✈❡ t❤❡ ❣r❛♣❤s✳❚❤❡ s♣r❡❛❞s❤❡❡t ❛❧s♦ s❤♦✇s ❡✐❣❡♥s♣❛❝❡s ❛s t✇♦ ✭♦r ♦♥❡✱ ♦r ♥♦♥❡✮ str❛✐❣❤t ❧✐♥❡s❀ t❤❡② r❡♠❛✐♥ ✐♥ ♣❧❛❝❡ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❧❡❛r♥ ❤♦✇ t♦ ♣r❡❞✐❝t t❤❡ ♦✉t❝♦♠❡ ❜② ❡①❛♠✐♥✐♥❣ ♦♥❧② ✐ts ♠❛tr✐①✳
✷✳✼✳
❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs
✶✽✵
❇❡❧♦✇ ✐s ❛ ❢❛♠✐❧✐❛r ❢❛❝t t❤❛t ✇✐❧❧ t❛❦❡ ✉s ❞♦✇♥ t❤❛t r♦❛❞✿
❚❤❡♦r❡♠ ✷✳✼✳✶✿ Pr❡✐♠❛❣❡s ♦❢ ❩❡r♦
V 6= 0 ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r F kV ✳
■❢ t❤❡ ✐♠❛❣❡ ♦❢ ✐ts ♠✉❧t✐♣❧❡s
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✇✐t❤
V
✐s ③❡r♦✱ t❤❡♥ s♦ ✐s t❤❛t ♦❢ ❛♥② ♦❢
❛s ✐ts ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ✐s ❝♦❧❧❛♣s❡❞ t♦
0
❜②
F✿
❊①❛♠♣❧❡ ✷✳✼✳✷✿ ❝♦❧❧❛♣s❡ ♦♥ ❛①✐s ❲❡ st❛rt ✇✐t❤ ❛ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡✿
u = 2x v =0
✐s r❡✲✇r✐tt❡♥ ❛s
2 0 x u . = 0 0 y v
❊✈❡♥ ✇✐t❤♦✉t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥✱ ✇❡ ❝❛♥ ❣✉❡ss t❤❡ ❡✐❣❡♥✈❛❧✉❡✲❡✐❣❡♥✈❡❝t♦r ♣❛✐rs✿
2 0 0 0
1 1 , =2 0 0
2 0 0 0
0 0 . =0 1 1
❇❡❧♦✇✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤✐s ♦♣❡r❛t♦r ♣r♦ ❥❡❝ts t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡
❚❤❡ ♦♣❡r❛t♦r ❝♦❧❧❛♣s❡s t❤❡
y ✲❛①✐s
t♦
0✱
✇❤✐❧❡ t❤❡
x✲❛①✐s
x✲❛①✐s✿
✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r ♦❢
2✳
❚❤❡ st❛♥❞❛r❞
❜❛s✐s ✈❡❝t♦rs ❤❛♣♣❡♥ t♦ ❜❡ ❡✐❣❡♥✈❡❝t♦rs✦ ❚❤❛t✬s t❤❡ r❡❛s♦♥ ✇❤② t❤❡ ♠❛tr✐① ✐s s♦ s✐♠♣❧❡✳
✷✳✼✳
❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs
✶✽✶
❊①❛♠♣❧❡ ✷✳✼✳✸✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✐ts ♠❛tr✐①
u = 2x v = 4y
❛♥❞
2 0 . F = 0 4
❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥ t♦ s✉❣❣❡st t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✭❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✮✿
2 0 0 0
1 1 , =2 0 0
2 0 0 0
0 0 . =4 1 1
❆s ✐t t✉r♥s ♦✉t✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ tr❛❝❦ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs✱ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ❛r❡ s❡❡♥ ❛s ❛
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥
♦❢ t❤❡s❡✿
❚❤❡ r❡st ♦❢ t❤❡ ✈❡❝t♦rs t✉r♥ ♥♦♥✲✉♥✐❢♦r♠❧②❀ ✐✳❡✳✱ t❤❡② ✏❢❛♥ ♦✉t✑✿
X =< x, y >✿ 0 1 0 2 2 0 x = 2xe1 + 4ye2 . + 4y = 2x +y =x FX = 1 0 4 0 0 4 y
❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r
❚❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦
st❛♥❞❛r❞
❜❛s✐s ✈❡❝t♦rs✳ ❚❤❡ ♠✐❞❞❧❡✱
❤♦✇❡✈❡r✱ ✐s ❛❧s♦ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❜✉t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡s❡ t✇♦ ✈❡❝t♦rs✿
2 V1 = 0
❛♥❞
0 . V2 = 4
❚❤❡② ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❢♦r♠✐♥❣ ❛ ✏♥♦♥✲st❛♥❞❛r❞✑ ❜❛s✐s✳ ❚❤♦✉❣❤ ♥♦t ✉♥✐t ✈❡❝t♦rs ❛s t❤❡ st❛♥❞❛r❞ ♦♥❡s✱ t❤❡② ❛r❡ st✐❧❧ ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❛①❡s✳ ◆♦✇✱ ✇❤❛t ✐s t❤❡ ♣♦✐♥t❄ ❊✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s
✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs
✶✽✷
❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦✿
y x V1 + V2 . 2 4 ❋✉rt❤❡r♠♦r❡✱ t♦ ❦♥♦✇ ✇❤❡r❡ ❛♥② X ❣♦❡s ✉♥❞❡r F ✱ ✇❡ ♥❡❡❞ ♦♥❧② t♦ ❦♥♦✇ ✇❤❡r❡ t❤❡s❡ t✇♦ ❣♦✿ ■t✬s ♣✉r❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✦ ❲❡ ✇✐❧❧ s❡❡ t❤❛t ❛♥② ♣❛✐r ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✕ ✇❤❡♥ ♥♦t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ✕ ✇♦✉❧❞ ❞♦✳ < x, y >=
❊①❛♠♣❧❡ ✷✳✼✳✹✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s
❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ♦♣❡r❛t♦r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ u = −x v = 4y
−1 0 . ❛♥❞ F = 0 4
■t ✐s st✐❧❧ s✐♠♣❧❡ ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s r❡♠❛✐♥ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❆s ❛ r❡s✉❧t✱ t❤❡ t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①❡s ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿
❚❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ♣r♦❞✉❝❡s ❛ ❞✐✛❡r❡♥t ♣❛tt❡r♥✿ ❲❡ s❡❡ t❤❡ r❡✈❡rs❛❧ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❛s ❜❡❢♦r❡✿ −1 0 x −1 0 1 0 FX = =x +y = −x = −xe1 + 4ye2 . + 4y 1 0 4 y 0 4 0 ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✱ ✇❤✐❧❡ t❤❡ ♠✐❞❞❧❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❜✉t ✇✐t❤ r❡s♣❡❝t t♦ ❛♥♦t❤❡r ❜❛s✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿ 0 −1 . ❛♥❞ V2 = V1 = 4 0 ❲❤❛t ✐❢ t❤❡ ♠❛tr✐① ✐s♥✬t ❞✐❛❣♦♥❛❧❄ ❊①❡r❝✐s❡ ✷✳✼✳✺
❙❤♦✇ t❤❛t t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs e1 , e2 ❛r❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳ ■♥st❡❛❞ ♦❢ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ♦♣❡r❛t♦r ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t t❤❡ ♦♣❡r❛t♦r ❞♦❡s✳ ❚♦ ✐❧❧✉str❛t❡✱ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♣✐❝t✉r❡ ♦♥ t❤❡ ❧❡❢t ❤❛s ❜❡❡♥ s❦❡✇❡❞✱ r❡s✉❧t✐♥❣ ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ r✐❣❤t✿
✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs
✶✽✸
❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ♠❛tr✐① ✇✐❧❧ s❡r✈❡ ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s✳ ❊①❛♠♣❧❡ ✷✳✼✳✻✿ ❝♦❧❧❛♣s❡
▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦r✿
u = x + 2y v = 2x + 4y
1 2 . =⇒ F = 2 4
■t ❛♣♣❡❛rs t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ str❡t❝❤✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❛ ❝♦❧❧❛♣s❡ ✐♥ ❛♥♦t❤❡r✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ▲✐♥❡❛r ❛❧❣❡❜r❛ ❣✐✈❡s t❤❡ ❛♥s✇❡r✳ ❊✈❡♥ ✇✐t❤♦✉t ❧♦♦❦✐♥❣ ❢♦r ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ③❡r♦ ✿ 1 2 = 1 · 4 − 2 · 2 = 0. det F = det 2 4
■t✬s ♥♦t ♦♥❡✲t♦✲♦♥❡ ❛♥❞✱ ✐♥ ❢❛❝t✱ t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚♦ ✜♥❞ ✐t✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤✐s s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿
x + 2y = 0 2x + 4y = 0
=⇒ x = −2y .
❚❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t ❛♥❞ r❡♣r❡s❡♥t t❤❡ s❛♠❡ ❧✐♥❡✳ ❲❡ ❤❛✈❡✱ ✐♥❞✐r❡❝t❧②✱ ❢♦✉♥❞ t❤❡ ❡✐❣❡♥s♣❛❝❡ ❛♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ λ1 = 0✳ ❲❡ ❝❛♥ t❛❦❡ t❤✐s ❡✐❣❡♥✈❡❝t♦r ❢♦r ❢✉rt❤❡r ✉s❡✿ V1 =
2 −1
=⇒ F V1 = 0 .
▲❡t✬s ✐♥st❡❛❞ t✉r♥ t♦ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ 1−λ 2 = λ2 − 5λ = λ(λ − 5) =⇒ λ1 = 0, λ2 = 5 . det(F − λI) = det 2 4−λ
▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❢♦r λ2 = 5✳ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ✈❡❝t♦r ❡q✉❛t✐♦♥✿ F V = 5V ,
✷✳✽✳
❇❛s❡s
✶✽✹
✐✳❡✳✱
1 2 FV = 2 4
x x . =5 y y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ t✇♦ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✭✐t✬s t❤❡ s❛♠❡ ❡q✉❛t✐♦♥✮✿
x + 2y = 5x ❆◆❉ 2x + 4y = 5y
=⇒
−4x + 2y = 0 ❆◆❉ 2x − y =0
=⇒ y = 2x .
❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ ❛ ✈❡❝t♦r ❛❧♦♥❣ t❤✐s ❧✐♥❡ t♦ ❛s t❤❡ ❡✐❣❡♥✈❡❝t♦r ❢♦r ❢✉rt❤❡r ✉s❡✿
1 . V2 = 2 ❲❡ s✉♠♠❛r✐③❡ ✇❤❛t
• •
F
❞♦❡s✿
❆ ♣r♦❥❡❝t✐♦♥ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r ❆ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢
5
< 2, −1 >✿
x = −2y ✐s ❝♦❧❧❛♣s❡❞ t♦ 0✳ < 1, 2 >✿ ❚❤❡ ❧✐♥❡ y = 2x ✐s str❡t❝❤❡❞
❚❤❡ ❧✐♥❡
❛❧♦♥❣ t❤❡ ✈❡❝t♦r
✇✐t❤♦✉t ❛♥②
r♦t❛t✐♦♥✳
❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❡ ✐❧❧✉str❛t✐♦♥ ❛❜♦✈❡✦ ❋✉rt❤❡r♠♦r❡✱ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❛t ✐s ✇❤② ❡✈❡r② ✈❡❝t♦r ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ✈❛❧✉❡ ✉♥❞❡r
F
✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡
❡✐❣❡♥✈❡❝t♦rs t♦♦✿
X = xV1 + yV2 =⇒ F X = x · 0 · V1 + y · 5 · V2 . ❲❡ ❞❡r✐✈❡ ✇❤❡r❡
X
❣♦❡s ❢r♦♠ t❤❡ ❛❜♦✈❡ s✉♠♠❛r②✦
❊①❡r❝✐s❡ ✷✳✼✳✼
❋✐♥❞ t❤❡ ❧✐♥❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥✳
❲❡ ❛r❡ ❛❜❧❡ t♦ s✉♠♠❛r✐③❡ ✇❤❛t t❤❡ ♦♣❡r❛t♦r ❞♦❡s ❢r♦♠ t❤❡ ❛❧❣❡❜r❛ ♦♥❧②✳ ❚❤❡ ✐❞❡❛ ✐s ✉♥❝♦♠♣❧✐❝❛t❡❞✿
◮
❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r
✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡ ✐s ✏ 1✲❞✐♠❡♥s✐♦♥❛❧✑❀ ✐t ❝❛♥ t❤❡♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛
s✐♥❣❧❡ ♥✉♠❜❡r✳ ❚❤✐s ♥✉♠❜❡r✱ t❤❡ str❡t❝❤✲s❤r✐♥❦ ❢❛❝t♦r✱ ✐s ♦❢ ❝♦✉rs❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡✳ ■❢ ✇❡ ❦♥♦✇ t❤❡s❡ t✇♦ ♥✉♠❜❡rs✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r❄ ■♥ t✇♦ st❡♣s✳ ❋✐rst✱ ❡✈❡r② ✉♥❞❡r
F
X
t❤❛t ❧✐❡s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡✱ ✇❤✐❝❤ ✐s ❛ ❧✐♥❡✱ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦r✱ ❛♥❞ ✐ts ✈❛❧✉❡
❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞✿
X = rV =⇒ U = F (rV ) = rF V = rλV . ❙❡❝♦♥❞✱ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② ❢♦❧❧♦✇✐♥❣ t❤✐s ✐❞❡❛✿ ❚r② t♦ ❡①♣r❡ss t❤❡ ✈❛❧✉❡ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ✈❛❧✉❡s ❢♦✉♥❞ s♦ ❢❛r✳ ▲❡t✬s ♣r♦✈✐❞❡ ❛ ❢♦✉♥❞❛t✐♦♥ ❢♦r t❤✐s ✐❞❡❛✳
✷✳✽✳ ❇❛s❡s
❚❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐s ❞❡t❡r♠✐♥❡❞ ❜② ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳
❖♥
t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❛t ✐t ❞♦❡s ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ✇✐t❤ s✉❝❤ ✇♦r❞s ❛s ✏str❡t❝❤✑✱ ✏r♦t❛t✐♦♥✑✱ ✏✢✐♣✑✱ ❡t❝✳ ❚❤❡s❡
✷✳✽✳
❇❛s❡s
✶✽✺
❞❡s❝r✐♣t✐♦♥s ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❆♥❞ ♥❡✐t❤❡r ❞♦ s✉❝❤ ❛❧❣❡❜r❛✐❝ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ♦♣❡r❛t♦r ❛s t❤❡ tr❛❝❡✱ t❤❡ ❞❡t❡r♠✐♥❛♥t✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ ❛r❡ ♦♥ t❤❡ r✐❣❤t tr❛❝❦✦ ❖♥❝❡ ❛❣❛✐♥✱ ❞❡❛❧✐♥❣ ✇✐t❤ ✈❡❝t♦rs ✐♥st❡❛❞ ♦❢ ♣♦✐♥ts r❡q✉✐r❡s ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✳ ❚❤❡ st❛♥❞❛r❞
❜❛s✐s ♦❢ R2 ✱ ❛s ❜❡❢♦r❡✱ ❝♦♥s✐sts ♦❢ t❤❡s❡ t✇♦✿ e1 =< 1, 0 >, e2 =< 0, 1 > .
❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦r X =< a, b > ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❜❛s✐s ❛r❡✱ ❛s ❜❡❢♦r❡✱ a, b✿ < a, b >= a < 1, 0 > +b < 0, 1 >= ae1 + be2 .
■♥ ♦t❤❡r ✇♦r❞s✱ ❡✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❧✐♥❡❛r
❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs✳
❍♦✇❡✈❡r✱ t❤❡② ❛r❡♥✬t t❤❡ ♦♥❧② ♦♥❡s ✇✐t❤ t❤✐s ♣r♦♣❡rt②✦ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s r❡✇r✐t❡ t❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥✿ < a, b >= a < 1, 0 > +b < 0, 1 >= ae1 + (−b)(−e2 ) .
❙♦✱ t❤✐s ✈❡❝t♦r ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ✈❡❝t♦rs V1 = e1 ❛♥❞ V2 = −e2 ✳
❊①❡r❝✐s❡ ✷✳✽✳✶
❘❡♣r❡s❡♥t ✈❡❝t♦r < a, b > ✐♥ t❡r♠s ♦❢ e1 ❛♥❞ e1 + e2 ✳ ❆❧❧ ♦❢ t❤❡s❡ ♣❛✐rs ♦❢ ✈❡❝t♦rs ♠❛② s❡r✈❡ ✐♥ s✉❝❤ r❡♣r❡s❡♥t❛t✐♦♥s✿
❊①❡r❝✐s❡ ✷✳✽✳✷ ❙❤♦✇ t❤❛t ✈❡❝t♦rs t❤❛t ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ❝❛♥✬t ❜❡ ✉s❡❞ ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❆ ✈❡r② ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t ❜❡❧♦✇ ❝❛♣t✉r❡s t❤✐s ✐❞❡❛✿
❉❡✜♥✐t✐♦♥ ✷✳✽✳✸✿ ❜❛s✐s ❆ ❜❛s✐s ♦❢ R2 ✐s ❛♥② s✉❝❤ ♣❛✐r ♦❢ ✈❡❝t♦rs V1 , V2 t❤❛t ❡✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ r❡♣r❡✲ s❡♥t❡❞ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ✈❡❝t♦rs✿ X = r1 V 1 + r2 V 2 .
❚❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts r1 , r2 ❛r❡ ❝❛❧❧❡❞ t❤❡ ❜❛s✐s✳
❝♦♠♣♦♥❡♥ts ♦❢ X ✇✐t❤ r❡s♣❡❝t t♦ t❤❡
❊①❡r❝✐s❡ ✷✳✽✳✹ Pr♦✈❡ t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ r❡❛s♦♥✐♥❣ ✐s t❤❛t t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✇❛s ❡st❛❜❧✐s❤❡❞ ❜❡❢♦r❡ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✇❛s ❛❞❞❡❞ t♦ t❤❡ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ ❜❡❢♦r❡ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ❛❞❞ ✈❡❝t♦rs✿
✷✳✽✳
❇❛s❡s
✶✽✻
❊①❛♠♣❧❡ ✷✳✽✳✺✿ ❝♦♠♣♦♥❡♥ts
❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ❛r❡ ❢♦✉♥❞ ✈✐❛ t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥s✿
❆ ✈❡r② ❞✐✛❡r❡♥t ❝❤♦✐❝❡ ♦❢ ❜❛s✐s U, V ✐s s❤♦✇♥ ❜❡❧♦✇✿
❍❡r❡✱ ✈❡❝t♦r X ❤❛s ❝♦♠♣♦♥❡♥ts 2 ❛♥❞ 2 ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❜❛s✐s✿ X = 2U + 2V . ❊①❛♠♣❧❡ ✷✳✽✳✻✿ ❝♦♠♣♦♥❡♥t ❛❧❣❡❜r❛
▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ❛r❡ ❜❡❤✐♥❞ t❤❡ ❝♦♠♣♦♥❡♥ts ❛❧❣❡❜r❛✳ ❈♦❧❧❡❝t✐♥❣ ❝♦♠♠♦♥ t❡r♠s ❛❢t❡r ❛❞❞✐t✐♦♥ ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❝♦♠♣♦♥❡♥ts✿ ❝♦♠♣♦♥❡♥ts
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥
U
=< 2, 3 >
= 2e1 + 3e2
W
=< −1, 2 >
= −1e1 + 2e2
U +V
=< 2, 3 > + < −1, 2 > = (2e1 + 3e2 ) + (−1e1 + 2e2 ) =< 2 − 1, 3 + 2 >
= (2e1 − 1e1 ) + (3e2 + 2e2 )
=< 1, 5 >
= 1e1 + 5e2
❘❡♣❧❛❝✐♥❣ e1 , e2 ✇✐t❤✱ s❛②✱ V1 , V2 ✇♦♥✬t ❝❤❛♥❣❡ ❛♥②t❤✐♥❣ ✐♥ t❤✐s ❝♦♠♣✉t❛t✐♦♥✳ ❙❛♠❡ ❢♦r s❝❛❧❛r ♠✉❧t✐✲
✷✳✽✳
❇❛s❡s
✶✽✼
♣❧✐❝❛t✐♦♥✿ ❝♦♠♣♦♥❡♥ts
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥
U
=< 2, 3 >
= 2e1 + 3e2
r
= −3
rU = (−3) < 2, 3 >
= (−3)(2e1 + 3e2 )
=< (−3)2, (−3)3 > = (−3)2e1 + (−3)(1e1 ) =< −6, −9 >
= −6e1 − 9e2
❊①❛♠♣❧❡ ✷✳✽✳✼✿ ♥♦♥✲❜❛s✐s
❲❡ ♥❡❡❞ t♦ ❝♦✈❡r ❛❧❧ t❤❡ ✈❡❝t♦rs✿
▲❡t✬s tr② V1 =< 1, 0 >, V2 =< 2, 0 >✳ ❲❤❛t ❛r❡ ❛❧❧ ♣♦ss✐❜❧❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❄ ❋♦r ❛❧❧ ♣❛✐rs r1 , r2 ✱ ✇❡ ❤❛✈❡ X = r1 V1 + r2 V2 = r1 < 1, 0 > +r2 < 2, 0 >=< r1 + 2r2 , 0 > .
❚❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ✇✐❧❧ r❡♠❛✐♥ 0 ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡✦ ❚❤✐s ✐s ♥♦t ❛ ❜❛s✐s ❜❡❝❛✉s❡ ✇❡ ❝❛♥✬t r❡♣r❡s❡♥t s♦♠❡ ♦❢ t❤❡ ✈❡❝t♦rs✳ ❚❤❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✷✳✽✳✽✿ ❇❛s✐s ♦♥ t❤❡ P❧❛♥❡ ❆♥② t✇♦ ✈❡❝t♦rs t❤❛t ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ ♦♥❧② t❤❡② ❢♦r♠ ❛ ❜❛s✐s ♦❢
R2 ❀
✐✳❡✳✱
V1 = rV2 ⇐⇒ {V1 , V2 }
✐s ♥♦t ❛ ❜❛s✐s✳
❊①❡r❝✐s❡ ✷✳✽✳✾
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✷✳✽✳✶✵✿ ❧✐♥❡❛r s②st❡♠
❘❡❝❛❧❧ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r ❤♦✇ t❤❡ s♦❧✉t✐♦♥ t♦ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇❛s s❡❡♥ ❛s ✐❢ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ✇❡r❡ ❡q✉❛t✐♦♥s ❛❜♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts✱ x ❛♥❞ y ✱ ♦❢ ✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡✿ 6 1 1 . = +y x 14 3 2
❚♦ s♦❧✈❡ t❤❡ s②st❡♠ ✐s t♦ ✜♥❞ ❛ ✇❛② t♦ str❡t❝❤ t❤❡s❡ t✇♦ ✈❡❝t♦rs s♦ t❤❛t ❛❢t❡r ❛❞❞✐♥❣ t❤❡♠ t❤❡ r❡s✉❧t ✐s t❤❡ ✈❡❝t♦r ♦♥ t❤❡ r✐❣❤t✿
✷✳✽✳
❇❛s❡s
✶✽✽
◆♦✇ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✇❡ ❝❛♥ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥ ♦♥❧② ✇❤❡♥ t❤❡ t✇♦ ✈❡❝t♦rs ❢♦r♠ ❛ ❜❛s✐s✦ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ♠✐①t✉r❡ ♣r♦❜❧❡♠ ❞♦❡s♥✬t ❤❛✈❡ ❛ s♦❧✉t✐♦♥✿ 6 2 1 . = +y x 14 6 2 ❊①❛♠♣❧❡ ✷✳✽✳✶✶✿ ❝♦♦r❞✐♥❛t❡s
❈♦♥s✐❞❡r ❛♥ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ❛❧♦♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ♦♥❡✿
■t✬s ❡❛s② t♦ ❡①♣r❡ss t❤❡ ♥❡✇ ✈❡❝t♦rs ✐♥ t❡r♠s ♦❢ ♦❧❞✿ V1 =< 1, 1 >, V2 =< −1, 1 > .
❇✉t ✇❤❛t ❛❜♦✉t ✈✐❝❡ ✈❡rs❛❄ ■t✬s ❤❛r❞❡r❀ ✇❡ ♥❡❡❞ t♦ ✜♥❞ a ❛♥❞ b s♦ t❤❛t e1 = aV1 + bV2 ,
❛♥❞ t❤❡ s❛♠❡ ❢♦r e2 ✳ ❲❡ ❞r❛✇ ❛ ❣r✐❞ ❢♦r t❤❡ ♥❡✇ s②st❡♠ t♦ ❤❡❧♣✿
❚❤❡ ❛♥s✇❡r ✐s✿
1 1 1 1 e1 = V1 − V2 , e2 = V1 + V2 . 2 2 2 2
✷✳✽✳
❇❛s❡s
✶✽✾
❲❡ ❝❛♥ t❤❡♥ r❡✲✇r✐t❡ t❤❡s❡ ✈❡❝t♦rs ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❝♦♠♣♦♥❡♥ts ✇✐t❤ e1 =
1 1 ,− 2 2
1 = < 1, −1 >, e2 = 2
1 1 , 2 2
=
r❡s♣❡❝t t♦ t❤❡ ♥❡✇ ❜❛s✐s ✿
1 < 1, 1 > . 2
■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t ❝♦❡✣❝✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s❛♠❡ ✈❡❝t♦rs ✇✐t❤ r❡s♣❡❝t t♦ ❞✐✛❡r❡♥t ❜❛s❡s✿ ❜❛s❡s✿ {e1 , e2 } V1 = V2 =
{V1 , V2 } < 1, 1 > =< 1, 0 > < −1, 1 > =< 0, 1 >
❛♥❞ ❜❛s❡s✿ {e1 , e2 } e1 = e2 =
{V1 , V2 } + * 1 1 < 1, 0 > = ,− 2 2 * + 1 1 < 0, 1 > = ,− 2 2
■♥ ❣❡♥❡r❛❧✱ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ♠✐❣❤t ❤❛✈❡ ❛♥② ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠ ✭❛s ❧♦♥❣ ❛s ✐t✬s ♥♦t ③❡r♦✮✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ s❦❡✇❡❞ ❣r✐❞✿
❚❤✉s✱ t❤❡ ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛❧❣❡❜r❛ ✐s ❢✉❧❧② ♦♣❡r❛t✐♦♥❛❧ ✇❤❛t❡✈❡r ❜❛s✐s ✇❡ ❝❤♦♦s❡✿ < a, b > + < c, d >=< a + c, b + d >
❛♥❞ r < a, b >=< ra, rb > . ❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ t❤❡ ❛❧❣❡❜r❛✱ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ r❡❧✐❡s ♦♥ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳
❆s ❛
r❡s✉❧t✱ t❤❡ ❢♦r♠✉❧❛s ❢♦r ♠❛❣♥✐t✉❞❡s ❛♥❞ t❤❡ ❞♦t ♣r♦❞✉❝ts ❢❛✐❧ ✐♥ t❤❡ ❝✉rr❡♥t ❢♦r♠ ✐❢ ✉s❡❞ ✇✐t❤ ♥♦♥✲ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛s❡s✳
◆♦✇✱ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳ ❚❤❡② ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛♣❛rt ❢r♦♠ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✭t❤❛t ✇❛s ❛❞❞❡❞ t♦ t❤❡ ✈❡❝t♦r s♣❛❝❡✮✱ ✐✳❡✳✱ r♦t❛t✐♦♥✱ str❡t❝❤✐♥❣✱ ❡t❝✳✿
✷✳✽✳
❇❛s❡s
✶✾✵
❍♦✇❡✈❡r✱ ❥✉st ❛s ✈❡❝t♦r ❛❧❣❡❜r❛ ✇♦r❦s ❝♦♠♣♦♥❡♥t✇✐s❡✱ s♦ ❞♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ❛♣♣r♦❛❝❤ ✇♦r❦s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② ❜❛s✐s✿ ❚❤❡♦r❡♠ ✷✳✽✳✶✷✿ ▲✐♥❡❛r ❖♣❡r❛t♦r ✐♥ ❚❡r♠s ♦❢ ❇❛s✐s
❙✉♣♣♦s❡ {V1, V2} ✐s ❛ ❜❛s✐s✳ ❚❤❡♥✱ ❛❧❧ ✈❛❧✉❡s ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r Y = F (X) ❛r❡ ❡①♣r❡ss❡❞ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐ts ✈❛❧✉❡s ♦♥ t❤❡s❡ ✈❡❝t♦rs❀ ✐✳❡✳✱ ❢♦r ❛♥② ♣❛✐r ♦❢ r❡❛❧ ❝♦❡✣❝✐❡♥ts r1 ❛♥❞ r2✱ ✇❡ ❤❛✈❡✿ X = r1 V1 + r2 V2 =⇒ F (X) = r1 F (V1 ) + r2 F (V2 ) . ❊①❡r❝✐s❡ ✷✳✽✳✶✸
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦♣❡r❛t♦r ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❜❛s✐s ✈❡❝t♦rs ✕ ❥✉st ❛s ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❜❛s✐s✳
❇✉t✱ ❥✉st ❛s ✇✐t❤ ✈❡❝t♦rs✱ ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t ♠❛tr✐❝❡s ❢♦r t❤❡ s❛♠❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ r❡s♣❡❝t t♦
❞✐✛❡r❡♥t ❜❛s❡s✳ ❚❤❡ ❝♦❧✉♠♥s ♦❢ t❤❡ ♠❛tr✐① ♦❢
A
❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤❡ ♦♣❡r❛t♦r✿
A(V1 )
❛♥❞
A(V2 ) .
❊①❛♠♣❧❡ ✷✳✽✳✶✹✿ ♠❛tr✐❝❡s
❲❡✱ ❛❣❛✐♥✱ ❝♦♥s✐❞❡r t❤✐s ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ❛❧♦♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ♦♥❡✿
❚❤❡ ♠✉t✉❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡✿
1 1 1 1 e1 = V1 − V2 , e2 = V1 + V2 , 2 2 2 2 ❛♥❞
V1 = e1 + e2 , V2 = −e1 + e2 .
A str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❛❧♦♥❣ t❤❡ x✲❛①✐s ♦❢ A ✇✐t❤ r❡s♣❡❝t t♦ {e1 , e2 } ✐s✿ 2 0 . A= 0 1
❙✉♣♣♦s❡ ❛♥ ♦♣❡r❛t♦r
2✳
❚❤❡♥ t❤❡ ♠❛tr✐①
❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ❜❛s✐s✱
{V1 , V2 }❄
✭✐♥ ♦t❤❡r ✇♦r❞s✱ ❛❧♦♥❣
e1 ✮
❜② ❛ ❢❛❝t♦r ♦❢
■t ✐s ❤❛r❞ t♦ ❣✉❡ss t❤✐s t✐♠❡ ❜❡❝❛✉s❡ ✕ ✉♥❧✐❦❡ ❢♦r t❤❡ st❛♥❞❛r❞
❜❛s✐s ✕ t❤❡ ❝❤❛♥❣❡ ✐s ♥♦t ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❜❛s✐s ✈❡❝t♦rs✳
▲❡t✬s ✉s❡ t❤❡ ✏❝♦♥✈❡♥✐❡♥t✑ ❜❛s✐s ❛♥❞ t❤❡♥
s✇✐t❝❤ t♦ t❤❡ ♦♥❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ❲❡ ✇r✐t❡ t❤❡ ❢♦r♠✉❧❛s ✐♥ t❡r♠s ♦❢
A(V1 ) = A(e1 + e2 )
= A(e1 ) + A(e2 )
e 1 , e2
✜rst✿
= 2e1 + e2
A(V2 ) = A(−e1 + e2 ) = −A(e1 ) + A(e2 ) = −2e1 + e2 ❚❤❡♥ ✇❡ s✉❜st✐t✉t❡ t❤❡ ✈❛❧✉❡s ♦❢
A(V1 ) = 2e1 + e2 A(V2 ) = −2e1 + e2
e 1 , e2
V1 , V2 ❛s ✇r✐tt❡♥ ❛❜♦✈❡✿ 3 1 1 1 1 1 = V1 − V2 V1 − V2 + V1 + V2 =2 2 2 2 2 2 2 1 1 1 1 3 1 = − V1 + V2 V1 − V2 + V1 + V2 = −2 2 2 2 2 2 2 ✐♥ t❡r♠s ♦❢
✷✳✾✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
❚❤❡r❡❢♦r❡✱ t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ r❡s♣❡❝t t♦
{V1 , V2 }
✶✾✶
✐s✿
3 1 − 2 2 A= 1 3 . − 2 2 ❲❤❛t ✐❢ t❤❡ str❡t❝❤ ✇❛s ❛❧♦♥❣
V1 ❄
❚❤✐s ♦♣❡r❛t♦r✬s ♠❛tr✐① ✇✐t❤ r❡s♣❡❝t t♦
{V1 , V2 }
✐s s✐♠♣❧❡✿
2 0 . B= 0 1
❊①❡r❝✐s❡ ✷✳✽✳✶✺ ❘❡✇r✐t❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❝♦♠♣♦♥❡♥ts✳
✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s ❲❡ ❛♣♣❧② t❤❡ ❧❛st t❤❡♦r❡♠ t♦ ❡✐❣❡♥✈❡❝t♦rs✳
❈♦r♦❧❧❛r② ✷✳✾✳✶✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ✐♥ ❚❡r♠s ♦❢ ❊✐❣❡♥✈❡❝t♦rs
❙✉♣♣♦s❡ V1 ❛♥❞ V2 ❛r❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r F t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ✭♣♦ss✐❜❧② ❡q✉❛❧✮ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t V1 ❛♥❞ V2 ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡♥✱ ❛❧❧ ✈❛❧✉❡s ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r Y = F (X) ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿ X = r1 V1 + r2 V2 =⇒ F (X) = r1 λ1 V1 + r2 λ2 V2 ,
✇✐t❤ s♦♠❡ r❡❛❧ ❝♦❡✣❝✐❡♥ts r1 ❛♥❞ r2✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♠❛tr✐① ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {V1, V2} ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✐s ❞✐❛❣♦♥❛❧ ✇✐t❤ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✿
λ1 0 F = . 0 λ2 ❊①❛♠♣❧❡ ✷✳✾✳✷✿ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥✿
u = −x − 2y v =x − 4y
−1 −2 . =⇒ F = 1 −4
✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✶✾✷
▲❡t✬s ❝♦♥✜r♠ ✇❤❛t ✐s s❤♦✇♥ ❛❜♦✈❡✳ ❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ −1 − λ −2 = λ2 − 5λ + 6 . det(F − λI) = det 1 −4 − λ
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡✿ λ1 = −3, λ2 = −2 .
❚♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ s♦❧✈❡ t❤❡ t✇♦ ✈❡❝t♦r ❡q✉❛t✐♦♥s✿
F Vi = λi Vi , i = 1, 2 .
❚❤❡ ✜rst✱ λ1 = −3✿
−1 −2 F V1 = 1 −4
x x . = −3 y y
❚❤✐s ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
−x − 2y = −3x x − 4y = −3y
=⇒
2x − 2y = 0 x − y =0
=⇒ x = y .
❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✱ ❛❣❛✐♥✱ t❤❛t t❤✐s ✐s t❤❡ s❛♠❡ ❡q✉❛t✐♦♥❀ t❤✐s ❧✐♥❡ ❣✐✈❡s ✉s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ ♦♥❡ ❡✐❣❡♥✈❡❝t♦r✿ 1 . 1
V1 =
❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿
−1 −2 F V2 = 1 −4
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ✭s❛♠❡ ❡q✉❛t✐♦♥✮✿
−x − 2y = −2x x − 4y = −2y
=⇒
x x . = −2 y y
x − 2y = 0 x − 2y = 0
=⇒ x = 2y .
❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ λ2 = −2✳ ❲❡ ❝❤♦♦s❡ ♦♥❡ ❡✐❣❡♥✈❡❝t♦r✿ 2 . V2 = 1
❚❤❡ ♣❛✐r {V1 , V2 } ✐s ❛ ❜❛s✐s✦ ❲❡ s✉♠♠❛r✐③❡ ✇❤❛t F ❞♦❡s✿ ✶✳ ❆ ✢✐♣ ❛♥❞ str❡t❝❤ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 1, 1 >✿ ❚❤❡ ❧✐♥❡ y = x r❡♠❛✐♥s ✐♥t❛❝t✳ ✷✳ ❆ ✢✐♣ ❛♥❞ str❡t❝❤ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 2, 1 >✿ ❚❤❡ ❧✐♥❡ x = 2y r❡♠❛✐♥s ✐♥t❛❝t✳
✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✶✾✸
❲❡ ❛❧s♦ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡r❡ ✐s ♥♦ ❝❤❛♥❣❡ ♦❢ ♦r✐❡♥t❛t✐♦♥✳ ❙tr❡t❝❤✐♥❣ ❛s✐❞❡✱ t❤✐s ❧♦♦❦s ❧✐❦❡ ❝❡♥tr❛❧ s②♠♠❡tr②✿
❲❡ ♦❜s❡r✈❡ ❢❛♥♥✐♥❣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❧✐♥❡s✳ ❋♦r t❤❡ r❡st ♦❢ t❤❡ ✈❡❝t♦rs✱ ✇❡ ❤❛✈❡✿ X = xV1 + yV2
1 2 . − 2y =⇒ F X = −3x 2 −1
❚❤❡r❡❢♦r❡✱ ♠❛tr✐① ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {V1 , V2 } ✐s ❞✐❛❣♦♥❛❧✿
−3 0 F = . 0 −2 ❊①❛♠♣❧❡ ✷✳✾✳✸✿ str❡t❝❤✲s❤r✐♥❦
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿
u = x + 2y v = 3x + 2y
1 2 . =⇒ F = 3 2
▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿ 1−λ 2 = λ2 − 3λ − 4 . det(F − λI) = det 3 2−λ
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡✿
λ1 = −1, λ2 = 4 .
◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿
F Vi = λi Vi , i = 1, 2 .
❚❤❡ ✜rst✿
1 2 F V1 = 3 2
x x . = −1 y y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
x + 2y = −x 3x + 2y = −y
=⇒
2x + 2y = 0 3x + 3y = 0
=⇒ x = −y .
✷✳✾✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
❲❡ ❝❤♦♦s❡✿
✶✾✹
1 . V1 = −1
y = −x✮
❊✈❡r② ✈❛❧✉❡ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡
✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❡✐❣❡♥✈❡❝t♦r✿
1 . X = λ1 V 1 = − −1
❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿
1 2 F V2 = 3 2
x x . =4 y y
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
x + 2y = 4x 3x + 2y = 4y
=⇒
❲❡ ❝❤♦♦s❡✿
−3x + 2y = 0 3x − 2y = 0
1 . V2 = 3/2
y = 3x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡ 1 . X = λ2 V 2 = 4 3/2
❊✈❡r② ✈❛❧✉❡ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡
❚❤❡ ♣❛✐r
=⇒ x = 2y/3 .
{V1 , V2 }
♦❢ t❤✐s ❡✐❣❡♥✈❡❝t♦r✿
✐s ❛ ❜❛s✐s✦ ❚❤❡♥✱
X = xV1 + yV2 =⇒ U = F (X) = −xV1 + 4yV2 . ❚❤❡ ♠❛tr✐① ♦❢
F
✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s
{V1 , V2 } ✐s✿ −1 0 . F = 0 4
❲❡ ♦❜s❡r✈❡ ❢❛♥♥✐♥❣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❧✐♥❡s✳
▲❡t✬s s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts✳
❚❤❡♦r❡♠ ✷✳✾✳✹✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✕ ❘❡❛❧ ❊✐❣❡♥✈❛❧✉❡s F ❤❛s t✇♦ r❡❛❧ ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2 ✳ ❚❤❡♥✱ t❤❡ ❢✉♥❝t✐♦♥ U = F (X) str❡t❝❤❡s✴s❤r✐♥❦s t❤❡ t✇♦ ❡✐❣❡♥s♣❛❝❡ ❜② ❢❛❝t♦rs |λ1 | ❛♥❞ |λ2 | ❙✉♣♣♦s❡ ♠❛tr✐①
r❡s♣❡❝t✐✈❡❧② ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✿
•
■❢
•
■❢
λ1
λ2
❛♥❞
❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✱ ✐t ♣r❡s❡r✈❡s t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞
❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳
λ1
❛♥❞
λ2
❤❛✈❡ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥s✱ ✐t r❡✈❡rs❡s t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞
❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳
❊①❡r❝✐s❡ ✷✳✾✳✺ ❆♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✳
❊①❛♠♣❧❡ ✷✳✾✳✻✿ s❦❡✇✐♥❣✲s❤❡❛r✐♥❣ ❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿
1 1 . F = 0 1
✷✳✾✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✶✾✺
❇❡❧♦✇✱ ✇❡ r❡♣❧❛❝❡ ❛ ❝✐r❝❧❡ ✇✐t❤ ❛♥ ❡❧❧✐♣s❡ t♦ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ ✐t ✉♥❞❡r s✉❝❤ ❛ ❢✉♥❝t✐♦♥✿
❚❤❡r❡ ✐s st✐❧❧ ❛♥❣✉❧❛r str❡t❝❤✲s❤r✐♥❦ ❜✉t t❤✐s t✐♠❡ ✐t ✐s ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ s❛♠❡ ❧✐♥❡✳ ❲❡ s❡❡ ✏❢❛♥♥✐♥❣ ♦✉t✑ ❛❣❛✐♥✿
❚❤✐s t✐♠❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❢❛♥ ✐s ❢✉❧❧② ♦♣❡♥✦ ■t ♠❛❦❡s ❛ ❞✐✛❡r❡♥❝❡ t❤❛t t❤❡ ❢❛♥♥✐♥❣ ❤❛♣♣❡♥s t♦ ❛ ✇❤♦❧❡ ❤❛❧❢✲♣❧❛♥❡✳ ❚♦ s❡❡ ♠♦r❡ ❝❧❡❛r❧②✱ ❝♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s t♦ ❛ sq✉❛r❡✿
❚❤✐s ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿
1−λ 1 = (1 − λ)2 . det(F − λI) = det 0 1−λ
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡
λ1 = λ2 = 1 . ❲❤❛t ❛r❡ t❤❡ ❡✐❣❡♥✈❡❝t♦rs❄
1 1 FV = 0 1
x x . =1 y y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
x + y = x ❆◆❉ y =y
=⇒ x
❛♥②,
y = 0.
✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✶✾✻
❚❤❡ ♦♥❧② ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❤♦r✐③♦♥t❛❧✦ ❚❤❡r❡❢♦r❡✱ ♦✉r ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧②✳ ❚❤❡r❡ ✐s ♥♦ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ❢♦r t❤✐s ♦♣❡r❛t♦r✳ ❊①❛♠♣❧❡ ✷✳✾✳✼✿ r♦t❛t✐♦♥s
❚❤❡r❡ ❛r❡ ♦t❤❡r ♦✉t❝♦♠❡s t❤❛t t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❝♦✈❡r✳ ❘❡❝❛❧❧ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ♠❛tr✐① A ♦❢ t❤❡ 90✲❞❡❣r❡❡ r♦t❛t✐♦♥✿ −λ −1 = λ2 + 1 . χA (λ) = det 1 −λ
❇✉t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥✱
x2 + 1 = 0 ,
❤❛s ♥♦ s♦❧✉t✐♦♥s✦ ❆r❡ ✇❡ ❞♦♥❡ t❤❡♥❄ ◆♦t ✐❢ ✇❡ ❛r❡ ✇✐❧❧✐♥❣ t♦ ✉s❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✭♥❡①t ❝❤❛♣t❡r✮✿ λ1,2 = i ❛♥❞ − i .
❚❤✐s ✐s t❤❡ ❡✛❡❝t ♦❢ r♦t❛t✐♦♥✿
▲❡t✬s ❝♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛r❜✐tr❛r② ❛♥❣❧❡ θ✿ cos θ − sin θ x u . = y sin θ cos θ v
❚❤❡ ❛♥❣❧❡ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ χA (λ) = (cos θ − λ)2 + sin2 θ = cos2 θ − 2 cos θ λ + λ2 + sin2 θ = λ2 − 2 cos θ λ + 1 .
❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤✐s ♣♦❧②♥♦♠✐❛❧ ✐s ♥❡❣❛t✐✈❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ❤❛s ♥♦ r❡❛❧ r♦♦ts✳ ❚❤❡ r❡s✉❧t ♠❛❦❡s s❡♥s❡✿ ❆ r♦t❛t✐♦♥ ❝❛♥♥♦t ♣♦ss✐❜❧② ❤❛✈❡ ❡✐❣❡♥✈❡❝t♦rs ❜❡❝❛✉s❡ ❛❧❧ ✈❡❝t♦rs ❛r❡ r♦t❛t❡❞✦
✷✳✾✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s
✶✾✼
❊①❛♠♣❧❡ ✷✳✾✳✽✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿
u = 3x −13y , v = 5x +y ,
❛♥❞
3 −13 . F = 5 1
❇❡❧♦✇ ✇❡ ❝❛♥ r❡❝♦❣♥✐③❡ ❜♦t❤ r♦t❛t✐♦♥ ❛♥❞ r❡✲s❝❛❧✐♥❣✿
❚❤✐s ✐s ♦✉r ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿
3 − λ −13 = λ2 − 4λ + 68 . χ(λ) = det(F − λI) = det 5 1−λ
❊①❡r❝✐s❡ ✷✳✾✳✾ ❲❤❛t ❞♦❡s ✐t t❡❧❧ ✉s❄
❖✉r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❡r♠s ♦❢ t❤❡ tr❛❝❡ ♦❢ t❤❡ ♠❛tr✐①✿
χ(λ) = λ2 − tr F λ + det F . ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢♦r t❤❡ ❝❛s❡ ♦❢ ♥♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✿
❈♦r♦❧❧❛r② ✷✳✾✳✶✵✿ ❚r❛❝❡ ❛♥❞ ❉✐s❝r✐♠✐♥❛♥t
❙✉♣♣♦s❡ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛ ♠❛tr✐① F s❛t✐s✜❡s✿ D = (tr F )2 − 4 det F ≤ 0 .
❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡✿ θ = sin−1
1 2
r
4 − (tr F )2 det F
!
.
✷✳ ■t r❡✲s❝❛❧❡s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ s= ❊①❡r❝✐s❡ ✷✳✾✳✶✶ ❆♣♣❧② t❤❡ ❝♦r♦❧❧❛r② t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✳
√
det F .
❈❤❛♣t❡r ✸✿ ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s
❈♦♥t❡♥ts
✸✳✶ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✸✳✻ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✸✳✼ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✾✽ ✷✵✹ ✷✵✾ ✷✶✻ ✷✶✾ ✷✷✸ ✷✷✽ ✷✸✸ ✷✸✻ ✷✹✷ ✷✹✹ ✷✺✵
✸✳✶✳ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
❲❡ ✇✐❧❧ t❛❦❡ ❛ ❜r♦❛❞❡r ✈✐❡✇ ❛t ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✐♥❝❧✉❞❡ t❤❡ ❧♦✇❡r ❞✐♠❡♥s✐♦♥s✳ ♣♦ss✐❜✐❧✐t✐❡s✿
❚❤❡s❡ ❛r❡ t❤❡ ❢♦✉r
✸✳✶✳
❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
✶✾✾
❲❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡s❡ ♦♣❡r❛t♦rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♠❛tr✐❝❡s ❛♥❞ ❤♦✇ t❤❡s❡ ♠❛tr✐❝❡s ❛r❡ ❝♦♠❜✐♥❡❞ t♦ ♣r♦❞✉❝❡ ❝♦♠♣♦s✐t✐♦♥s✳ ❚❤❡ r✉❧❡ r❡♠❛✐♥s✿ ◮ ❚❤❡ ✈❛❧✉❡ ✉♥❞❡r F ♦❢ ❡❛❝❤ ❜❛s✐s ✈❡❝t♦r ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ F ❜❡❝♦♠❡s ❛ ❝♦❧✉♠♥ ✐♥ t❤❡ ♠❛tr✐① ♦❢ F ✳
▲❡t✬s ❛♣♣❧② t❤❡ r✉❧❡ t♦ t❤❡s❡ ❢♦✉r s✐t✉❛t✐♦♥s ✉s✐♥❣ t❤❡ st❛♥❞❛r❞ ❜❛s❡s✳ ❊①❛♠♣❧❡ ✸✳✶✳✶✿ ❞✐♠❡♥s✐♦♥s ✶ ❛♥❞ ✶
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x .
■t ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ x✲❛①✐s ✐s < 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ f (< 1 >) = 3 < 1 > .
❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s f = [3] . ❊①❛♠♣❧❡ ✸✳✶✳✷✿ ❞✐♠❡♥s✐♦♥s ✶ ❛♥❞ ✷
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ F : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > .
■t str❡t❝❤❡s t❤❡ x✲❛①✐s ♦♥ t❤❡ uv ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 3, 2 >✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ x✲❛①✐s ✐s < 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ uv ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ F (< 1 >) =< 3, 2 > .
❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s
3 . F = 2
❊①❛♠♣❧❡ ✸✳✶✳✸✿ ❞✐♠❡♥s✐♦♥s ✷ ❛♥❞ ✶
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ f : R2 → R ❞❡✜♥❡❞ ❜② f (x, y) = 3x + 2y .
■t r♦❧❧s t❤❡ xy ✲♣❧❛♥❡ ♦♥ t❤❡ u✲❛①✐s✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ f (< 1, 0 >) =< 3 >
❛♥❞ f (< 0, 1 >) =< 2 > .
❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s f = [3 2] . ❊①❛♠♣❧❡ ✸✳✶✳✹✿ ❞✐♠❡♥s✐♦♥s ✷ ❛♥❞ ✷
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✿ F : R2 → R2 ❞❡✜♥❡❞ ❜② f (x, y) =< 3x + 2y, 5x − y > .
❲❡✬❞ ♥❡❡❞ t❤❡ ❡✐❣❡♥✈❡❝t♦r ❛♥❛❧②s✐s ✐♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ✇❤❛t ✐t ❞♦❡s✳✳✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡
✸✳✶✳
❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
✷✵✵
❜❛s✐s ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1, 0 > ❛♥❞ < 0, 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ F (< 1, 0 >) =< 3, 5 >
❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s
❛♥❞ F (< 0, 1 >) =< 2, −1 > .
3 2 F = . 5 −1
❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥ ❥✉♠♣ ❛❤❡❛❞ ♦❢ t❤❡ r✉❧❡ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ ❛♥❞ ✇r✐t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♣r❡s❡♥t ✐♥ t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r str❛✐❣❤t ✐♥t♦ t❤❡ ♠❛tr✐①✳
❲❛r♥✐♥❣✦ ■ts ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳
❊①❡r❝✐s❡ ✸✳✶✳✺ ■♥❝❧✉❞❡ t❤❡ ❞✐♠❡♥s✐♦♥ 0 ❢♦r ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s ✐♥ t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s✳ ❲❤❡♥❡✈❡r t❤❡r❡ ✐s ❛❧❣❡❜r❛ ✐♥ t❤❡ ♦✉t♣✉t s♣❛❝❡✱ ✇❡ ❝❛♥ ✉s❡ ✐t t♦ ❞♦ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳ ■❢ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥s ✐s ❛ ✈❡❝t♦r s♣❛❝❡✱ ✇❡ ❝❛♥ ❛❞❞ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧② t❤❡♠ ❜② ❛ ❝♦♥st❛♥t✳ ❲❡ ❥✉st ♥❛rr♦✇ ❞♦✇♥ t❤✐s ✐❞❡❛ t♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✿
❉❡✜♥✐t✐♦♥ ✸✳✶✳✻✿ ❛❞❞✐t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ●✐✈❡♥ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✿
t❤❡✐r
s✉♠ ✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿
F, G : Rn → Rm ,
F + G : Rn → Rm ,
❞❡✜♥❡❞ ❜②✿ (F + G)(x) = F (x) + G(x) .
❲❡ ✐❧❧✉str❛t❡ t❤✐s ♦♣❡r❛t✐♦♥ ❥✉st ❛s ❜❡❢♦r❡✿
❇✉t ✐❢ t❤❡s❡ ❛r❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r ♠❛tr✐❝❡s❄ ❲❡ ❣♦ t❤♦✉❣❤ t❤❡ s❛♠❡ ❢♦✉r ❝❛s❡s ❜❡❧♦✇✳
✸✳✶✳
❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
❊①❛♠♣❧❡ ✸✳✶✳✼✿ ❞✐♠❡♥s✐♦♥s
1
❛♥❞
✷✵✶
1
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✮✿
f, g : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x ❛♥❞ g(x) = 2x . ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
f = [3] ❛♥❞ g = [2] .
❲❤❛t ❛❜♦✉t t❤❡✐r s✉♠❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ f + g : R → R ❞❡✜♥❡❞ ❜② (f + g)(x) = f (x) + g(x) = 3x + 2x = 5x .
■ts ♠❛tr✐① ✐s
f + g = [5] . ❖❢ ❝♦✉rs❡✱ t❤✐s ♥❡✇ ♥✉♠❜❡r ✐s ❥✉st t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ♦r✐❣✐♥❛❧ ♥✉♠❜❡rs✳ ❊①❛♠♣❧❡ ✸✳✶✳✽✿ ❞✐♠❡♥s✐♦♥s
1
❛♥❞
2
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮✿
F, G : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
3 F = 2
❛♥❞ G(x) =< 5x, −x > .
5 . ❛♥❞ G = −1
❲❤❛t ❛❜♦✉t t❤❡✐r s✉♠❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿
F + G : R → R2 ❞❡✜♥❡❞ ❜② (F + G)(x) = F (x) + G(x) =< 3x, 2x > + < 5x, −x >=< 8x, x > . ■ts ♠❛tr✐① ✐s
8 . F +G= 1
❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❥✉st t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ❛s ✐❢ t❤❡② ✇❡r❡ ✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✸✳✶✳✾✿ ❞✐♠❡♥s✐♦♥s
2
❛♥❞
1
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✮✿
f, g : R2 → R ❞❡✜♥❡❞ ❜② f (x, y) = 3x + 2y ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
❛♥❞ g(x, y) = 5x − 2y .
f = [3, 2] ❛♥❞ g = [5, −2] .
❚❤❡✐r s✉♠ ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ✭t❤✐s ✐s ✈❡❝t♦r ❛❞❞✐t✐♦♥✮✿
f + g : R2 → R ❞❡✜♥❡❞ ❜② (f + g)(x, y) = f (x, y) + g(x, y) = 3x + 2y + 5x − 2y = 8x . ■ts ♠❛tr✐① ✐s
f = [8, 0] , t❤❡ s✉♠ ✕ ❝♦♠♣♦♥❡♥t✇✐s❡ ✕ ♦❢ t❤❡ t✇♦✳
✸✳✶✳
❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
✷✵✷
❊①❛♠♣❧❡ ✸✳✶✳✶✵✿ ❞✐♠❡♥s✐♦♥s 2 ❛♥❞ 2 ●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡✮✿
F, G : R2 → R2 ❞❡✜♥❡❞ ❜② F (x, y) =< 3x + 2y, 5x − y > ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
3 2 F = 5 −1
❛♥❞ G(x, y) =< 5x + y, x + y > .
5 1 . ❛♥❞ G = 1 1
❚❤❡✐r s✉♠ ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ✭t❤✐s ✐s ✈❡❝t♦r ❛❞❞✐t✐♦♥✮✿
F +G : R2 → R2 ❞❡✜♥❡❞ ❜② (F +G)(x, y) =< 3x+2y, 5x−y > + < 5x+y, x+y >=< 8x+3y, 6x > . ■ts ♠❛tr✐① ✐s
8 3 , F +G= 6 0
t❤❡ s✉♠ ✕ ❝♦♠♣♦♥❡♥t✇✐s❡ ✕ ♦❢ t❤❡ t✇♦✿ 8 3 3+5 2+1 5 1 3 2 . = = + F +G= 6 0 5 + 1 (−1) + 1 1 1 5 −1 ❚❤✐s ♦♣❡r❛t✐♦♥ ♦❢ ♠❛tr✐① ❛❞❞✐t✐♦♥ ✐s ❝♦♠♣♦♥❡♥t✇✐s❡✿ ❚❤❡ t✇♦ ♦♣❡r❛t♦rs ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥✱ s♦ t❤❛t t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ♠❛tr✐❝❡s ❛r❡ ❡q✉❛❧ t♦♦✳ ❚❤❡② ❝❛♥ t❤❡♥ ❜❡ ♦✈❡r❧❛♣♣❡❞ s♦ t❤❛t t❤❡ ❡♥tr✐❡s ❛r❡ ❛❧✐❣♥❡❞ ❛♥❞ ❛❞❞❡❞ ❛❝❝♦r❞✐♥❣❧②✿
❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✶✿ ❛❞❞✐t✐♦♥ ♦❢ ♠❛tr✐❝❡s ❙✉♣♣♦s❡ A ❛♥❞ B ❛r❡ t✇♦ m × n ♠❛tr✐❝❡s✳ ❚❤❡♥ t❤❡✐r ❞❡♥♦t❡❞ ❜②✿
s✉♠ ✐s t❤❡ m × n ♠❛tr✐①✱
A+B t❤❡ ij ✲❡♥tr② ♦❢ ✇❤✐❝❤ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ij ✲❡♥tr✐❡s ♦❢ A ❛♥❞ B ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ A = aij ✱ B = bij ✱ ❛♥❞ C = A + B = cij ✱ t❤❡♥
cij = aij + bij , ❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳
✸✳✶✳
❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s
✷✵✸
■t ✐s s✐♠♣❧❡r ❢♦r s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿
❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✷✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦r ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ✐ts s❝❛❧❛r
F : Rn → R m ,
♣r♦❞✉❝t ✇✐t❤ ❛ r❡❛❧ ♥✉♠❜❡r r ✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿ rF : Rn → Rm ,
❞❡✜♥❡❞ ❜②✿ (rF )(x) = rF (x) .
❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛❣❛✐♥✿ ❊✈❡r② ❡♥tr② ✐s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ s❛♠❡ ♥✉♠❜❡r✳
❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✸✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♠❛tr✐❝❡s ❙✉♣♣♦s❡ A ✐s ❛♥ m × n ♠❛tr✐①✳ ❚❤❡♥ ✐ts ❞❡♥♦t❡❞ ❜②✿
s❝❛❧❛r ♠✉❧t✐♣❧❡
❜② ❛ r❡❛❧ ♥✉♠❜❡r r✱
rA
✐s ❛♥ m × n ♠❛tr✐①✱ t❤❡ ij ✲❡♥tr② ♦❢ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ij ✲❡♥tr② ♦❢ A ❜② r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ A = aij ❛♥❞ C = rA = cij ✱ t❤❡♥ cij = raij ,
❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳
❲❛r♥✐♥❣✦ ❚❤❡ ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❆❝❝♦r❞✐♥❣❧②✱ t❤❡ ♠❛tr✐① ♦♣❡r✲ ❛t✐♦♥s ❛r❡ ❥✉st ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ♦♣❡r❛t✐♦♥s ♦♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳
✸✳✷✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✵✹
✸✳✷✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
▲❡t✬s t❛❦❡ t❤❡ ♣r♦❜❧❡♠ ❛❜♦✉t ♠✐①t✉r❡s t♦ t❤❡ ♥❡①t ❧❡✈❡❧✳ ❲❡ ❤❛✈❡✿ ✶✳ n ✐♥❣r❡❞✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ n ✉♥❦♥♦✇♥s ♦r ✈❛r✐❛❜❧❡s x1, ..., xn r❡♣r❡s❡♥t✐♥❣ t❤❡ ❛♠♦✉♥ts ♦❢ ❡❛❝❤❀ ❛♥❞ ✷✳ m r❡q✉✐r❡♠❡♥ts ♦r r❡str✐❝t✐♦♥s✱ ✐✳❡✳✱ m ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ t❤❡s❡ ✈❛r✐❛❜❧❡s ✭k = 1, 2, ..., m✮✿ ak1 x1 + ... + akn xn = bk .
❚❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐s ✈❡r② ❝✉♠❜❡rs♦♠❡✳ ❆s ❜❡❢♦r❡✱ ✇❡ tr❛♥s❧❛t❡ t❤✐s ❛ s②st❡♠ ✐♥t♦ ❛ ✈❡❝t♦r✲♠❛tr✐① ❡q✉❛t✐♦♥✿ FX = B
✇❤❡r❡ ✶✳ X =< x1, ..., xn > ✐s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ✉♥❦♥♦✇♥s✱ ✷✳ B =< b1, ..., bm > ✐s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ t♦t❛❧s✱ ❛♥❞ ✸✳ F = aij ✐s t❤❡ m × n ♠❛tr✐① ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥t✱ ✇❡ ♣r❡❢❡r t♦ ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✐✳❡✳✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿
F (X) = B F : Rn → Rm .
❆♥❞ t❤❡ ❡q✉❛t✐♦♥ ♥❡❡❞s t♦ ❜❡ s♦❧✈❡❞✦ ■♥ ❞✐♠❡♥s✐♦♥ 1✱ t❤❡ ❡q✉❛t✐♦♥ kx = b ✐s s♦❧✈❡❞ ❜② ✉♥❞♦✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② k ❜② ❞✐✈✐s✐♦♥ ❜② k✿ kx = b =⇒ x =
❙✐♠✐❧❛r❧②✱ ✇❡ ♥❡❡❞ t❤❡ ✐♥✈❡rs❡ ♦❢ F t♦ s♦❧✈❡ ♦✉r ❡q✉❛t✐♦♥✿
b . k
F −1 : Rm → Rn F (X) = B =⇒ X = F −1 (B)
❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ t❤❡ ♦♣❡r❛t♦r F tr❛♥s❢♦r♠s t❤❡ xy✲♣❧❛♥❡ ✐♥t♦ t❤❡✱ s❛②✱ uv✲♣❧❛♥❡✿
✸✳✷✳
❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✵✺
❖♥❡ ♣❛rt✐❝✉❧❛r ✈❡❝t♦r ✐♥ t❤❡ uv ✲♣❧❛♥❡✱ B ✱ ♥❡❡❞s t♦ ❜❡ tr❛❝❡❞ ❜❛❝❦ t♦ t❤❡ xy ✲♣❧❛♥❡✳ ❖❢ ❝♦✉rs❡✱ ✐❢ ✇❡ ❤❛✈❡ F −1 ✱ ✇❡✬❧❧ ✜♥❞ t❤❡ ❝♦✉♥t❡r♣❛rts ❢♦r ❛❧❧ B ✬s✳ ❊①❛♠♣❧❡ ✸✳✷✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡
❲❡ ❦♥♦✇ s♦♠❡ ♦❢ t❤❡ ❛♥s✇❡rs✿ ✶✳ ■❢ F ✐s t❤❡ ✉♥✐❢♦r♠ str❡t❝❤ ❜② 2✱ t❤❡♥ F −1 ✐s t❤❡ ✉♥✐❢♦r♠ s❤r✐♥❦ ❜② 2✳ ✷✳ ■❢ F ✐s t❤❡ str❡t❝❤ ❜② 2 ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r e1 =< 1, 0 >✱ t❤❡♥ F −1 ✐s t❤❡ ✉♥✐❢♦r♠ s❤r✐♥❦ ❜② 2 ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ e1 ✳ ✸✳ ■❢ F ✐s t❤❡ r♦t❛t✐♦♥ ❜② 90 ❞❡❣r❡❡s ❝❧♦❝❦✇✐s❡✱ t❤❡♥ F −1 ✐s t❤❡ r♦t❛t✐♦♥ ❜② 90 ❞❡❣r❡❡s ❝♦✉♥t❡r❝❧♦❝❦✲ ✇✐s❡✳ ✹✳ ■❢ F ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ x✲❛①✐s✱ t❤❡♥ F −1 ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ x✲❛①✐s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✶✳ F (X) = 2X =⇒ F −1 (Y ) = 21 Y
✷✳ F =
✸✳ F =
✹✳ F =
2 0 0 1 0
1
−1 0 1
0
0 −1
=⇒ F −1 =
1 2
0 1
0 −1 1
0
1
0
=⇒ F −1 =
0
=⇒ F −1 =
0 −1
▲❡t✬s r❡❝❛❧❧ t❤❛t t❤❡ ✐♥✈❡rs❡ ✐s ❞❡✜♥❡❞ ✈✐❛ ❝♦♠♣♦s✐t✐♦♥s✳ ■t ♠✉st s❛t✐s❢②✿ F (F −1 (Y )) = Y ,
❢♦r ❛❧❧ Y ✱ ❛♥❞
F −1 (F (X)) = X ,
❢♦r ❛❧❧ X ✳ ■♥ ♦t❤❡r ✇♦r❞s✱
F ◦ F −1 = I ,
❛♥❞
F −1 ◦ F = I ,
✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❲❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ ❝♦♠♣♦s✐t✐♦♥s ❜❡tt❡r✳
❲❡ ❦♥♦✇ ❤♦✇ t♦ ❝♦♠♣✉t❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥✿ F
G
Rn −−−−→ Rm −−−−→ Rk
■t ✐s ❝♦♠♣✉t❡❞ ✈✐❛ s✉❜st✐t✉t✐♦♥✿ (G ◦ F )(X) = G(F (X)) .
❇✉t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♠❛tr✐❝❡s❄ ❍♦✇ ❛r❡ t❤❡ ♠❛tr✐❝❡s ♦❢ F ❛♥❞ G ❝♦♠❜✐♥❡❞ t♦ ♣r♦❞✉❝❡ t❤❡ ♠❛tr✐① ♦❢ G ◦ F ❄ ■t ✐s ❝❛❧❧❡❞ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❍❡r❡ ✐s ✇❤②✳ ❊①❛♠♣❧❡ ✸✳✷✳✷✿ ❝♦♠♣♦s✐t✐♦♥s
R1 → R1 → R1
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✮✿ ❛♥❞
f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x , g : R → R ❞❡✜♥❡❞ ❜② g(y) = 2y .
✸✳✷✳
❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
✷✵✻
f = [3] ❛♥❞ g = [2] .
❲❤❛t ❛❜♦✉t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥❄ ❚❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r ♠❛t❝❤✦ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ g ◦ f : R → R ❞❡✜♥❡❞ ❜② (g ◦ f )(x) = g(f (x)) = 2(3x) = 6x .
■ts ♠❛tr✐① ✐s g ◦ f = [6] .
❖❢ ❝♦✉rs❡✱ t❤✐s ♥❡✇ ♥✉♠❜❡r ✐s ❥✉st t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ♦r✐❣✐♥❛❧ ♥✉♠❜❡rs✳ ❊①❛♠♣❧❡ ✸✳✷✳✸✿ ❝♦♠♣♦s✐t✐♦♥s
R1 → R1 → R2
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮✿ f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x ,
❛♥❞ ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
G : R → R2 ❞❡✜♥❡❞ ❜② G(y) =< 3y, 2y > . 3 . f = [3] ❛♥❞ G = 2
❲❤❛t ❛❜♦✉t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✭❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ G ◦ f : R → R2 ,
❞❡✜♥❡❞ ❜② ■ts ♠❛tr✐① ✐s
(G ◦ f )(x) = G(f (x)) =< 3(3x), 2(3x) >=< 9x, 6x > . 9 . G◦f = 6
❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❥✉st t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❛s ✐❢ t❤❡ ✜rst ✐s ❛ ♥✉♠❜❡r ❛♥❞ t❤❡ s❡❝♦♥❞ ❛ ✈❡❝t♦r✿ 9 3·3 3 . = [3] = Gf = 6 2·3 2 ❊①❛♠♣❧❡ ✸✳✷✳✹✿ ❝♦♠♣♦s✐t✐♦♥s
R1 → R2 → R1
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✮✿ ❛♥❞ ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
F : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > , g : R2 → R ❞❡✜♥❡❞ ❜② g(u, v) = 5u − 2v . 3 F = 2
❛♥❞ g = [5, −2] .
❚❤❡✐r s✉♠ ✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ❞❡✜♥❡❞ ❜②
g ◦ F : R → R, (g ◦ F )(x) = g(F (x)) = 5(3x) − 2(2x) = 11x .
✸✳✷✳
❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✵✼
■ts ♠❛tr✐① ✐s g ◦ F = [11] .
■t✬s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ✈❡❝t♦r✲❧✐❦❡ ♠❛tr✐❝❡s✿
3 = [5 · 3 + (−2) · 2] = [11] . gF = [5, −2] 2 ❊①❛♠♣❧❡ ✸✳✷✳✺✿ ❝♦♠♣♦s✐t✐♦♥s
R2 → R2 → R2
●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡s✮✿ F, G : R2 → R2
❞❡✜♥❡❞ ❜② F (x, y) =< u, v >=< 3x + 2y, 5x − y >
❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿
3 2 F = 5 −1
❛♥❞ G(u, v) =< 5u + v, u + v > .
5 1 . ❛♥❞ G = 1 1
❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s ❛♥ ♦♣❡r❛t♦r ❝♦♠♣✉t❡❞ ❜② s✉❜st✐t✉t✐♦♥✿ G ◦ F : R2 → R 2 ,
❞❡✜♥❡❞ ❜② (G ◦ F )(x, y) =< 5(3x + 2y) + (5x − y), (3x + 2y) + (5x − y) >=< 20x + 9y, 8x + y > .
■ts ♠❛tr✐① ✐s
20 9 . G◦F = 8 1
■t ✐s s❡❡♥ ❛s ❝♦♠♣✉t❡❞ ✈✐❛ ❢♦✉r ❞♦t ♣r♦❞✉❝ts ♦❢ t❤❡ ❢♦✉r ♣❛✐rs ♦❢ r♦✇s ✭❢r♦♠ t❤❡ ✜rst ♠❛tr✐①✮ ❛♥❞ ❝♦❧✉♠♥s ✭❢r♦♠ t❤❡ s❡❝♦♥❞✮✿
5 1 5 1 5 1 5 1
1 3 · 1 5 1 3 · 1 5 1 3 · 1 5 1 3 · 1 5
2 → −1 2 → −1 2 → −1 2 → −1
20 → 8 10 5 · 2 + 1 · (−1) = 9 → 8 10 1·3+1·5=8 → 8 10 1 · 2 + 1 · (−1) = 1 → 8 5 · 3 + 1 · 5 = 20
■♥ ❣❡♥❡r❛❧✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛✿
ae + bg af + bh e f a b = · ce + dg cf + dh g h c d
9 1 9 1 9 1 9 1
✸✳✷✳
❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✵✽
❲❛r♥✐♥❣✦ ❆s t❤❡ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛✲ t♦rs✱ ✐t ✐s s❡❝♦♥❞❛r② t♦ t❤❡ s✉❜st✐t✉t✐♦♥ ♦❢ t❤❡ ❢♦r✲ ♠✉❧❛s ✐t ❝♦♠❡s ❢r♦♠✳
◆♦✇ ✇❡ ❝♦♥s✐❞❡r ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐♥ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥s✿
R n → Rm → R k . ❋♦r t❤❡✐r ♠❛tr✐❝❡s✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦❧✉♠♥ ✐♥ t❤❡ ✜rst ♠✉st ❜❡ ❡q✉❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r♦✇ ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❛t✬s
m✦
❲❡ ❞♦ t❤✐s
❚❤❡♥✱ ✇❡ ❝❛♥ ❝❛rr② ♦✉t ❛ ❞♦t ♣r♦❞✉❝t✿
nk
t✐♠❡s ❛♥❞ ♣r♦❞✉❝❡ ❛♥
n×k
♠❛tr✐①✳
❊①❡r❝✐s❡ ✸✳✷✳✻ ▼✉❧t✐♣❧②✿
1 −1 1 2 3 4 5 6 · −1 1 . 1 −1 7 8 9 ❊①❛♠♣❧❡ ✸✳✷✳✼✿ s♣r❡❛❞s❤❡❡t ❖♥❡ ❝❛♥ ✉t✐❧✐③❡ ❛ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦t❤❡r s♦❢t✇❛r❡ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥s✳ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ✇♦r❦✱ t❤❡ s❡❝♦♥❞ ♠❛tr✐①
❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r t❤❡ tr❛♥s♣♦s❡ ♦❢
B
❤❛s t♦ ❜❡ ✏tr❛♥s♣♦s❡❞✑ ✭❜♦tt♦♠✮✿
X✿
❂❚❘❆◆❙P❖❙❊✭❘❬✲✺❪❈✿❘❬✲✸❪❈❬✶❪✮ ❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r
Y✿
❂❙❯▼P❘❖❉❯❈❚✭❘❈✷✿❘❈✹✱❘✽❈❬✲✸❪✿❘✽❈❬✲✶❪✮
❋✐♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ♠❛tr✐①✱ ❡s♣❡❝✐❛❧❧② ♦❢ ❤✐❣❤ ❞✐♠❡♥s✐♦♥✱ ✐s ❛ ❝❤❛❧❧❡♥❣✐♥❣ ♣r♦❜❧❡♠✳ ❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡
2×2
♠❛tr✐❝❡s✿
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✵✾
❚❤❡♦r❡♠ ✸✳✷✳✽✿ ■♥✈❡rs❡ ♦❢
2×2
❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐①
a b c d
−1
▼❛tr✐①
A
✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿
1 d −b = det A −c a
❊①❡r❝✐s❡ ✸✳✷✳✾ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳
❊①❡r❝✐s❡ ✸✳✷✳✶✵
$3 ♣❡r ❝❤✐❧❞ ❛♥❞ $3.20 ♣❡r ❛❞✉❧t ❢♦r ❛ t♦t❛❧ ♦❢ $118.40✳ ❚❤❡② t♦♦❦ t❤❡ tr❛✐♥ ❜❛❝❦ ❛t $3.50 ♣❡r ❝❤✐❧❞ ❛♥❞ $3.60 ♣❡r ❛❞✉❧t ❢♦r ❛ t♦t❛❧ ♦❢ $135.20✳ ❍♦✇ ♠❛♥② ❝❤✐❧❞r❡♥✱ ❛♥❞ ❤♦✇ ♠❛♥② ❛❞✉❧ts❄ ❯s❡ t❤❡ t❤❡♦r❡♠ t♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❆ t♦✉r✐st ❣r♦✉♣ t♦♦❦ ❛ tr❛✐♥ tr✐♣ ❛t
❊①❡r❝✐s❡ ✸✳✷✳✶✶ ❯s❡ t❤❡ t❤❡♦r❡♠ t♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❆ t♦✉r✐st ❣r♦✉♣ ✇✐t❤ ❛ tr❛✐♥ tr✐♣ ❢♦r ❛ t♦t❛❧ ♦❢ ❢♦r ❛ t♦t❛❧ ♦❢
$145✳
$110✳
❆♥♦t❤❡r t♦✉r✐st ❣r♦✉♣ ✇✐t❤
15
10
❝❤✐❧❞r❡♥ ❛♥❞
❝❤✐❧❞r❡♥ ❛♥❞
25
20
❛❞✉❧ts t♦♦❦
❛❞✉❧ts t♦♦❦ ❛ tr❛✐♥ tr✐♣
❲❤❛t ✇❡r❡ t❤❡ t✐❝❦❡t ♣r✐❝❡s❄
❚♦ s✉♠♠❛r✐③❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♣♣❧✐❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿
G
F
Rn −−−−→ Rm −−−−→ Rk ❲❡ ❤❛✈❡ ✐♠♣❧✐❝✐t❧② ✉s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿
❚❤❡♦r❡♠ ✸✳✷✳✶✷✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳
❊①❡r❝✐s❡ ✸✳✷✳✶✸ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳
❲❡ ❤❛✈❡ ❛❧s♦ ✐♠♣❧✐❝✐t❧② ✉s❡❞ t❤❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧t✿
❚❤❡♦r❡♠ ✸✳✷✳✶✹✿ ▼❛tr✐① ♦❢ ❈♦♠♣♦s✐t✐♦♥ ❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❛tr✐❝❡s t❤❛t r❡♣r❡s❡♥t t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥✳
✸✳✸✳ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
❚❤❡ ❡q✉❛t✐♦♥
x2 + 1 = 0 ❤❛s ♥♦ s♦❧✉t✐♦♥s✳ ■♥❞❡❡❞✱ ✇❡ ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
x2 ≥ 0 =⇒ x2 + 1 > 0 =⇒ x2 + 1 6= 0 .
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✶✵
■❢ ✇❡ tr② t♦ s♦❧✈❡ ✐t t❤❡ ✉s✉❛❧ ✇❛②✱ ✇❡ ❣❡t t❤❡s❡✿
x= ❚❤❡r❡ ❛r❡ ♥♦ s✉❝❤
√
r❡❛❧ ♥✉♠❜❡rs✳
−1
❛♥❞
√ x = − −1 .
❍♦✇❡✈❡r✱ ❧❡t✬s ✐❣♥♦r❡ t❤✐s ❢❛❝t ❢♦r ❛ ♠♦♠❡♥t✳ ▲❡t✬s s✉❜st✐t✉t❡ ✇❤❛t ✇❡ ❤❛✈❡ ❜❛❝❦ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✕ ❜❧✐♥❞❧② ✕ ❢♦❧❧♦✇ t❤❡ r✉❧❡s ♦❢ ❛❧❣❡❜r❛✳ ❲❡ ✏❝♦♥✜r♠✑ t❤❛t t❤✐s ✏♥✉♠❜❡r✑ ✐s ❛ ✏s♦❧✉t✐♦♥✑✿
√ x2 + 1 = ( −1)2 + 1 = (−1) + 1 = 0 . ❲❡ ❝❛❧❧ t❤✐s ❡♥t✐t② t❤❡
✐♠❛❣✐♥❛r② ✉♥✐t✱ ❞❡♥♦t❡❞ ❜② i✳
❲❡ ❥✉st ❛❞❞ t❤✐s ✏♥✉♠❜❡r✑ t♦ t❤❡ s❡t ♦❢ ♥✉♠❜❡rs ✇❡ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤✿
❆♥❞ s❡❡ ✇❤❛t ❤❛♣♣❡♥s✳✳✳ ▼❛❦✐♥❣
i
❛ ♣❛rt ♦❢ ❛❧❣❡❜r❛ ✇✐❧❧ ♦♥❧② r❡q✉✐r❡ t❤✐s t❤r❡❡✲♣❛rt ❝♦♥✈❡♥t✐♦♥✿
✶✳
i
✷✳
i ❝❛♥ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ✭❢♦✉r✮ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ r❡❛❧ ♥✉♠❜❡rs ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ r✉❧❡s❀
✸✳
i2 = −1✳
✐s ♥♦t ❛ r❡❛❧ ♥✉♠❜❡r ✭❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱
i 6= 0✮✱
❜✉t ❛❧s♦
❲❤❛t ❛❧❣❡❜r❛✐❝ r✉❧❡s ❛r❡ t❤♦s❡❄ ❆ ❢❡✇ ✈❡r② ❜❛s✐❝ ♦♥❡s✿
x + y = y + x, x · y = y · x, x(y + z) = xy + xz, ❲❡ ❛❧❧♦✇ ♦♥❡ ♦r s❡✈❡r❛❧ ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs t♦ ❜❡
i✳
❡t❝✳
❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡✿
i + y = y + i, i · y = y · i, i(y + z) = iy + iz,
❡t❝✳
❲❤❛t ♠❛❦❡s t❤✐s ❡①tr❛ ❡✛♦rt ✇♦rt❤✇❤✐❧❡ ✐s ❛ ♥❡✇ ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ♠❛② ❢❛❝t♦r ♦♥❡✿
x2 − 1 = (x − 1)(x + 1) . ❚❤❡♥
x=1
❛♥❞
x = −1
❛r❡ t❤❡
❇✉t s♦♠❡ ♣♦❧②♥♦♠✐❛❧s✱ ❝❛❧❧❡❞
x✲✐♥t❡r❝❡♣ts
♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✿
✐rr❡❞✉❝✐❜❧❡✱ ❝❛♥♥♦t ❜❡ ❢❛❝t♦r❡❞❀ t❤❡r❡ ❛r❡ ♥♦ a, b s✉❝❤ t❤❛t✿ x2 + 1 = (x − a)(x − b) .
❚❤❡r❡ ❛r❡ ♥♦
r❡❛❧ a, b✱ t❤❛t ✐s✦ ❯s✐♥❣ ♦✉r r✉❧❡s✱ ✇❡ ❞✐s❝♦✈❡r✿
(x − i)(x + i) = x2 − ix + ix − i2 = x2 + 1 .
✸✳✸✳ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
❖❢ ❝♦✉rs❡✱ t❤❡ ♥✉♠❜❡r
i
✐s ♥♦t ❛♥
✷✶✶
x✲✐♥t❡r❝❡♣t
♦❢
f (x) = x2 + 1
❛s t❤❡
x✲❛①✐s
✭✏t❤❡ r❡❛❧ ❧✐♥❡✑✮ ❝♦♥s✐sts ♦❢ ♦♥❧②
✭❛♥❞ ❛❧❧✮ r❡❛❧ ♥✉♠❜❡rs✳ ❙♦✱ ♠✉❧t✐♣❧❡s ♦❢
i
❛♣♣❡❛r ✐♠♠❡❞✐❛t❡❧② ❛s ✇❡ st❛rt ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐t✳
❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✿ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ❚❤❡ r❡❛❧ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱ ✐✳❡✳✱
z = ri, r
r❡❛❧,
❛r❡ ❝❛❧❧❡❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳
❲❡ ❤❛✈❡ ❝r❡❛t❡❞ ❛ ✇❤♦❧❡ ❝❧❛ss ♦❢ ♥♦♥✲r❡❛❧ ♥✉♠❜❡rs✦ ❖❢ ❝♦✉rs❡✱
ri✱
✇❤❡r❡
r
✐s r❡❛❧✱ ❝❛♥✬t ❜❡ r❡❛❧✿
(ri)2 = r2 i2 = −r2 < 0 . ❚❤❡ ♦♥❧② ❡①❝❡♣t✐♦♥ ✐s
0i = 0❀
✐t✬s r❡❛❧✦
❚❤❡r❡ ❛r❡ ❛s ♠❛♥② ♦❢ t❤❡♠ ❛s t❤❡ r❡❛❧ ♥✉♠❜❡rs✿
❊①❛♠♣❧❡ ✸✳✸✳✷✿ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❚❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ t❤❡ s✐♠♣❧❡st q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥
x2 + 4 = 0 ❣✐✈❡s ✉s ✈✐❛ ♦✉r s✉❜st✐t✉t✐♦♥✿
■♥❞❡❡❞✱ ✐❢ ✇❡ s✉❜st✐t✉t❡
p √ √ √ x = ± −4 = ± 4(−1) = ± 4 −1 = ±2i .
x = 2i
✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ❤❛✈❡✿
(2i)2 + 4 = (2)2 (i)2 + 4 = 4(−1) + 4 = 0 . ▼♦r❡ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳
■♠❛❣✐♥❛r② ♥✉♠❜❡rs ♦❜❡② t❤❡ ❧❛✇s ♦❢ ❛❧❣❡❜r❛ ❛s ✇❡ ❦♥♦✇ t❤❡♠✦ ■❢ ✇❡ ♥❡❡❞ t♦ s✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥✱ ✇❡ tr② t♦ ♠❛♥✐♣✉❧❛t❡ ✐t ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ r❡❛❧ ✇❤✐❧❡ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❥✉st ❢❛❝t♦r
i
i
✐s ♣✉s❤❡❞ ❛s✐❞❡✳
♦✉t ♦❢ ❛❧❧ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥✿
5i + 3i = (5 + 3)i = 8i . ■t ❧♦♦❦s ❡①❛❝t❧② ❧✐❦❡ ♠✐❞❞❧❡ s❝❤♦♦❧ ❛❧❣❡❜r❛✿
5x + 3x = (5 + 3)x = 8x . ❆❢t❡r ❛❧❧✱
x
❝♦✉❧❞ ❜❡ i✳ ❆♥♦t❤❡r s✐♠✐❧❛r✐t② ✐s ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❤❛✈❡ ✉♥✐ts✿
5
✐♥✳
+3
✐♥✳
= (5 + 3)
✐♥✳
=8
✐♥✳ .
❙♦✱ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✉♥✐t ❞♦❡s♥✬t ♠❛tt❡r ✭✐❢ ✇❡ ❝❛♥ ♣✉s❤ ✐t ❛s✐❞❡✮✳ ❊✈❡♥ s✐♠♣❧❡r✿
5
❛♣♣❧❡s
+3
❛♣♣❧❡s
= (5 + 3)
❛♣♣❧❡s
=8
❛♣♣❧❡s
.
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✶✷
■t✬s ✏ 8 ❛♣♣❧❡s✑ ♥♦t ✏ 8✑✦ ❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♠✉❧t✐♣❧② ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❜② ❛ r❡❛❧ ♥✉♠❜❡r✿
2 · (3i) = (2 · 3)i = 6i . ❲❡ ❤❛✈❡ ❛ ♥❡✇ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r✳ ❍♦✇ ❞♦ ✇❡ ♠✉❧t✐♣❧② t✇♦ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs❄ ■t✬s ❞✐✛❡r❡♥t❀ ❛❢t❡r ❛❧❧✱ ✇❡ ❞♦♥✬t ✉s✉❛❧❧② ♠✉❧t✐♣❧② ❛♣♣❧❡s ❜② ❛♣♣❧❡s✦ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❛❜♦✈❡✱ ❡✈❡♥ t❤♦✉❣❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❢♦❧❧♦✇ t❤❡ s❛♠❡ r✉❧❡ ❛s ❛❧✇❛②s✱ ✇❡ ❝❛♥✱ ✇❤❡♥ ♥❡❝❡ss❛r②✱ ❛♥❞ ♦❢t❡♥ ❤❛✈❡ t♦✱ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ♦❢ ♦✉r ❛❧❣❡❜r❛ ✉s✐♥❣ ♦✉r
❢✉♥❞❛♠❡♥t❛❧ ✐❞❡♥t✐t② ✿
i2 = −1 . ❋♦r ❡①❛♠♣❧❡✿
(5i) · (3i) = (5 · 3)(i · i) = 15i2 = 15(−1) = −15 . ■t✬s r❡❛❧✦ ❲❡ ❛❧s♦ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ❜② ✉s✐♥❣ t❤❡ ♦t❤❡r
❢✉♥❞❛♠❡♥t❛❧ ❢❛❝t
❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✿
i 6= 0 . ❲❡ ❝❛♥ ❞✐✈✐❞❡ ❜②
i✦
❋♦r ❡①❛♠♣❧❡✱
5i 5 5 5i = = ·1= . 3i 3i 3 3
❆s ②♦✉ ❝❛♥ s❡❡✱ ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✇✐❧❧ ♦❢t❡♥ ❜r✐♥❣ ✉s ❜❛❝❦ t♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ t✇♦ ❝❧❛ss❡s ♦❢ ♥✉♠❜❡rs ❝❛♥♥♦t ❜❡ s❡♣❛r❛t❡❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r✦ ❚❤❡② ❛r❡♥✬t✳ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❡q✉❛t✐♦♥
ax2 + bx + c = 0, a 6= 0 , ✐s s♦❧✈❡❞ ✇✐t❤ t❤❡ ❢❛♠✐❧✐❛r
◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ x=
−b ±
√
b2 − 4ac . 2a
▲❡t✬s ❝♦♥s✐❞❡r
x2 + 2x + 10 = 0 . ❚❤❡♥ t❤❡ r♦♦ts ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡✿
√
22 − 4 · 10 √2 −2 ± −36 = 2 √ = −1 ± −9 √ √ = −1 ± 9 −1
x =
−2 ±
❚❤❡r❡ ✐s ♥♦ r❡❛❧ s♦❧✉t✐♦♥✦ ❇✉t ✇❡ ❣♦ ♦♥✳
= −1 ± 3i . ❲❡ ❡♥❞ ✉♣ ❛❞❞✐♥❣ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✦ ❆s t❤❡r❡ ✐s ♥♦ ✇❛② t♦ s✐♠♣❧✐❢② t❤✐s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
◮
❆ ♥✉♠❜❡r
a + bi✱
✇❤❡r❡
a, b 6= 0
❛r❡ r❡❛❧✱ ✐s ♥❡✐t❤❡r r❡❛❧ ♥♦r ✐♠❛❣✐♥❛r②✳
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✶✸
❊①❡r❝✐s❡ ✸✳✸✳✸ ❊①♣❧❛✐♥ ✇❤②✳
❚❤✐s ❛❞❞✐t✐♦♥ ✐s ♥♦t ❧✐t❡r❛❧✳ ■t✬s ❧✐❦❡ ✏❛❞❞✐♥❣✑ ❛♣♣❧❡s t♦ ♦r❛♥❣❡s✿
5 ■t✬s ♥♦t
❛♣♣❧❡s
+3
♦r❛♥❣❡s
= ...
8 ❛♥❞ ✐t✬s ♥♦t 8 ❢r✉✐t ❜❡❝❛✉s❡ ✇❡ ✇♦✉❧❞♥✬t ❜❡ ❛❜❧❡ t♦ r❡❛❞ t❤✐s ❡q✉❛❧✐t② ❜❛❝❦✇❛r❞s✳
❚❤❡ ❛❧❣❡❜r❛ ✇✐❧❧✱
❤♦✇❡✈❡r✱ ❜❡ ♠❡❛♥✐♥❣❢✉❧✿
(5a + 3o) + (2a + 4o) = (5 + 3)a + (3 + 4)o = 8a + 7o . ■t ✐s ❛s ✐❢ ✇❡ ❝♦❧❧❡❝t
s✐♠✐❧❛r t❡r♠s✱ ❧✐❦❡ t❤✐s✿ (5 + 3x) + (2 + 4x) = (5 + 2) + (3 + 4)x = 8 + 7x .
❚❤✐s ✐❞❡❛ ❡♥❛❜❧❡s ✉s t♦ ❞♦ t❤✐s✿
(5 + 3i) + (2 + 4i) = (5 + 3) + (3 + 4)i = 8 + 7i . ❊❛❝❤ ♦❢ t❤❡ ♥✉♠❜❡rs ✇❡ ❛r❡ ❢❛❝✐♥❣ ❝♦♥t❛✐♥ ❜♦t❤ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ❚❤✐s ❢❛❝t ♠❛❦❡s t❤❡♠ ✏❝♦♠♣❧❡①✑✳✳✳
❉❡✜♥✐t✐♦♥ ✸✳✸✳✹✿ ❝♦♠♣❧❡① ♥✉♠❜❡r ❆♥② s✉♠ ♦❢ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ❛
❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡ s❡t ♦❢
❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿
C = {z = a + bi : a, b
r❡❛❧}
❲❛r♥✐♥❣✦ ❆❧❧ r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠♣❧❡① ✭b
= 0✮✳
❆❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❛r❡ ❡❛s②❀ ✇❡ ❥✉st ❝♦♠❜✐♥❡ s✐♠✐❧❛r t❡r♠s ❥✉st ❧✐❦❡ ✐♥ ♠✐❞❞❧❡ s❝❤♦♦❧✳ ❋♦r ❡①❛♠♣❧❡✱ (1 + 5i) + (3 − i) = 1 + 5i + 3 − i = (1 + 3) + (5i − i) = 4 + 4i . ❚♦ s✐♠♣❧✐❢②
♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ❡①♣❛♥❞ ❛♥❞ t❤❡♥ ✉s❡ i2 = −1✱ ❛s ❢♦❧❧♦✇s✿ (1 + 5i) · (3 − i) = 1 · 3 + 5i · 3 + 1 · (−i) + 5i · (−i) = 3 + 15i − i − 5i2 = (3 + 5) + (15i − i) = 8 + 14i .
■t✬s ❛ ❜✐t tr✐❝❦✐❡r ✇✐t❤
❞✐✈✐s✐♦♥ ✿
1 + 5i 1 + 5i 3 + i = 3−i 3−i 3+i (1 + 5i)(3 + i) = (3 − i)(3 + i) −2 + 8i = 2 3 − i2 −2 + 8i = 2 3 +1 1 = (−2 + 8i) 10 = −0.2 + 0.8i .
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✶✹
❚❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ tr✐❝❦ ♦❢ ♠✉❧t✐♣❧②✐♥❣ ❜② 3 + i✳ ■t ✐s t❤❡ s❛♠❡ tr✐❝❦ ✇❡ ✉s❡❞ ✐♥ ❱♦❧✉♠❡ ✶ t♦ s✐♠♣❧✐❢② ❢r❛❝t✐♦♥s ✇✐t❤ r♦♦ts t♦ ❝♦♠♣✉t❡ t❤❡✐r ❧✐♠✐ts✿ √ √ 1 1 1+ x 1+ x √ = √ √ = . 1−x 1− x 1− x1+ x
❉❡✜♥✐t✐♦♥ ✸✳✸✳✺✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❚❤❡
❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ z = a + bi ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ z¯ = a + bi = a − bi .
❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✳
❚❤❡♦r❡♠ ✸✳✸✳✻✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❚❤❡ r✉❧❡s ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤♦s❡ ♦❢ r❡❛❧ ♥✉♠✲ ❜❡rs✿
• • • • •
z+u=u+z ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿ (z + u) + v = z + (u + v) ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ z · u = u · z ❆ss♦❝✐❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ (z · u) · v = z · (u · v) ❉✐str✐❜✉t✐✈✐t②✿ z · (u + v) = z · u + z · v ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿
❚❤✐s ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠ ❀ ✐t ❢♦❧❧♦✇s t❤❡ r✉❧❡s ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳ ❚❤✐s t❤❡♦r❡♠ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ ❜✉✐❧❞ ❝❛❧❝✉❧✉s ❢♦r ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s t❤❛t ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ ❝❛❧❝✉❧✉s ❢♦r r❡❛❧ ❢✉♥❝t✐♦♥s ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳
❉❡✜♥✐t✐♦♥ ✸✳✸✳✼✿ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r ❊✈❡r② ❝♦♠♣❧❡① ♥✉♠❜❡r x ❤❛s t❤❡ st❛♥❞❛r❞
r❡♣r❡s❡♥t❛t✐♦♥ ✿
z = a + bi ,
✇❤❡r❡ a ❛♥❞ b ❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ❛r❡ ♥❛♠❡❞ ❛s ❢♦❧❧♦✇s✿ • a ✐s t❤❡ r❡❛❧ ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ a = Re(z) ; • bi ✐s t❤❡
✐♠❛❣✐♥❛r② ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ b = Im(z) .
❚❤❡♥✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛❜♦✈❡ ✇❛s t♦ ✜♥❞ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❝♦♠❡s ❢r♦♠ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ ♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚❤❡② ✇❡r❡ ❧✐t❡r❛❧❧② s✐♠♣❧✐✜❝❛t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿
❚❤❡♦r❡♠ ✸✳✸✳✽✿ ❙t❛♥❞❛r❞ ❋♦r♠ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡r ❚✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡✐r r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ❛r❡ ❡q✉❛❧✳
❙♦✱ ✇❡ ❤❛✈❡✿ z = Re(z) + Im(z)i .
■♥ ♦r❞❡r t♦ s❡❡ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠❜✐♥❡ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❧✐♥❡✳ ❍♦✇❄ ❲❡ r❡❛❧✐③❡ t❤❛t t❤❡② ❤❛✈❡ ♥♦t❤✐♥❣ ✐♥ ❝♦♠♠♦♥✳✳✳ ❡①❝❡♣t 0 = 0i
✸✳✸✳
❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡
✷✶✺
❜❡❧♦♥❣s t♦ ❜♦t❤✿
❲❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ❧✐❦❡ t❤❛t✱ ♦r ❧✐❦❡ t❤✐s✿
❖r ✇❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ✇❡ ❜✉✐❧t t❤❡
xy ✲♣❧❛♥❡✿
❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❤❡❧♣s ✉s ✉♥❞❡rst❛♥❞ t❤❡ ♠❛✐♥ ✐❞❡❛✿
◮ ■❢
❈♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ✉♥✐t✱
z = a + bi✱
t❤❡♥
a
❛♥❞
b
1✱
❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦r
❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱
z
i✳
✐♥ t❤❡ ♣❧❛♥❡✳ ❲❡ ❤❛✈❡ ❛ ♦♥❡✲t♦✲♦♥❡
❝♦rr❡s♣♦♥❞❡♥❝❡✿
C ←→ R2 , ❣✐✈❡♥ ❜②
a + bi ←→ < a, b > . ❚❤❡♥ t❤❡
x✲❛①✐s
■t ✐s ❝❛❧❧❡❞ t❤❡
♦❢ t❤✐s ♣❧❛♥❡ ❝♦♥s✐sts ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❡
y ✲❛①✐s
♦❢ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳
❝♦♠♣❧❡① ♣❧❛♥❡
✳
❲❛r♥✐♥❣✦ ❚❤✐s ✐s ❥✉st ❛ ✈✐s✉❛❧✐③❛t✐♦♥✳
✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s
✷✶✻
❚❤❡♥ t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ z ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r ✇✐t❤ t❤❡ s❛♠❡ r❡❛❧ ♣❛rt ❛s z ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥✿ Re(¯ z ) = Re(z) ❛♥❞ Im(¯ z ) = − Im(z) .
❲❛r♥✐♥❣✦ ❆❧❧ ♥✉♠❜❡rs ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ s♦ ❢❛r ❛r❡ r❡❛❧ ♥♦♥✲❝♦♠♣❧❡①✱ ❛♥❞ s♦ ❛r❡ ❛❧❧ q✉❛♥t✐t✐❡s ♦♥❡ ❝❛♥ ❡♥✲ ❝♦✉♥t❡r ✐♥ ❞❛②✲t♦✲❞❛② ❧✐❢❡ ♦r s❝✐❡♥❝❡✿ t✐♠❡✱ ❧♦❝❛t✐♦♥✱ ❧❡♥❣t❤✱ ❛r❡❛✱ ✈♦❧✉♠❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ ❡t❝✳
✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✱ ax2 + bx + c = 0, a 6= 0 ,
❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞✳ ▲❡t✬s ❞✐✈✐❞❡ ❜② a ❛♥❞ st✉❞② t❤❡ r❡s✉❧t✐♥❣ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✿ f (x) = x2 + px + q ,
✇❤❡r❡ p = b/a ❛♥❞ q = c/a✳ ❚❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ t❤❡♥ ♣r♦✈✐❞❡s t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ p x=− ± 2
p p2 − 4q . 2
❖❢ ❝♦✉rs❡✱ t❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ t❤❡ r❡❛❧ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❛t ✐s ✇❤② t❤❡ r❡s✉❧t ♦♥❧② ♠❛❦❡s s❡♥s❡ ✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ D = p2 − 4q ,
✐s ♥♦♥✲♥❡❣❛t✐✈❡✳ ◆♦✇✱ ✐♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ♣❛ss t❤❡ x✲❛①✐s ❡♥t✐r❡❧②✳ ❲❡ ❝❛♥ ♦❜s❡r✈❡ ❤♦✇ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿
✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s
✷✶✼
❚❤✐s ♣r♦❝❡ss ✐s ❡①♣❧❛✐♥❡❞ ❜② ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✱ ✇✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ q ✱ t♦ t❤❡ r♦♦ts ❣✐✈❡♥ ❜② t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ √ p D x1,2 = − ± . 2 2 √ D ❞❡❝r❡❛s❡s❀ t❤❡♥ • ❙t❛rt✐♥❣ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡✱ D ❞❡❝r❡❛s❡s ❛♥❞ 2 √ D = 0❀ t❤❡♥ • D = 0 ❛♥❞ 2 √ √ D −D • D ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡ ❛♥❞ ❜❡❝♦♠❡s ✐♠❛❣✐♥❛r② ✭❜✉t ✐s r❡❛❧✮✳ ❚❤❡ r♦♦ts ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✿ 2 2
❞✐s❝r✐♠✐♥❛♥t D>0 D=0 D 0✮ ❛r❡ ✉♥r❡❧❛t❡❞ ✇❤✐❧❡ t❤❡ ❝♦♠♣❧❡① ♦♥❡s ✭D < 0✮ ❛r❡ ❧✐♥❦❡❞ s♦ ♠✉❝❤ t❤❛t ❦♥♦✇✐♥❣ ♦♥❡ t❡❧❧s ✉s ✇❤❛t t❤❡ ♦♥❡ ✐s✿ ❥✉st ✢✐♣ t❤❡ s✐❣♥❀ t❤❡② ❛r❡ ❝♦♥❥✉❣❛t❡ ♦❢ ❡❛❝❤ ♦t❤❡r✿
❚❤❡② ❛❧✇❛②s ❝♦♠❡ ✐♥ ♣❛✐rs✦ ❆s ❛ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡ r♦♦ts ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❡r♠s ♦❢ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t✳
❚❤❡♦r❡♠ ✸✳✹✳✶✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■
❚❤❡ t✇♦ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts ❛r❡✿ • ❞✐st✐♥❝t r❡❛❧ ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ♣♦s✐t✐✈❡❀ • ❡q✉❛❧ r❡❛❧ ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ③❡r♦❀ • ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ♥❡❣❛t✐✈❡✳
✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s
✷✶✽
❚♦ ✉♥❞❡rst❛♥❞ ❖❉❊s✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛ ♠♦r❡ ♣r❡❝✐s❡ ✇❛② t♦ ❝❧❛ss✐❢② t❤❡ ♣♦❧②♥♦♠✐❛❧s✿ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✐❣♥s ♦❢ t❤❡ r❡❛❧ ♣❛rts ♦❢ t❤❡✐r r♦♦ts✳ ❚❤❡ s✐❣♥s ✇✐❧❧ ❞❡t❡r♠✐♥❡ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r ♦❢ ❝❡rt❛✐♥ s♦❧✉t✐♦♥s✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡s❡ ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s✿
❚❤❡♦r❡♠ ✸✳✹✳✷✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■■
❙✉♣♣♦s❡ x1 ❛♥❞ x2 ❛r❡ t❤❡ t✇♦ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ f (x) = x2 +px+q ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✳ ❚❤❡♥ t❤❡ s✐❣♥s ♦❢ t❤❡ r❡❛❧ ♣❛rts Re(x1) ❛♥❞ Re(x2) ♦❢ x1 ❛♥❞ x2 ❛r❡✿ • s❛♠❡ ✇❤❡♥ p2 > 4q ❛♥❞ q ≥ 0❀ • ♦♣♣♦s✐t❡ ✇❤❡♥ p2 > 4q ❛♥❞ q < 0❀ • s❛♠❡ ❛♥❞ ♦♣♣♦s✐t❡ ♦❢ t❤❛t ♦❢ p ✇❤❡♥ p2 ≤ 4q ✳ Pr♦♦❢✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ p2 ≤ 4q ✐s ❡q✉✐✈❛❧❡♥t t♦ D ≤ 0✳ ❲❡ ❝❛♥ s❡❡ ✐♥ t❤❡ t❛❜❧❡ ❛❜♦✈❡ t❤❛t✱ ✐♥ t❤❛t ❝❛s❡✱ ✇❡ p ❤❛✈❡ Re(x1 ) = Re(x2 ) = − ✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ ❝❛s❡ D > 0 ❛♥❞ r❡❛❧ r♦♦ts✳ ❚❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧ s✐❣♥s 2 ✐s s❡♣❛r❛t❡❞ ❢r♦♠ t❤❡ ❝❛s❡ ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s ♦❢ x1 ❛♥❞ x2 ❜② t❤❡ ❝❛s❡ ✇❤❡♥ ❜♦t❤ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦✿ x1 = x2 = 0✳ ❲❡ s♦❧✈❡✿ √ p p D − − = 0 =⇒ p = − p2 − 4q =⇒ p2 = p2 − 4q =⇒ q = 0 . 2 2
❊①❡r❝✐s❡ ✸✳✹✳✸
❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ▲❡t✬s ✈✐s✉❛❧✐③❡ ♦✉r ❝♦♥❝❧✉s✐♦♥✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ s❤♦✇ t❤❡ ♠❛✐♥ s❝❡♥❛r✐♦s ♦❢ ✇❤❛t ❦✐♥❞s ♦❢ r♦♦ts t❤❡ ♣♦❧②♥♦♠✐❛❧ ♠✐❣❤t ❤❛✈❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ✐ts t✇♦ ❝♦❡✣❝✐❡♥ts✱ p ❛♥❞ q ✳ ❋✐rst✱ ❤♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ❄ ❆s ♣♦✐♥ts ♦♥ ❛ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ♦❢ ❝♦✉rs❡✳✳✳ ❜✉t ♦♥❧② ✇❤❡♥ t❤❡② ❛r❡ r❡❛❧✳ ❙✉♣♣♦s❡ ❢♦r ♥♦✇ t❤❛t t❤❡② ❛r❡✳ ❲❡ st❛rt ✇✐t❤ ❛ ♣❧❛♥❡✱ t❤❡ x1 x2 ✲♣❧❛♥❡ t♦ ❜❡ ❡①❛❝t✱ ❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ r❡❛❧ r♦♦ts ✭❧❡❢t✮✳ ❚❤❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤✐s ♣❧❛♥❡ r❡♣r❡s❡♥ts t❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧ ✭❛♥❞ st✐❧❧ r❡❛❧✮ r♦♦ts✱ x1 = x2 ✱ ✐✳❡✳✱ D = 0✳ ❙✐♥❝❡ t❤❡ ♦r❞❡r ♦❢ t❤❡ r♦♦ts ❞♦❡s♥✬t ♠❛tt❡r ✕ (x1 , x2 ) ✐s ❛s ❣♦♦❞ ❛s (x2 , x1 ) ✕ ✇❡ ♥❡❡❞ ♦♥❧② ❤❛❧❢ ♦❢ t❤❡ ♣❧❛♥❡✳ ❲❡ ❢♦❧❞ t❤❡ ♣❧❛♥❡ ❛❧♦♥❣ t❤❡ ❞✐❛❣♦♥❛❧ ✭♠✐❞❞❧❡✮✳
✸✳✺✳
❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2
✷✶✾
❚❤❡ ❞✐❛❣♦♥❛❧ ✕ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥ D = 0 ✕ ❡①♣♦s❡❞ t❤✐s ✇❛② ❝❛♥ ♥♦✇ s❡r✈❡ ✐ts ♣✉r♣♦s❡ ♦❢ s❡♣❛✲ r❛t✐♥❣ t❤❡ ❝❛s❡ ♦❢ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① r♦♦ts✳ ◆♦✇✱ ❧❡t✬s ❣♦ t♦ t❤❡ pq ✲♣❧❛♥❡✳ ❍❡r❡✱ t❤❡ ♣❛r❛❜♦❧❛ p2 = 4q ❛❧s♦ r❡♣r❡s❡♥ts t❤❡ ❡q✉❛t✐♦♥ D = 0✳ ▲❡t✬s ❜r✐♥❣ t❤❡♠ t♦❣❡t❤❡r✦ ❲❡ t❛❦❡ ♦✉r ❤❛❧❢✲♣❧❛♥❡ ❛♥❞ ❜❡♥❞ ✐ts ❞✐❛❣♦♥❛❧ ❡❞❣❡ ✐♥t♦ t❤❡ ♣❛r❛❜♦❧❛ p2 = 4q ✭r✐❣❤t✮✳ ❈❧❛ss✐❢②✐♥❣ ♣♦❧②♥♦♠✐❛❧s t❤✐s ✇❛② ❛❧❧♦✇s ♦♥❡ t♦ ❝❧❛ss✐❢② ♠❛tr✐❝❡s ❛♥❞ ✉♥❞❡rst❛♥❞ ✇❤❛t ❡❛❝❤ ♦❢ t❤❡♠ ❞♦❡s ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✱ ✇❤✐❝❤ ✐♥ t✉r♥ ✇✐❧❧ ❤❡❧♣ ✉s ✉♥❞❡rst❛♥❞ s②st❡♠s ♦❢ ❖❉❊s✳
✸✳✺✳ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡
C
✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡
R2
♥✉♠❜❡rs✱ t❤❡② ♠✉st ❜❡ s✉❜❥❡❝t t♦ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❲❡ ✇✐❧❧ ✐♥✐t✐❛❧❧② ❧♦♦❦ ❛t t❤❡♠ t❤r♦✉❣❤ t❤❡ ❧❡♥s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ t❤❡ ♣❧❛♥❡ R2 ✳ ❆ ❝♦♠♣❧❡① ♥✉♠❜❡r z ❤❛s t❤❡ st❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ✿ ■❢ ✇❡ ❝❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡r
z = a + bi , ✇❤❡r❡ a ❛♥❞ b ❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ t✇♦ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥✉♠❜❡rs✿
❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥
♦❢ ❝♦♠♣❧❡①
❚❤❡r❡❢♦r❡✱ a ❛♥❞ b ❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ z ❛s ❛ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡✳ ❇✉t ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ✐s ♥♦t ♦♥❧② ❛ ♣♦✐♥t ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ❜✉t ❛❧s♦ ❛ ✈❡❝t♦r✳ ❲❡ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿
C ←→ R2 , ❣✐✈❡♥ ❜②
a + bi ←→ < a, b >
✸✳✺✳
❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2
✷✷✵
❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤✐s t❤❛♥ ❥✉st ❛ ♠❛t❝❤❀ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✐♥
R2
❛♣♣❧✐❡s✦
❲❛r♥✐♥❣✦
■♥ s♣✐t❡ ♦❢ t❤✐s ❢✉♥❞❛♠❡♥t❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ t❤✐♥❦ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛s ♥✉♠❜❡rs ✭❛♥❞ ✉s❡ t❤❡ ❧♦✇❡r ❝❛s❡ ❧❡tt❡rs✮✳ ▲❡t✬s s❡❡ ❤♦✇ t❤✐s ❛❧❣❡❜r❛ ♦❢ ♥✉♠❜❡rs ✇♦r❦s ✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❋✐rst✱ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞♦♥❡
2✲✈❡❝t♦rs✳
❝♦♠♣♦♥❡♥t✇✐s❡ ✿
(a + bi) + (c + di) = (a + c) + (b + d)i < a, b > + < c, d > = < a + c , b + d > ■t ❝♦rr❡s♣♦♥❞s t♦ ❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs✿
❙❡❝♦♥❞✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♠✉❧t✐♣❧② ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜② r❡❛❧ ♦♥❡s✿
(a + bi) c = (ac) + (bc)i < a, b > c = < ac , bc > ■t ❝♦rr❡s♣♦♥❞s t♦ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs✳
❲❛r♥✐♥❣✦
❱❡❝t♦r ❛❧❣❡❜r❛ ♦❢ R2 ✐s ❝♦♠♣❧❡① ❛❧❣❡❜r❛✱ ❜✉t ♥♦t ✈✐❝❡ ✈❡rs❛✳ ❈♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ✇❤❛t ♠❛❦❡s ✐t ❞✐✛❡r❡♥t✳ ❊①❛♠♣❧❡ ✸✳✺✳✶✿ ❝✐r❝❧❡ ❲❡ ❝❛♥ ❡❛s✐❧② r❡♣r❡s❡♥t ❝✐r❝❧❡s ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿
z = r cos θ + r sin θ · i .
❖✉r st✉❞② ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs st❛rts ✇✐t❤ t❤❡ st✉❞② ♦❢ t❤❡ t♦♣♦❧♦❣② ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢
t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡
R
2
✦
t♦♣♦❧♦❣②
♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳ ❚❤✐s
✸✳✺✳
❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2
❏✉st ❛s ❜❡❢♦r❡✱ ❡✈❡r② ❢✉♥❝t✐♦♥
z = f (t)
✷✷✶
✇✐t❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥ ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡✿
zk = f (k) . ❆ ❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠♣❧❡① ✈❛❧✉❡s ❞❡✜♥❡❞ ♦♥ ❛ r❛② ✐♥ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱
s❡q✉❡♥❝❡✱ ♦r s✐♠♣❧② s❡q✉❡♥❝❡✳
{p, p + 1, ...}✱
✐s ❝❛❧❧❡❞ ❛♥
✐♥✜♥✐t❡
❊①❛♠♣❧❡ ✸✳✺✳✷✿ s♣✐r❛❧
❆ ❣♦♦❞ ❡①❛♠♣❧❡ ✐s t❤❛t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♠❛❞❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿
zk = ■t
cos k sin k + i. k k
t❡♥❞s t♦ 0 ✇❤✐❧❡ s♣✐r❛❧✐♥❣ ❛r♦✉♥❞ ✐t✳
❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛
♥✉♠❜❡rs
s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡①
✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✐ts r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦r✱ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣♦✐♥ts
✭♦r ✈❡❝t♦rs✮ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ s❡❡♥ ❛s ❛♥② ♣❧❛♥❡✿ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡
k t❤ ♣♦✐♥t t♦ t❤❡ ❧✐♠✐t ✐s ❣❡tt✐♥❣
s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳
❲❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ✈❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡ ❜② s✐♠♣❧② r❡♣❧❛❝✐♥❣ ✈❡❝t♦rs ✇✐t❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ ✏♠❛❣♥✐t✉❞❡✑ ✇✐t❤ ✏♠♦❞✉❧✉s✑✳
✸✳✺✳
❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2
✷✷✷
❉❡✜♥✐t✐♦♥ ✸✳✺✳✸✿ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❙✉♣♣♦s❡
C✳
{zk : k = 1, 2, 3, ...}
✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐✳❡✳✱ ♣♦✐♥ts ✐♥
❝♦♥✈❡r❣❡s
❲❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡
♣♦✐♥t ✐♥
C✱
❝❛❧❧❡❞ t❤❡
❧✐♠✐t
t♦ ❛♥♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡r
z✱
✐✳❡✳✱ ❛
♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢✿
||zk − z|| → 0
❛s
k → ∞,
❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿
zk → z
❛s
k → ∞,
♦r
z = lim zk . k→∞
❝♦♥✈❡r❣❡♥t ❞✐✈❡r❣❡s✳
■❢ ❛ s❡q✉❡♥❝❡ ❤❛s ❛ ❧✐♠✐t✱ ✇❡ ❝❛❧❧ t❤❡ s❡q✉❡♥❝❡
✈❡r❣❡s ❀ ♦t❤❡r✇✐s❡ ✐t ✐s ❞✐✈❡r❣❡♥t
❛♥❞ ✇❡ s❛② ✐t
❛♥❞ s❛② t❤❛t ✐t
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦✐♥ts st❛rt t♦ ❛❝❝✉♠✉❧❛t❡ ✐♥ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❝✐r❝❧❡s ❛r♦✉♥❞ ❛ tr❡♥❞ ✐♥ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s t♦ ❡♥❝❧♦s❡ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐♥ ❛
❞✐s❦ ✿
z✳
❝♦♥✲
❆ ✇❛② t♦ ✈✐s✉❛❧✐③❡
❚❤❡♦r❡♠ ✸✳✺✳✹✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ❆ s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ a ❛♥❞ b ❛r❡ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ s❡q✉❡♥❝❡✱ t❤❡♥ a = b✳
❉❡✜♥✐t✐♦♥ ✸✳✺✳✺✿ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t② ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ♥✉♠❜❡r
k > N✱
zk t❡♥❞s t♦ ✐♥✜♥✐t②
✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ❋♦r
R✱ t❤❡r❡ ❡①✐sts s✉❝❤ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N
t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧
✇❡ ❤❛✈❡
||zk || > R . ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿
zk → ∞
❛s
k → ∞.
❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥♦t❤❡r ❛♥❛❧♦❣ ♦❢ ❛ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ❛❜♦✉t t❤❡ t♦♣♦❧♦❣② ♦❢ t❤❡ ♣❧❛♥❡✳
❚❤❡♦r❡♠ ✸✳✺✳✻✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s ❆ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs zk ✐♥ C ❝♦♥✈❡r❣❡s t♦ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ zk ❝♦♥✈❡r❣❡ t♦ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ z r❡s♣❡❝t✐✈❡❧②❀ ✐✳❡✳✱ zk → z ⇐⇒ Re(zk ) → Re(z) ❛♥❞ Im(zk ) → Im(z) .
C
✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
✐s♥✬t ❥✉st
R2
✷✷✸
❚❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✇✐❧❧ ❛❧s♦ ❧♦♦❦ ❢❛♠✐❧✐❛r✿
❚❤❡♦r❡♠ ✸✳✺✳✼✿ ❙✉♠ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s
zk , uk
❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s
z k + uk ✱
❛♥❞ ✇❡ ❤❛✈❡✿
lim (zk + uk ) = lim zk + lim uk .
k→∞
k→∞
k→∞
❚❤❡♦r❡♠ ✸✳✺✳✽✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡
zk
❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s
czk
❢♦r ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r
c✱
❛♥❞ ✇❡
❤❛✈❡✿
lim c zk = c · lim zk .
k→∞
k→∞
❲♦✉❧❞♥✬t ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜❡ ❥✉st ❛ ❝♦♣② ♦❢ ❝❛❧❝✉❧✉s ♦♥ t❤❡ ♣❧❛♥❡❄ ◆♦✱ ♥♦t ✇✐t❤ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳
✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
❙♦✱ t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ ❏✉st ❧✐❦❡ ✐♥ ♣❧❛♥❡
C✳
R2 ✱
R2
✐s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❝♦♠♣❧❡① ❛❧❣❡❜r❛ ♦❢
♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ♥✉♠❜❡r
r
▲❡t✬s st❛rt ✇✐t❤ ❞❡❣r❡❡s✿
1
1
❜❡❝♦♠❡s
i✱
✇❤✐❧❡
c
R2
❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤❡ ❧❛tt❡r✳
✇✐❧❧ ❛❧s♦ r♦t❛t❡ ❡❛❝❤ ✈❡❝t♦r✳
i
i s❡✈❡r❛❧ t✐♠❡s✳ i ❜❡❝♦♠❡s −1✱ ❡t❝✳✿
❛♥❞ ♠✉❧t✐♣❧② ✐t ❜②
✐s♥✬t ❥✉st
✇✐❧❧ str❡t❝❤✴s❤r✐♥❦ ❛❧❧ ✈❡❝t♦rs ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝♦♠♣❧❡①
❍♦✇❡✈❡r✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r
❊①❛♠♣❧❡ ✸✳✻✳✶✿ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②
C✳
C
▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜②
i
r♦t❛t❡s t❤❡ ♥✉♠❜❡r ❜②
90
✸✳✻✳
▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
C
✐s♥✬t ❥✉st
1·i=i i · i = i2 = −1 −1 · i = −i −i · i = −i2 = 1
✷✷✹
R2
r♦t❛t✐♦♥ ❢r♦♠ ✵ ❞❡❣r❡❡s t♦ ✾✵ r♦t❛t✐♦♥ ❢r♦♠ ✾✵ ❞❡❣r❡❡s t♦ ✶✽✵ r♦t❛t✐♦♥ ❢r♦♠ ✶✽✵ ❞❡❣r❡❡s t♦ ✷✼✵ r♦t❛t✐♦♥ ❢r♦♠ ✷✼✵ ❞❡❣r❡❡s t♦ ✸✻✵ ❛♥❞ s♦ ♦♥✳
❊①❛♠♣❧❡ ✸✳✻✳✷✿ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥
❆ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡✿
u = 1 + 2i v =2+i uv = 2 + 4i + i + 2i2 = (2 − 2) + (4 + 1)i = 0 + 5i
❚❤❡ r♦t❛t✐♦♥ ♦❢ v ✐s ✈✐s✐❜❧❡✿
■♥ ❝♦♥tr❛st✱ ✇❡ ❝❛♥ s❡❡ t❤❡ r❡s✉❧t ♦❢ ♠✉❧t✐♣❧②✐♥❣ v ❜② w = 2✿ ♥♦ r♦t❛t✐♦♥✳ ❙♦✱ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ c ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r r♦t❛t✐♦♥✳ ❍♦✇ ❞♦❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛✛❡❝t t♦♣♦❧♦❣②❄
❚❤❡♦r❡♠ ✸✳✻✳✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s
zk , uk
❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s
z k · uk ✱
❛♥❞
lim (zk · uk ) = lim zk · lim uk .
k→∞
k→∞
k→∞
Pr♦♦❢✳
❙✉♣♣♦s❡
zk = ak + bk i → a + bi ❛♥❞ uk = pk + qk i = p + qi .
✸✳✻✳
▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
C
❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♠♣♦♥❡♥t✇✐s❡
✐s♥✬t ❥✉st
✷✷✺
R2
❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠
❛❜♦✈❡✱ ✇❡ ❤❛✈❡✿
ak → a, bk → b ❛♥❞ pk → p, qk → q .
❚❤❡♥✱ ❜② t❤❡ Pr♦❞✉❝t
✱ ✇❡ ❤❛✈❡✿
❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s
ak pk → ap, ak qk → aq, bk pk → bp, bk qk → bq .
❚❤❡♥✱ ❛s ✇❡ ❦♥♦✇✱ zk · uk = (ak pk − bk qk ) + (ak qk + bk pk )i → (ap − bq) + (aq + bq)i = (a + bi)(p + qi) ,
❜② t❤❡ ❙✉♠
✳
❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s
❚❤❡♦r❡♠ ✸✳✻✳✹✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s
z k , uk
❝♦♥✈❡r❣❡ ✭✇✐t❤
lim
k→∞
uk 6= 0✮✱
t❤❡♥ s♦ ❞♦❡s
zk /uk ✱
❛♥❞
limk→∞ zk zk = , uk limk→∞ uk
♣r♦✈✐❞❡❞
lim uk 6= 0 .
k→∞
❏✉st ❧✐❦❡ r❡❛❧ ♥✉♠❜❡rs✦
❊①❡r❝✐s❡ ✸✳✻✳✺ Pr♦✈❡ t❤❡ ❧❛st t❤❡♦r❡♠✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞✱ ❈❛rt❡s✐❛♥✱ r❡♣r❡s❡♥t❛t✐♦♥✱ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r x = a + bi ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✳
❲❡ ❥✉st ❛♣♣❡♥❞ ♦✉r ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ❛ ♥❡✇ ♦♥❡✿ a + bi ←→ (a, b) ←→ (θ, r)
❚❤❡ t✇♦ q✉❛♥t✐t✐❡s θ ❛♥❞ r ❜❡❝♦♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
❉❡✜♥✐t✐♦♥ ✸✳✻✳✻✿ ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t ❙✉♣♣♦s❡ z ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ✶✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ z ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ t♦ t❤❡ ♦r✐❣✐♥ O ✐s ❝❛❧❧❡❞ t❤❡ ♠♦❞✉❧✉s ♦❢ z ❞❡♥♦t❡❞ ❜②✿ ||z||
✷✳ ❚❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t z ❢r♦♠ t❤❡ ♦r✐❣✐♥ O ✇✐t❤ t❤❡ x✲❛①✐s
✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
C
✐s♥✬t ❥✉st
R2
✐s ❝❛❧❧❡❞ t❤❡ ❛r❣✉♠❡♥t ♦❢
✷✷✻
z
❞❡♥♦t❡❞ ❜②✿
Arg(z)
❆ s✐♠♣❧❡ ❡①❛♠✐♥❛t✐♦♥ t❡❧❧s ✉s ❤♦✇ t♦ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠s✿
❚❤❡♦r❡♠ ✸✳✻✳✼✿ ❈♦♥✈❡rs✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❙✉♣♣♦s❡
x = a + bi
✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿
✶✳ ❚❤❡ ♠♦❞✉❧✉s ♦❢
z
✐s ❢♦✉♥❞ ❜②✿
||z|| = ✷✳ ❚❤❡ ❛r❣✉♠❡♥t ♦❢
z
√
a2 + b2
✐s ❢♦✉♥❞ ❜②✿
Arg(z) = arctan
❆♥② t✇♦ r❡❛❧ ♥✉♠❜❡rs t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r✿
r≥0
❛♥❞
0 ≤ θ < 2π
b a
❝❛♥ s❡r✈❡ ❛s t❤♦s❡✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢
z = r cos θ + i sin θ ❚❤❡ ❛❧❣❡❜r❛ t❛❦❡s ❛ ♥❡✇ ❢♦r♠ t♦♦✳ ❲❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ t♦ ❝♦♠♣✉t❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐✲ ♣❧✐❝❛t✐♦♥ ❜② r❡❛❧ ♥✉♠❜❡rs✱ ❜✉t ✇❡ ♥❡❡❞ ✐t ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
❈♦♥s✐❞❡r✿
z1 = r1 cos ϕ1 + i sin ϕ1
❛♥❞
z2 = r2 cos ϕ2 + i sin ϕ2 ?
z1 z2 = r1 cos ϕ1 + i sin ϕ1 ) · r2 cos ϕ2 + i sin ϕ2 = r1 r2 cos ϕ1 + i sin ϕ1 · cos ϕ2 + i sin ϕ2 = r1 r2 cos ϕ1 cos ϕ2 + i sin ϕ1 cos ϕ2 + cos ϕ1 sin ϕ2 + i2 sin ϕ1 sin ϕ2 .
❲❡ ✉t✐❧✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ✐❞❡♥t✐t✐❡s ✭❱♦❧✉♠❡ ✶✮✿
cos a cos b − sin a sin b = cos(a + b)
❛♥❞
cos a sin b + sin a cos b = sin(a + b) .
❚❤❡♥✱
z1 z2 = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) .
✸✳✻✳
▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
C
✐s♥✬t ❥✉st
R2
✷✷✼
❊①❛♠♣❧❡ ✸✳✻✳✽✿ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥
❲❡ ❝❛♥ s❡❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿
❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✸✳✻✳✾✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❲❤❡♥ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ♠✉❧t✐♣❧✐❡❞✱ t❤❡✐r ♠♦❞✉❧✐ ❛r❡ ♠✉❧t✐♣❧✐❡❞ ❛♥❞ t❤❡ ❛r❣✉♠❡♥ts ❛r❡ ❛❞❞❡❞✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿
cos ϕ + i sin ϕ ϕ + i sin ϕ ) · r r1 cos 2 2 1 1 2 = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) ❊①❡r❝✐s❡ ✸✳✻✳✶✵
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ♠♦❞✉❧✐ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❛r❣✉♠❡♥ts ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡② ❝♦♥✈❡r❣❡✳ ❊①❡r❝✐s❡ ✸✳✻✳✶✶
✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿
(2 + 3i)(−1 + 2i)✳
■♥❞✐❝❛t❡ t❤❡ r❡❛❧
❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ✭❜✮ ❋✐♥❞ ✐ts ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t✳ ❊①❡r❝✐s❡ ✸✳✻✳✶✷
❙✐♠♣❧✐❢②
(1 + i)2 ✳
❊①❡r❝✐s❡ ✸✳✻✳✶✸
✭❛✮ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧
x2 + 2x + 2✳
✭❜✮ ❋✐♥❞ ✐ts
x✲✐♥t❡r❝❡♣ts✳
✭❝✮ ❋✐♥❞ ✐ts ❢❛❝t♦rs✳
❊①❡r❝✐s❡ ✸✳✻✳✶✹
❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s❄
✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
✷✷✽
✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
❆ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ✐s s✐♠♣❧② ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜♦t❤ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ F : C → C.
❍♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ ❢✉♥❝t✐♦♥s❄ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ F ❧✐❡s ✐♥ t❤❡ 4✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❛♥❞ ✐s♥✬t ♦❢ ♠✉❝❤ ❤❡❧♣✦ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❡ ❝❛♥ r❡❝❛st s♦♠❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛s ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✼✳✶✿ ❝♦♠♣❧❡① ❛❞❞✐t✐♦♥
❚❤❡ s❤✐❢t ❜② ✈❡❝t♦r V =< a, b > ❜❡❝♦♠❡s ❛❞❞✐t✐♦♥ ♦❢ ❛ ✜①❡❞ ❝♦♠♣❧❡① ♥✉♠❜❡r✿ F (x, y) = (x + a, y + b) ✱ r❡✲✇r✐tt❡♥ F (z) = z + z0 ,
✇❤❡r❡ z0 = a + bi✳
❊①❛♠♣❧❡ ✸✳✼✳✷✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t✐♦♥
❚❤❡ ✈❡rt✐❝❛❧ ✢✐♣ ❜❡❝♦♠❡s ❝♦♥❥✉❣❛t✐♦♥✿ F (x, y) = (x, −y) ✱ r❡✲✇r✐tt❡♥ F (z) = z¯ .
✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
✷✷✾
❊①❡r❝✐s❡ ✸✳✼✳✸
❋✐♥❞ ❛ ❝♦♠♣❧❡① ❢♦r♠✉❧❛ ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✿ F (x, y) = (−x, y) .
❊①❛♠♣❧❡ ✸✳✼✳✹✿ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥
❚❤❡ ✉♥✐❢♦r♠ str❡t❝❤ ❜❡❝♦♠❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ♥✉♠❜❡r✿ F (x, y) = (kx, ky) , r❡✲✇r✐tt❡♥ F (z) = kz .
❖❢ ❝♦✉rs❡✱ t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ♦r✐❣✐♥ ✐s ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② −1✳ ❊①❡r❝✐s❡ ✸✳✼✳✺
❋✐♥❞ ❝♦♠♣❧❡① ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤ ✿ F (x, y) = (x, ky)
❛♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ str❡t❝❤✿ F (x, y) = (kx, y) . ❊①❛♠♣❧❡ ✸✳✼✳✻✿ r♦t❛t✐♦♥
❆ r♦t❛t✐♦♥ ✐s ❝❛rr✐❡❞ ♦✉t ✈✐❛ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ✜①❡❞ ❝♦♠♣❧❡① ♥✉♠❜❡r✿ F (z) = z0 z .
❙♣❡❝✐✜❝❛❧❧②✱ ✐t ❤❛s t♦ ❜❡ ❛ ♥✉♠❜❡r ✇✐t❤ ♠♦❞✉❧✉s ❡q✉❛❧ t♦ 1✿ z0 = cos α + i sin α .
▼❡❛♥✇❤✐❧❡✱ ✇✐t❤ z0 = 2i✱ ✇❡ ❤❛✈❡ t❤❡ 90 ❞❡❣r❡❡s r♦t❛t✐♦♥ ✇✐t❤ ❛ str❡t❝❤ ✇✐t❤ ❛ ❢❛❝t♦r ♦❢ 2✿
✸✳✼✳
❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
✷✸✵
❋♦r ❛♥② ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✱ ✇❡ r❡♣r❡s❡♥t ❜♦t❤ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐♥ t❡r♠s ♦❢ t❤❡✐r r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✱ ❥✉st ❛s ✈❡❝t♦r ❢✉♥❝t✐♦♥s✳ ❋✐rst✿ x = u + iv ,
✇❤❡r❡ u, v ❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❙❡❝♦♥❞✿ z = F (x) = f (u, v) + ig(u, v) , f (u, v), g(u, v) ❛r❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳
❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱ f ❛♥❞ g ✱ ❝❛♥ ❜❡ ♣❧♦tt❡❞✳ ❆♥❞ s♦ ❝❛♥ t❤❡ ❛r❣✉♠❡♥t arg F ❛♥❞ t❤❡ ♠♦❞✉❧❡
||F || ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳
❊①❛♠♣❧❡ ✸✳✼✳✼✿ ♣r♦❥❡❝t✐♦♥s
❚❤❡ ♣r♦❥❡❝t✐♦♥s ♦♥ t❤❡ x✲ ❛♥❞ y ✲❛①❡s ❛r❡ t❤❡s❡✿ F (z) = Re z ❛♥❞ F (z) = Im z .
❊①❛♠♣❧❡ ✸✳✼✳✽✿ q✉❛❞r❛t✐❝
▲❡t✬s ❝♦♥s✐❞❡r ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❛❣❛✐♥✿ f (x) = x2 + px + q .
❘❡❝❛❧❧ t❤❛t ✐♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞✿ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿
✸✳✼✳
❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
✷✸✶
◆♦t❡ t❤❛t ❛♥ ✐❞❡♥t✐❝❛❧ r❡s✉❧t ✐s s❡❡♥ ✐♥ ❛ s❡❡♠✐♥❣❧② ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛❜♦❧♦✐❞✱ t❤❡♥ ✐t ♣r♦❞✉❝❡s ❛ ♣❛r❛❜♦❧❛ ❛s ✐t ✐s ❝✉t ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡✳ ❙✉♣♣♦s❡ t❤❡ ♣❛r❛❜♦❧♦✐❞ ✐s ♠♦✈✐♥❣ ❤♦r✐✲ ③♦♥t❛❧❧②✳ ■❢ ✐t ✐s ❢❛❞✐♥❣ ❛✇❛②✱ t❤❡ ♣❛r❛❜♦❧❛ ✐s ♠♦✈✐♥❣ ✉♣✇❛r❞✳
❚❤✐s ✐❧❧✉str❛t❡s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ♦✉r q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s s❡❡♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡✳ ❲❡ ❛r❡ ♣❧♦tt✐♥❣ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❛r♥✐♥❣✦
❱✐s✉❛❧✐③✐♥❣ F (x) ✈✐❛ t❤♦s❡ ❢✉♥❝t✐♦♥s✱ ♦r ❛s ❛ ✈❡❝t♦r ✜❡❧❞✱ ♠❛② ❜❡ ♠✐s❧❡❛❞✐♥❣✳
❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✦ ❊①❛♠♣❧❡ ✸✳✼✳✾✿ ❝♦♠♣❧❡① ♣♦✇❡rs
❚❤✐s ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦✈❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❋♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ z ✱ t❤❡ ✈❛❧✉❡s z, z 2 , z 3 , ...
❛r❡ ♣❧♦tt❡❞ ❛s s❡q✉❡♥❝❡s✳
❖♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ r❡❛❧ ♣❛rt ♦❢ z ♠❛❦❡s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② z str❡t❝❤ ♦r s❤r✐♥❦ t❤❡ ♥✉♠❜❡r ✇❤✐❧❡
✸✳✼✳
❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s
✷✸✷
t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ x ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r r♦t❛t✐♥❣ t❤❡ ♥✉♠❜❡r ❛r♦✉♥❞ 0✳ ❆ s♣❡❝✐❛❧✱ sq✉❛r❡ ♣❛t❤ ✐s ♣r♦❞✉❝❡❞ ❜② z = i✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛ ❝♦♣② ♦❢ t❤❡ ♦♥❡ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✷✳
❉❡✜♥✐t✐♦♥ ✸✳✼✳✶✵✿ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❚❤❡
❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ z = F (x) ❛t ❛ ♣♦✐♥t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t lim F (xn )
n→∞
❝♦♥s✐❞❡r❡❞ ❢♦r ❛❧❧ s❡q✉❡♥❝❡s {xn } ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ F ❡①❝❧✉❞✐♥❣ a t❤❛t ❝♦♥✲ ✈❡r❣❡ t♦ a✱ a 6= xn → a ❛s n → ∞ ,
✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿ lim F (x) .
x→a
❖t❤❡r✇✐s❡✱
t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳
❚❤❡♦r❡♠ ✸✳✼✳✶✶✿ ▲♦❝❛❧✐t② ❙✉♣♣♦s❡ t✇♦ ❢✉♥❝t✐♦♥s
f
❛♥❞
g
❝♦✐♥❝✐❞❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ♣♦✐♥t
f (x) = g(x) ❢♦r s♦♠❡
ε > 0✳
❢♦r ❛❧❧
❚❤❡♥✱ t❤❡✐r ❧✐♠✐ts ❛t
a
x
✇✐t❤
a✿
||x − a|| < ε ,
❝♦✐♥❝✐❞❡ t♦♦✿
lim f (x) = lim g(x) .
x→a
x→a
▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳✳✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s t♦ ♣r♦✈❡ ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✳ ▲❡t✬s r❡✲✇r✐t❡ t❤❡ ♠❛✐♥ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ✉s✐♥❣ t❤❡ ❛❧t❡r♥❛t✐✈❡ ♥♦t❛t✐♦♥✳
❚❤❡♦r❡♠ ✸✳✼✳✶✷✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❙✉♣♣♦s❡
an → a
❙❘✿ P❘✿
❛♥❞
bn → b✳
an + bn → a + b an · bn → ab
❚❤❡♥
❈▼❘✿ ◗❘✿
c · an → ca an /bn → a/b
❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞
c
b 6= 0
❊❛❝❤ ♣r♦♣❡rt② ✐s ♠❛t❝❤❡❞ ❜② ✐ts ❛♥❛❧♦❣ ❢♦r ❢✉♥❝t✐♦♥s✳
❚❤❡♦r❡♠ ✸✳✼✳✶✸✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ❙✉♣♣♦s❡
❙❘✿ P❘✿
f (x) → F
❛♥❞
g(x) → G
f (x) + g(x) → F + G f (x) · g(x) → F G
❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ♥❡①t ❝♦♥❝❡♣t ✐s ❝♦♥t✐♥✉✐t②✿
❛s
❈▼❘✿ ◗❘✿
x → a✳
❚❤❡♥
c · f (x) → cF f (x)/g(x) → F/G
❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞
G 6= 0
c
✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✸✸
❉❡✜♥✐t✐♦♥ ✸✳✼✳✶✹✿ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t a ✐❢✿ • ❚❤❡ ❢✉♥❝t✐♦♥ f (x) ✐s ❞❡✜♥❡❞ ❛t x = a✳ • ❚❤❡ ❧✐♠✐t ♦❢ f ❡①✐sts ❛t a✳ • ❚❤❡ t✇♦ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿
lim f (x) = f (a) .
x→a
❚❤✉s✱ t❤❡ ❧✐♠✐ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❛ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t a ✐❢
lim f (xn ) = f (a) ,
n→∞
❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a✳
❆ t②♣✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝❧❛ss ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✸✳✼✳✶✺✿ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s ❊✈❡r② ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t✳
❯♥❧✐❦❡ ✈❡❝t♦r ❢✉♥❝t✐♦♥s✱ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s ❤❛✈❡ ♠♦r❡ ♦♣❡r❛t✐♦♥s t♦ ✇♦rr② ❛❜♦✉t✳ ❚❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ r❡s✉❧t✳
❚❤❡♦r❡♠ ✸✳✼✳✶✻✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♥t✐♥✉✐t② ❙✉♣♣♦s❡
f
❛♥❞
g
❛r❡ ❝♦♥t✐♥✉♦✉s ❛t
❙❘✿ P❘✿
f +g f ·g
x = a✳
❈▼❘✿ ◗❘✿
c·f f /g
❚❤❡♥ s♦ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿
c g(a) 6= 0
❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞
✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs
▲❡t✬s ❝♦♥s✐❞❡r ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❣❛✐♥✳ ❙✉♣♣♦s❡ A ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ 2 × 2 ♠❛tr✐① ✇✐t❤ r❡❛❧ ❡♥tr✐❡s✳ ❙✉♣♣♦s❡ D ✐s t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ A✳ ❲❤❡♥ D > 0✱ ✇❡ ❤❛✈❡ t✇♦ ❞✐st✐♥❝t r❡❛❧ r♦♦ts ❝♦✈❡r❡❞ ❜② t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✳ ❲❡ ❛❧s♦ s❛✇ t❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ ✇❤❡♥ D = 0✳ ❚❤✉s✱ ✇❡ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝❛s❡ D < 0 ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ ❛s r♦♦ts ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✕ ❛r❡ ❝♦♠♣❧❡①✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❥✉st ❛s ❣♦♦❞ ✭♦r ❜❡tt❡r✦✮ t❤❛♥ t❤❡ r❡❛❧✱ s♦ ✇❤② ♥♦t ✐♥❝❧✉❞❡ t❤✐s ♣♦ss✐❜✐❧✐t②❄
❊①❛♠♣❧❡ ✸✳✽✳✶✿ r♦t❛t✐♦♥ ❚❤✐s ✐s ❤♦✇ ✇❡ ✜♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❢♦r t❤❡ r♦t❛t✐♦♥ ❛♥❞ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❜② s♦❧✈✐♥❣ ✐t✿ −λ −1 1 0 0 −1 0 −1 = λ2 + 1 = 0 =⇒ λ = ±i . = det −λ =⇒ det A= 1 −λ 0 1 1 0 1 0
❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ✐♠❛❣✐♥❛r②✦ ▲❡t✬s ♥♦t✐❝❡ t❤♦✉❣❤ t❤❛t ♥♦ r❡❛❧ ✈❡❝t♦r ♠✉❧t✐♣❧✐❡❞ ❜② ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❝❛♥ ♣r♦❞✉❝❡ ❛ r❡❛❧ ✈❡❝t♦r✳✳✳ ■♥❞❡❡❞✱ ❧❡t✬s tr② t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs❀ s♦❧✈❡ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥
✸✳✽✳
✷✸✹
❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs
AV = λV ❢♦r V ✐♥ R2 ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ r❡❛❧✭✦✮ x, y t❤❛t s❛t✐s❢②✿ −y = ix x 0 −1 x =i =⇒ x = iy y y 1 0
❯♥❧❡ss ❜♦t❤ ③❡r♦✱ x ❛♥❞ y ❝❛♥✬t ❜❡ ❜♦t❤ r❡❛❧✳✳✳ ❙♦✱ ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡♥✬t r❡❛❧✱ t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ ❡✐❣❡♥✈❡❝t♦rs✳ ❇✉t ✇❤② st♦♣ ❤❡r❡❄ ❲❤② ♥♦t ❤❛✈❡ ❛❧❧ t❤❡ ♥✉♠❜❡rs ❛♥❞ ✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s ❝♦♠♣❧❡①❄ ❏✉st ❛s ❛ r❡❛❧ 2✲✈❡❝t♦r ✐s ❛ ♣❛✐r ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ❛ ❡①❛♠♣❧❡✱
❝♦♠♣❧❡①
2✲✈❡❝t♦r ✐s ❛ ♣❛✐r ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs❀ ❢♦r
2 + i V = . −1 + 2i
❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ✈✐❛ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts✿
❍❡r❡✱ ✇❡ s❡❡ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ❢♦r t❤❡ ✜rst ❛♥❞ t❤❡♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ❢♦r t❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❝t♦r V ✳ ❚❤✐s ✐s ✇❤② ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♠♣❧❡① 2✲✈❡❝t♦rs ❜② C2 ✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ✈❡❝t♦r V ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ ✐ts r❡❛❧
❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ✿
1 2 2 + i V = . +i = 2 −1 −1 + 2i
❊❛❝❤ ♦❢ t❤❡s❡ ✐s ❛ r❡❛❧ ✈❡❝t♦r ❛♥❞ t❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❛❝❝♦r❞✐♥❣❧②✿
❚❤❡ ❢♦r♠❡r ✐s ♦❢ t❤❡ ♠❛✐♥ ✐♥t❡r❡st ❛♥❞ ✐t ✐s ❧♦❝❛t❡❞ ✐♥ t❤❡ ❢❛♠✐❧✐❛r r❡❛❧ ♣❧❛♥❡ R2 ✳ ◆❡①t✱ ❛ ❝♦♠♣❧❡① 2 × 2
♠❛tr✐①
A ✐s s✐♠♣❧② ❛ 2 × 2 t❛❜❧❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs❀ ❢♦r ❡①❛♠♣❧❡✿ 0 1+i . A= i 2 − 3i
✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs
✷✸✺
❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✕ ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛❧❧♦✇s ✉s t♦ ❝❛rr② ♦✉t t❤❡ ♦♣❡r❛t✐♦♥s ✕ ❛❞❞✐t✐♦♥✱ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✕ ♦♥ t❤❡s❡ ✈❡❝t♦rs ❛♥❞ t❤❡s❡ ♠❛tr✐❝❡s✦ ❚❤❡♥✱ ❛ ❝♦♠♣❧❡① ♠❛tr✐① A ❞❡✜♥❡s ❛ ❢✉♥❝t✐♦♥✿
A : C2 → C2 , t❤r♦✉❣❤ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ A(X) = AX ✳ ▼♦r❡♦✈❡r✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ A(αX + βY ) = αA(X) + βA(Y ) , ❢♦r ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡rs α ❛♥❞ β ❛♥❞ ❛♥② ❝♦♠♣❧❡① ✈❡❝t♦rs X ❛♥❞ Y ✳ ❇✉t ❤♦✇ ❝❛♥ ✇❡ ✉s❡ t❤❡s❡ ❢✉♥❝t✐♦♥s t♦ ✉♥❞❡rst❛♥❞ r❡❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦rs❄ ■ts ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❛r❡ ❝♦♠♣❧❡① ✈❡❝t♦rs ❛♥❞ t❤❡② ❤❛✈❡ r❡❛❧ ♣❛rts ❛s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❙♦✱ ✇❡ ❝❛♥ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r t♦ t❤❡ r❡❛❧ ♣❧❛♥❡ ✜rst ❛♥❞ t❤❡♥ t❛❦❡ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ♦✉t♣✉t✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❢❛♠✐❧✐❛r r❡❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ B : R2 → R2 , t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r A✳
▲❡t✬s r❡✈✐❡✇ ♦✉r t❤❡♦r② ❣❡♥❡r❛❧✐③❡❞ t❤✐s ✇❛②✳ ❚❤❡r❡ ✐s ♥♦ ❞✐✛❡r❡♥❝❡✿
❉❡✜♥✐t✐♦♥ ✸✳✽✳✷✿ ❞❡t❡r♠✐♥❛♥t ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ❝♦♠♣❧❡① 2×2 ♠❛tr✐① ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r✿ a b = ad − bc det c d
❚❤❡♦r❡♠ ✸✳✽✳✸✿ ◆♦♥✲③❡r♦ ❉❡t❡r♠✐♥❛♥t ❙✉♣♣♦s❡
A ✐s ❛ ❝♦♠♣❧❡① 2 × 2 ♠❛tr✐①✳ ❚❤❡♥✱ det A 6= 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s♦❧✉t✐♦♥ AX = 0 ❝♦♥s✐sts ♦❢ ♦♥❧② 0✳
s❡t ♦❢ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥
❉❡✜♥✐t✐♦♥ ✸✳✽✳✹✿ ❡✐❣❡♥✈❛❧✉❡ ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A : C2 → C2 ✱ ❛ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡r λ ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ A ✐❢
A(V ) = λV ❢♦r s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r V ✐♥ C2 ✳ ❚❤❡♥✱ V ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ A ❝♦rr❡✲ s♣♦♥❞✐♥❣ t♦ λ✳
❉❡✜♥✐t✐♦♥ ✸✳✽✳✺✿ ❡✐❣❡♥s♣❛❝❡ ❋♦r ❛ ✭❝♦♠♣❧❡①✮ ❡✐❣❡♥✈❛❧✉❡ λ ♦❢ A✱ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ ❛ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐♥ C2 ✿
E(λ) = {V : A(V ) = λV } ■t ✐s ❛ ✈❡r② ✐♠♣♦rt❛♥t ❢❛❝t t❤❛t ❛❧❧ t❤❡ ❝♦♠♣✉t❛t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ♣❡r❢♦r♠❡❞ ♦♥ r❡❛❧ ♠❛tr✐❝❡s ❛♥❞ ✈❡❝t♦rs r❡♠❛✐♥ ✈❛❧✐❞✦ ❆♠♦♥❣ t❤❡ r❡s✉❧ts t❤❛t r❡♠❛✐♥ ✈❛❧✐❞ ✐s t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs r❡♣r❡s❡♥t❡❞ ❜② r❡❛❧ ♠❛tr✐❝❡s ❤♦✇❡✈❡r ✇✐❧❧ r❡♠❛✐♥ ♦✉r ❡①❝❧✉s✐✈❡ ✐♥t❡r❡st✳
✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✸✻
❊✈❡r② ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♣r❡s❡♥t❡❞ ❜② ❛ r❡❛❧ ♠❛tr✐① A ✐s st✐❧❧ ❥✉st ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ❝♦♠♣❧❡① ♦♣❡r❛t♦r A : C2 → C2 ✳ ■♥ ❢❛❝t✱ ✐t ✇✐❧❧ ❛❧✇❛②s ❤❛✈❡ s♦♠❡ ❝♦♠♣❧❡① ✭♥♦♥✲r❡❛❧✮ ✈❡❝t♦rs ❛♠♦♥❣ ✐ts ✈❛❧✉❡s✱ ✉♥❧❡ss A = 0✳ ■ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❤❛s r❡❛❧ ❝♦❡✣❝✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛♣♣❧✐❡s✱ ❛s ❢♦❧❧♦✇s✳
❚❤❡♦r❡♠ ✸✳✽✳✻✿ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ❊✐❣❡♥✈❛❧✉❡s ❙✉♣♣♦s❡
A
✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♣r❡s❡♥t❡❞ ❜② ❛ r❡❛❧
2×2
t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡♥ t❤❡
A
D ✐s ❡✐❣❡♥✈❛❧✉❡s λ1 , λ2 ♦❢ ♠❛tr✐①
A
❛♥❞
❢❛❧❧ ✐♥t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝❛t❡❣♦r✐❡s✿ ✶✳ ■❢ ✷✳ ■❢ ✸✳ ■❢
D > 0✱ D = 0✱ D < 0✱
t❤❡♥ t❤❡♥ t❤❡♥
λ1 , λ 2 λ1 , λ 2 λ1 , λ 2
❛r❡ ❞✐st✐♥❝t r❡❛❧✳ ❛r❡ ❡q✉❛❧ r❡❛❧✳ ❛r❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡✳
❲❡ ♦♥❧② ♥❡❡❞ t♦ ❛❞❞r❡ss t❤❡ ❧❛st ❝❛s❡✳
✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
❆❧❧ ♥✉♠❜❡rs ❜❡❧♦✇ ❛r❡ ❝♦♠♣❧❡① ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ❚❤❡ ✜rst t❤✐♥❣ ✇❡ ♥♦t✐❝❡ ❛❜♦✉t t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛r❡♥✬t r❡❛❧ ✐s t❤❛t t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ ❡✐❣❡♥s♣❛❝❡s s❤♦✇♥ ✐♥ t❤❡ r❡❛❧ ♣❧❛♥❡ s❤♦✇♥ ✐♥ t❤❡ ❡①❛♠♣❧❡s ❜❡❧♦✇✳
❊①❛♠♣❧❡ ✸✳✾✳✶✿ r♦t❛t✐♦♥s ❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 90 ❞❡❣r❡❡s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❛❣❛✐♥✿ 0 −1 x u =⇒ λ1,2 = ±i . = y 1 0 v
❚♦ ✜♥❞ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r✱ ✇❡ s♦❧✈❡✿
0 −1 F V1 = 1 0
x x . =i y y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
❲❡ ❝❤♦♦s❡ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r✿
x
−y = ix, ❆◆❉ = iy
=⇒ y = −ix .
1 , V1 = −i
✸✳✾✳
▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✸✼
❛♥❞ s✐♠✐❧❛r❧② ❛♥ ❡✐❣❡♥✈❡❝t♦r ❢♦r t❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿ 1 . V2 = i ❚❤❡ r❡st ❛r❡ ✭❝♦♠♣❧❡①✮ ♠✉❧t✐♣❧❡s ♦❢ t❤❡s❡✳ ❙♦✱ ✉♥❞❡r F ✱ ❝♦♠♣❧❡① ✈❡❝t♦r Vk ✐s ♠✉❧t✐♣❧✐❡❞ ❜② λk ✱ ❛♥❞ s♦ ✐s ❡✈❡r② ♦❢ ✐ts ♠✉❧t✐♣❧❡s✱ k = 1, 2✳ ❙✐♥❝❡ t❤❡s❡ ♠✉❧t✐♣❧❡s ❛r❡ ❝♦♠♣❧❡①✱ t❤✐s ♠✉❧t✐♣❧✐❝❛t✐♦♥ r♦t❛t❡s ✭t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢✮ ❡✐t❤❡r ❝♦♠♣♦♥❡♥t ♦❢ t❤✐s ✈❡❝t♦r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤✐s ✈❡❝t♦r ✐s ❛❧s♦ r♦t❛t❡❞ ✕ ♦♥ t❤❡ r❡❛❧ ♣❧❛♥❡✳ ■t ✐s t❤✐s r♦t❛t✐♦♥ t❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ■t ✐s s❤♦✇♥ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇ ❜❡❧♦✇✿
❚❤❡ ❣❡♥❡r❛❧ ✈❛❧✉❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✕ ♦✈❡r t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ♦❢ ♦✉r t✇♦ ❡✐❣❡♥✈❡❝t♦rs✿ 1 1 . + β(−i) X = α V1 + β V2 =⇒ F X = αi i −i ▲❡t✬s ❝♦♥s✐❞❡r ❛
r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛r❜✐tr❛r② ❛♥❣❧❡ θ✿ cos θ − sin θ x u . = y sin θ cos θ v
❚❤❡♥✱ ✇❡ ❤❛✈❡ ❚❤❡r❡❢♦r❡✱
χA (λ) = (cos θ − λ)2 + sin2 θ = cos2 θ − 2 cos θ λ + λ2 + sin2 θ = λ2 − 2 cos θ λ + 1 .
λ1,2 =
2 cos θ ±
p √ p (2 cos θ)2 − 4 2 cos θ ± 2 cos2 θ − 1 = = cos θ ± − sin2 θ = cosθ ± i sin θ . 2 2
✸✳✾✳
❙♦✱
▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✸✽
t❤❡ ❛r❣✉♠❡♥t ♦❢ t❤❡ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✐s ❡q✉❛❧ t♦ t❤❡ ❛♥❣❧❡ ♦❢ r♦t❛t✐♦♥
✭✉♣ t♦ ❛ s✐❣♥✮✿
| arg λ1,2 | = |θ| . ❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿
1 V1 = −i
❛♥❞
1 . V2 = i
■♥❞❡❡❞✱ t❤❡② ❛r❡ r♦t❛t❡❞ t❤r♦✉❣❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ ❡✐❣❡♥✈❛❧✉❡s✿
1 cos θ − sin θ cos θ ± i sin θ 1 . = = λk Vk = (cosθ ± i sin θ) ∓i sin θ cos θ sin θ ∓ i cos θ ∓i
❖♥❝❡ ❛❣❛✐♥✱ ✇❡ s❡❡ ❤♦✇ t❤❡ r❡❛❧ ♣❛rt ♦❢ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r ✐s r♦t❛t❡❞ ✈✐❛ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿
❲❛r♥✐♥❣✦ ❚❤✐s ❞♦❡s ♥♦t ❛♣♣❧② t♦ ✈❡❝t♦rs ✐♥ ❡✐❣❡♥✈❡❝t♦rs✳
❊①❛♠♣❧❡ ✸✳✾✳✷✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿
u = 3x −13y v = 5x +y
❛♥❞
3 −13 . F = 5 1
C2
t❤❛t ❛r❡♥✬t
✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✸✾
❖✉r ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ 3 − λ −13 = λ2 − 4λ + 68 . χ(λ) = det(F − λI) = det 5 1−λ
❲❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❢r♦♠ t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿
λ1,2 = 2 ± 8i .
◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿ F Vk = λk Vk , k = 1, 2 .
❚❤❡ ✜rst✿
3 −13 F V1 = 5 1
x x . = (2 + 8i) y y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
3x −13y = (2 + 8i)x, ❆◆❉ 5x +y = (2 + 8i)y
=⇒
❲❡ ❝❤♦♦s❡ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿
(1 − 8i)x −13y =0 5x +(−1 − 8i)y = 0
1 + 8i . V1 = 5
3 −13 F V2 = 5 1
3x −13y = (2 − 8i)x 5x +y = (2 − 8i)y
=⇒
❲❡ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿
x x . = (2 − 8i) y y
(1 + 8i)x −13y =0 5x +(−1 + 8i)y = 0
=⇒ x =
(1 − 8i) y. 5
1 − 8i . V2 = 5
❚❤❡ ❣❡♥❡r❛❧ ❝♦♠♣❧❡① ✈❛❧✉❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦✿ X = α V1 + β V2
(1 + 8i) y. 5
❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡ s❛t✐s✜❡s✿
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
=⇒ x =
1 − 8i 1 + 8i . + β(2 − 8i) =⇒ F X = α(2 + 8i) 5 5
❲❡ ❦♥♦✇ t❤❡ ❡✛❡❝t ♦❢ F ♦♥ t❤❡s❡ t✇♦ ✈❡❝t♦rs✿ t❤❡② ❛r❡ r♦t❛t❡❞ ❛♥❞ str❡t❝❤❡❞✳
✸✳✾✳
▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✹✵
❆s ②♦✉ ❝❛♥ s❡❡✱ str❡t❝❤✐♥❣ ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ ❝♦♠♣♦♥❡♥ts ❛♥❞ ❜♦t❤ ❜❛s✐s ✈❡❝t♦rs✳ ❚❤❛t ✐s ✇❤② ✇❡ ❤❛✈❡ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r♦t❛t✐♦♥ ✕ ❛ ✉♥✐❢♦r♠ str❡t❝❤ ✭r❡✲s❝❛❧✐♥❣✮ ❢♦r t❤❡ r❡❛❧ ♣❧❛♥❡✳ ❚❤✐s ✐s s❤♦✇♥ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛ ❢♦r t❤❡ ❛❜♦✈❡ ✐❧❧✉str❛t✐♦♥✿
−62 + 24i 1 + 8i = λ1 V1 = (2 + 8i) 10 + 40i 5
❆❝❝♦r❞✐♥❣ t♦ t❤❡
−62 − 24i 1 − 8i . = λ2 V2 = (2 − 8i) 10 − 40i 5
❛♥❞
❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s✱
✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t
D < 0✱
t❤❡ t✇♦
r♦♦ts ❛r❡ ❝♦♥❥✉❣❛t❡✿
λ1,2 = a ± bi . ❚❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♠♦❞✉❧✉s✿
||λ1,2 || =
√
a2 + b2 .
❚❤✐s ✐s ✇❤② ♠✉❧t✐♣❧②✐♥❣ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ❜② ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❛t❡ 2 2 ♦❢ str❡t❝❤✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t C ❛♥❞✱ ❡s♣❡❝✐❛❧❧②✱ t❤❡ r❡❛❧ ♣❧❛♥❡ R ❛r❡ str❡t❝❤❡❞ ✦ ❚❤✐s ✐s t❤❡
✉♥✐❢♦r♠❧②
s✉♠♠❛r② ♦❢ ♦✉r ❛♥❛❧②s✐s✳
❚❤❡♦r❡♠ ✸✳✾✳✸✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ ❈♦♠♣❧❡① ❊✐❣❡♥✈❛❧✉❡s ❙✉♣♣♦s❡ ❛ r❡❛❧ ♠❛tr✐① F ❤❛s t✇♦ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2 ✳ ❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❛♥❣❧❡ θ t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ | sin θ| = | arg λ1 | = | arg λ2 | . • ■t str❡t❝❤❡s✲s❤r✐♥❦s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ s = ||λ1 || = ||λ2 || . ❲❡ ❝❛♥ ❛❧s♦ r❡❝❛st t❤✐s t❤❡♦r❡♠ ✐♥ ❡①❝❧✉s✐✈❡❧② ✏r❡❛❧✑ t❡r♠s✳ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿
❚❤❡♦r❡♠ ✸✳✾✳✹✿ ◆♦♥✲♣♦s✐t✐✈❡ ❉✐s❝r✐♠✐♥❛♥t ■❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱
x2 + px + q ,
❤❛s ❛ ♥♦♥✲♣♦s✐t✐✈❡ ❞✐s❝r✐♠✐♥❛♥t✱ D = p2 − 4q ≤ 0 ,
t❤❡♥ ❡✐t❤❡r ♦❢ ✐ts r♦♦ts✱
√ 1 x = (−p ± D) , 2
❤❛s ✐ts ❛r❣✉♠❡♥t s❛t✐s❢②✿
sin arg x =
❛♥❞ ✐ts ♠♦❞✉❧❡ s❛t✐s❢②✿
1 2
s
||x||2 = q .
4−
p2 , q
✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s
✷✹✶
Pr♦♦❢✳
❚❤❡ ♠♦❞✉❧✉s ♦❢ t❤❡ r♦♦ts ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ||x||2 = Re x
2
+ Im x
2
1 1 1 = (p2 + |D|) = (p2 − D) = p2 − (p2 − 4q) = q . 4 4 4
❆♥❞ t❤❡✐r ❛r❣✉♠❡♥t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿
s s s √ 1 2 − 4q D p p2 Im x −D 1 1 1 − 4− . sin arg x = = 2√ = = = ||x|| q 2 q 2 q 2 q
❲❡ ❞❡✜♥❡❞ t❤❡ tr❛❝❡ ♦❢ ❛ ♠❛tr✐① ❛s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts✿ a b = a + d. tr c d
❚❤❡♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t❛❦❡s t❤✐s ❢♦r♠✿ a−λ b χ(λ) = det c d−λ = ad − aλ − λd + λ2 − bc = λ2 − (a + d)λ + (ad − bc) = λ2 − tr F λ + det F.
❲❡ ♠❛t❝❤ t❤✐s t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✿ p = − tr F, q = det F, D = (tr F )2 − 4 det F .
❚❤❡ r♦♦ts ❛r❡ λ1,2 =
√ 1 tr F ± D . 2
❲❤❡♥ t❤❡ r♦♦ts ❛r❡ ❝♦♠♣❧❡①✱ t❤❡ ♠♦❞✉❧✉s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ||λ1,2 || =
√
det F .
❆♥❞ t❤❡✐r ❛r❣✉♠❡♥t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ sin arg λ1,2
❚❤✐s ✐s t❤❡ ✜♥❛❧ r❡s✉❧t✳
1 = 2
r
(tr F )2 − 4. det F
❈♦r♦❧❧❛r② ✸✳✾✳✺✿ ◆♦♥✲♣♦s✐t✐✈❡ ❉✐s❝r✐♠✐♥❛♥t
❙✉♣♣♦s❡ ❛ r❡❛❧ ♠❛tr✐① F s❛t✐s✜❡s✿ D = (tr F )2 − 4 det F ≤ 0 .
❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡✿ θ = sin−1
1 2
r
4 − (tr F )2 det F
!
.
• ■t str❡t❝❤❡s✲s❤r✐♥❦s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ √ s = det F .
✸✳✶✵✳
❈♦♠♣❧❡① ❝❛❧❝✉❧✉s
✷✹✷
❨❡s✱ ✇❡ ❤❛✈❡ ✐♥❝❧✉❞❡❞ t❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ D = 0✦ ■♥❞❡❡❞✱ ✐t ❤❛s t❤❡ s❛♠❡ str❡t❝❤ ❜✉t ♥♦ r♦t❛t✐♦♥✳ ❲❡ ✜♥❛❧❧② ♣✉t t♦❣❡t❤❡r ♦✉r t✇♦ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠s✿ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐♥ t❡r♠s ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ r❡❛❧ ♦r ❝♦♠♣❧❡① ✕ ♦❢ t❤❡✐r ♠❛tr✐❝❡s✳ ❲❡ ✐❧❧✉str❛t❡ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ❜❡❧♦✇ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s ✿
✸✳✶✵✳ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s
❊✈❡♥ t❤♦✉❣❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♣❧❛♥❡ ✈❡❝t♦rs✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❣r❛❞✐❡♥t✳ ■t r❛t❤❡r ❢♦❧❧♦✇s✱ ❛♥❞ ✐s ✐❞❡♥t✐❝❛❧ t♦✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ✉s✉❛❧ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳ ❲❡ r❡❧② ♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❛❧❣❡❜r❛ ✐s t❤❡ s❛♠❡ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ♥✉♠❜❡rs ✐s ❞✐✛❡r❡♥t✿
❉❡✜♥✐t✐♦♥ ✸✳✶✵✳✶✿ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ u = f (x) ❛t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛t x = a ❛s t❤❡ ✐♥❝r❡♠❡♥t ∆x ✐s ❛♣♣r♦❛❝❤✐♥❣ 0✱ ❞❡♥♦t❡❞ ❜②✿
f ′ (x) = lim
x→a
f (x) − f (a) f (x) − f (a) = lim ||x−a||→0 x−a x−a
✐♥ t❤❛t ❝❛s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞
❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳
❚❤✐s ❢♦r♠✉❧❛ ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥✳
✸✳✶✵✳
❈♦♠♣❧❡① ❝❛❧❝✉❧✉s
✷✹✸
❲❛r♥✐♥❣✦ ❚❤❡ ❢♦r♠✉❧❛ ✐s ♥♦t t❤❡ s❛♠❡ ❛s t❤✐s✿
f (x) − f (a) . x→a ||x − a|| lim
❲❛r♥✐♥❣✦ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s♥✬t ❛❜♦✉t t❤❡ s❧♦♣❡✱ r✐s❡ ♦✈❡r t❤❡ r✉♥✱ ❛♥②♠♦r❡✳
❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s st✐❧❧ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ♦✉t♣✉t r❡❧❛t✐✈❡ t♦ t❤❡ ✐♥♣✉t✳ ▼❛♥② r❡s✉❧ts ❛r❡ ❢❛♠✐❧✐❛r s✉❝❤ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✿
❚❤❡♦r❡♠ ✸✳✶✵✳✷✿ ❉✐✛ ❂❃ ❈♦♥t ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✱ ✐t ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s✳
❊①❛♠♣❧❡ ✸✳✶✵✳✸✿ ❞❡r✐✈❛t✐✈❡ ❈♦♠♣✉t❛t✐♦♥s ✇♦r❦ ♦✉t ✐♥ t❤❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ▲❡t✬s ❝♦♠♣✉t❡ f ′ (2) ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r f (x) = −x2 − x.
❉❡✜♥✐t✐♦♥✿
f (2 + h) − f (2) . h→0 h
f ′ (2) = lim
❚♦ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✇❡ ♥❡❡❞ t♦ s✉❜st✐t✉t❡ t✇✐❝❡✿ f (2 + h) = −(2 + h)2 − (2 + h), f (2) = −22 − 2 .
◆♦✇✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❞❡✜♥✐t✐♦♥✿ [−(2 − h)2 − (2 − h)] − [−22 − 2] h→0 h 2 −4 − 4h − h − 2 − h + 4 + 2 = lim h→0 h −5h − h2 = lim h→0 h = lim (−5 − h)
f ′ (2) = lim
h→0
= −5 − 0 = 5.
❚❤❡♦r❡♠ ✸✳✶✵✳✹✿ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ (xn )′ = nxn−1
Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ r❡❧✐❡s ❡♥t✐r❡❧② ♦♥ t❤❡ ❢♦r♠✉❧❛✿ an − bn = (a − b)(an−1 + an−2 b + ... + abn−2 + bn−1 ) .
✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
✷✹✹
❊①❡r❝✐s❡ ✸✳✶✵✳✺
❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳
❚❤❡r❡ ✐s ❛ ❝♦✉♥t❡r♣❛rt ❢♦r ❡❛❝❤ r✉❧❡ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✦ ❚❤❡♦r❡♠ ✸✳✶✵✳✻✿ ❆❧❣❡❜r❛ ♦❢ ❉❡r✐✈❛t✐✈❡s
❲❤❡r❡✈❡r ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙❘✿
(f + g)′ = f ′ + g ′ ❈▼❘✿
P❘✿
(f g)′ = f ′ g + f g ′
◗❘✿
(cf )′ = cf ′ f ′g − f g′ (f /g)′ = g2
❢♦r ❛♥② ❝♦♠♣❧❡① c ♣r♦✈✐❞❡❞ g 6= 0
❲❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ❛♥② ♣♦❧②♥♦♠✐❛❧ ❡❛s✐❧② ♥♦✇✳ ❚❤❡♦r❡♠ ✸✳✶✵✳✼✿ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧
❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ❛♥❞ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡rs a0 , ..., an ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ an x n + an−1 xn−1 +an−2 xn−2 +... +a2 x2 +a1 x +a0 = an nxn−1 + an−1 (n − 1)xn−2 +an−2 (n − 2)xn−3 +... +a2 2x +a1
′
❆s ❢❛r ❛s ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ❝♦♥❝❡r♥❡❞✱ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❛ r❡❧❛t✐♦♥ ❛s ✏❧❡ss✑ ♦r ✏♠♦r❡✑ ❛♠♦♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡rs✦ ❚❤❛t✬s ✇❤② ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ✇♦rr② ❛❜♦✉t✿ ✶✳ ♠♦♥♦t♦♥✐❝✐t② ✷✳ ❡①tr❡♠❡ ♣♦✐♥ts ✸✳ ❝♦♥❝❛✈✐t②✱ ❡t❝✳ ❏✉st ❛s ❜❡❢♦r❡✱ r❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ❝❛❧❧❡❞ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❛♥t✐❞❡r✐✈❛✲ ′ t✐✈❡s ❀ F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✐❢ F = f ✳ ❚❤❡r❡ ✐s ❛ ❝♦✉♥t❡r♣❛rt ❢♦r ❡❛❝❤ r✉❧❡ ♦❢ ✐♥t❡❣r❛t✐♦♥✦ ❚❤❡♦r❡♠ ✸✳✶✵✳✽✿ ❆❧❣❡❜r❛ ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s
❲❤❡r❡✈❡r f ❛♥❞ g ❛r❡ ✐♥t❡❣r❛❜❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙❘✿ P❘✿
Z
Z
(f + g) dx = f dg = f g −
Z
Z
f dx + g df
Z
Z
Z
g dx ❈▼❘✿ (cf ) dx = c f dx Z Z 1 ▲❈❘✿ f (mx + b) dx = m f (t) dt
❢ t=mx+b
✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
❆❧❧ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❝♦♥t✐♥✉❡ t♦ ❜❡ ✐❞❡♥t✐❝❛❧ ♦r ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡s ✐♥ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✺✮✳
♣
✸✳✶✶✳
✷✹✺
❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✿ s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♣❛rt✐❛❧ s✉♠s ❋♦r ❛ ❣✐✈❡♥ s❡q✉❡♥❝❡ {zn } = {zn : n = s, s + 1, s + 2, ...} ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♦r ♣❛rt✐❛❧ s✉♠s✱ {pn : n = s, s + 1, s + 2, ...} ✐s ❛ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ ps = z s ,
pn+1 = pn + zn , n = s, s + 1, s + 2, ...
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✷✿ s✉♠ ♦❢ s❡r✐❡s ❋♦r ❛ s❡q✉❡♥❝❡ {zn }✱ t❤❡ ❧✐♠✐t S ♦❢ ✐ts s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s {pn } ✐s ❝❛❧❧❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦r✱ ♠♦r❡ ❝♦♠♠♦♥❧②✱ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✱ ❞❡♥♦t❡❞ ❜②✿ S=
∞ X
zi = lim
n→∞
i=s
n X
zi
i=s
❚❤✐s ❧✐♠✐t ♠✐❣❤t ❛❧s♦ ❜❡ ✐♥✜♥✐t❡✳ ❋r♦♠ t❤❡ ❯♥✐q✉❡♥❡ss
♦❢ ▲✐♠✐t
✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✸✳✶✶✳✸✿ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♠ ❆ s❡r✐❡s ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ a ❛♥❞ b ❛r❡ s✉♠s ♦❢ t❤❡ s❛♠❡ s❡r✐❡s✱ t❤❡♥ a = b✳ ❋r♦♠ t❤❡ ❈♦♠♣♦♥❡♥t✲✇✐s❡
❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s
✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✸✳✶✶✳✹✿ ❈♦♠♣♦♥❡♥t✲✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡r✐❡s ❆ s❡r✐❡s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs
∞ X
zi ✐ ❝♦♥✈❡r❣❡s t♦ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✐❢ ❛♥❞
i=s
♦♥❧② ✐❢ ❜♦t❤ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✭♣❛rt✐❛❧✮ s✉♠s ♦❢ zk ❝♦♥✈❡r❣❡ t♦ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ z r❡s♣❡❝t✐✈❡❧②❀ ✐✳❡✳✱ ∞ X i=s
zi = z ⇐⇒ Re
∞ X
zi = Re(z) ❛♥❞ Im
i=s
i=s
❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤❡ s❛♠❡ ♦r❞❡r ❛s ✇✐t❤ r❡❛❧✲✈❛❧✉❡s s❡r✐❡s✿
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✺✿ ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ❋♦r ❛ s❡q✉❡♥❝❡ {zn }✱ t❤❡ s❡r✐❡s✿
n X
zi ,
i=∞ ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②
✐❢ t❤❡ s❡r✐❡s ♦❢ ✐ts ♠♦❞✉❧✐✱ n X
i=∞
❝♦♥✈❡r❣❡s✳ ◆♦✇✱ t❤❡ ❛❧❣❡❜r❛ ♦❢ s❡r✐❡s✳
∞ X
||zi || ,
zi = Im(z) .
✸✳✶✶✳
❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝❛♥
✷✹✻
♠✉❧t✐♣❧② ❛ ❝♦♥✈❡r❣❡♥t s❡r✐❡s ❜② ❛ ♥✉♠❜❡r t❡r♠ ❜② t❡r♠✳
❚❤❡♦r❡♠ ✸✳✶✶✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡
{sn }
✐s ❛ s❡q✉❡♥❝❡✳ ❋♦r ❛♥② ✐♥t❡❣❡r
∞ X a
(c · sn ) = c ·
a
❛♥❞ ❛♥② ❝♦♠♣❧❡①
∞ X
c✱
✇❡ ❤❛✈❡✿
sn
a
♣r♦✈✐❞❡❞ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳
❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝❛♥
❛❞❞ t✇♦ ❝♦♥✈❡r❣❡♥t s❡r✐❡s t❡r♠ ❜② t❡r♠✳
❚❤❡♦r❡♠ ✸✳✶✶✳✼✿ ❙✉♠ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡
{sn }
❛♥❞
{tn }
❛r❡ s❡q✉❡♥❝❡s✳ ❋♦r ❛♥② ✐♥t❡❣❡r
∞ X
(sn + tn ) =
a
∞ X a
sn +
∞ X
a✱
✇❡ ❤❛✈❡✿
tn
a
♣r♦✈✐❞❡❞ t❤❡ t✇♦ s❡r✐❡s ❝♦♥✈❡r❣❡✳
❊①❡r❝✐s❡ ✸✳✶✶✳✽ Pr♦✈❡ t❤❡s❡ t❤❡♦r❡♠s✳
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✾✿ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❚❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❝♦♠♣❧❡① ♥✉♠❜❡r✮ ∞ X
an = a · rn
✇✐t❤ r❛t✐♦
r
✭❛
arn = ar1 + ar2 + ar3 + ar4 + ... + arn + ...
n=s
✐s ❝❛❧❧❡❞ t❤❡
❣❡♦♠❡tr✐❝ s❡r✐❡s
✇✐t❤ r❛t✐♦
r✳
❊①❛♠♣❧❡ ✸✳✶✶✳✶✵✿ ❝♦♠♣❧❡① ♣♦✇❡r s❡r✐❡s ❚❤✐s ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ♦✈❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ s❡r✐❡s✿
z + z 2 + z 3 + ... ❋♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ z ✱ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ t❤❡♥ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ s❡r✐❡s ❛r❡ ♣❧♦tt❡❞✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ st❛rt✐♥❣ ❢r♦♠ ❛ ♣♦✐♥t ✐♥s✐❞❡ t❤❡ ❞✐s❦
||z|| ≤ 1
♣r♦❞✉❝❡s ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ♦✉ts✐❞❡ ♣r♦❞✉❝❡s ❞✐✈❡r❣❡♥❝❡✿
✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
✷✹✼
■♥❞❡❡❞✱ t❤❡ s❡r✐❡s
1 + z + z 2 + z 3 + ...
✐s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ r❛t✐♦ r = z ✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝♦♥✈❡r❣❡s ❢♦r ❛❧❧ ||z|| < 1✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ❛♥❞ ❞✐✈❡r❣❡s ❢♦r ❛❧❧ ||z|| > 1✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ❝♦♥✈❡r❣❡s ♦♥ t❤❡ ❞✐s❦ ♦❢ r❛❞✐✉s 1 ❝❡♥t❡r❡❞ ❛t 0 ✐♥ C ❛♥❞ ❞✐✈❡r❣❡s ♦✉ts✐❞❡ ♦❢ ✐t✳ ❚❤✐s ❝✐r❝❧❡ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ s❡r✐❡s✳ ❲❡ ❡✈❡♥ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ 1 + z + z 2 + z 3 + .. =
❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✐s ✐♥ t❤❡ ❞♦♠❛✐♥✳
1 . 1−z
■ts s✉♠ ✐s ❢♦✉♥❞ t❤❡ s❛♠❡ ✇❛② ❛s ❢♦r r❡❛❧ ✈❛r✐❛❜❧❡✳ ❚❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ♠♦❞✉❧✉s✿ ❚❤❡♦r❡♠ ✸✳✶✶✳✶✶✿ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ ❙❡r✐❡s ❚❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦ ✇❤❡♥
||r|| > 1✳
r ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ✇❤❡♥ ||r|| < 1 ❛♥❞ ❞✐✈❡r❣❡s
■♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ t❤❡ s✉♠ ✐s✿
∞ X
arn =
n=0
a . 1−r
❊①❡r❝✐s❡ ✸✳✶✶✳✶✷
Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✸✳✶✶✳✶✸✿ tr✐❣ r❡♣r❡s❡♥t❛t✐♦♥
❚❤❡r❡ ✐s ❛ ❤✐♥t ❛t t❤❡ ✐❞❡❛ ♦❢ ❝♦♠♣❧❡① ♣♦✇❡r s❡r✐❡s ✐♥ ❈❤❛♣t❡r ✹❍❉✲✶✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢ t❤❡ s✐♥❡✱ t❤❡ ❝♦s✐♥❡✱ ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❚❤❡ s✐♥❡ ✐s ♦❞❞✱ ❛♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s ♦♥❧② ✐♥❝❧✉❞❡s ♦❞❞ t❡r♠s✿ ∞ sin x =
X (−1)k x2k+1 . (2k + 1)! k=0
❚❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✱ ❛♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s ♦♥❧② ✐♥❝❧✉❞❡s ❡✈❡♥ t❡r♠s✿ cos x =
∞ X (−1)k k=0
(2k)!
x2k .
■❢ ✇❡ ✇r✐t❡ t❤❡♠ ♦♥❡ ✉♥❞❡r t❤❡ ♦t❤❡r✱ ✇❡ s❡❡ ❤♦✇ t❤❡② ❝♦♠♣❧❡♠❡♥t ❡❛❝❤ ♦t❤❡r✿ n 0 1 sin x
2
x
cos x 1
x2 − 2!
3 −
x3 3!
4
x4 4!
5
...
x5 ... 5! ...
❲❤❛t ✐❢ ✇❡ ❛❞❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ t♦ t❤✐s❄ n 0 1
2
sin x 0 x
0 x2 2! x2 2!
cos x 1 0 − ex 1 x
3
4
5
x3 3!
0
x5 ... 5!
−
0 x3 3!
x4 4! x4 4!
0
...
...
x5 ... 5!
✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
✷✹✽
❚❤✐s ❧♦♦❦s ❛❧♠♦st ❧✐❦❡ ❛❞❞✐t✐♦♥✦✳✳ ❡①❝❡♣t ❢♦r t❤♦s❡ ♠✐♥✉s s✐❣♥s✳ ❚❤❛t✬s ✇❤❡r❡ i ❝♦♠❡s ✐♥✳ ▲❡t✬s s✉❜st✐t✉t❡ x = it✿ 3 4 5 2 eit
(it) (it) (it) (it) + + + + ··· 2! 3! 4! 5! t2 it3 t4 it5 + + − ··· = 1 + it − − 2! 3! 4! 5! t 3 x5 t2 t4 = 1 − + − ··· + i t − + − ··· 2! 4! 3! 5! = cos t + i sin t .
= 1 + it +
■t ✐s ❝❛❧❧❡❞ ❊✉❧❡r✬s ❢♦r♠✉❧❛✳ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ea+bi = ea (cos b + i sin b) .
■♥ ❝♦♠♣❧❡① ❝❛❧❝✉❧✉s✱ ❢✉♥❝t✐♦♥s ❛r❡ ♠♦r❡ ✐♥t❡rr❡❧❛t❡❞✦
❊①❡r❝✐s❡ ✸✳✶✶✳✶✹ ❙❤♦✇ t❤❛t sin2 x + cos2 x = 1✳
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✺✿ ♣♦✇❡r s❡r✐❡s ❆ s❡q✉❡♥❝❡ {qn } ♦❢ ♣♦❧②♥♦♠✐❛❧s ❣✐✈❡♥ ❜② ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ qk+1 (x) = qk (x) + ck+1 (x − a)k+1 , k = 0, 1, 2, ... ,
❢♦r s♦♠❡ ✜①❡❞ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡r a ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡rs {ck }✱ ✐s ❝❛❧❧❡❞ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t a✳ ❚❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❧✐♠✐t ♦❢ qn ✐s ❝❛❧❧❡❞ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✱ ✇r✐tt❡♥ ❛s✿ f (x) = c0 + c1 (x − a) + c2 (x − a)2 + ... =
X k
ck (x − a)k = lim qk (x) k→∞
❢♦r ❛❧❧ x ❢♦r ✇❤✐❝❤ t❤❡ ❧✐♠✐t ❡①✐sts✳ ❚❤❡♥ t❤❡ t❤r❡❡ ♣♦✇❡r s❡r✐❡s ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡ ♠❛② s❡r✈❡ ❛s t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦❢ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡✿ n 0 sin x =
1
2
x
3 −
4
x3 3!
5 +
...
x5 + ... 5!
x2 x4 + − ... 2! 4! x2 x3 x4 x5 + + + + ..., ex = 1 +x + 2! 3! 4! 5!
cos x = 1
−
♣❡♥❞✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡✐r ❝♦♥✈❡r❣❡♥❝❡✳ ❲❡ ✇✐❧❧ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✳
❚❤❡♦r❡♠ ✸✳✶✶✳✶✻✿ ❲❡✐❡rstr❛ss ▼✲❚❡st
❈♦♥s✐❞❡r t❤❡ ♣♦✇❡r s❡r✐❡s
∞ X n=0
cn (z − a)n .
✸✳✶✶✳
❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s
✷✹✾
❙✉♣♣♦s❡ t❤❡r❡ ❡①✐sts s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡rs
{Mn }
t❤❛t
|cn (z − a)n | ≤ Mn , n = 0, 1, 2, ... , ❢♦r ❛❧❧
z
✐♥ ❛♥ ♦♣❡♥ ❞✐s❦
D
❛r♦✉♥❞
a
❛♥❞ t❤❡ s❡r✐❡s
∞ X n=0
❝♦♥✈❡r❣❡s✳ ❚❤❡♥ t❤❡ ♣♦✇❡r s❡r✐❡s ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ♦♥
Mn
✭♦❢ r❡❛❧ ♥✉♠❜❡rs✮
D✳
❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❤r❡❡ s❡r✐❡s ❛❜♦✈❡ ❢♦❧❧♦✇s✳
❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✼✿ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❚❤❡ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞
r
✭t❤❛t ❝♦✉❧❞ ❜❡ ✐♥✜♥✐t❡✮ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠
❛ ♣♦✐♥t ❢♦r ✇❤✐❝❤ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t
❝♦♥✈❡r❣❡♥❝❡
a
❞✐✈❡r❣❡s ✐s ❝❛❧❧❡❞ t❤❡
a
t♦
r❛❞✐✉s ♦❢
♦❢ t❤❡ s❡r✐❡s✳
❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❧❡❣✐t✐♠❛t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡
❊①✐st❡♥❝❡ ♦❢ sup ❚❤❡♦r❡♠
❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳
❲❡ ✜♥❛❧❧② ❝❛♥ s❡❡ ✇❤❡r❡ t❤❡ ✇♦r❞ ✏r❛❞✐✉s✑ ❝♦♠❡s ❢r♦♠✳
❚❤❡♦r❡♠ ✸✳✶✶✳✶✽✿ ❘❛❞✐✉s ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ❙✉♣♣♦s❡
r
✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ∞ X n=0
cn (x − a)n .
❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ ❲❤❡♥
r < ∞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s ❛ ❞✐s❦ B(a, R) ✐♥ C ♦❢ r❛❞✐✉s r a ✇✐t❤ s♦♠❡ ♣♦✐♥ts ♦♥ ✐ts ❜♦✉♥❞❛r② ♣♦ss✐❜❧② ✐♥❝❧✉❞❡❞✳ r = ∞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s t❤❡ ✇❤♦❧❡ C✳
❝❡♥t❡r❡❞ ❛t ✷✳ ❲❤❡♥
✸✳✶✷✳
❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s
✷✺✵
❊①❛♠♣❧❡ ✸✳✶✶✳✶✾✿ ❞♦♠❛✐♥s ❆ ❢❡✇ ❜❛s✐❝ s❡r✐❡s ❛s ❢✉♥❝t✐♦♥s✿
s❡r✐❡s
∞ X
s✉♠
❞♦♠❛✐♥
1 1−x
xk
=
(−1)k k x k!
= ex
k=0 ∞ X k=0
B(0, 1) C
∞ X (−1)k 2k+1 x = sin x (2k + 1)! k=0 ∞ X (−1)k 2k x = cos x (2k)! k=0
C C
✸✳✶✷✳ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s
❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✸■❈✲✺ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧ts✳
❚❤❡♦r❡♠ ✸✳✶✷✳✶✿ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s ■❢ t✇♦ ♣♦✇❡r s❡r✐❡s ❛r❡ ❡q✉❛❧✱ ❛s ❢✉♥❝t✐♦♥s✱ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ (a−r, a+r), r > 0✱ t❤❡♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧✱ ✐✳❡✳✱ ∞ X n=0
cn (x − a)
n
=
∞ X n=0
dn (x − a)n ❢♦r ❛❧❧ a − r < x < a + r
cn = dn ❢♦r ❛❧❧ n = 0, 1, 2, 3, ...
=⇒
❚❤❡♦r❡♠ ✸✳✶✷✳✷✿ ❚❡r♠✲❜②✲❚❡r♠ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ■♥t❡❣r❛t✐♦♥ ❙✉♣♣♦s❡ R > 0 ✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s f (x) =
∞ X n=0
an (x − a)n .
❚❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥t❡❞ ❜② t❤✐s ♣♦✇❡r s❡r✐❡s ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ✭❛♥❞✱ t❤❡r❡✲ ❢♦r❡✱ ✐♥t❡❣r❛❜❧❡✮ ♦♥ t❤❡ ♦♣❡♥ ❞✐s❦ |x−a| < R ❛♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❛♥❞ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡ ❝♦♥✈❡r❣❡ ♦♥ t❤✐s ❞✐s❦ ❛♥❞ ❛r❡ ❢♦✉♥❞ ❜② t❡r♠ ❜② t❡r♠ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s ♦❢ f ✱ ✐✳❡✳✱ ′
f (x) =
∞ X n=0
❛♥❞ Z
f (x) dx =
Z
cn (x − a)
∞ X n=0
n
!′
cn (x − a)
n
=
!
∞ X n=0
n ′
(cn (x − a) ) =
dx =
∞ Z X n=0
∞ X n=1
ncn (x − a)n−1 ,
∞ X cn (x−a)n+1 . cn (x−a) dx = n+1 n=0 n
✸✳✶✷✳
❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s
✷✺✶
❊①❛♠♣❧❡ ✸✳✶✷✳✸✿ ❖❉❊s ♦❢ ✜rst ♦r❞❡r ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ✭✇❡ ♣r❡t❡♥❞ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ❛♥s✇❡r✮✿ y ′ = ky, y(0) = y0 .
❚❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✸■❈✲✺✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦✇❡r s❡r✐❡s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s❡r✐❡s ❛♥❞ t❤❡♥ ♠❛t❝❤ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✿ y y′ =⇒ y′ || k·y
= =
c0
=
c1 || = kc0
+ c2 x2 + 2c2 x ւ ւ + 2c2 x + 3c3 x2 || || + kc1 x + kc2 x2 +
c1 x c1
+ c3 x3 + 3c3 x2 ւ + ... +
...
+ + ... +
... ...
ncn xn−1 || + kcn−1 xn−1
+ cn xn + ... n−1 + ncn x + ... ւ + (n + 1)cn+1 xn +... || + kcn xn +...
❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s✱ t❤❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ♠❛t❝❤✦ ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❡q✉❛t✐♦♥s✿ c1 2c2 3c3 ... (n + 1)cn+1 +... || || || || kc0 kc1 kc2 ... kcn +...
❲❡ ❝❛♥ st❛rt s♦❧✈✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿ c1 =⇒ c1 = kc0 2c2 =⇒ c2 = k 2 c0 /2 3c3 ... || || || kc0 =⇒ kc1 = k 2 c0 =⇒ kc2 = k 3 c0 /2 ...
❚❤❡ ❝♦♥❞✐t✐♦♥ y(0) = y0 ♠❡❛♥s t❤❛t c0 = y0 ✳ ❚❤❡r❡❢♦r❡✱ cn = y0
kn . n!
❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ ∞ X 1 kn n (kx)n = y0 ekx . y= y0 x = y0 n! n! n=0 n=0 ∞ X
◆♦t❡ t♦ ❣❡t t❤❡ nt❤ t❡r♠ ✇❡ s♦❧✈❡ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ❛✉❣♠❡♥t❡❞ ♠❛tr✐①✿ k −1 0 0 0 ... 0 k −2 0 0 ... 0 0 k −3 0 ... ... ... ... ... ... ... 0 0 0 0 0 ...
0 0 0 0 0 0 ... ... k −n
0 0 0 ... 0
.
❲❛r♥✐♥❣✦ ❘❡❝♦❣♥✐③✐♥❣ ♣❡❝t❡❞✳
❊①❡r❝✐s❡ ✸✳✶✷✳✹ ❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿
y ′ = ky + 1, y(0) = y0 .
t❤❡
r❡s✉❧t✐♥❣
s❡r✐❡s
✐s♥✬t
t♦
❜❡
❡①✲
✸✳✶✷✳
❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s
✷✺✷
❊①❛♠♣❧❡ ✸✳✶✷✳✺✿ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r
❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿ y ′′ = −y, y(0) = y0 , y ′ (0) = v0 .
❆❣❛✐♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦✇❡r s❡r✐❡s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s❡r✐❡s t✇✐❝❡ ❛♥❞ t❤❡♥ ♠❛t❝❤ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✿ y y′ y ′′ =⇒ y ′′ || −y
= = =
c0
+
c1 x c1
+ + +
c2 x2 2c2 x 2c2
c3 x3 3c3 x2 3 · 2c3 x
+ + +
+ c4 x4 + c5 x5 + ... 3 + 4cn x + 5cn x4 + ... + 4 · 3cn x2 + 5 · 4cn x3 + ...
= 2c2 + 3 · 2c3 x + 4 · 3c4 x2 + 5 · 4cn x3 + || || || = −c0 + −c1 x + −c2 x2 + −c3 x3 +
... ...
❚❤❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ♠❛t❝❤✿ 2c2 3 · 2c3 4 · 3c4 5 · 4c5 ... n(n − 1)cn ... || || || || −c0 −c1 −c2 −c3 ... −cn−2 ...
❲❡ ❝❛♥ st❛rt s♦❧✈✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ♦❞❞ s❡♣❛r❛t❡ ❢r♦♠ ❡✈❡♥✳ ❋✐rst✱ ❢♦r ❡✈❡♥ n✿ 2c2 =⇒ c2 = c0 /2 4 · 3c4 =⇒ c4 = −c0 /(4 · 3 · 2) ... =⇒ cn = ±c0 /n! ... || || −c0 =⇒ −c2 = −c0 /2
❚❤❡ ❝♦♥❞✐t✐♦♥ y(0) = y0 ♠❡❛♥s t❤❛t c0 = y0 ✳ ❚❤❡r❡❢♦r❡✱ n ❡✈❡♥ =⇒ cn = (−1)n/2+1
y0 . n!
❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ ∞ X
❡✈❡♥✱
(−1)
n/2+1 y0
n=0
n!
n
x = y0
∞ X
❡✈❡♥
(−1)n/2+1
n=0
1 n x = y0 cos x . n!
❙❡❝♦♥❞✱ ❢♦r ♦❞❞ n✿ 3 · 2c3 ⇒ c3 = c1 /(3 · 2) 5 · 4c5 ⇒ c5 = −c1 /(5 · 4 · 3 · 2) ... ⇒ cn = ±c1 /n! || || −c1 ⇒ −c3 = −c1 /(3 · 2)
❚❤❡ ❝♦♥❞✐t✐♦♥ y ′ (0) = v0 ♠❡❛♥s t❤❛t c1 = v0 ✳ ❚❤❡r❡❢♦r❡✱ n ♦❞❞ =⇒ cn = (−1)(n+1)/2
v0 . n!
❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ y=
∞ X
♦❞❞
n=1
(−1)
(n+1)/2 v0
n!
n
x = v0
∞ X
♦❞❞✱
(−1)(n+1)/2
n=1
❚❤✐s ✐s t❤❡ r❡s✉❧t✿ y = y0 cos x + v0 sin x .
1 n x = v0 sin x . n!
✸✳✶✷✳
❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s
✷✺✸
❊①❡r❝✐s❡ ✸✳✶✷✳✻ Pr♦✈✐❞❡ t❤❡ ❛✉❣♠❡♥t❡❞ ♠❛tr✐① ❢♦r t❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳
❊①❡r❝✐s❡ ✸✳✶✷✳✼ ❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿
y ′′ + xy ′ + y = 0, y(0) = 1, y ′ (0) = 1 . ❲❤❡♥ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s ✐s♥✬t r❡❝♦❣♥✐③❛❜❧❡✱ ✐t ✐s ✉s❡❞ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❛♥s✇❡r ✈✐❛ ✐ts ♣❛rt✐❛❧ s✉♠s✳ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❡rr♦r ❜♦✉♥❞ ❢♦r ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❣✐✈❡♥ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✶✱ ❛s ❢♦❧❧♦✇s✳
❚❤❡♦r❡♠ ✸✳✶✷✳✽✿ ❊rr♦r ❇♦✉♥❞
❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = y(x) ✐s (n + 1) t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ❡❛❝❤ i = 0, 1, 2, ..., n + 1✱ ✇❡ ❤❛✈❡ |y (i) (t)| < Ki ❢♦r ❛❧❧ t ❜❡t✇❡❡♥ a ❛♥❞ x ,
❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r Ki ✳ ❚❤❡♥ en (x) = |y(x) − Tn (x)| ≤ Kn+1
✇❤❡r❡ Tn ✐s t❤❡ nt❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ y ✳
|x − a|n+1 , (n + 1)!
❈❤❛♣t❡r ✹✿ ❙②st❡♠s ♦❢ ❖❉❊s
❈♦♥t❡♥ts
✹✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹ ✹✳✷ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸ ✹✳✸ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✼ ✹✳✹ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✵ ✹✳✺ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✸ ✹✳✻ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✾ ✹✳✼ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✸ ✹✳✽ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✼ ✹✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✸ ✹✳✶✵ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✽
✹✳✶✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s
❋✉♥❝t✐♦♥s ♣r♦❝❡ss ❛♥ ✐♥♣✉t ♦❢ ❛♥② ♥❛t✉r❡ ❛♥❞ ♣r♦❞✉❝❡ ❛♥ ♦✉t♣✉t ♦❢ ❛♥② ♥❛t✉r❡✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t x
7→
f
7→
y
❋✉♥❝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♣❛❝❡✳ ■♥ ❢❛❝t✱ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ s✉❜s❡t ♦❢ Rn ✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s ❢♦r n = 1, 2 ❛♥❞ m = 1, 2✿
✹✳✶✳
✷✺✺
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✉s✉❛❧ ❢✉♥❝t✐♦♥s✱ ✇❡ s❡❡ • ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P : t 7→ (x, y) ♦r P : t 7→< x, y >✱
• ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s F : (x, y) 7→ z ♦r F :< x, y >7→ z ✱ ❛♥❞
• ❛ ✈❡❝t♦r ✜❡❧❞ ♦♥ t❤❡ ♣❧❛♥❡ V : (x, y) 7→< u, v >✳
❲❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❤♦✇✱ ✐♥st❡❛❞ ♦❢ tr❡❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥❡ ❛①✐s ❛t ❛ t✐♠❡✱ t♦ st✉❞② t❤❡♠ ❛s ❢✉♥❝t✐♦♥s ✇✐t❤ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❱❡❝t♦r ❛❧❣❡❜r❛ ✇✐❧❧ ❜❡ ❡s♣❡❝✐❛❧❧② ✉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...✳
■♥ 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❤❡ ♣❧❛♥❡❀ t❤❡ ✜rst ♠❡t❤♦❞ ✐s t❤❡ s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠ ✿ y = mx + b .
❚❤✐s ♠❡t❤♦❞ ❞♦❡s ♥♦t ✐♥❝❧✉❞❡ ✈❡rt✐❝❛❧ ❧✐♥❡s✿ t❤❡ s❧♦♣❡ ✐s ✐♥✜♥✐t❡✦ ■♥ ♦✉r st✉❞② ♦❢ ❝✉r✈❡s ✭s♣❡❝✐✜❝❛❧❧② t♦ r❡♣r❡s❡♥t ♠♦t✐♦♥✮ ♦♥ t❤❡ ♣❧❛♥❡✱ t❤❡r❡ ❛r❡ ♥♦ ♣r❡❢❡rr❡❞ ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡♥ ✐t ✐s ✉♥❛❝❝❡♣t❛❜❧❡ t♦ ❡①❝❧✉❞❡ ❛♥② str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ ✐s ✐♠♣❧✐❝✐t ✿ px + qy = r .
❚❤❡ ❝❛s❡ ♦❢ p 6= 0, q = 0 ❣✐✈❡s ✉s ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✳ ❚❤❡ t❤✐r❞ ♠❡t❤♦❞ ✐s ✐♥t❡r♣r❡t❛t✐♦♥✳
♣❛r❛♠❡tr✐❝✳
■t ❤❛s ❛ ❞②♥❛♠✐❝
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
✷✺✻
❊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 >✳
❲❡ ✉s❡ ♠♦t✐♦♥ ❛s ❛ st❛rt✐♥❣ ♣♦✐♥t ❛♥❞ ❛s ✇❡❧❧ ❛s ❛ ♠❡t❛♣❤♦r ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ st❛rt ♠♦✈✐♥❣ ✭t = 0✮ • ❢r♦♠ t❤❡ ♣♦✐♥t P0 = (1, 3)✱ • ✉♥❞❡r ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ♦❢ V =< 2, 3 >✳ ❲❡ ♠♦✈❡ • 2 ❢❡❡t ♣❡r s❡❝♦♥❞ ❤♦r✐③♦♥t❛❧❧②✱ ❛♥❞ • 3 ❢❡❡t ♣❡r s❡❝♦♥❞ ✈❡rt✐❝❛❧❧②✳
■♥ 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❤❡♥ ❤❛✈❡ ❛❧r❡❛❞② t✇♦ ♣♦✐♥ts ♦♥ ♦✉r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P ✿ P (0) = P0 = (1, 3) ❛♥❞ P (1) = P1 = (3, 6) .
❖❢ ❝♦✉rs❡✱ t❤❡s❡ ❛r❡ ❛❧s♦ t❤❡ ✈❛❧✉❡s ♦❢ ♦✉r ❢✉♥❝t✐♦♥✳ ▲❡t✬s ✜♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤✐s ❢✉♥❝t✐♦♥✳ ❊❛r❧② ♦♥✱ ✐t✬s ❖❑ t♦ ❞♦ t❤✐s ❝♦♠♣♦♥❡♥t✲✇✐s❡✳ ❚❤❡♥ P (t) = (x(t), y(t)) ,
❛♥❞
x(0) = 1, x(1) = 3 ❛♥❞ y(0) = 3, y(1) = 6 .
❚❤❡s❡ ❢✉♥❝t✐♦♥s ♠✉st ❜❡ ❧✐♥❡❛r❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ x(t) = 1 + 2t ❛♥❞ y(t) = 3 + 3t .
❚❤✐s ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳✳✳ ❜✉t ♥♦t ❛♥ ❛❝❝❡♣t❛❜❧❡ ❛♥s✇❡r ✐❢ ✇❡ ❛r❡ t♦ ❧❡❛r♥ ❤♦✇ t♦ ✉s❡ ✈❡❝t♦rs✦ ❚❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts✱ ♦❢ ❝♦✉rs❡✱ ❝♦♠❡ ❢r♦♠ t❤❡ s♣❡❝✐✜❝ ♥✉♠❜❡rs t❤❛t ❣✐✈❡ ✉s P0 ❛♥❞ V ✳ ▲❡t✬s ❛ss❡♠❜❧❡ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ ❢✉♥❝t✐♦♥ ✐♥t♦ ♦♥❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ P (t) = (x(t), y(t)) = (1 + 2t, 3 + 3t) .
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
✷✺✼
❚❤✐s ✐s st✐❧❧ ♥♦t ❣♦♦❞ ❡♥♦✉❣❤❀ ✇❡ st✐❧❧ ❝❛♥✬t s❡❡ t❤❡ P0 ❛♥❞ V ❞✐r❡❝t❧②✦ ❲❡ ❝♦♥t✐♥✉❡ ❜② ✉s✐♥❣ ✈❡❝t♦r ❛❧❣❡❜r❛✿ ❲❡ ✉s❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ P (t) = (1 + 2t, 3 + 3t) = (1, 3)+ < 2t, 3t > ❚❤❡♥ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ = (1, 3) + t < 2, 3 > ❆♥❞ ✜♥❛❧❧② s✉❜st✐t✉t❡✳ = P0 + tV.
❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ❛ ✈❡❝t♦r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ str❛✐❣❤t ♠♦t✐♦♥✿ P (t) = P0 + tV ,
✇❤❡r❡ P0 ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ V ✐s t❤❡ ✭❝♦♥st❛♥t✮ ✈❡❧♦❝✐t②✳
❲❛r♥✐♥❣✦ ❖♥❡ ❝❛♥✱ ♦❢ ❝♦✉rs❡✱ ♠♦✈❡ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛
✈❛r✐❛❜❧❡
✈❡❧♦❝✐t②✳
❙♦✱ ♣♦s✐t✐♦♥ ❛t t✐♠❡ t = ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ + t · ✈❡❧♦❝✐t② ❲❡ st❛t❡❞ t❤✐s ❝♦♥❝❧✉s✐♦♥ ❜❡❢♦r❡✦ ❚❤✐s t✐♠❡ ♦♥❧② t❤❡ ❝♦♥t❡①t ❤❛s ❝❤❛♥❣❡❞✳ ❚❤❡ ♣❛tt❡r♥ ✐s ❝❧❡❛r✿ t❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t (a, b) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < u, v > ✐s r❡♣r❡s❡♥t❡❞ ♣❛r❛♠❡tr✐❝❛❧❧② ❛s✿ P (t) = (a, b) + t < u, v > .
❙✐♠✐❧❛r ❢♦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✿ 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✐♦♥ ❙✉♣♣♦s❡ P0 ✐s ❛ ♣♦✐♥t ✐♥ Rm ❛♥❞ V ✐s ❛ ✈❡❝t♦r✳ ❚❤❡♥ t❤❡
♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢
t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥ t❤r♦✉❣❤ P0 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ V ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ P (t) = P0 + tV
❚❤❡♥✱ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ P0 ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳
✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢
V ✐s t❤❡ ♣❛t❤ ✭✐♠❛❣❡✮ ♦❢ t❤✐s
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
❙t❛t❡❞ ❢♦r
m = 1✱
✷✺✽
t❤❡ ❞❡✜♥✐t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❢❛♠✐❧✐❛r ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✦
❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ s✐♥❣❧❡
♥✉♠❜❡r ✭t❤❡ s❧♦♣❡✮ ❜❡❝❛✉s❡ t❤❡ ❝❤❛♥❣❡ ✐s ❡♥t✐r❡❧② ✇✐t❤✐♥ t❤❡ y ✲❛①✐s✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ ✐s t❤❡ ❝♦♥t❡①t✿ 2 t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ✐♥ R ❢♦r ❝❤❛♥❣❡✳ ❚❤❛t ✐s ✇❤② t❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ ✈❡❝t♦r✳ ❇✉t t❤❡ ❡q✉❛t✐♦♥ ❧♦♦❦s ❡①❛❝t❧② t❤❡ s❛♠❡✳✳✳ ❡✈❡♥ t❤♦✉❣❤ ❡❛❝❤ ❧❡tt❡r ♠❛② ❝♦♥t❛✐♥ ✉♥❧✐♠✐t❡❞ ❛♠♦✉♥t ♦❢ ✐♥❢♦r♠❛t✐♦♥✦ ❊①❛♠♣❧❡ ✹✳✶✳✸✿ ♣r✐❝❡s
❚❤❡ ❞❡✜♥✐t✐♦♥ ❛♣♣❧✐❡s t♦ t❤❡ ❛❜str❛❝t s♣❛❝❡s✳ ■❢ ✇❡ ♠✐❣❤t ❤❛✈❡
m = 10, 000✳
Rm
✐s t❤❡ s♣❛❝❡ ♦❢ ♣r✐❝❡s ✭♦❢ st♦❝❦s ♦r ❝♦♠♠♦❞✐t✐❡s✮✱
❚❤❡ ♣r✐❝❡s r❡❝♦r❞❡❞ ❝♦♥t✐♥✉♦✉s❧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✳ ❆❧s♦✱ ✐♥ t❤❡ s❤♦rt t❡r♠ t❤✐s ❝✉r✈❡ ✐s ❧✐❦❡❧② t♦ ❧♦♦❦ ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✿ t❤❡ ♠♦st r❡❝❡♥t ❝❤❛♥❣❡ ♦❢ ❡❛❝❤ ♣r✐❝❡ ✐s r❡❝♦r❞❡❞ ✐s t❤❡♥ t❤❡ s❛♠❡ ❝❤❛♥❣❡ ✐s ♣r❡❞✐❝t❡❞ ❢♦r t❤❡ ♥❡①t t✐♠❡ ♣❡r✐♦❞✳ ■♥ ❡❛❝❤ ❝♦❧✉♠♥ ✇❡ ✉s❡ t❤❡ s❛♠❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡
k t❤
♣r✐❝❡✿
xk (t + ∆t) = xk (t) + vk ∆t , ✇❤❡r❡
vk
✐s t❤❡
k t❤
❚❤❡ t❛❜❧❡ ✐s ♦✉r
r❛t❡ ♦❢ ❝❤❛♥❣❡✳
10, 000✲❞✐♠❡♥s✐♦♥❛❧
❝✉r✈❡✦ ❈❛♥ ✇❡ ✈✐s✉❛❧✐③❡ s✉❝❤ ❛ ❝✉r✈❡ ✐♥ ❛♥② ✇❛②❄ ❲❡ ♣✐❝❦ t✇♦
❝♦❧✉♠♥s ❛t ❛ t✐♠❡ ❛♥❞ ♣❧♦t t❤❛t ❝✉r✈❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❙✐♥❝❡ t❤❡s❡ ❝♦❧✉♠♥s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛①❡s✱ ✇❡ ❛r❡ ♣❧♦tt✐♥❣ ❛ ✏s❤❛❞♦✇✑ ♦❢ ♦✉r ❝✉r✈❡ ❝❛st ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✳ ❚❤❡② ❛r❡ ❛❧❧ str❛✐❣❤t ❧✐♥❡s✳ ❊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❡ ♥♦t ❢♦r❝❡s ❛t ♣❧❛②✳
❊✈❡♥ ❛
❝♦♥st❛♥t ❢♦r❝❡ ❧❡❛❞s t♦ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤✐❝❤ ♠❛② ❝❤❛♥❣❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✶✳✺✿ ❝♦♥st❛♥t ✈❡❧♦❝✐t②
❘❡❝❛❧❧ t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s t❤❛t ❣✐✈❡s 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
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
✷✺✾
❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ♥♦✇ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✿
Pk = (pk , qk ) , ❛♥❞ t❤❡ ❡q✉❛t✐♦♥s t❛❦❡ ❛ ✈❡❝t♦r ❢♦r♠ t♦♦✿
Pk+1 = Pk + Vk ∆t . ❊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✳
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
✷✻✵
❚❤❡♥ ✇❡ ❤❛✈❡ 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 pk+1 y : uk+1 qk+1
= v0 = pk +vk ∆t = vk −g∆t = qk +uk ∆t
❚❤❡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✱ ′′ x = 0, y ′′ = −g, ✇❡ ✐♥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 , y = − 21 gt2 + vy t + py ,
❚❤✉s✱ ✇❡ ❤❛✈❡✿ ❞❡♣t❤ = ❤❡✐❣❤t =
✐♥✐t✐❛❧ ❞❡♣t❤ ✐♥✐t✐❛❧ ❤❡✐❣❤t
+ +
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 + y py vy −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✿
P (t) = P0 + V0 · t + A ·
t2 2
✹✳✶✳
P❛r❛♠❡tr✐❝ ❝✉r✈❡s
✷✻✷
❲❡ ❤❛✈❡ ❛♥ ❡①tr❛ t❡r♠✱ t❤❛t ❞✐s❛♣♣❡❛rs ✇❤❡♥
1✲❞✐♠❡♥s✐♦♥❛❧
A = 0✱
✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥✳ ❏✉st ❛s ✐♥ t❤❡
❝❛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✈❡ ❧✐❡ ✐♥
✈❡❝t♦rs✳
Rm
❛s
♣♦✐♥ts
❜✉t ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s
❋♦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✿
■♥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✳
❊①❛♠♣❧❡ ✹✳✶✳✶✶✿ ❝✐r❝❧❡ tr❛♥s❢♦r♠❡❞ ❘❡❝❛❧❧ ❤♦✇ ✇❡ ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✉s✐♥❣ t❤❡ ❛♥❣❧❡ ❛s t❤❡ ♣❛r❛♠❡t❡r✳ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ❛t ❛♥❣❧❡
t
✐s
cos t
❛♥❞
sin t
❍❡r❡✱ 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
Q(t) = 2R(t) = 2 < cos t, sin t > , 2✳
❝❛s❡
2 ♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s str❡t❝❤✐♥❣ r❛❞✐❛❧❧②
t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❲❡ t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿
✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s
1✲❞✐♠❡♥s✐♦♥❛❧
✹✳✷✳
❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✷✻✸
❙✐♠✐❧❛r❧②✱ ✉s✐♥❣ ✈❡❝t♦r ❛❞❞✐t✐♦♥ ✇✐t❤
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
❆♥❞ s♦ ♦♥ ✇✐t❤ ♦t❤❡r
tr❛♥s❢♦r♠❛t✐♦♥s
2
❝❡♥t❡r❡❞ ❛t
(1, 2)✳
♦❢ t❤❡ ♣❧❛♥❡✳
✹✳✷✳ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
❚❤✐s ✐s t❤❡ ■❱P ✇❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞ s♦ ❢❛r✿
∆y = f (t, y), y(t0 ) = y0 , ∆t ❛♥❞
y ′ = f (t, y), y(t0 ) = y0 . ❚❤❡ ❡q✉❛t✐♦♥s s❤♦✇ ❤♦✇ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡
t✇♦
y
❞❡♣❡♥❞s ♦♥
✈❛r✐❛❜❧❡ q✉❛♥t✐t✐❡s ❞❡♣❡♥❞❡♥t ♦♥
❚❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ✐s ❛s ❢♦❧❧♦✇s✿
• x
✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ ❛♥❞
• y
✐s t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✳
t❄
t
❛♥❞
y✳
s❤✐❢t✐♥❣
t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❜②
✹✳✷✳ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✷✻✹
❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤✐s s✐♠♣❧❡st s❡tt✐♥❣ ♦❢ ❢r❡❡ ❢❛❧❧✿
∆x = vx , ∆t ∆y = vy − gt , ∆t
❛♥❞
x′ = vx ,
y ′ = v − gt . y
■t ✐s ❥✉st ❛s s✐♠♣❧❡ ✇❤❡♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥s ❛r❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✭t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s s♦❧✈❡❞ ❜② ✐♥t❡❣r❛t✐♦♥✮✳ ❍❡r❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ t✐♠❡
t
♦♥❧②✳
▼♦r❡ ❝♦♠♣❧❡① ✐s t❤❡ s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❧♦❝❛t✐♦♥✳ ❲❤❡♥ t❤❡ ❢♦r♠❡r ❞❡♣❡♥❞s ♦♥❧② ♦♥ ✐ts ♦✇♥ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥✱ t❤❡ ♠♦t✐♦♥ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤✐s ♣❛✐r ♦❢ ❖❉❊s✿
∆x = g(x) , ∆t ∆y = h(y) , ∆t
❛♥❞
x′ = g(x) , y ′ = h(y) .
❚❤❡ s♦❧✉t✐♦♥ ❝♦♥s✐sts ♦❢ t✇♦ s♦❧✉t✐♦♥s t♦ t❤❡ t✇♦✱ ✉♥r❡❧❛t❡❞✱ ❖❉❊s✳
❲❡ ❝❛♥ t❤❡♥ ❛♣♣❧② t❤❡ ♠❡t❤♦❞s ♦❢
❈❤❛♣t❡r ✸✳ ❆s ❛♥ ❡①❛♠♣❧❡ ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❞♦ ✐♥t❡r❛❝t✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧✳ ▲❡t
• x
❜❡ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ❛♥❞
• y
❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐♥ t❤❡ ❢♦r❡st✳
▲❡t
• ∆t
❜❡ t❤❡ ✜①❡❞ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳
❊✈❡♥ t❤♦✉❣❤ t✐♠❡
t
✐s ♥♦✇ ❞✐s❝r❡t❡✱ t❤❡ ✏♥✉♠❜❡r✑ ♦❢ r❛❜❜✐ts
♦✉r ♠♦❞❡❧✳ ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ ✐s ♦♥❡ s♦♠❡✇❤❡r❡ ✐s
0.1
x
♦r ❢♦①❡s
y
✐s♥✬t✳ ❚❤♦s❡ ❛r❡ r❡❛❧ ♥✉♠❜❡rs ✐♥
r❛❜❜✐ts ❛s ✐❢ t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡r ✐s ✉♥❦♥♦✇♥ ❜✉t t❤❡ ❧✐❦❡❧✐❤♦♦❞ t❤❛t t❤❡r❡
10%✳
❲❡ ❜❡❣✐♥ ✇✐t❤ t❤❡ r❛❜❜✐ts✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢❛❝t♦rs ❛✛❡❝t✐♥❣ t❤❡✐r ♣♦♣✉❧❛t✐♦♥✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡② ❤❛✈❡ ❛♥ ✉♥❧✐♠✐t❡❞ ❢♦♦❞ s✉♣♣❧② ❛♥❞ r❡♣r♦❞✉❝❡ ✐♥ ❛ ♠❛♥♥❡r ❞❡s❝r✐❜❡❞ ♣r❡✈✐♦✉s❧② ✕ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♣r❡❞❛t♦r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❣❛✐♥ ♦❢ t❤❡ r❛❜❜✐t ♣♦♣✉❧❛t✐♦♥ ♣❡r ✉♥✐t ♦❢ t✐♠❡ t❤r♦✉❣❤ t❤❡✐r ♥❛t✉r❛❧ r❡♣r♦❞✉❝t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❘❛❜❜✐ts✬ ❣❛✐♥ ❢♦r s♦♠❡
= α · x · ∆t ,
α > 0✳
❙❡❝♦♥❞✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ r❛t❡ ♦❢ ♣r❡❞❛t✐♦♥ ✉♣♦♥ t❤❡ r❛❜❜✐ts t♦ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ t❤❡ r❛❜❜✐ts ❛♥❞ t❤❡ ❢♦①❡s ♠❡❡t✱ ✇❤✐❝❤✱ ✐♥ t✉r♥✱ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡s ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥s✱
y✳
❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❘❛❜❜✐ts✬ ❧♦ss
= β · x · y · ∆t ,
x
❛♥❞
✹✳✷✳
✷✻✺
❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
❢♦r s♦♠❡ β > 0✳ ❈♦♠❜✐♥❡❞✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛❜❜✐t ♣♦♣✉❧❛t✐♦♥ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❧❡♥❣t❤ ∆t ✐s✿
∆x = αx∆t − βxy∆t . ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❢♦①❡s✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢❛❝t♦rs ❛✛❡❝t✐♥❣ t❤❡✐r ♣♦♣✉❧❛t✐♦♥✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦①❡s ❤❛✈❡ ♦♥❧② ❛ ❧✐♠✐t❡❞ ❢♦♦❞ s✉♣♣❧②✱ ✐✳❡✳✱ t❤❡ r❛❜❜✐ts✳ ❚❤❡ ❢♦①❡s ❞✐❡ ♦✉t ❣❡♦♠❡tr✐❝❛❧❧② ✐♥ ❛ ♠❛♥♥❡r ❞❡s❝r✐❜❡❞ ♣r❡✈✐♦✉s❧② ✕ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♣r❡②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧♦ss ♦❢ t❤❡ ❢♦① ♣♦♣✉❧❛t✐♦♥ ♣❡r ✉♥✐t ♦❢ t✐♠❡ t❤r♦✉❣❤ t❤❡✐r ♥❛t✉r❛❧ ❞❡❛t❤ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❋♦①❡s✬ ❧♦ss = γ · y · ∆t , ❢♦r s♦♠❡ γ > 0✳
❙❡❝♦♥❞✱ ✇❡ ❛❣❛✐♥ ❛ss✉♠❡ t❤❛t t❤❡ r❛t❡ ♦❢ r❡♣r♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❢♦①❡s ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❛t❡ ♦❢ t❤❡✐r ♣r❡❞❛t✐♦♥ ✉♣♦♥ t❤❡ r❛❜❜✐ts ✇❤✐❝❤ ✐s✱ ❛s ✇❡ ❦♥♦✇✱ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡s ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥s✱ x ❛♥❞ y ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❋♦①❡s✬ ❣❛✐♥ = δ · x · y · ∆t , ❢♦r s♦♠❡ δ > 0✳
❈♦♠❜✐♥❡❞✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢♦① ♣♦♣✉❧❛t✐♦♥ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❧❡♥❣t❤ ∆t ✐s✿
∆y = δxy∆t − γy∆t . P✉tt✐♥❣ t❤❡s❡ t✇♦ t♦❣❡t❤❡r ❣✐✈❡s ✉s ❛
❞✐s❝r❡t❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✿
∆x = (αx − βxy) ∆t, ∆y = (δxy − γy) ∆t.
❚❤❡♥ t❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛s ❛r❡ ❢♦r x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✿
❂❘❬✲✶❪❈✰❘✷❈✸✯❘❬✲✶❪❈✯❘✸❈✶✲❘✷❈✹✯❘❬✲✶❪❈✯❘❬✲✶❪❈❬✶❪✯❘✸❈✶ ❂❘❬✲✶❪❈✲❘✷❈✺✯❘❬✲✶❪❈✯❘✸❈✶✰❘✷❈✻✯❘❬✲✶❪❈✯❘❬✲✶❪❈❬✲✶❪✯❘✸❈✶ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♣♦ss✐❜❧❡ ❞②♥❛♠✐❝s ✭α = 0.10✱ β = 0.50✱ γ = 0.20✱ δ = 0.20✱ h = 0.2✮✿
✹✳✷✳
❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✷✻✻
❚❤✐s ✐s ✇❤❛t ✇❡ s❡❡✳ ■♥✐t✐❛❧❧②✱ t❤❡r❡ ❛r❡ ♠❛♥② r❛❜❜✐ts ❛♥❞✱ ✇✐t❤ t❤✐s ♠✉❝❤ ❢♦♦❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✇❛s
↑✳ տ✳
←✳
❣r♦✇✐♥❣✿
❚❤✐s ✇❛s ❝❛✉s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts t♦ ❞❡❝❧✐♥❡✿
s②st❡♠✿
▲❛t❡r✱ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ❞❡❝❧✐♥❡s s♦ ♠✉❝❤ t❤❛t✱ ✇✐t❤ s♦ ❧✐tt❧❡ ❢♦♦❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ❛❧s♦
st❛rt❡❞ t♦ ❞❡❝❧✐♥❡✿
↓✳
❈♦♠❜✐♥❡❞✱ t❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡
❆t t❤❡ ❡♥❞✱ ❜♦t❤ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥s s❡❡♠ t♦ ❤❛✈❡ ❞✐s❛♣♣❡❛r❡❞✳✳✳
❆♥♦t❤❡r ❡①♣❡r✐♠❡♥t s❤♦✇s t❤❛t t❤❡② ❝❛♥ r❡❝♦✈❡r ✭α
= 3✱ β = 2✱ γ = 3✱ δ = 1✱ h = 0.03✮✿
■♥ ❢❛❝t✱ ✇❡ ❝❛♥ s❡❡ ❛ r❡♣❡❛t✐♥❣ ♣❛tt❡r♥✳ ❋✉rt❤❡r♠♦r❡✱ ✇✐t❤
∆t → 0✱
✇❡ ❤❛✈❡ t✇♦✱ r❡❧❛t❡❞✱ ❖❉❊s✱ ❛
❝♦♥t✐♥✉♦✉s ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✿
dx = αx − βxy , dt dy = δxy − γy . dt
■t ❛♣♣r♦①✐♠❛t❡s t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❚❤❡ ❡q✉❛t✐♦♥s ❛r❡ ❦♥♦✇♥ ❛s t❤❡
▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s✳
✹✳✸✳
✷✻✼
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✹✳✸✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
❚♦ ❝♦♥✜r♠ ♦✉r ♦❜s❡r✈❛t✐♦♥s✱ ✇❡ ✇✐❧❧ ❝❛rr② ♦✉t q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s✳ ■t ✐s ❡q✉❛❧❧② ❛♣♣❧✐❝❛❜❧❡ t♦ ❜♦t❤ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❛♥❞ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✳ ■♥❞❡❡❞✱ t❤❡② ❜♦t❤ ❤❛✈❡ t❤❡ s❛♠❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ ∆x = αx − βxy , ∆t ∆y = δxy − γy , ∆t
❛♥❞
dx = αx − βxy , dt dy = δxy − γy . dt
❲❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ t❤❡ ❞②♥❛♠✐❝s ❛t ❛❧❧ ❧♦❝❛t✐♦♥s ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ♦❢ t❤❡ tx✲♣❧❛♥❡✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ x ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡✱ ✐✳❡✳✱ x′ = 0✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❧❡❛❞s t♦ ♥♦ ❝❤❛♥❣❡ ✐♥ x✱ ✐✳❡✳✱ ∆x = 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s✿ αx − βxy = 0 .
❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿
x = 0 ♦r y = α/β .
❲❡ ❞✐s❝♦✈❡r t❤❛t✱ ✜rst✱ • x = 0 ✐s ❛ s♦❧✉t✐♦♥✱ ❛♥❞✱ s❡❝♦♥❞✱
• t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ y = α/β ✐s ❝r♦ss❡❞ ✈❡rt✐❝❛❧❧② ❜② t❤❡ s♦❧✉t✐♦♥s✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✜rst✱ t❤❡ ❢♦①❡s ❛r❡ ❞②✐♥❣ ♦✉t ✇✐t❤ ♥♦ r❛❜❜✐ts ❛♥❞✱ s❡❝♦♥❞✱ t❤❡r❡ ♠❛② ❜❡ ❛ r❡✈❡rs❛❧ ✐♥ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ r❛❜❜✐ts ❛t ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ ❢♦①❡s✳ ❚♦ ✜♥❞ ♦✉t✱ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿ x′ > 0 ♦r ∆x > 0 =⇒ αx − βxy > 0 =⇒ y < α/β .
■t ❢♦❧❧♦✇s t❤❛t✱ ✐♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ✐♥❝r❡❛s❡s ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐s ❜❡❧♦✇ α/β ✱ ♦t❤❡r✇✐s❡ ✐t ❞❡❝r❡❛s❡s✳
✹✳✸✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✷✻✽
❙❡❝♦♥❞✱ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ y ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡✱ ✐✳❡✳✱ y ′ = 0✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❧❡❛❞s t♦ ♥♦ ❝❤❛♥❣❡ ✐♥ y ✱ ✐✳❡✳✱ ∆y = 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s✿ δxy − γy = 0 .
❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿
y = 0 ♦r x = γ/δ .
❲❡ ❞✐s❝♦✈❡r t❤❛t✱ ✜rst✱ • y = 0 ✐s ❛ s♦❧✉t✐♦♥✱ ❛♥❞✱ s❡❝♦♥❞✱
• t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ x = γ/δ ✐s ❝r♦ss❡❞ ❤♦r✐③♦♥t❛❧❧② ❜② t❤❡ s♦❧✉t✐♦♥s✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✜rst✱ t❤❡ r❛❜❜✐ts t❤r✐✈❡ ✇✐t❤ ♥♦ ❢♦①❡s ❛♥❞✱ s❡❝♦♥❞✱ t❤❡r❡ ♠❛② ❜❡ ❛ r❡✈❡rs❛❧ ✐♥ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❢♦①❡s ❛t ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts✳ ❚♦ ✜♥❞ ♦✉t✱ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿ y ′ > 0 ♦r ∆y > 0 =⇒ δxy − γy > 0 =⇒ x > γ/δ .
■t ❢♦❧❧♦✇s t❤❛t✱ ✐♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐♥❝r❡❛s❡s ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ✐s ❛❜♦✈❡ γ/δ ✱ ♦t❤❡r✇✐s❡ ✐t ❞❡❝r❡❛s❡s✳
◆♦✇ ✇❡ ♣✉t t❤❡s❡ t✇♦ ♣❛rts t♦❣❡t❤❡r✳ ❲❡ ❤❛✈❡ ❢♦✉r s❡❝t♦rs ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢♦✉r ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ t❤❡ s✐❣♥s ♦❢ x′ ❛♥❞ y ′ ✱ ♦r ∆x ❛♥❞ ∆y ✿
✹✳✸✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧
✷✻✾
■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ r❡s✉❧t ✐s ❛ r♦✉❣❤ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❞②♥❛♠✐❝s ♦♥ t❤❡ ❧♦❝❛❧ ❧❡✈❡❧✿ ✐❢ t❤✐s ✐s t❤❡ ❝✉rr❡♥t st❛t❡✱ t❤❡♥ t❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ✐t ✐s ❣♦✐♥❣✳ ■t ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ✦ ❲❡ ✈✐s✉❛❧✐③❡ t❤✐s ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs ❛s ❜❡❢♦r❡✿
❚❤❡ ❛rr♦✇s ❛r❡♥✬t ♠❡❛♥t ②❡t t♦ ❜❡ ❝♦♥♥❡❝t❡❞ ✐♥t♦ ❝✉r✈❡s t♦ ♣r♦❞✉❝❡ s♦❧✉t✐♦♥s✳ ❚❤❡ ♦♥❧② ❢♦✉r ❞✐st✐♥❝t s♦❧✉t✐♦♥s ✇❡ ❦♥♦✇ ❢♦r s✉r❡ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿
• t❤❡ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ❢♦①❡s ✐♥ t❤❡ ❛❜s❡♥❝❡ r❛❜❜✐ts ✕ ♦♥ t❤❡ y ✲❛①✐s❀
• t❤❡ ❡①♣❧♦s✐♦♥ ♦❢ t❤❡ r❛❜❜✐ts ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❢♦①❡s ✕ ♦♥ t❤❡ x✲❛①✐s❀
• t❤❡ ❢r❡❡③✐♥❣ ♦❢ ❜♦t❤ r❛❜❜✐ts ❛♥❞ ❢♦①❡s ❛t t❤❡ s♣❡❝✐❛❧ ❧❡✈❡❧s ✕ ✐♥ t❤❡ ♠✐❞❞❧❡❀ ❛♥❞ ❛❧s♦ • t❤❡ ❢r❡❡③✐♥❣ ♦❢ ❜♦t❤ r❛❜❜✐ts ❛♥❞ ❢♦①❡s ❛t t❤❡ ③❡r♦ ❧❡✈❡❧✳
❊✐t❤❡r ♦❢ t❤❡ ❧❛st t✇♦ ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥ ❝❛❧❧❡❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤❡ ♠❛✐♥✱ ♥♦♥✲③❡r♦✱ ❡q✉✐❧✐❜r✐✉♠ ✐s✿
x(t) = γ/δ, y(t) = α/β . ❲❤❛t ❛❜♦✉t t❤❡ r❡st ♦❢ t❤❡ s♦❧✉t✐♦♥s❄ ■♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❝✐r❝❧❡ t❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠✱ ✇❡ ♥❡❡❞ ❛ ♠♦r❡ ♣r❡❝✐s❡ ❛♥❛❧②s✐s✳ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r s❡❝t♦rs✱ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ❤❛s ❜❡❡♥ ♣r♦✈❡♥✳ ❍♦✇❡✈❡r✱ ✐t ✐s st✐❧❧ ♣♦ss✐❜❧❡ t❤❛t s♦♠❡ s♦❧✉t✐♦♥s ✇✐❧❧ st❛② ✇✐t❤✐♥ t❤❡ s❡❝t♦r ✇❤❡♥ ♦♥❡ ♦r ❜♦t❤ ♦❢ x ❛♥❞ y ❜❡❤❛✈❡ ❛s②♠♣t♦t✐❝❛❧❧②✿
x(t) → a ❛♥❞✴♦r y(t) → b ❛s t → ∞ . ❙✐♥❝❡ ❜♦t❤ ❢✉♥❝t✐♦♥s ❛r❡ ♠♦♥♦t♦♥✐❝✱ t❤✐s ✐♠♣❧✐❡s t❤❛t
x′ (t) → 0 ❛♥❞✴♦r y ′ (t) → 0 ❛s t → ∞ , ❛♥❞ t❤❡ s❛♠❡ ❢♦r ∆x ❛♥❞ ∆y ✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t t❤✐s ✐s ✐♠♣♦ss✐❜❧❡✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ st❛rt ✐♥ t❤❡ ❜♦tt♦♠ r✐❣❤t s❡❝t♦r✱ ✐✳❡✳✱ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡✿
• x(0) = p > γ/δ ❀
• y(0) = q < α/β ✳
✹✳✹✳
❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s
✷✼✵
❚❤❡♥✱ ❢♦r ❛s ❧♦♥❣ ❛s t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥ t❤✐s s❡❝t♦r✱ ✇❡ ❤❛✈❡
• x′ > 0 =⇒ x > p • y ′ > 0 =⇒ y > q
❚❤❡r❡❢♦r❡✱
❚❤✐s ♥✉♠❜❡r ✐s ❛ ❣❛♣ ❜❡t✇❡❡♥
y′
y ′ = y(δx − γ) > q(δp − γ) > 0 . ❛♥❞
0✳
❚❤❡r❡❢♦r❡✱
❢♦❧❧♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✇✐❧❧ r❡❛❝❤ t❤❡ ❧✐♥❡
y = α/β
y′
❝❛♥♥♦t ❞✐♠✐♥✐s❤ t♦
0✱
❛♥❞ t❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r
∆y ✳
■t
✏✐♥ ✜♥✐t❡ t✐♠❡✑✳
❊①❡r❝✐s❡ ✹✳✸✳✶
Pr♦✈❡ t❤❡ ❛♥❛❧♦❣♦✉s ❢❛❝ts ❛❜♦✉t t❤❡ t❤r❡❡ r❡♠❛✐♥✐♥❣ s❡❝t♦rs✳
❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ❛ s♦❧✉t✐♦♥ ✇✐❧❧ ❣♦ ❛r♦✉♥❞ t❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠✱ ❜✉t ✇❤❡♥ ✐t ❝♦♠❡s ❜❛❝❦✱ ✇✐❧❧ ✐t ❜❡ ❝❧♦s❡r t♦ t❤❡ ❝❡♥t❡r✱ ❢❛rt❤❡r✱ ♦r ✇✐❧❧ ✐t ❝♦♠❡ t♦ t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥❄
❚❤❡ ✜rst ♦♣t✐♦♥ ✐s ✐♥❞✐❝❛t❡❞ ❜② ♦✉r s♣r❡❛❞s❤❡❡t r❡s✉❧t✳ ◆❡①t✱ ✇❡ s❡t t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❛s✐❞❡ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ s♦❧✈✐♥❣ ❛♥❛❧②t✐❝❛❧❧② ♦✉r s②st❡♠ ♦❢ ❖❉❊s t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✱ ✐s t❤✐s
❛ ❝②❝❧❡ ♦r ❛ s♣✐r❛❧ ❄
✹✳✹✳ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s
❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦
❡❧✐♠✐♥❛t❡ t✐♠❡
❢r♦♠ t❤❡ ❡q✉❛t✐♦♥s ✭x
❚❤✐s st❡♣ ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣
> 0, y > 0✮✿
dx = αx − βxy, dt dy = δxy − γy. dt tr✐❝❦✳
❲❡ ✐♥t❡r♣r❡t t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧
❢♦r♠s✿
dx αx − βxy dy =⇒ dt = δxy − γy
dx = (αx − βxy)dt =⇒ dt = dy = (δxy − γy)dt ❚❤❡r❡❢♦r❡✱
dt = ❲❡ ♥❡①t
s❡♣❛r❛t❡ ✈❛r✐❛❜❧❡s ✿
dx dy = . αx − βxy δxy − γy
α − βy δx − γ dx = dy . x y
✹✳✹✳
❲❡
❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✐♥t❡❣r❛t❡ ✿
Z
❛♥❞ ✇❡ ❤❛✈❡✿
❚❤❡ s②st❡♠ ✐s
✷✼✶
γ dx = δ− x
Z
α − β dy , y
δx − γ ln x = α ln y − βy + C .
s♦❧✈❡❞ ✦
❇✉t ✇❤❛t ❞♦❡s t❤✐s ❡q✉❛t✐♦♥ ♠❡❛♥❄ ❊✈❡r② s♦❧✉t✐♦♥
x = x(t)
❛♥❞
y = y(t)✱
✇❤❡♥ s✉❜st✐t✉t❡❞ ✐♥t♦ t❤❡ ❢✉♥❝t✐♦♥
G(x, y) = δx − γ ln x − α ln y + βy , ♣r♦❞✉❝❡s ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛
❧❡✈❡❧ ❝✉r✈❡
♦❢
z = G(x, y)✳
❊①❡r❝✐s❡ ✹✳✹✳✶ Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳
❖♥❝❡ ✇❡ ❤❛✈❡ ♥♦ ❞❡r✐✈❛t✐✈❡s ❧❡❢t✱ ✇❡ ❞❡❝❧❛r❡ t❤❡ s②st❡♠ s♦❧✈❡❞ ❡✈❡♥ t❤♦✉❣❤ ♦♥❧② ✐♠♣❧✐❝✐t❧②✳ ✇❡ ❞♦♥✬t ❤❛✈❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r
x = x(t)
♦r
y = y(t)✱
❊✈❡♥ t❤♦✉❣❤
✇❡ ❝❛♥ ✉s❡ ✇❤❛t ✇❡ ❤❛✈❡ t♦ ❢✉rt❤❡r st✉❞② t❤❡
q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s②st❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s ❛ ❧❡✈❡❧ ❝✉r✈❡ ❛❧r❡❛❞② s✉❣❣❡sts t❤❛t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s
♥♦t ❛ s♣✐r❛❧ ✿
❏✉st tr② t♦ ✐♠❛❣✐♥❡ s✉❝❤ ❛ s✉r❢❛❝❡ t❤❛t ✐ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ s♣✐r❛❧s✳ ❲❡ t✉r♥ ✐♥st❡❛❞ t♦ t❤❡ ❛❝t✉❛❧ ❢✉♥❝t✐♦♥✳
α = 3✱ β = 2✱ γ = 3✱ δ = 1✮✿
❋✐rst✱ ✇❡ ♣❧♦t ✐t ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ✭
❚❤❡ ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ✈✐s✐❜❧❡✳
❙♦♠❡ ♦❢ t❤❡♠ ❛r❡ ❝❧❡❛r❧② ❝✐r❝✉❧❛r ❛♥❞ ♦t❤❡rs ❛r❡♥✬t✳
❛r❡♥✬t s❤♦✇♥ ❛❧❧ t❤❡ ✇❛② t♦ t❤❡ ❛①❡s ❜❡❝❛✉s❡ t❤❡ ✈❛❧✉❡ ♦❢
G
❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡②
r✐s❡s s♦ q✉✐❝❦❧② ✭✐♥ ❢❛❝t✱ ❛s②♠♣t♦t✐❝❛❧❧②✮✳
❆s ❡①♣❡❝t❡❞✱ t❤❡ s✉r❢❛❝❡ s❡❡♠s t♦ ❤❛✈❡ ❛ s✐♥❣❧❡ ♠✐♥✐♠✉♠ ♣♦✐♥t✳ ▲❡t✬s ♣r♦✈❡ t❤❛t ❛❧❣❡❜r❛✐❝❛❧❧②✿
∂G (x, y) = δ − γ/x, ∂x ∂G (x, y) = −α/y + β. ∂y
✹✳✹✳ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s
✷✼✷
❲❡ ♥❡①t ✜♥❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ G✳ ❲❡ s❡t t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s ❡q✉❛❧ t♦ ③❡r♦ ❛♥❞ s♦❧✈❡ ❢♦r x ❛♥❞ y ✿ γ ∂G (x, y) = δ − γ/x = 0 =⇒ x = ∂x δ ∂G α (x, y) = −α/y + β = 0 =⇒ y = ∂y β ❚❤✐s ♣♦✐♥t ✐s ✐♥❞❡❡❞ ♦✉r ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t✳ ❚❤❡ s✉r❢❛❝❡ ❤❡r❡ ❤❛s ❛ ❤♦r✐③♦♥t❛❧ t❛♥❣❡♥t ♣❧❛♥❡✳ ❲❡ ❤❛✈❡
❛❧s♦ ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡rs ♣♦✐♥ts ❧✐❦❡ t❤❛t✦
❇✉t ❝♦✉❧❞ t❤✐s ❜❡ ❛ ♠❛①✐♠✉♠ ♣♦✐♥t❄ ❏✉st ❛s y = x3 ❝r♦ss❡s t❤❡ x✲❛①✐s ❛t 0 ❞❡❣r❡❡s✱ ❛ s✉r❢❛❝❡ ❝❛♥ ❝r♦ss ✐ts t❛♥❣❡♥t ♣❧❛♥❡✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿ ∂ 2G 1 (x, y) = γ 2 > 0, 2 ∂x x 1 ∂ 2G (x, y) = α > 0. ∂y 2 y2
❇♦t❤ ❛r❡ ♣♦s✐t✐✈❡ t❤r♦✉❣❤♦✉t✱ t❤❡r❡❢♦r❡✱ ❡✐t❤❡r ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ s✉r❢❛❝❡ ❛❧♦♥❣ t❤❡ ❛①❡s ❤❛✈❡ ❛ ♠✐♥✐♠✉♠ ♣♦✐♥t ❤❡r❡ ❛♥❞ ✐t ❤❛s t♦ st❛② ♦♥ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ♣❧❛♥❡✳ ❍♦✇❡✈❡r✱ ✐t ♠✐❣❤t ❝r♦ss ✐t ❛t ♦t❤❡r ❞✐r❡❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s st✐❧❧ ♠✐❣❤t ❜❡ ❛ s❛❞❞❧❡ ♣♦✐♥t✦ ❲❡ ✐♥✈♦❦❡ t❤❡ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ❚❡st ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✺ t♦ r❡s♦❧✈❡ t❤✐s✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❍❡ss✐❛♥ ♠❛tr✐① ✭❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✸✮ ♦❢ G✳ ■t ✐s t❤❡ 2 × 2 ♠❛tr✐① ♦❢ t❤❡ ❢♦✉r ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ G✿ ∂ 2G ∂x2 H(x, y) = ∂ 2G ∂y∂x
∂ 2G 1 0 γ x 2 ∂x∂y = . 1 ∂ 2G 0 α 2 y ∂y 2
❍❡r❡✱ ✐♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿
∂ 2G ∂ 2G (x, y) = (x, y) = 0 . ∂x∂y ∂y∂x
◆❡①t✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ❍❡ss✐❛♥✿ D(x, y) = det(H(x, y)) =
■t✬s ♣♦s✐t✐✈❡✦ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣♦✐♥t ✐s ❛ ♠✐♥✐♠✉♠✳
1 γ 2 x
1 αγ · α 2 = 2 2 > 0. y xy
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❡✈❡r② ❧❡✈❡❧ ❝✉r✈❡ ♦❢ G✱ ✐✳❡✳✱ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠✱ ❣♦❡s ❛r♦✉♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ ❝♦♠❡s ❜❛❝❦ t♦ ❝r❡❛t❡ ❛ ❝②❝❧❡✳ ❆♥ ❡❛s✐❡r✱ ❜✉t ♠♦r❡ ❛❞ ❤♦❝✱ ✇❛② t♦ r❡❛❝❤ t❤✐s ❝♦♥❝❧✉s✐♦♥ ✐s t♦ ✐♠❛❣✐♥❡ t❤❛t ❛ s♦❧✉t✐♦♥ st❛rts ♦♥✱ s❛②✱ t❤❡ ❧✐♥❡ y = α/β ❛t x = x0 t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ t❤❡♥ ❝♦♠❡s ❜❛❝❦ t♦ t❤❡ ❧✐♥❡ ❛t x = x1 ✳ ❙✐♥❝❡ t❤✐s ✐s ❛ ❧❡✈❡❧ ❝✉r✈❡✱ ✇❡ ❤❛✈❡ G(x0 , α/β) = G(x1 , α/β)✳ ❆❝❝♦r❞✐♥❣ t♦ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✺✱ t❤✐s > 0 ❛❧♦♥❣ t❤✐s ❧✐♥❡✳ ❝♦♥tr❛❞✐❝ts ♦✉r ❝♦♥❝❧✉s✐♦♥ t❤❛t ∂G ∂x P❧♦tt❡❞ ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs✱ t❤✐s ✐s ✇❤❛t t❤❡s❡ ❝✉r✈❡s ❧♦♦❦ ❧✐❦❡✿
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✸
❊①❡r❝✐s❡ ✹✳✹✳✷
Pr♦✈❡ t❤❛t t❤❡ ♣r❡❞❛t♦r✲♣r❡② ❞✐s❝r❡t❡ ♠♦❞❡❧✱ ✐✳❡✳✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ❢♦r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✱ ♣r♦❞✉❝❡s s♦❧✉t✐♦♥s t❤❛t s♣✐r❛❧ ❛✇❛② ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ ❍✐♥t✿
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ◆✉♠❡r♦✉s ♣r♦❝❡ss❡s ❛r❡ ♠♦❞❡❧❡❞ ❜② s②st❡♠s ♦❢ ❖❉❊s✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ ❧✐tt❧❡ ✢❛❣s ❛r❡ ♣❧❛❝❡❞ ♦♥ t❤❡ ❧❛✇♥✱ t❤❡♥ t❤❡✐r ❞✐r❡❝t✐♦♥s t❛❦❡♥ t♦❣❡t❤❡r r❡♣r❡s❡♥t ❛ s②st❡♠ ♦❢ ❖❉❊s✱ ✇❤✐❧❡ t❤❡ ✭✐♥✈✐s✐❜❧❡✮ ❛✐r ✢♦✇ ✐s t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤✐s s②st❡♠✳
❆ s✐♠✐❧❛r ✐❞❡❛ ✐s ✉s❡❞ t♦ ♠♦❞❡❧ ❛ ✢✉✐❞ ✢♦✇✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❡❛❝❤ ♣❛rt✐❝❧❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇✱ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ t❤❡ s❛♠❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳
❚♦ s♦❧✈❡ s✉❝❤ ❛ s②st❡♠ ✇♦✉❧❞ r❡q✉✐r❡ tr❛❝✐♥❣ t❤❡ ♣❛t❤ ♦❢ ❡✈❡r② ♣❛rt✐❝❧❡ ♦❢ t❤❡ ❧✐q✉✐❞✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ❛ ✢♦✇✿ ❣✐✈❡♥ ❛ ✢♦✇ ♦♥ ❛ ♣❧❛♥❡✱ tr❛❝❡ ❛ s✐♥❣❧❡ ♣❛rt✐❝❧❡ ♦❢ t❤✐s str❡❛♠✳
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✹
❋♦r ❜♦t❤ ❝♦♦r❞✐♥❛t❡s✱ x ❛♥❞ y ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ ✐s ❜❡✐♥❣ ❜✉✐❧t✳ ❚❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿ tn , vn , pn , n = 1, 2, 3, ... ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❛r❡ ✉s❡❞✿
tn+1 = tn + ∆t ❛♥❞ ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿
pn+1 = pn + vn+1 · ∆t .
✐t❡r❛t✐♦♥ n t✐♠❡ tn ✐♥✐t✐❛❧✿ 0 3.5 1 3.6 ... ... 1000 103.5 ... ...
✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn −− 22 33 25.3 ... ... 4 336 ... ...
❙♦✱ ✐♥st❡❛❞ ♦❢ t✇♦ ✭✈❡❧♦❝✐t② ✕ ❧♦❝❛t✐♦♥✱ ❛s ❜❡❢♦r❡✮✱ t❤❡r❡ ✇✐❧❧ ❜❡ ❢♦✉r ♠❛✐♥ ❝♦❧✉♠♥s ✇❤❡♥ t❤❡ ♠♦t✐♦♥ ✐s t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ s✐① ✇❤❡♥ ✐t ✐s t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧✿ t✐♠❡ ❤♦r✐③✳ ❤♦r✐③✳ ✈❡rt✳ ✈❡rt✳ t ✈❡❧✳ x′ ❧♦❝✳ x ✈❡❧✳ y ′ ❧♦❝✳ y n 0 3.5 −− 22 −− 3 3.6 33 25.3 4 3.5 1 ... ... ... ... ... ... 1000 103.5 4 336 66 4 ... ... ... ... ... ...
... ... ... ... ... ... ...
❊①❛♠♣❧❡ ✹✳✺✳✶✿ s✐♠✉❧❛t✐♦♥
❘❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡s s✉❝❤ ✢♦✇s✳ ■❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥✿
vn+1 = 0.2 · pn , ❢♦r ❜♦t❤ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✱ t❤❡ r❡s✉❧t ✐s ♣❛rt✐❝❧❡s ✢②✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✱ ❢❛st❡r ❛♥❞ ❢❛st❡r✿
■❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ t❤❡ r❡s✉❧t r❡s❡♠❜❧❡s r♦t❛t✐♦♥✿
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✺
◆♦✇✱ s✉♣♣♦s❡ t❤❡ ✈❡❧♦❝✐t② ❝♦♠❡s ❢r♦♠ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❧♦❝❛t✐♦♥✿ u = f (x, y), v = g(x, y) ,
❛r❡ t❤❡r❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿ x = x(t), y = y(t)?
❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ♦❢ ♦✉r r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥✱ pn+1 = pn + vn+1 · ∆t ,
❢♦r ❡✈❡r② ∆t > 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❤❡♥ ♦✉r t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢② ❢♦r x✿ vn = f (pn ) ❛♥❞ pn = x(tn ) ,
❛♥❞ ❢♦r y ✿
vn = g(pn ) ❛♥❞ pn = y(tn ) ,
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦✱ ❛s ✇❡❧❧ ❛s t = tn ✱ ✐♥t♦ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r pn+1 ❢♦r x✿ ❛♥❞ ❢♦r y ✿ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❢♦r x✿ ❛♥❞ ❢♦r y ✿
x(t + ∆t) = x(t) + f (x(t + ∆t), y(t + ∆t)) · ∆t , y(t + ∆t) = y(t) + g(x(t + ∆t), y(t + ∆t)) · ∆t . x(t + ∆t) − x(t) = f (x(t + ∆t), y(t + ∆t)) , ∆t y(t + ∆t) − y(t) = g(x(t + ∆t), y(t + ∆t)) . ∆t
❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r ∆t → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
x′ (t) = f (x(t), y(t)), y ′ (t) = g(x(t), y(t)),
♣r♦✈✐❞❡❞ x = x(t), y = y(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ u = f (x, y), v = g(x, y) ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t (x(t), y(t))✳ ◆♦✇✱ ❧❡t✬s r❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✹❍❉✲✻ t❤❛t ❛ ✈❡❝t♦r ✜❡❧❞ s✉♣♣❧✐❡s ❛ ❞✐r❡❝t✐♦♥ t♦ ❡✈❡r② ❧♦❝❛t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡ ✐s ❛ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡✿ point 7→ vector .
❆ ✈❡❝t♦r ✜❡❧❞ ✐♥ ❞✐♠❡♥s✐♦♥ 2 ✐s ✐♥ ❢❛❝t ❛♥② ❢✉♥❝t✐♦♥✿ ❣✐✈❡♥ ❜② t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿
F : R2 → R2 ,
F (x, y) =< f (x, y), g(x, y) > .
❚❤❡♥✱ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡s ❛ s②st❡♠ ♦❢ t✇♦ ✭t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✮ ❖❉❊s✳
✹✳✺✳
❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✻
❉❡✜♥✐t✐♦♥ ✹✳✺✳✷✿ s♦❧✉t✐♦♥ ♦❢ s②st❡♠ ♦❢ ❖❉❊s ❆
s♦❧✉t✐♦♥
♦❢ ❛ s②st❡♠ ♦❢ ❖❉❊s ✐s ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s
x = x(t)
❛♥❞
♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ✇✐t❤ ❡✐t❤❡r ♦♥❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❢♦r ❡✈❡r②
t
✐♥
I
y = y(t) ✭❛ I s✉❝❤ t❤❛t
✇❡ ❤❛✈❡✿
x′ (t) = f (x(t), y(t)) y ′ (t) = g(x(t), y(t))
♦r ❛❜❜r❡✈✐❛t❡❞✿
x′ = f (x, y) y ′ = g(x, y)
❚❤❡ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤✐s s②st❡♠ ✐s t❤❡ s❧♦♣❡ ✜❡❧❞ ♦❢ t❤❡ ❖❉❊✳ ❍♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ s✉❝❤ ❛ s②st❡♠❄ ❲✐t❤ ❛ s✐♥❣❧❡ ❖❉❊✱
y ′ = f (t) =⇒ y = y(t) ,
❣r❛♣❤s✱
ty ✲♣❧❛♥❡✳ ❚❤✐s t✐♠❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✦ ❚❤❡✐r ❣r❛♣❤s✱ ✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ (t, x(t), y(t)) ❧✐❡ ✐♥ t❤❡ 3✲ ❞✐♠❡♥s✐♦♥❛❧ txy ✲s♣❛❝❡✳ ❚❤❛t ✐s ✇❤②✱ ✇❡✱ ✐♥st❡❛❞✱ ♣❧♦t t❤❡✐r ✐♠❛❣❡s✱ ✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ ♣♦✐♥ts (x(t), y(t)) ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ■♥ t❤❡ t❤❡♦r② ♦❢ ❖❉❊s✱ t❤❡② ❛r❡ ❛❧s♦ ❦♥♦✇♥ ❛s tr❛❥❡❝t♦r✐❡s✱ ♦r ♣❛t❤s✳
✇❡ s✐♠♣❧② ♣❧♦t t❤❡✐r
✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ ♣♦✐♥ts
(t, y(t))✱
♦❢ s♦♠❡ ♦❢ t❤❡♠ ♦♥ t❤❡
❚❤❡♥ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡s❡ tr❛❥❡❝t♦r✐❡s✳ ❙✐♥❝❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✱ s✉❝❤ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♦❢t❡♥ s✉✣❝✐❡♥t ❛s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳
❚❤❡♦r❡♠ ✹✳✺✳✸✿ ❙❤✐❢t❡❞ ❙♦❧✉t✐♦♥s x = x(t), y = y(t) ✐s ❛ s♦❧✉t✐♦♥ x(t + s), y = y(t + s) ❢♦r ❛♥② r❡❛❧ s✳
■❢
♦❢ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✱ t❤❡♥ s♦ ✐s
x =
■t ✐s ❛❧s♦ ✐♠♣♦rt❛♥t t♦ ❜❡ ❛✇❛r❡ ♦❢ t❤❡ ❢❛❝t t❤❛t t❤❡ t❤❡♦r② ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✏✐♥❝❧✉❞❡s✑ t❤❡ t❤❡♦r② ♦❢ s✐♥❣❧❡ ❖❉❊s✳ ❘❡❝❛❧❧✱ ✜rst✱ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❡✈❡r② ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ tr❛❥❡❝t♦r② ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿
y = r(x) −→
x = t, y = r(x).
❙✐♠✐❧❛r❧②✱ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❡✈❡r② t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❖❉❊ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ ❛ s②st❡♠ ♦❢ t✇♦ ❖❉❊s✿
′
y = g(x) −→
x′ = 1, y ′ = g(y).
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✼
❉❡✜♥✐t✐♦♥ ✹✳✺✳✹✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ s②st❡♠ ♦❢ ❖❉❊s ❛♥❞ ❛ ❣✐✈❡♥ tr✐♣❧❡ (t0 , x0 , y0 )✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜✲ ❧❡♠✱ ♦r ❛♥ ■❱P✱ ✐s x′ = f (x, y), y ′ = g(x, y);
x(t0 ) = x0 , y(t0 ) = y0 ;
❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡✳ ❇② t❤❡ ❧❛st t❤❡♦r❡♠✱ t❤❡ ✈❛❧✉❡ ♦❢ t0 ❞♦❡s♥✬t ♠❛tt❡r❀ ✐t ❝❛♥ ❛❧✇❛②s ❜❡ ❝❤♦s❡♥ t♦ ❜❡ 0✳
❉❡✜♥✐t✐♦♥ ✹✳✺✳✺✿ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❲❡ s❛② t❤❛t ❛ s②st❡♠ ♦❢ ❖❉❊s s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , x0 , y0 ) ✇❤❡♥ t❤❡ ■❱P ❛❜♦✈❡ ❤❛s ❛ s♦❧✉t✐♦♥✳ ■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② ❡①✐st❡♥❝❡✱ ✐t r❡✢❡❝ts ❧✐♠✐t❛t✐♦♥s ♦❢ ②♦✉r ♠♦❞❡❧✳ ■t ✐s ❛s ✐❢ t❤❡ ♣r♦❝❡ss st❛rts ❜✉t ♥❡✈❡r ❝♦♥t✐♥✉❡s✳
❉❡✜♥✐t✐♦♥ ✹✳✺✳✻✿ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , x0 , y0 ) ✐❢ ❡✈❡r② ♣❛✐r ♦❢ s♦❧✉t✐♦♥s✱ (x1 , y1 ), (x2 , y2 )✱ ♦❢ t❤❡ ■❱P ❛❜♦✈❡ ❛r❡ ❡q✉❛❧✱ x1 (t) = x2 (t), y1 (t) = y2 (t) ,
❢♦r ❡✈❡r② t ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s t0 ✳
■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② ✉♥✐q✉❡♥❡ss✱ ✐t r❡✢❡❝ts ❧✐♠✐t❛t✐♦♥s ♦❢ ②♦✉r ♠♦❞❡❧✳ ■t✬s ❛s ✐❢ ②♦✉ ❤❛✈❡ ❛❧❧ t❤❡ ❞❛t❛ ❜✉t ❝❛♥✬t ♣r❡❞✐❝t ❡✈❡♥ t❤❡ ♥❡❛r❡st ❢✉t✉r❡✳ ❚❤✉s s②st❡♠s ♦❢ ❖❉❊s ♣r♦❞✉❝❡ ❢❛♠✐❧✐❡s ♦❢ ❝✉r✈❡s ❛s t❤❡ s❡ts ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s ✐s ❣✐✈❡♥ ❜② ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ s✐♥❣❧❡ ♣❛r❛♠❡t❡r✱ ✇❡ ♠❛② ❜❡ ❛❜❧❡ t♦ ✜♥❞ ❛ s②st❡♠ ♦❢ ❖❉❊s ❢♦r ✐t✳
❊①❛♠♣❧❡ ✹✳✺✳✼✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ❣r❛♣❤s✱
y = x2 + C ,
❝r❡❛t❡s ❛♥ ❖❉❊ ✐❢ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ t✿ y ′ = 2xx′ .
❙✐♥❝❡ t❤❡s❡ ❛r❡ ❥✉st ❢✉♥❝t✐♦♥s ♦❢ x✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ x = t✳ ❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿
x′ = 1, y ′ = 2x.
✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s
✷✼✽
❊①❛♠♣❧❡ ✹✳✺✳✽✿ ❡①♣♦♥❡♥t✐❛❧ ❝❛s❡
❚❤❡ ❢❛♠✐❧② ♦❢ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ❣r❛♣❤s✱ y = Cex ,
❝r❡❛t❡s ❛♥ ❖❉❊s✿
y ′ = Cex x′ .
❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿
x′ = 1, y ′ = Cex .
❊①❛♠♣❧❡ ✹✳✺✳✾✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s
❲❤❛t ❛❜♦✉t t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s❄
✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ ♦r✐❣✐♥✳✮ ❚❤❡② ❛r❡ ❣✐✈❡♥ ❜② x2 + y 2 = C ≥ 0 .
❲❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ t✿ 2xx′ + 2yy ′ = 0 .
❲❡ ❝❤♦♦s❡ ✇❤❛t x′ , y ′ ♠✐❣❤t ❜❡ ❡q✉❛❧ t♦ ✐♥ ♦r❞❡r ❢♦r t❤❡ t✇♦ t❡r♠s t♦ ❝❛♥❝❡❧✳ ❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿ x′ = −y, y ′ = x.
❊①❛♠♣❧❡ ✹✳✺✳✶✵✿ ❢❛♠✐❧② ♦❢ ❤②♣❡r❜♦❧❛s
❚❤❡s❡ ❤②♣❡r❜♦❧❛s ❛r❡ ❣✐✈❡♥ ❜② t❤❡s❡ ❡q✉❛t✐♦♥s✿ xy = C .
✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ❛①❡s✳✮
❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❖❉❊✿
x′ y + xy ′ = 0 .
✹✳✻✳
❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s
❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿
✷✼✾
x′ = x, y ′ = −y.
◆♦♥❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ❤❛✈❡ ♣r♦❜❧❡♠s ✇✐t❤ ❡✐t❤❡r ❡①✐st❡♥❝❡ ♦r ✉♥✐q✉❡♥❡ss ✕ ✐♥ ❝♦♥tr❛st t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❖❉❊s✳ ❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠s ❧✐❡ ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡♦r❡♠ ✹✳✺✳✶✶✿ ❊①✐st❡♥❝❡
❙✉♣♣♦s❡ (x0 , y0 ) ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ s✉♣♣♦s❡✿ • H ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s x0 ✳ • G ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s y0 ✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢✉♥❝t✐♦♥s z = f (x, y) ❛♥❞ z = g(x, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ♦♥ H × G✳ ❚❤❡♥ t❤❡ s②st❡♠ ♦❢ ❖❉❊✱
x′ = f (x, y), y ′ = g(x, y).
s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t (t0 , x0 , y0 ) ❢♦r ❛♥② t0 ✳
❚❤❡♦r❡♠ ✹✳✺✳✶✷✿ ❯♥✐q✉❡♥❡ss
❙✉♣♣♦s❡ (x0 , y0 ) ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ s✉♣♣♦s❡✿ • H ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s x0 ✳ • G ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s y0 ✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢✉♥❝t✐♦♥ z = f (x, y) ❛♥❞ z = g(x, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ♦♥ H × G✳ ❚❤❡♥ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✱
x′ = f (x, y), y ′ = g(x, y).
s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t (t0 , x0 , y0 ) ❢♦r ❛♥② t0 ✳
✹✳✻✳ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s
❉✐s❝r❡t❡ ❖❉❊s ❛♣♣r♦①✐♠❛t❡ ❛♥❞ ❛r❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❝♦♥t✐♥✉♦✉s ❖❉❊s✳ ❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r s②st❡♠s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❞✐s❝r❡t❡ s②st❡♠ ❢♦r t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ♣r♦❞✉❝❡s t❤✐s ❛❧♠♦st ❡①❛❝t❧② ❝②❝❧✐❝ ♣❛t❤✿
✹✳✻✳
❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s
✷✽✵
■♥ ♦t❤❡r ✇♦r❞s✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s ❝❛♣❛❜❧❡ ♦❢ tr❛❝✐♥❣ s♦❧✉t✐♦♥s ✈❡r② ❝❧♦s❡ t❤❡ ♦♥❡s ♦❢ t❤❡ ❖❉❊ ✐t ❝❛♠❡ ❢r♦♠✳ ❏✉st ❛s ✐♥ t❤❡
1✲❞✐♠❡♥s✐♦♥❛❧
❝❛s❡✱ t❤❡ ■❱P t❡❧❧s ✉s✿
•
✇❤❡r❡ ✇❡ ❛r❡ ✭t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞
•
t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ❛r❡ ❣♦✐♥❣ ✭t❤❡ ❖❉❊✮✳
❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ✉♥❦♥♦✇♥ s♦❧✉t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✐ts
❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥
✳
❊①❛♠♣❧❡ ✹✳✻✳✶✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✿
❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊s✿
❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢
t✿
x′ = y , y ′ = −x . ∆t = 1 .
❲❡ st❛rt ✇✐t❤ t❤✐s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
t0 = 0,
x0 = 0,
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s✿
x′ = 2, y ′ = 0;
y0 = 2 .
✹✳✻✳
❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s
✷✽✶
❚❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡
∆x = 2 · ∆t = 2 · 1 = 2 , ∆y = 0 · ∆t = 0 · 1 = 0 .
❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿
x1 = x0 + ∆x = 0 + 2 = 2 , y1 = y0 + ∆y = 2 + 0 = 2 .
❆ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣❡❛rs✿ x0 = 2,
y0 = 2 .
❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s✿
x′ = 2 , y ′ = −2 .
♣r♦❞✉❝✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡
∆x = 2 · ∆t = 2 · 1 = 2 , ∆y = −2 · ∆t = −2 · 1 = −2 .
❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿
x2 = x1 + ∆x = 2 + 2 = 4 , y2 = y1 + ∆y = 2 + (−2) = 0 .
❖♥❡ ♠♦r❡ ■❱P✿ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡
x2 = 0, y2 = −4 .
∆x = 0 · ∆t = 0 · 1 = 0 , ∆y = −4 · ∆t = −4 · 1 = −4 .
x3 = x2 + ∆x = 4 + 0 = 4 , y3 = y2 + ∆y = 0 − 4 = −4 .
❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿
❚❤❡s❡ ❢♦✉r ♣♦✐♥ts ❢♦r♠ ❛ ✈❡r② ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♦♥❡ ♦❢ ♦✉r ❝✐r❝✉❧❛r s♦❧✉t✐♦♥s✿
❚❤❡② ❛r❡ ❝❧❡❛r❧② s♣✐r❛❧✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ t❤✐s ✐s ♦✉r ♣❧❛♥✿ ◮ ❆t ♦✉r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t t✐♠❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❖❉❊ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥
♠♦✈❡ ❛❝❝♦r❞✐♥❣❧② t♦ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✳
✹✳✻✳
❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s
✷✽✷
❉❡✜♥✐t✐♦♥ ✹✳✻✳✷✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡
❊✉❧❡r s♦❧✉t✐♦♥
✇✐t❤ ✐♥❝r❡♠❡♥t
x′ = f (x, y) y ′ = g(x, y)
✐s t❤❡ t✇♦ s❡q✉❡♥❝❡s
{xn }
✇❤❡r❡
∆t > 0
❛♥❞
{yn }
♦❢ t❤❡ ■❱P
x(t0 ) = x0 y(t0 ) = y0
♦❢ r❡❛❧ ♥✉♠❜❡rs ❣✐✈❡♥ ❜②✿
xn+1 = xn + f (xn , yn ) · ∆t yn+1 = yn + g(xn , yn ) · ∆t
tn+1 = tn + ∆t✳
❖♥❝❡ ❛❣❛✐♥✱ ✐❢ ✇❡ ❞❡r✐✈❡❞ ♦✉r ❖❉❊s ❢r♦♠ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✭✈✐❛ ❜❛❝❦ t♦ ✐t✿
∆t → 0✮✱
❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐❧❧ ❜r✐♥❣ ✉s r✐❣❤t
❖❉❊s
ր ❞✐s❝r❡t❡ ♠♦❞❡❧
same!
←−−−−−−−
ց ❊✉❧❡r✬s ♠❡t❤♦❞
❊①❛♠♣❧❡ ✹✳✻✳✸✿ s♣r❡❛❞s❤❡❡t ▲❡t✬s ♥♦✇ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ ❢♦r♠✉❧❛s ❢♦r
❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✶❪✯❘✸❈✶
❛♥❞
xn
❛♥❞
yn
❛r❡ r❡s♣❡❝t✐✈❡❧②✿
❂❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✯❘✸❈✶
❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿
■♥ ❝♦♥tr❛st t♦ t❤❡ ❝❛s❡ ♦❢ ❛ s✐♥❣❧❡ ❖❉❊✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❞♦ ♥♦t ❜❡❤❛✈❡ ❡rr❛t✐❝❛❧❧② ❝❧♦s❡ t♦ t❤❡
x✲❛①✐s✳
❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡r❡ ✐s ♥♦ ❞✐✈✐s✐♦♥ ❜②
y
❛♥②♠♦r❡✳
❊①❛♠♣❧❡ ✹✳✻✳✹✿ ❢❛♠✐❧② ♦❢ ❤②♣❡r❜♦❧❛s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❤②♣❡r❜♦❧❛s✿
xy = C . ❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠✿
❆♥ ❊✉❧❡r s♦❧✉t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿
x′ = x , y ′ = −y .
✹✳✼✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s
✷✽✸
❍♦✇❡✈❡r✱ ✐s t❤✐s ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡ x✲❛①✐s ♦r ❞♦ t❤❡② ♠❡r❣❡❄
✹✳✼✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s
❙✐♥❝❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ h✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✱ t❤❡♥✱ ❡✈❡♥ ✇✐t❤ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ✈❛❧✉❡s ♦❢ h✱ t❤❡ r❡s✉❧t r❡♠❛✐♥s ❛ ♠❡r❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ▼❡❛♥✇❤✐❧❡✱ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ❝♦❧❧❡❝ts ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤♦✉t s♦❧✈✐♥❣ t❤❡ s②st❡♠ ✕ ❡✐t❤❡r ❛♥❛❧②t✐❝❛❧❧② ♦r ♥✉♠❡r✐❝❛❧❧②✳ ❚❤❡ r❡s✉❧t ✐s ❢✉❧❧② ❛❝❝✉r❛t❡ ❜✉t ✈❡r② ❜r♦❛❞ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s✳ ❊①❛♠♣❧❡ ✹✳✼✳✶✿ ✶❞ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s
▲❡t✬s r❡✈✐❡✇ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ s✐♥❣❧❡ ❖❉❊ ❢r♦♠ ❧❛st s❡❝t✐♦♥✿
y ′ = − tan(y) . ❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t t❤❡ str✐♣ [−π/2, π/2]✿ • ❋♦r −π/2 < y < 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y > 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ր✳ • ❋♦r y = 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y = 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ • ❋♦r 0 < y < π/2✱ ✇❡ ❤❛✈❡ y ′ = − tan y < 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ց✳ ■♥ ❢❛❝t✱ ❡✈❡r② s♦❧✉t✐♦♥ y ✐s ❞❡❝r❡❛s✐♥❣ ✭♦r ✐♥❝r❡❛s✐♥❣✮ t❤r♦✉❣❤♦✉t ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿
❲❡ ❝❛♥ ♠❛t❝❤ t❤✐s ❖❉❊ ✇✐t❤ ❛ s②st❡♠✿
x′ = 1 , y ′ = − tan y .
■ts s♦❧✉t✐♦♥s ❤❛✈❡ t❤❡s❡ tr❛❥❡❝t♦r✐❡s ❛s s♦❧✉t✐♦♥s✳
✹✳✼✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s
✷✽✹
❚❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛r❡ ✐ts t❛♥❣❡♥t ✈❡❝t♦rs✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✿
x′ = f (x, y) , y ′ = g(x, y) ,
❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ s✐❣♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿
x′ = f (x, y) < 0 x′ = f (x, y) = 0 x′ = f (x, y) > 0 ← • → ′ y = g(x, y) < 0 ↓ ւ ↓ ց ′ y = g(x, y) = 0 • ← • → y ′ = g(x, y) > 0 ↑ տ ↑ ր ❊①❛♠♣❧❡ ✹✳✼✳✷✿ ♠♦r❡ ❝♦♠♣❧❡①
❈♦♥s✐❞❡r ♥❡①t✿
y ′ = cos y ·
p
|y| .
✇❡ ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ✈❛r✐❡s ✇✐t❤ t❤❡ ❝♦s✐♥❡✿
❲❡ ❝❛♥ s❡❡ t❤❛t ❛❧❧ s♦❧✉t✐♦♥s ♣r♦❣r❡ss ❢♦r✇❛r❞ ❛❧♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢
x❄
❈♦♥s✐❞❡r✿
x✲❛①✐s✳
❲❤❛t ✐❢ ✇❡ ❛❞❞ ✈❛r✐❛❜✐❧✐t② ♦❢ t❤❡
x′ = sin y , y ′ = cos y .
❲❡ ❝♦♥❞✉❝t t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ❢♦r ❜♦t❤ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ ♣❛t❤ y x′ x = x(t) 2π 0 − x↓ ←←←← π 0 + x↑ →→→→ 0 0 − x↓ ←←←← −π 0
y y ′ y = y(t) ♣❛t❤ 3π/2 0 − y↓ ↓↓↓↓ π/2 0 + y↑ ↑↑↑↑ −π/2 0
◆♦✇ ♣✉t t❤❡♠ t♦❣❡t❤❡r✿
→ ր ↑ տ ←
→ ր ↑ տ ←
❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿
→ ր ↑ տ ←
→ ր ↑ տ ←
✹✳✼✳
◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s
✷✽✺
❊①❛♠♣❧❡ ✹✳✼✳✸✿ tr✐❣
❚❤❡ ♥❡①t ♦♥❡✿
x′ = sin x , y ′ = cos y .
❲❡ ❝♦♥❞✉❝t t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿
y 3π/2 π/2 −π/2 −3π/2
x −π π 2π 3π ′ y |x 0 + 0 − 0 + 0 0 • → • ← • → • − ↓ ց ↓ ւ ↓ ց ↓ 0 • → • ← • → • + ↑ ր ↑ տ ↑ ր ↑ 0 • → • ← • → • − ↓ ց ↓ ւ ↓ ց ↓ 0 • → • ← • → • ′
❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿
❊①❛♠♣❧❡ ✹✳✼✳✹✿ ❞✐s❝♦♥t✐♥✉♦✉s ❘❍❙
❚❤❡ ♥❡①t ♦♥❡ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✿
❋♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ✇❡ ✉s❡ t❤❡
❋▲❖❖❘
x′ = x − y, y ′ = [x + y] . ❢✉♥❝t✐♦♥✿
✹✳✼✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s
✷✽✻
❊①❡r❝✐s❡ ✹✳✼✳✺
❈♦♥✜r♠ t❤❡ ♣❧♦t ❜❡❧♦✇ ❜② ❛♥❛❧②③✐♥❣ t❤✐s s②st❡♠✿ x′ = y, y ′ = sin y · y .
❙✉♣♣♦s❡ t❤❡ s②st❡♠ ✐s t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✱ x′ = f (x, y), y ′ = g(x, y) .
❚❤❡♥ ✐t ✐s t❤♦✉❣❤t ♦❢ ❛s ❛ ✢♦✇✿ ❧✐q✉✐❞ ✐♥ ❛ ♣♦♥❞ ♦r t❤❡ ❛✐r ♦✈❡r ❛ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳
❚❤❡♥✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✇✐t❤ ♦♥❧② t❤r❡❡ ♠❛✐♥ ♣♦ss✐❜✐❧✐t✐❡s✱ ✇❡ ✇✐❧❧ s❡❡ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ❛r♦✉♥❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡♥ t❤❡ s♣❛❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✐s ❛ ♣❧❛♥❡✳ ❲✐t❤ s✉❝❤ ❛ s②st❡♠✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✐s ♠✉❝❤ s✐♠♣❧❡r✳ ■♥ ❢❛❝t✱ t❤❡ ♦♥❡s ❛❜♦✈❡ ❡①❤✐❜✐t ♠♦st ♦❢ t❤❡ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ♦❢ ❧♦❝❛❧ ❜❡❤❛✈✐♦r✳ ❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❣✐✈❡♥ ❧♦❝❛t✐♦♥ (x, y) = (a, b)✳
✹✳✽✳ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✽✼
❚❤❡ ✜rst ♠❛✐♥ ♣♦ss✐❜✐❧✐t② ✐s
f (a, b) 6= 0
♦r
g(a, b) 6= 0 ,
✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦
F (a, b) =< f (a, b) , g(a, b) >6= 0 . ❚❤❡♥✱ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢
(a, b)✳
f
g ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ♥❡✐t❤❡r ❝❤❛♥❣❡s ✐ts s✐❣♥ ✐♥ s♦♠❡ ❞✐s❦ D t❤❛t ❝♦♥t❛✐♥s ♣❛t❤s ❧♦❝❛t❡❞ ✇✐t❤✐♥ D ♣r♦❝❡❡❞ ✐♥ ❛♥ ❛❜♦✉t t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✿
❛♥❞
❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥s ✇✐t❤
❚❤❡ ❜❡❤❛✈✐♦r ✐s ✏❣❡♥❡r✐❝✑✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❜❡❤❛✈✐♦rs ❛r❡ s❡❡♥ ❛r♦✉♥❞ ❛ ③❡r♦ ♦❢
◮ F (a, b) = 0 =⇒ (x, y) = (a, b)
F✿
✐s ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ✭❛♥ ❡q✉✐❧✐❜r✐✉♠✮✳
❚❤❡♥ t❤❡ ♣❛tt❡r♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✱ ✐✳❡✳✱ ❛♥ ♦♣❡♥ ❞✐s❦ ♦❢
f
♦r
g✱
D✱
❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r t❤✐s ✐s ❛ ♠❛①✐♠✉♠
♦r ❛ ♠✐♥✐♠✉♠✱ ♦r ♥❡✐t❤❡r✳ ❙♦♠❡ ♦❢ t❤❡ ✐❞❡❛s ❝♦♠❡ ❢r♦♠ ❞✐♠❡♥s✐♦♥
1✳
❋♦r ❡①❛♠♣❧❡✱ ❛ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱
y→a
❛s
x → +∞ ,
y→a
❛s
x → −∞ ,
✐s ❛ s✐♥❦✿ ✢♦✇ ✐♥ ♦♥❧②✳ ❆♥ ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱
✐s ❛ s♦✉r❝❡✿ ✢♦✇ ♦✉t ♦♥❧②✳
❆♥ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦✉❧❞ ♠❡❛♥ t❤❛t s♦♠❡ t❤❡ s♦❧✉t✐♦♥s ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ ♦t❤❡rs ❞♦ ♥♦t✳ ❆s ②♦✉ ❝❛♥ s❡❡ t❤❡r❡ ❛r❡ ♠❛♥② ♠♦r❡ ♣♦ss✐❜✐❧✐t✐❡s t❤❛♥ ✐♥ ❞✐♠❡♥s✐♦♥
✹✳✽✳ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
❲❡ ❝❛♥ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s t♦ ❢♦r♠ ❛ ♣♦✐♥t ♦r ❛ ✈❡❝t♦r ♦♥ t❤❡ ♣❧❛♥❡✿
X = (x, y)
X =< x, y > .
♦r
❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❝♦❧✉♠♥✲✈❡❝t♦r ♥♦t❛t✐♦♥✿
X=
x y
.
1✳
✹✳✽✳
❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✽✽
◆❡①t✱ t❤❡ s❛♠❡ ❤❛♣♣❡♥s t♦ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛s ✈❡❝t♦rs✿
∆x ∆X ∆x ∆y ∆t . = , = ∆t ∆t ∆t ∆y ∆t
❚❤❡♥ t❤❡ s❡t✉♣ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ ❢♦r r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s r❡❛♣♣❡❛rs✿
∆X = F (X), X(t0 ) = X0 . ∆t ■♥ t❤✐s s❡❝t✐♦♥✱ ♦✉r ♠❛✐♥ ❝♦♥❝❡r♥ ✇✐❧❧ ❜❡ ❖❉❊s ✇✐t❤ r❡s♣❡❝t t♦
′
′
′
X =< x , y >=
x′ y′
❞❡r✐✈❛t✐✈❡s ✿
,
❛♥❞
X ′ = F (X), X(t0 ) = X0 . ❆❧❧ t❤❡ ♣❧♦tt✐♥❣✱ ❤♦✇❡✈❡r✱ ✐s ❞♦♥❡ ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❖❉❊s✳
2 ❚❤❡ ✏♣❤❛s❡ s♣❛❝❡✑ R ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❧♦❝❛t✐♦♥s✳ ❚❤❡♥ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ❣✐✈❡♥ ♣❛rt✐❝❧❡ ✐s ❛ ❢✉♥❝t✐♦♥ + 2 X : R → R ♦❢ t✐♠❡ t ≥ 0✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇✱ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ t❤❡ s❛♠❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✿ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♣❛rt✐❝❧❡ ✐❢ ✐t ❤❛♣♣❡♥s t♦ ❜❡ ❛t ♣♦✐♥t
X
✐s
F (X)✳
❚❤❡♥ ❡✐t❤❡r t❤❡ ♥❡①t ♣♦s✐t✐♦♥ ✐s ♣r❡❞✐❝t❡❞ t♦ ❜❡
X + F (X)
✕ t❤❛t✬s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✕ ♦r
F (X)
✐s ❥✉st ❛
t❛♥❣❡♥t ♦❢ t❤❡ tr❛❥❡❝t♦r② ✕ t❤❛t✬s ❛♥ ❖❉❊✳ ❆ ✈❡❝t♦r ✜❡❧❞ s✉♣♣❧✐❡s ❛
❞✐r❡❝t✐♦♥ t♦ ❡✈❡r② ❧♦❝❛t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡ ✐s ❛ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡✿ point 7→ vector .
❆
✈❡❝t♦r ✜❡❧❞
✐♥ ❞✐♠❡♥s✐♦♥
2
✐s ❛♥② ❢✉♥❝t✐♦♥✿
F : R2 → R2 . ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛s ❛
t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t
X ′ = F (X) . ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❱P ❛❞❞s ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿
X(t0 ) = X0 .
❖❉❊ ♦♥ t❤❡ ♣❧❛♥❡✿
✹✳✽✳
❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✽✾
❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✿ s♦❧✉t✐♦♥ ♦❢ s②st❡♠ ♦❢ ❖❉❊s ❆ s♦❧✉t✐♦♥ ♦❢ ❛ s②st❡♠ ♦❢ ❖❉❊s ✐s ❛ ❢✉♥❝t✐♦♥ u ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ✐♥ I ✇❡ ❤❛✈❡✿
X ′ (t) = F (X(t))
❉❡✜♥✐t✐♦♥ ✹✳✽✳✷✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ s②st❡♠ ♦❢ ❖❉❊s ❛♥❞ ❛ ❣✐✈❡♥ (t0 , X0 )✱ t❤❡ ❛♥ ■❱P✱ ✐s
X ′ = F (X),
✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✱ ♦r
X(t0 ) = X0
❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡✳
❉❡✜♥✐t✐♦♥ ✹✳✽✳✸✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡
❊✉❧❡r s♦❧✉t✐♦♥
✇✐t❤ ✐♥❝r❡♠❡♥t ∆t > 0 ♦❢ t❤❡ ■❱P✿
X ′ = F (X),
X(t0 ) = X0 ;
✐s ❛ s❡q✉❡♥❝❡ {Xn } ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿
Xn+1 = Xn + F (Xn ) · ∆t ✇❤❡r❡ tn+1 = tn + ∆t✳ ❆❧❧ t❤❡ ❞❡✜♥✐t✐♦♥s ❛❜♦✈❡ r❡♠❛✐♥ ✈❛❧✐❞ ✐❢ ✇❡ t❤✐♥❦ ♦❢ X ❛s ❛ ❧♦❝❛t✐♦♥ ✐♥ ❛♥ N ✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ RN ✳ ❲❡ ♥♦✇ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❧✐♥❡❛r s②st❡♠s ✭t❤❛t ♠❛② ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ ♥♦♥✲❧✐♥❡❛r ♦♥❡s ✈✐❛ ❧✐♥❡❛r✐③❛t✐♦♥✮✳ ❆❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞ ❛r❡ ❧✐♥❡❛r ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t✐❛❜❧❡❀ ❜♦t❤ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✦ ❚❤✐s ✐s t❤❡ ❝❛s❡ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ F ✳ ❆s s✉❝❤✱ ✐t ✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐① ❛♥❞ ✐s ❡✈❛❧✉❛t❡❞ ✈✐❛ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ax + by x a b . = F (X) = F X = cx + dy y c d ■t ✐s ✇r✐tt❡♥ s✐♠♣❧② ❛s
X′ = F X .
❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤✐s ♠❛tr✐① ✕ t❤❡ ❞❡t❡r♠✐♥❛♥t✱ t❤❡ tr❛❝❡✱ ❛♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ ✇✐❧❧ ❤❡❧♣ ✉s ❝❧❛ss✐❢② s✉❝❤ ❛ s②st❡♠✳ ❍♦✇❡✈❡r✱ t❤❡ ✜rst ♦❜s❡r✈❛t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡✿ X = 0 ✐s t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ s②st❡♠✳ ■♥ ❢❛❝t ✇❤❛t ✇❡✬✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t s②st❡♠s ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t❡❧❧s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✳
❚❤❡♦r❡♠ ✹✳✽✳✹✿ ❊q✉✐❧✐❜r✐✉♠ ♦❢ ▲✐♥❡❛r ❙②st❡♠ ❲❤❡♥ det F 6= 0✱ X = 0 ✐s t❤❡ ♦♥❧② ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ s②st❡♠ X ′ = F X ❀ ♦t❤❡r✇✐s❡✱ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s✉❝❤ ♣♦✐♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ♠❛tt❡r ✐s ✇❤❡t❤❡r F ✱ ❛s ❛ ❢✉♥❝t✐♦♥✱ ✐s ♦r ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳
✹✳✽✳
❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✾✵
❊①❛♠♣❧❡ ✹✳✽✳✺✿ ❞❡❣❡♥❡r❛t❡
❚❤❡ ❧❛tt❡r ✐s t❤❡ ✏❞❡❣❡♥❡r❛t❡✑ ❝❛s❡ s✉❝❤ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ✈❡r② s✐♠♣❧❡ s②st❡♠ ♦❢ ❖❉❊s✿
x′ = 2x y′ = 0
=⇒ =⇒
x = Ce2t y =K
■t ✐s ❡❛s② t♦ s♦❧✈❡✱ ♦♥❡ ❡q✉❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❲❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ t❤❡ t❤❡
x✲❛①✐s
❛♥❞ ❝♦♥st❛♥t ♦♥
y ✲❛①✐s✳
❊①❛♠♣❧❡ ✹✳✽✳✻✿ s❛❞❞❧❡
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿
x′ = −x y ′ = 4y
=⇒ =⇒
x = Ce−t y = Ke4t
❲❡ s♦❧✈❡ ✐t ✐♥st❛♥t❧② ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ t✇♦ ❖❉❊s ❛s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠ t❤❛t ❧✐✈❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛①❡s✿
❲❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ t❤❡
x✲❛①✐s
❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦♥ t❤❡
y ✲❛①✐s✳
❚❤❡ r❡st ♦❢ t❤❡
s♦❧✉t✐♦♥s ❛r❡ s❡❡♥ t♦ t❡♥❞ t♦✇❛r❞ ♦♥❡ ♦❢ t❤❡s❡✳ ❙✐♥❝❡ ♥♦t ❛❧❧ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❣♦ t♦✇❛r❞ t❤❡ ♦r✐❣✐♥✱ ✐t ✐s
✉♥st❛❜❧❡✳
❚❤✐s ♣❛tt❡r♥ ✐s ❝❛❧❧❡❞ ❛ ✏s❛❞❞❧❡✑ ❜❡❝❛✉s❡ t❤❡ ❝✉r✈❡s ❧♦♦❦ ❧✐❦❡ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r♦✉♥❞ ❛ s❛❞❞❧❡ ♣♦✐♥t✳ ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢
F =
F
−1 0 0 4
✐s ❞✐❛❣♦♥❛❧✿
.
✹✳✽✳
❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✾✶
❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿
X=
x y
=
Ce−t Ke4t
= Ce
−t
1 0
+ Ke
4t
0 1
.
❲❡ ❤❛✈❡ ❡①♣r❡ss❡❞ t❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs✦ ❊①❛♠♣❧❡ ✹✳✽✳✼✿ ♥♦❞❡
❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t s②st❡♠ ✐s✿
x′ = 2x y ′ = 4y
❍❡r❡✱
F =
=⇒ =⇒
x = Ce2t y = Ke4t
2 0 0 4
.
❖♥❝❡ ❛❣❛✐♥✱ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ t✇♦ ❖❉❊s ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠ t❤❛t ❧✐✈❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛①❡s ✭❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ ❜♦t❤ ♦❢ t❤❡ ❛①❡s✮ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s❡❡♥ t♦ t❡♥❞ t♦✇❛r❞ ♦♥❡ ♦❢ t❤❡s❡✳ ❚❤❡ s❧✐❣❤t ❝❤❛♥❣❡ t♦ t❤❡ s②st❡♠ ♣r♦❞✉❝❡s ❛ ✈❡r② ❞✐✛❡r❡♥t ♣❛tt❡r♥✿
❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿
2t Ce x 4t 0 2t 1 . + Ke = = Ce X= 1 Ke4t 0 y
❚❤✐s ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✇✐t❤ t✐♠❡✲❞❡♣❡♥❞❡♥t ✇❡✐❣❤ts✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ✉♥st❛❜❧❡ ❜✉t ❝❤❛♥❣✐♥❣ 2 ❛♥❞ 4 t♦ −2 ❛♥❞ −4 ✇✐❧❧ r❡✈❡rs❡ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡s ❛♥❞ ♠❛❦❡ ✐t st❛❜❧❡✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✐s ❢❛st❡r ❛❧♦♥❣ t❤❡ y ✲❛①✐s❀ t❤❛t ✐s ✇❤② t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ ❜❡ t❛♥❣❡♥t t♦ t❤❡ x✲❛①✐s✳ ■♥ ❢❛❝t✱ ❡❧✐♠✐♥❛t✐♥❣ t ❣✐✈❡s ✉s y = x2 ❛♥❞ s✐♠✐❧❛r ❣r❛♣❤s✳ ❲❤❡♥ t❤❡ t✇♦ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧✱ t❤❡ ❣r♦✇t❤ ✐s ✐❞❡♥t✐❝❛❧ ❛♥❞ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✐♠♣❧② str❛✐❣❤t ❧✐♥❡s✿
❲❤❛t ✐❢ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡♥✬t s❡♣❛r❛t❡❞❄ ❚❤❡ ✐♥s✐❣❤t ✐s t❤❛t t❤❡② ❝❛♥ ❜❡ ✕ ❛❧♦♥❣ t❤❡
❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡
✹✳✽✳
❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s
✷✾✷
♠❛tr✐①✳ ■♥❞❡❡❞✱ t❤❡ ❜❛s✐s ✈❡❝t♦rs ❛r❡ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡s❡ t✇♦ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳ ◆♦✇ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ t✇♦ ♣✐❝t✉r❡s ❛r❡ s❦❡✇❡❞✿
▲❡t✬s ❧♦♦❦ ❛t t❤❡♠ ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤❡ ✐❞❡❛ ✐s ✉♥❝♦♠♣❧✐❝❛t❡❞✿ t❤❡ s②st❡♠ ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡ ✐s 1✲ ❞✐♠❡♥s✐♦♥❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✐s ❛ s✐♥❣❧❡ ❖❉❊ ❛♥❞ ❝❛♥ ❜❡ s♦❧✈❡❞ t❤❡ ✉s✉❛❧ ✇❛②✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ✜♥❞ s♦❧✉t✐♦♥s✳ ❊✈❡r② s♦❧✉t✐♦♥ X t❤❛t ❧✐❡s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡✱ ✇❤✐❝❤ ✐s ❛ ❧✐♥❡✱ ✐s ❛ t✲❞❡♣❡♥❞❡♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦r V ✿ X = rV
=⇒ X ′ = (rV )′ = F (rV ) = rF V = rλV =⇒ r′ V = λrV =⇒ r′ = λr =⇒ r = eλt
❚❤❡♦r❡♠ ✹✳✽✳✽✿ ❊✐❣❡♥s♣❛❝❡ ❙♦❧✉t✐♦♥s ■❢
λ
✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ❛♥❞
V
❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦r ♦❢ ❛ ♠❛tr✐①
F✱
t❤❡♥
X = eλt V ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠
X′ = F X✳
Pr♦♦❢✳
❚♦ ✈❡r✐❢②✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✉s❡ ❧✐♥❡❛r✐t② ✭♦❢ ❜♦t❤ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✛❡r✲ ❡♥t✐❛t✐♦♥✮✿ X = eλt V
=⇒ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ X ′ = (eλt V )′ = (eλt )′ V = λeλt V r✐❣❤t✲❤❛♥❞ s✐❞❡✿ F X = F (eλt V ) = eλt F V = eλt λV
❚❤❡ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ❝♦♠♣❧❡① ✦ ❚❤❡ s❡❝♦♥❞ ✐❞❡❛ ✐s t♦ tr② t♦ ❡①♣r❡ss t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❧✐♥❡❛r s②st❡♠ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ s♦❧✉t✐♦♥s ❢♦✉♥❞ t❤✐s ✇❛②✳
✹✳✾✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s
✷✾✸
❚❤❡♦r❡♠ ✹✳✽✳✾✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ■♥ ❚❡r♠s ♦❢ ❊✐❣❡♥s♦❧✉t✐♦♥s
❙✉♣♣♦s❡ V1 ❛♥❞ V2 ❛r❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❛ ♠❛tr✐① F t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ′ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡♥ ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠ X = F X ❛r❡ ❣✐✈❡♥ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡s✿ X = Ceλ1 t V1 + Keλ2 t V2 ,
✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts C ❛♥❞ K ✳ Pr♦♦❢✳ ❙✐♥❝❡ t❤❡s❡ s♦❧✉t✐♦♥s ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt②✳
❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✵✿ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥ ❋♦r ❛ ❣✐✈❡♥ ❡✐❣❡♥✈❡❝t♦r
s♦❧✉t✐♦♥✳
V
✇✐t❤ ❡✐❣❡♥✈❛❧✉❡
λ✱
✇❡ ✇✐❧❧ ❝❛❧❧
eλt V
❛
❝❤❛r❛❝t❡r✐st✐❝
❛❧❧
t❤❡ s♦❧✉t✐♦♥s✳
❊①❡r❝✐s❡ ✹✳✽✳✶✶ ❙❤♦✇ t❤❛t ✇❤❡♥ ❛❧❧ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✱ t❤❡ ❢♦r♠✉❧❛ ✇♦♥✬t ❣✐✈❡ ✉s
✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ❲❡ ❝♦♥s✐❞❡r ❛ ❢❡✇ s②st❡♠s ✇✐t❤
♥♦♥✲❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳
❚❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs
❝♦♠❡ ❢r♦♠ ❈❤❛♣t❡r ✸✳
❊①❛♠♣❧❡ ✹✳✾✳✶✿ ❞❡❣❡♥❡r❛t❡ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s②st❡♠ ♦❢ ❖❉❊s✿
x′ = x +2y, y ′ = 2x +4y,
=⇒ F =
1 2 2 4
.
❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣ s♦❧✉t✐♦♥s✿
■t ❛♣♣❡❛rs t❤❛t t❤❡ s②st❡♠ ❤❛s ✭❡①♣♦♥❡♥t✐❛❧✮ ❣r♦✇t❤ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ✐♥ ❛♥♦t❤❡r✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ▲✐♥❡❛r ❛❧❣❡❜r❛ ❤❡❧♣s✳
✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s
✷✾✹
❋✐rst✱ t❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ③❡r♦ ✿ det F = det
1 2 2 4
= 1 · 4 − 2 · 2 = 0.
❚❤❛t✬s ✇❤② t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚❤❡s❡ ❛r❡ st❛t✐♦♥❛r② ♣♦✐♥ts✳ ❚♦ ✜♥❞ t❤❡♠✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿ x +2y = 0, 2x +4y = 0,
=⇒ x = −2y .
❲❡ ❤❛✈❡✱ t❤❡♥✱ ❡✐❣❡♥✈❡❝t♦rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ λ1 = 0✿ V1 =
2 −1
=⇒ F V1 = 0 .
❙♦✱ s❡❝♦♥❞✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛❧✉❡ ✿ det(F − λI) = det
1−λ 2 2 4−λ
= λ2 − 5λ = λ(λ − 5) .
▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❢♦r λ2 = 5✳ ❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ F V = λ2 V ,
❛s ❢♦❧❧♦✇s✿ FV =
1 2 2 4
x y
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
x +2y = 5x, 2x +4y = 5y,
=⇒
=5
x y
−4x +2y = 0, 2x −y = 0,
.
=⇒ y = 2x .
❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ t❤❡ ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿ V2 =
1 2
.
❊✈❡r② s♦❧✉t✐♦♥ st❛rts ♦✛ t❤❡ ❧✐♥❡ y = −x/2 ❛♥❞ ❝♦♥t✐♥✉❡s ❛❧♦♥❣ t❤✐s ✈❡❝t♦r✳ ■t ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs✿ X = CV1 + KV2 = C
2 −1
+ Ke5t
❊①❡r❝✐s❡ ✹✳✾✳✷
❋✐♥❞ t❤❡ ❧✐♥❡ ♦❢ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s✳ ❊①❛♠♣❧❡ ✹✳✾✳✸✿ s❛❞❞❧❡
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿ ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿
x′ = x +2y, y ′ = 3x +2y. F =
❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿
1 2 3 2
.
1 2
.
✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s
✷✾✺
❚❤❡ t✇♦ ❧✐♥❡s t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ ❝♦♥✈❡r❣❡ t♦ ❛r❡ t❤❡ ❡✐❣❡♥s♣❛❝❡s✳ ▲❡t✬s ✜♥❞ t❤❡♠✿ det(F − λI) = det
1−λ 2 3 2−λ
= λ2 − 3λ − 4 .
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = −1, λ2 = 4 .
◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿
F Vk = λk Vk , k = 1, 2 .
❚❤❡ ✜rst✿ F V1 =
1 2 3 2
x y
= −1
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
x +2y = −x, 3x +2y = −y,
=⇒
❲❡ ❝❤♦♦s❡ V1 =
x y
2x +2y = 0, 3x +3y = 0,
1 −1
.
1 −1
.
=⇒ x = −y .
❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = −x✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X1 = e
❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿ F V2 =
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
x +2y = 4x, 3x +2y = 4y,
λ1 t
V1 = e
1 2 3 2
=⇒
❲❡ ❝❤♦♦s❡ V2 =
−t
x y
=4
.
x y
−3x +2y = 0, 3x −2y = 0,
1 3/2
.
1 3/2
.
=⇒ x = 2y/3 .
❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = 3x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X2 = e
λ2 t
V2 = e
4t
.
❚❤❡ t✇♦ s♦❧✉t✐♦♥s X1 ❛♥❞ X2 ✱ ❛s ✇❡❧❧ ❛s −X1 ❛♥❞ −X2 ✱ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿
✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s
✷✾✻
❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❜❛s✐❝ s♦❧✉t✐♦♥s✿ X = Ce
λ1 t
V1 + Ke
λ2 t
V2 = Ce
✐✳❡✳✱
❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ✉♥st❛❜❧❡✳
−t
1 −1
+ Ke
4t
1 3/2
=
Ce−t + Ke4t −Ce−t + 3/2Ke4t
x = Ce−t +Ke4t , −t y = −Ce +3/2Ke4t .
❊①❛♠♣❧❡ ✹✳✾✳✹✿ ♥♦❞❡
▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿
❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿
x′ = −x −2y, y′ = x −4y.
F =
−1 −2 1 −4
.
❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿
❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ det(F − λI) = det
−1 − λ −2 1 −4 − λ
= λ2 − 5λ + 6 .
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = −3, λ2 = −2 .
❚♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿
F Vk = λk Vk , k = 1, 2 .
❚❤❡ ✜rst✿ F V1 =
−1 −2 1 −4
x y
= −1
x y
.
,
✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s
✷✾✼
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
−x −2y = −3x, x −4y = −3y,
=⇒
2x −2y = 0, x −y = 0,
V1 =
1 1
❲❡ ❝❤♦♦s❡
=⇒ x = y .
.
❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = x✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X1 = e
❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿
V1 = e
−1 −2 F V2 = 1 −4
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
λ1 t
−x −2y = −2x, x −4y = −2y,
=⇒
❲❡ ❝❤♦♦s❡
−3t
1 . 1
x x . = −2 y y
x −2y = 0, x −2y = 0,
=⇒ x = 2y .
2 . V2 = 1
❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X2 = e
λ2 t
V2 = e
−2t
2 . 1
❚❤❡ t✇♦ s♦❧✉t✐♦♥s X1 ❛♥❞ X2 ✱ ❛s ✇❡❧❧ ❛s −X1 ❛♥❞ −X2 ✱ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿
❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❜❛s✐❝ s♦❧✉t✐♦♥s✿ X = Ce
λ1 t
V1 + Ke
λ2 t
V2 = Ce
−3t
1 1
+ Ke
−2t
2 1
.
❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s st❛❜❧❡✳
❉❡✜♥✐t✐♦♥ ✹✳✾✳✺✿ st❛❜❧❡ ♥♦❞❡ ❋♦r ❛ ❧✐♥❡❛r s②st❡♠ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ st❛❜❧❡ ♥♦❞❡ ✐❢ ❡✈❡r② ♦t❤❡r s♦❧✉t✐♦♥ X s❛t✐s✜❡s✿ X(t) → 0 ❛s t → +∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → −∞;
✹✳✶✵✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✉♥st❛❜❧❡ ♥♦❞❡
❛♥❞ ❛♥
X(t) → 0 ♣r♦✈✐❞❡❞ ♥♦
X
✷✾✽
✐❢
❛s
t → −∞
❛♥❞
||X(t)|| → ∞
♠❛❦❡s ❛ ❢✉❧❧ r♦t❛t✐♦♥ ❛r♦✉♥❞
❛s
t → +∞;
0✳
❉❡✜♥✐t✐♦♥ ✹✳✾✳✻✿ s❛❞❞❧❡ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ s❛❞❞❧❡ X t❤❛t s❛t✐s❢②✿
❋♦r ❛ ❧✐♥❡❛r s②st❡♠ ✐❢ ✐t ❤❛s s♦❧✉t✐♦♥s
||X(t)|| → ∞
❛s
t → ±∞ .
❚❤❡♦r❡♠ ✹✳✾✳✼✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■ F λ2
❙✉♣♣♦s❡ ♠❛tr✐①
•
■❢
•
■❢
λ1
❛♥❞
❤❛s t✇♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s
λ1
❛♥❞
❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✱ t❤❡ s②st❡♠
λ2 ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ X ′ = F X ❤❛s ❛ ♥♦❞❡✱
st❛❜❧❡
✇❤❡♥ t❤✐s s✐❣♥ ✐s ♥❡❣❛t✐✈❡ ❛♥❞ ✉♥st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ✐s ♣♦s✐t✐✈❡✳
λ1
❛♥❞
λ2
❤❛✈❡ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥s✱ t❤❡ s②st❡♠
X′ = F X
❤❛s ❛ s❛❞❞❧❡✳
Pr♦♦❢✳ ❚❤❡ st❛❜✐❧✐t② ✐s s❡❡♥ ✐♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✱ ❛s
λt
λt
||X|| = ||e V || = e · ||V || →
∞ 0
✐❢ ✐❢
t → +∞✿
λ > 0, λ < 0.
❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✳ ❚❤❡♥✱ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇❡ ❤❛✈❡ ♦♥❡ ♦r t❤❡ ♦t❤❡r ♣❛tt❡r♥✱ ❛♥❞ ✐♥ t❤❡ ❧❛tt❡r✱ ❜♦t❤✳ ❚❤❡r❡ ❝❛♥ ❜❡ ♥♦ r♦t❛t✐♦♥ ❜❡❝❛✉s❡ ♥♦ s♦❧✉t✐♦♥ ❝❛♥ ✐♥t❡rs❡❝t ❛♥ ❡✐❣❡♥s♣❛❝❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt②✳
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ❲❤❛t ✐❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡
❝♦♠♣❧❡① ❄
❘❡❝❛❧❧ t❤❛t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ♠❛tr✐①
F
✐s
χ(λ) = det(F − λI) = λ2 − tr F · λ + det F . ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤✐s q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s
D = (tr F )2 − 4 det F . ❲❤❡♥
D > 0✱
✇❡ ❤❛✈❡ t✇♦ ❞✐st✐♥❝t r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✱ t❤❡ ❝❛s❡ ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳
✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✇❤❡♥❡✈❡r
D < 0✳
❚❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ ✐s
❲❡ ❛r❡ ❢❛❝❡❞
D = 0✳
❊①❛♠♣❧❡ ✹✳✶✵✳✶✿ ✐♠♣r♦♣❡r ♥♦❞❡ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❛ ♥♦❞❡ ♠❛② ❜❡ ♣r♦❞✉❝❡❞ ❜② ❛ ♠❛tr✐① ✇✐t❤ r❡♣❡❛t❡❞ ✭❛♥❞✱ t❤❡r❡❢♦r❡✱ r❡❛❧✮ ❡✐❣❡♥✈❛❧✉❡s✿
F = ❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿
−1 2 0 −1
.
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✷✾✾
❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ det(F − λI) = det
−1 − λ 2 0 −1 − λ
= (−1 − λ)2 .
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = λ2 = −1 .
❚❤❡ ♦♥❧② ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❤♦r✐③♦♥t❛❧✳ ❚❤❡ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜② X = Ce
−t
1 0
+ K te
−t
1 0
+e
−t
? 1
.
❊①❡r❝✐s❡ ✹✳✶✵✳✷
❋✐♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❤❡ ❡①❛♠♣❧❡✳ ❲❤❡♥ D < 0✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❝♦♠♣❧❡①✦ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❡❝t♦rs ✭♥♦t r❡❛❧ ♦♥❡s ❛♥②✇❛②✮✳ ❉♦❡s t❤❡ s②st❡♠ X ′ = F X ❡✈❡♥ ❤❛✈❡ s♦❧✉t✐♦♥s❄ ❚❤❡ t❤❡♦r❡♠ ❛❜♦✉t ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s s❛②s ②❡s✱ t❤❡② ❛r❡ ❝❡rt❛✐♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s✳✳✳ ❊①❛♠♣❧❡ ✹✳✶✵✳✸✿ ❝❡♥t❡r
❈♦♥s✐❞❡r
x′ = y, y ′ = −x.
❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✿ x′′ = (x′ )′ = y ′ = −x .
❚❤❡r❡❢♦r❡ t❤❡ s♦❧✉t✐♦♥s ❛r❡ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ sin t ❛♥❞ cos t✳ ❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞ ✭✇✐t❤ ❛ ❧✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ st❡♣ t♦ ♣r❡✈❡♥t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ s♣✐r❛❧ ♦✉t✮✿
❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r② ❛❜♦✈❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧ r❛t❤❡r t❤❛♥ tr✐❣♦♥♦♠❡tr✐❝✳ ❇✉t t❤❡ ❧❛tt❡r ❛r❡ ❥✉st ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✇✐t❤ ✐♠❛❣✐♥❛r② ❡①♣♦♥❡♥ts✳
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ▲❡t✬s ♠❛❦❡ t❤✐s s♣❡❝✐✜❝❀ ✇❡ ❤❛✈❡
F = ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✱
✸✵✵
0 1 −1 0
,
χ(λ) = λ2 + 1 ,
❤❛s t❤❡s❡ ❝♦♠♣❧❡① r♦♦ts✿ λ1,2 = ±i✳ ❚♦ ✜♥❞ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r✱ ✇❡ s♦❧✈❡✿
F V1 =
0 1 −1 0
x y
=i
x y
.
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ y = ix =⇒ y = ix . −x = iy ❲❡ ❝❤♦♦s❡ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r✿
V1 = ❛♥❞ s✐♠✐❧❛r❧②✿
V2 =
1 i 1 −i
,
,
❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✕ ♦✈❡r t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ♦❢ t❤❡s❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✿ 1 1 −it λ1 t λ2 t it . + Ke X = Ce V1 + Ke V2 = Ce −i i ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ✳✳✳✐♥ t❤❡ ❝♦♠♣❧❡① ❞♦♠❛✐♥✳ ❲❤❛t ✐s t❤❡ r❡❛❧ ♣❛rt ❄ ▲❡t K = 0✳ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ ✐s✿ it cos t + i sin t e cos t + i sin t 1 it . =C =C =C X = Ce − sin t + i cos t i(cos t + i sin t) i ieit ■ts r❡❛❧ ♣❛rt ✐s✿
Re X = C ❚❤❡s❡ ❛r❡ ❛❧❧ t❤❡ ❝✐r❝❧❡s✳
cos t − sin t
❊①❛♠♣❧❡ ✹✳✶✵✳✹✿ ❢♦❝✉s
▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❝♦♠♣❧❡① s②st❡♠ ❖❉❊s✿ ′ x = 3x −13y, y ′ = 5x +y. ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿
F = ❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿
3 −13 5 1
.
.
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✸✵✶
❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿
χ(λ) = det(F − λI) = det
3 − λ −13 5 1−λ
= λ2 − 4λ + 68 .
❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1,2 = 2 ± 8i .
◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿ F Vk = λk Vk , k = 1, 2 .
❚❤❡ ✜rst✿ F V1 =
3 −13 5 1
x y
= (2 + 8i)
❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿
3x −13y = (2 + 8i)x 5x +y = (2 + 8i)y
=⇒
V1 =
F V2 =
❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿
3x −13y = (2 − 8i)x 5x +y = (2 − 8i)y
3 −13 5 1
=⇒
x y
.
(1 − 8i)x −13y =0 5x +(−1 − 8i)y = 0
❲❡ ❝❤♦♦s❡ ❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿
1 + 8i 5 x y
V2 =
= (2 − 8i)
1 − 8i 5
1 + 8i y. 5
=⇒ x =
1 − 8i y. 5
.
x y
.
(1 + 8i)x −13y =0 5x +(−1 + 8i)y = 0
❲❡ ❝❤♦♦s❡
=⇒ x =
.
❚❤❡ ❣❡♥❡r❛❧ ❝♦♠♣❧❡① s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✿ Z = Ce
λ1 t
V1 + Ke
λ2 t
V2 = Ce
(2+8i)t
1 + 8i 5
+ Ke
(2−8i)t
1 − 8i 5
.
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✸✵✷
▲❡t✬s ♥♦✇ ❡①❛♠✐♥❡ ❛ s✐♠♣❧❡ r❡❛❧ s♦❧✉t✐♦♥✳ ❲❡ ❧❡t C = 1 ❛♥❞ K = 0✿
1 + 8i X = Re Z = Re e(2+8i)t 5 1 + 8i = e2t Re e8it 5
1 + 8i = e2t Re(cos 8t + i sin 8t) 5 (cos 8t + i sin 8t)(1 + 8i) = e2t Re (cos 8t + i sin 8t)5 cos 8t + i sin 8t + 8i cos 8t − 8 sin 8t = e2t Re 5 cos 8t + i5 sin 8t cos 8t − 8 sin 8t = e2t . 5 cos 8t
P❧♦tt✐♥❣ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❝♦♥✜r♠s ❊✉❧❡r✬s ♠❡t❤♦❞ r❡s✉❧t✿
❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✺✿ st❛❜❧❡ ❛♥❞ ✉♥st❛❜❧❡ ❢♦❝✉s ❋♦r ❛ ❧✐♥❡❛r s②st❡♠ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ st❛❜❧❡ ❢♦❝✉s ✐❢ ❡✈❡r② ♦t❤❡r s♦❧✉t✐♦♥ X s❛t✐s✜❡s✿ X(t) → 0 ❛s t → +∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → −∞;
❛♥❞ ❛♥ ✉♥st❛❜❧❡ ❢♦❝✉s ✐❢ X(t) → 0 ❛s t → −∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → +∞;
♣r♦✈✐❞❡❞ ❡✈❡r② s✉❝❤ X ♠❛❦❡s ❛ ❢✉❧❧ r♦t❛t✐♦♥ ❛r♦✉♥❞ 0✳
✹✳✶✵✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✸✵✸
❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✻✿ ❝❡♥t❡r ❋♦r ❛ ❧✐♥❡❛r s②st❡♠
X′ = F X✱
t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥
X0 = 0
✐s ❝❛❧❧❡❞ ❛
❝❡♥t❡r
✐❢ ❛❧❧ s♦❧✉t✐♦♥s ❛r❡ ❝②❝❧❡s✳
❚❤❡♦r❡♠ ✹✳✶✵✳✼✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■■ ❙✉♣♣♦s❡ ♠❛tr✐①
F
❤❛s t✇♦ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❡✐❣❡♥✈❛❧✉❡s
λ1
❛♥❞
λ2 ✳
❚❤❡♥ ✇❡
❤❛✈❡✿
•
■❢ t❤❡ r❡❛❧ ♣❛rt ♦❢
λ1
❛♥❞
λ2
✐s ♥♦♥✲③❡r♦✱ t❤❡ s②st❡♠
X′ = F X
❤❛s ❛ ❢♦❝✉s✱
st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ♦❢ t❤✐s ♥✉♠❜❡r ✐s ♥❡❣❛t✐✈❡ ❛♥❞ ✉♥st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ✐s ♣♦s✐t✐✈❡✳
•
■❢ t❤❡ r❡❛❧ ♣❛rt ♦❢
λ1
❛♥❞
λ2
✐s ③❡r♦✱ t❤❡ s②st❡♠
X′ = F X
❤❛s ❛ ❝❡♥t❡r✳
Pr♦♦❢✳ ❚❤❡ st❛❜✐❧✐t② ✐s s❡❡♥ ✐♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✱ ❛s
λt
||X|| = ||e V || = ||e
(a+bi)t
at
t → +∞✿ at
V || = e | cos bt + i sin bt| · ||V || = e ||V || →
∞ 0
✐❢ ✐❢
a > 0, a < 0.
❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠s ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿
❚❤❡r❡❜② ✇❡ ❝♦♠♣❧❡t❡ t❤❡ s❡q✉❡♥❝❡✿ ❡❧❡♠❡♥t❛r② ❛❧❣❡❜r❛
−→ ♠❛tr✐① ❛❧❣❡❜r❛ −→ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳
❚♦ s✉♠♠❛r✐③❡✱ ✐♥ ♦r❞❡r t♦ ❝❧❛ss✐❢② ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❖❉❊s ✈❡❝t♦r ♦♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ❝❧❛ss✐❢②
F
X′ = F X✱
✇❤❡r❡
F
✐s ❛
2×2
♠❛tr✐① ❛♥❞
X
✐s ❛
❛❝❝♦r❞✐♥❣ t♦ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ✈✐s✉❛❧✐③❡ ❤♦✇ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡s❡ t✇♦
♥✉♠❜❡rs ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ✐♥❞✐❝❛t❡ ✈❡r② ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs ♦❢ t❤❡ tr❛❥❡❝t♦r✐❡s✳ ✭❚❤❡ ♠✐ss✐♥❣ ♣❛tt❡r♥s ❛r❡ ❜❡tt❡r ✐❧❧✉str❛t❡❞ ❞②♥❛♠✐❝❛❧❧②✱ ❛s t❤❡ ❡①❛❝t ✏♠♦♠❡♥ts✑ ✇❤❡♥ ♦♥❡ ♣❛tt❡r♥ tr❛♥s✐t✐♦♥s ✐♥t♦ ❛♥♦t❤❡r✳✮
❊①❡r❝✐s❡ ✹✳✶✵✳✽ P♦✐♥t ♦✉t ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ ✐♠♣r♦♣❡r ♥♦❞❡✳
✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✸✵✹
❊①❡r❝✐s❡ ✹✳✶✵✳✾
❍♦✇ ❧✐❦❡❧② ✇♦✉❧❞ ❛ ❣✐✈❡♥ s②st❡♠ ❢❛❧❧ ✐♥t♦ ❡❛❝❤ ♦❢ t❤❡s❡ ✜✈❡ ❝❛t❡❣♦r✐❡s❄ ❲❤❛t ❛❜♦✉t t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ ✐♠♣r♦♣❡r ♥♦❞❡❄ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✵
❲❤❛t ♣❛r❛♠❡t❡rs ❞❡t❡r♠✐♥❡ t❤❡ ❝❧♦❝❦✇✐s❡ ✈s✳ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡ ❜❡❤❛✈✐♦r❄ ❊①❛♠♣❧❡ ✹✳✶✵✳✶✶✿ ♣r❡❞❛t♦r✲♣r❡②
▲❡t✬s ❝❧❛ss✐❢② t❤❡ ❡q✉✐❧✐❜r✐❛ ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✕ ✈✐❛ ❧✐♥❡❛r✐③❛t✐♦♥✳ ❖✉r ♥♦♥✲❧✐♥❡❛r s②st❡♠ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ′ x = αx −βxy, y ′ = δxy −γy, ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ ❝♦❡✣❝✐❡♥ts αx, β, δ, γ ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡
X ′ = G(X), ✇✐t❤ G(x, y) = (αx − βxy, δxy − γy) . ❚❤❡ ❏❛❝♦❜✐❛♥ ♦❢ G ✐s
∂ ∂ (αx − βxy) (αx − βxy) ∂x α − βy −βx ∂y = G′ (x, y) = . ∂ ∂ δy δx − γ (δxy − γy) (δxy − γy) ∂x ∂y
❚❤❡ ♠❛tr✐① ❞❡♣❡♥❞s ♦♥ (x, y) ❜❡❝❛✉s❡ t❤❡ s②st❡♠ ✐s ♥♦♥✲❧✐♥❡❛r✳ ❇② ✜①✐♥❣ ❧♦❝❛t✐♦♥s X = A✱ ✇❡ ❝r❡❛t❡ ❧✐♥❡❛r ✈❡❝t♦r ❖❉❊s✿ X ′ = G′ (A)X . ❋✐rst✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ③❡r♦ ❡q✉✐❧✐❜r✐✉♠✱
x = 0, y = 0 . ❍❡r❡✱ ′
F = G (0, 0) =
α − β · 0 −β · 0 δ·0 δ·0−γ
=
α 0 0 −γ
❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ α−λ 0 = (α − λ)(−γ − λ) = 0 . det 0 −γ − λ
.
❚❤❡r❡❢♦r❡✱
λ1 = α, λ2 = −γ .
❲❡ ❤❛✈❡ t✇♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s✳ ❚❤✐s ✐s ❛ s❛❞❞❧❡ ✦ ■♥❞❡❡❞✱ ❛r♦✉♥❞ t❤✐s ♣♦✐♥t t❤❡ ❢♦①❡s ❞❡❝❧✐♥❡ ✇❤✐❧❡ t❤❡ r❛❜❜✐ts ✐♥❝r❡❛s❡ ✐♥ ♥✉♠❜❡rs✳ ❚❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠ ✐s
x= ❍❡r❡✱
F = G′
γ α , δ β
α γ , y= . δ β
α γ βγ α−β· −β · 0 − β δ = δ = . α γ δα δ· δ· −γ 0 β δ β
✹✳✶✵✳
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞
✸✵✺
❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿
βγ −λ − δ 2 det δα = λ + αγ = 0 . −λ β
❚❤❡r❡❢♦r❡✱
λ1 =
√
❲❡ ❤❛✈❡ t✇♦ ♣✉r❡❧② ✐♠❛❣✐♥❛r② ❡✐❣❡♥✈❛❧✉❡s✳ ❝②❝❧✐❝ ❜❡❤❛✈✐♦r✳
❚❤❡ r❡s✉❧ts ♠❛t❝❤ ♦✉r ♣r❡✈✐♦✉s ❛♥❛❧②s✐s✳
√ αγ i, λ2 = − αγ i . ❚❤✐s ✐s ❛
❝❡♥t❡r ✦
■♥❞❡❡❞✱ ❛r♦✉♥❞ t❤✐s ♣♦✐♥t ✇❡ ❤❛✈❡ ❛
❈❤❛♣t❡r ✺✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s
❈♦♥t❡♥ts ✺✳✶ ✺✳✷ ✺✳✸ ✺✳✹ ✺✳✺ ✺✳✻ ✺✳✼ ✺✳✽ ✺✳✾
❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣ ❆ ♣❡♥❞✉❧✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❧❛♥❡t❛r② ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✸✵✻ ✸✵✼ ✸✶✷ ✸✶✺ ✸✷✶ ✸✷✹ ✸✸✵ ✸✸✻ ✸✸✾
✺✳✶✳ ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s
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❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐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✿
✺✳✷✳
❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
✸✵✼
❚❤❡ ❢♦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✿
❚❤❡ ❢♦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❡①t✳ ❲❡ ❥✉st ❛ss✉♠❡ t❤❛t ❛ s♣❛❝❡ ♦❢ ✐♥♣✉ts R ✇✐t❤ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ s♣❛❝❡ ♦❢ ♦✉t♣✉ts Rm ❛r❡ ❣✐✈❡♥✳
✺✳✷✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ r❡❢❡r t♦ s②st❡♠s ♦❢ ❖❉❊s ❛s ✈❡❝t♦r
❖❉❊s ♦r s✐♠♣❧② ❖❉❊s✳
■♠❛❣✐♥❡ ❛ ❤♦✉♥❞ r✉♥♥✐♥❣ ❛❢t❡r ❛ r❛❜❜✐t✳ ▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡✐r ♠✉t✉❛❧ ❞②♥❛♠✐❝s✳
❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤♦✉♥❞ ✐s ❛❧✇❛②s ❤❡❛❞✐♥❣ ❛t t❤❡ r❛❜❜✐t ✇❤✐❧❡ t❤❡ r❛❜❜✐t ✐s ❤❡❛❞✐♥❣ ❢♦r ✐ts ❤♦❧❡ ❛♥❞ ♥❡✈❡r ❝❤❛♥❣❡s ❞✐r❡❝t✐♦♥s✳
✺✳✷✳
❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
❲❡ ❜✉✐❧❞ ❛ ❞✐s❝r❡t❡
✸✵✽
♠♦❞❡❧✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✿ t0 = 0, tn+1 = tn + ∆t .
❚❤❡ ♠❛✐♥ ❛ss✉♠♣t✐♦♥ ✐s✿ ◮ ❉✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [tn , tn + ∆t]✱ t❤❡ ❤♦✉♥❞ ✇✐❧❧ ❜❡ r✉♥♥✐♥❣ t♦✇❛r❞ t❤❡ s♣♦t ✇❤❡r❡ t❤❡ r❛❜❜✐t ✇❛s ❛t t✐♠❡ tn ✳
❙✉♣♣♦s❡ t❤❡✐r ❧♦❝❛t✐♦♥s ❛r❡✿ • (x, y) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✱ ❛♥❞ • (p, q) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❤♦✉♥❞✳
❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♣❛r❛♠❡tr✐❝
❝✉r✈❡s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ st❛♥❞❛r❞ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✿
• (xn , yn ) = (x(tn ), y(tn )) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t ❛❢t❡r n ✐♥❝r❡♠❡♥ts✱ ❛♥❞ • (pn , qn ) = (p(tn ), q(tn )) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❤♦✉♥❞ ❛❢t❡r n ✐♥❝r❡♠❡♥ts✳
❖♥ t❤❡ ❧❡❢t ✇❡ ❤❛✈❡✱ ❥✉st ❛s ❜❡❢♦r❡✱ ❛ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✳ ❙✉♣♣♦s❡ • s ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ r❛❜❜✐t✱ ❛♥❞
• v ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ❤♦✉♥❞✱ v > s✳
❊①❛♠♣❧❡ ✺✳✷✳✶✿ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛t❤
❲❡ ❝r❡❛t❡ ❛ ♠♦❞❡❧ ❢♦r t❤❡ ❤♦✉♥❞ t❤❛t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛ s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢ t❤❡ ♣❛t❤ ♦❢ t❤❡ r❛❜❜✐t✿ ♦♥❡ st❡♣ ❛t ❛ t✐♠❡✳
❲❡ ❝♦♠♣✉t❡ ❝♦♥s❡❝✉t✐✈❡❧②✿ • t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦✿ xn − pn ❛♥❞ yn − qn ✱ ✇❤✐❝❤ ✐s t❤❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ t❤❡ ❤♦✉♥❞ t♦✇❛r❞s t❤❡ r❛❜❜✐t❀ • t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ✭❞✐✈✐❞✐♥❣ ❜② t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✮❀ • t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❤♦✉♥❞ ✭♠✉❧t✐♣❧②✐♥❣ t❤✐s ✈❡❝t♦r ❜② t❤❡ ❤♦✉♥❞s✬ s♣❡❡❞ v ✮❀ ❛♥❞ • t❤❡ ♥❡✇ ❧♦❝❛t✐♦♥ (pn+1 , qn+1 ) ♦❢ t❤❡ ❤♦✉♥❞ ✭❛❞❞✐♥❣ t❤❡ ✈❡❧♦❝✐t② t✐♠❡s t✐♠❡ ✐♥t❡r✈❛❧ h = ∆t t♦ t❤❡ ❧❛st ❧♦❝❛t✐♦♥✮✳
❆❜♦✈❡ ❛♥❞ ❜❡❧♦✇ ❧❡❢t s❤♦✇ t❤❡ ♣✉rs✉✐t ♦❢ ❛ r❛❜❜✐t ❢♦❧❧♦✇✐♥❣ ❛ ❝✐r❝✉❧❛r ♣❛t❤✳ ❇❡❧♦✇ r✐❣❤t✱ ✐t ❢♦❧❧♦✇s ❛ s✐♥✉s♦✐❞✿
✺✳✷✳
❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
✸✵✾
❊①❡r❝✐s❡ ✺✳✷✳✷
■♠♣❧❡♠❡♥t t❤✐s ♠♦❞❡❧ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ r❛❜❜✐t r✉♥♥✐♥❣ ♥♦rt❤✲✇❡st✿
❊①❡r❝✐s❡ ✺✳✷✳✸
■♠♣❧❡♠❡♥t t❤✐s ♠♦❞❡❧ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ r❛❜❜✐t ❢♦❧❧♦✇✐♥❣ ❛ ❝✐r❝❧❡✱ s✐♥✉s♦✐❞✱ s♣✐r❛❧✱ ❡t❝✳ ❲❡ ♥♦✇ ❛♣♣r♦❛❝❤ t❤✐s ❛♥❛❧②t✐❝❛❧❧②✳ ❚♦ ♠❛❦❡ t❤✐s ♣♦ss✐❜❧❡✱ ✇❡ ❝❤♦♦s❡ ❛ s✐♠♣❧❡ ❝❛s❡✿ t❤❡ r❛❜❜✐t ✐s r✉♥♥✐♥❣ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ❚❤❡♥✱
xn+1 = 0 , yn+1 = yn + s∆t .
❋✉rt❤❡r♠♦r❡✱ • ∆xn = ∆x (cn ) • ∆yn = ∆y (cn ) • ∆pn = ∆p (cn ) • ∆qn = ∆q (cn )
❍❡r❡ cn ❛r❡✱ s❛②✱ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❋♦r t❤❡ ❤♦✉♥❞✱ ✇❡ ❤❛✈❡✿
pn+1 = pn + ∆pn , qn+1 = qn + ∆qn .
❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤♦✉♥❞ ✐s ❧♦❝❛t❡❞ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ y ✲❛①✐s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♠♦✈✐♥❣ t♦ ✐ts ❧❡❢t✱ ∆xn < 0✳ ❆s t❤❡ ❤♦✉♥❞ ♠♦✈❡s ❢r♦♠ (pn , qn ) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ (xn , yn )✱ t❤❡ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛r❡ s✐♠✐❧❛r✿ • ✜rst✱ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s✐❞❡s pn − xn ❛♥❞ yn − qn ✱ ❛♥❞ • s❡❝♦♥❞✱ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s✐❞❡s −∆pn ❛♥❞ ∆qn ✳
✺✳✷✳
❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
✸✶✵
❚❤❡s❡ ♦❜s❡r✈❛t✐♦♥s ❣✐✈❡ ✉s t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❢♦r t❤❡ ❤♦✉♥❞✳ ❲❡ ❤❛✈❡ ❢♦r t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❜❛s❡ ❛♥❣❧❡ ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s✿ τn =
∆qn y n − qn = . pn − xn −∆pn
❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r ∆qn ✱
∆qn = −τn · ∆pn ,
✐❢ ✇❡ ❝❛♥ ❥✉st ✜♥❞ ∆pn ✳ ❚❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ t❤❡ ❧❛tt❡r tr✐❛♥❣❧❡ ✐s (∆pn )2 + (∆qn )2 = (v · ∆t)2 .
❲❡ ❞❡r✐✈❡ ❢r♦♠ t❤❡ ❧❛st ❡q✉❛t✐♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 1+
τn2
=1+
∆qn ∆pn
2
=
v · ∆t ∆pn
2
.
❙♦❧✈✐♥❣ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ∆pn ✱ ✇❡ ❝❤♦♦s❡ t❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✿ v ∆pn = − p ∆t . 1 + τn2 ❲❛r♥✐♥❣✦ ■❢ t❤❡ r❛❜❜✐t ❣❡ts t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❤♦✉♥❞✱ ♦♥❡ ❤❛s t♦ ❝❤❛♥❣❡ t❤❡ s✐❣♥✳
❊①❛♠♣❧❡ ✺✳✷✳✹✿ s♣r❡❛❞s❤❡❡t
▲❡t✬s ❝♦♥✜r♠ ♦✉r ❛♥❛❧②s✐s ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ s❡tt✐♥❣ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ∆t = 0.1 s❡❝♦♥❞s • s = 30 ❢❡❡t ♣❡r s❡❝♦♥❞ • v = 31 ❢❡❡t ♣❡r s❡❝♦♥❞ • (x0 , y0 ) = (0, 0) ❢❡❡t • (p0 , q0 ) = (100, 50) ❢❡❡t ❚❤❡ ♠❛✐♥ ❢♦r♠✉❧❛s ❛r❡ t❤❡ ♦♥❡s ❢♦r ∆pn ❛♥❞ ∆qn ✿ ❂✲✭❘✸❈✻✮✴✭❙◗❘❚✭✶✰❘❈❬✲✶❪✂ ✷✮✮✯❘✶❈✷
❚❤❡s❡ ✐s t❤❡ s✐♠✉❧❛t✐♦♥✿
❂✲❘❈❬✲✷❪✯❘❈❬✲✶❪
✺✳✷✳
❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s
✸✶✶
❚❤❡ r❡s✉❧ts ♠❛t❝❤ t❤♦s❡ ❛❝q✉✐r❡❞ ✈✐❛ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧✳
❋r♦♠ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✱ ✇❡ ❞❡r✐✈❡ ✈✐❛ s✉❜st✐t✉t✐♦♥✿
∆t → 0✱
❚❤❡♥✱ ✈✐❛
v v ∆pn ∆t =⇒ = −p ∆pn = − p 2 ∆t 1 + τn 1 + τn2 ∆qn vτn ∆qn = −τn · ∆pn =⇒ =p ∆t 1 + τn2 ✇❡ ❛rr✐✈❡ ❛t t✇♦ t✐♠❡✲❞❡♣❡♥❞❡♥t ❖❉❊s✿
dp v , = −√ dt 1 + τ2 dq vτ , =√ dt 1 + τ2
✇❤❡r❡
τ= ❛♥❞
(x, y)
✐s t❤❡
✜①❡❞
♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✳
◆♦✇✱ ❧❡t✬s s❡❡ ✇❤❛t t❤✐s ❛♥❛❧②s✐s ❧♦♦❦s ❧✐❦❡ ✐♥ t❤❡ t❤❡ ❤♦✉♥❞ ❛♥❞
∆H
♦❢
H
R
y−q , p−x
✈❡❝t♦r ♥♦t❛t✐♦♥✳
❙✉♣♣♦s❡✱ ❛s ❜❡❢♦r❡✱
H
✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢
✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✳ ❚❤❡ ❧❛tt❡r ✐s ❦♥♦✇♥✳ ❋♦r t❤❡ ❢♦r♠❡r✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r
❤❛s t♦ ❜❡ ❝♦❧❧✐♥❡❛r t♦ t❤❡ ✈❡❝t♦r ❢r♦♠
H
t♦
R✳
❚❤❡r❡❢♦r❡✱
∆H · HR = ||∆H|| · ||HR|| = S||HR|| . ❚❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s t♦ ❛♥② ♠✉t✉❛❧ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t ❛♥❞ t❤❡ ❤♦✉♥❞ ❛s ✇❡❧❧ t♦ ♣✉rs✉✐ts ✐♥ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳
✺✳✸✳
❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s
✸✶✷
❊①❡r❝✐s❡ ✺✳✷✳✺
❉❡r✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❖❉❊s ❢♦r t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ r❛❜❜✐t ❢♦❧❧♦✇s ❛ str❛✐❣❤t ♣❛t❤✳
✺✳✸✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s
❲❡ ♣r❡✈✐♦✉s❧② ❞✐s❝✉ss❡❞ ❤♦✇ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ♠♦❞❡❧ ♠♦t✐♦♥ ♦❢ ♦❜❥❡❝ts ❛✛❡❝t❡❞ ❜② ❢♦r❝❡s✳ ❆ s♣❡❝✐❛❧ ❝❛s❡ ✐s ❛ ❧✐♥❡❛r ❖❉❊✿ ∆x ∆2 x = a + bx ❛♥❞ x′′ = ax′ + bx . 2 ❍❡r❡✱ ✇❡ ❤❛✈❡✿
∆t
∆t
• x ✐s t❤❡ ♣♦s✐t✐♦♥✳
∆x ❛♥❞ x′ ✐s t❤❡ ✈❡❧♦❝✐t②✳ ∆t ∆2 x • ❛♥❞ x′′ ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ∆t2 ▼❡❛♥✇❤✐❧❡ a ❛♥❞ b ❛r❡ ❝♦♥st❛♥t ♥✉♠❜❡rs t❤❛t ❡①♣r❡ss t❤❡ ♣r♦♣♦rt✐♦♥❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❢♦r❝❡ ✭✐✳❡✳✱ t❤❡ •
❛❝❝❡❧❡r❛t✐♦♥✮ ♦♥ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♦❜❥❡❝t ♠❛② ❜❡ ❛tt❛❝❤❡❞ t♦ t❤❡ ✇❛❧❧ ❜② ❛ s♣r✐♥❣ ❛♥❞✱ ❛t t❤❡ s❛♠❡ t✐♠❡✱ ❜❡ ❛✛❡❝t❡❞ ❜② t❤❡ ❢r✐❝t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡✳
■♥ ❢❛❝t✱ ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ s♣r✐♥❣ ❤❛s ✐ts ❍♦♦❦❡✬s ❢♦r❝❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✱ ✇✐t❤ ❝♦❡✣❝✐❡♥t b✱ ❛♥❞ s✐♥❝❡ ✐t ♣✉❧❧s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t b ≤ 0❀ ❛♥❞
• ❚❤❡ ❢r✐❝t✐♦♥ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❧♦❝✐t②✱ ✇✐t❤ ❝♦❡✣❝✐❡♥t a✱ ❛♥❞ s✐♥❝❡ ✐t ♣✉❧❧s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ t♦ t❤❡ ✈❡❧♦❝✐t②✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t a ≤ 0✳
❚❤❡s❡ ✇✐❧❧ ❜❡ ♦✉r ❛ss✉♠♣t✐♦♥s✿
a ≤ 0, b < 0 .
Pr❡✈✐♦✉s❧②✱ ✇❡ ✉s❡❞ t❤❡ ❞✐s❝r❡t❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡ t♦ t❤❡ ♠♦t✐♦♥ ♦❢ s♣r✐♥❣ ❛♥❞ ♦t❤❡r ❞②♥❛♠✐❝s✳ ❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ❡q✉❛t✐♦♥❄ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡ a = 0✱ ✐t ✇❛s ❛❧s♦ s♦❧✈❡❞✳ ■♥ ♦✉r ❛tt❡♠♣t t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❡ r❡❛❧✐③❡ t❤❛t ✇❡ ❤❛✈❡ ♥♦ ♠❡t❤♦❞s ❞❡✈❡❧♦♣❡❞ ❢♦r ❡q✉❛t✐♦♥s ♦❢ s❡❝♦♥❞ ♦r❞❡r✦ ❲❤❡♥ ✇❡ ❡♥❝♦✉♥t❡r ❛ ❝♦♠♣❧❡t❡❧② ♥❡✇ s✐t✉❛t✐♦♥✱ ✇❡ s❤♦✉❧❞ ❛❧✇❛②s tr② t♦ r❡❞✉❝❡ ✐t t♦ s♦♠❡t❤✐♥❣ ❢❛♠✐❧✐❛r✳ ❲❡ ✇✐❧❧ ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❡✈❡r
tr✐❝❦✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❡①tr❛ ✭❞❡♣❡♥❞❡♥t✮ ✈❛r✐❛❜❧❡✿ y = x′ .
✺✳✸✳
❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s
✸✶✸
■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ t❤❡ tr✐❝❦ ✐s ♥♦t❤✐♥❣ ❜✉t r❡✲✐♥tr♦❞✉❝✐♥❣ t❤❡ ✈❡❧♦❝✐t② ❜❛❝❦ ✐♥t♦ t❤❡ ♣✐❝t✉r❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ s②st❡♠ ♦❢ t✇♦ ❖❉❊s✿ ′ x = y, ′ y = bx + ay. ❚❤❡ r❡s✉❧t ✐s ❛ tr❛❞❡✲♦✛✿ ✐♥❝r❡❛s✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥ ✕ ❢r♦♠ 1 t♦ 2 ✕ ❜✉t ❞❡❝r❡❛s✐♥❣ t❤❡ ♦r❞❡r ✕ ❢r♦♠ 2 ❢♦r 1 ✕ ♦❢ t❤❡ s②st❡♠✳ ❚❤✐s 2 × 2 ❧✐♥❡❛r ✈❡❝t♦r ❖❉❊✱ 0 1 ′ , X = F X, ✇✐t❤ F = b a ❝❛♥ ♥♦✇ ❜❡ s✉❜❥❡❝t❡❞ t♦ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❛❧②s✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✳ ❋✐rst✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s✿
χ(λ) = λ2 − aλ − b .
❙✉♣♣♦s❡ λ1 ❛♥❞ λ2 ❛r❡ ✐ts t✇♦ r♦♦ts✿
√ a2 + 4b a . λ1,2 = ± 2 2 ❚❤❡♥ t❤❡ ♦✉t❝♦♠❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t✱ D = a2 + 4b . ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ D ≥ 0✳ ❙✐♥❝❡ a ≤ 0✱ ✇❡ ❤❛✈❡
a λ1 = − 2
√
a2 + 4b < 0. 2
❙✐♥❝❡ b < 0✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t
a λ1 = + 2
√
√ a a −a a2 + 4b a2 < + = + =0 2 2 2 2 2
❚❤❡r❡❢♦r❡✱ ❛♥ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t
t❤❡ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❛❧✇❛②s ♥❡❣❛t✐✈❡✳
❆❝❝♦r❞✐♥❣ t♦ ♦✉r ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ ❝❛s❡s✿
• ❈❛s❡ ✶✿ t✇♦ ❞✐st✐♥❝t r❡❛❧ r♦♦ts❀
• ❈❛s❡ ✷✿ ♦♥❡ r❡❛❧ r❡♣❡❛t❡❞ r♦♦t❀
• ❈❛s❡ ✸✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ r♦♦ts✳
❈❛s❡ ✶ ✐s t❤❡ ❝❛s❡ ♦❢
❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②
D = a2 + 4b > 0 . x = Ceλ1 t + Keλ2 t .
❆s λ1 , λ2 < 0✱ ✇❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ✭❛ st❛❜❧❡ ♥♦❞❡ ♦❢ t❤❡ ✈❡❝t♦r ❖❉❊✮✿ t❤❡ ❢r✐❝t✐♦♥ ❜r✐♥❣s t❤❡ ❖❉❊ ❜❛❝❦ t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✇✐t❤♦✉t ♦s❝✐❧❧❛t✐♥❣✳ ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤ a = −8, b = −1✿
✺✳✸✳
❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s
✸✶✹
❈❛s❡ ✷ ✐s t❤❡ ❝❛s❡ ♦❢
D = a2 + 4b = 0 . ❆❣❛✐♥✱ ❛s
λ = λ1 = λ2 < 0 ✱
✇❡ s❡❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ✭❛ st❛❜❧❡ ✐♠♣r♦♣❡r ♥♦❞❡ ♦❢ t❤❡ ❖❉❊✮✿ t❤❡ ❖❉❊✬s
♠♦t✐♦♥ ❞✐❡s ♦✉t✱ ❡✈❡♥ t❤♦✉❣❤ ✐t ♠✐❣❤t ♦✈❡rs❤♦♦t✳ ❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②
x = (C + Kt)eλt . ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤
a = −2, b = −1✿
❈❛s❡ ✸ ✐s t❤❡ ❝❛s❡ ♦❢
D = a2 + 4b < 0 . ❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②
x = Ceαt cos(βt) + Keαt sin(βt) , ✇❤❡r❡
α = Re λ1,2 = ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ❜② ❛ss✉♠♣t✐♦♥ ❛♥❞
β = Im λ1,2 = ± ✐s ♥♦♥✲③❡r♦ ❜❡❝❛✉s❡
D 6= 0✳
❲❤❡♥
a 2 √
D 2
a < 0✱ ✇❡ s❡❡ ❛♥ ♦s❝✐❧❧❛t✐♥❣ ❞❡❝❛② ✭❛ st❛❜❧❡ ❢♦❝✉s ♦❢ t❤❡ ❖❉❊✮✿ t❤❡ ❢r✐❝t✐♦♥
✐s str♦♥❣ ❡♥♦✉❣❤ t♦ ❜r✐♥❣ t❤❡ ❖❉❊ t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❡✈❡♥t✉❛❧❧② ❜✉t ✐s♥✬t str♦♥❣ ❡♥♦✉❣❤ t♦ st♦♣ t❤❡ ❖❉❊ ❢r♦♠ ♦s❝✐❧❧❛t✐♥❣✳ ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤
a = 0✱ ✇❡ a = 0, b = −1✿
❲❤❡♥
a = −.5, b = −1✿
s❡❡ ♣✉r❡ ♦s❝✐❧❧❛t✐♦♥ ✭❛ ❝❡♥t❡r ♦❢ t❤❡ ❖❉❊✮✿ ♥♦ ❢r✐❝t✐♦♥✳
❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤
✺✳✹✳
✸✶✺
❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
▲♦♦❦✐♥❣ ❛t t❤❡s❡ ✐❧❧✉str❛t✐♦♥ ❛s ❛ s❡q✉❡♥❝❡ r❡✈❡❛❧s t❤❡ ❡✛❡❝t ♦❢ ❣r♦✇✐♥❣ ♦❢ t❤❡ ❢r✐❝t✐♦♥ ❝♦❡✣❝✐❡♥t b✳ ❊①❡r❝✐s❡ ✺✳✸✳✶
P♦✐♥t ♦✉t ✐♥ ✇❤❛t ✇❛② t❤❡ ❧❛st ✐❧❧✉str❛t✐♦♥ ❞♦❡s♥✬t ♠❛t❝❤ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❛♥❞ ❡①♣❧❛✐♥ ✇❤②✳
✺✳✹✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
❲❤❛t ♦❢ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❢♦r❝❡s r❡♠❛✐♥s t❤❡ s❛♠❡ ❜✉t t❤❡ ♠♦t✐♦♥ ✐s
♦♥ t❤❡ ♣❧❛♥❡
❄
■♥ t❤❡ s✐♠♣❧❡st s✐t✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s✿
• ❚✇♦ ❞✐✛❡r❡♥t s♣r✐♥❣s ❛r❡ ❛tt❛❝❤❡❞ t♦ t❤❡ ♦❜❥❡❝t ❛❧♦♥❣ t❤❡ t✇♦ ❛①❡s✳ • ❚✇♦ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ❢r✐❝t✐♦♥ ❛r❡ ♣r♦❞✉❝❡❞ ❛❧♦♥❣ t❤❡ t✇♦ ❛①❡s✳
❆s t❤❡ ♦❜❥❡❝t ♠♦✈❡s✱ ✇❡ ❤❛✈❡ t✇♦ ❡✛❡❝ts✿
• ❚✇♦ s♣r✐♥❣s ❡①❡rt t✇♦ ❢♦r❝❡s ❛❧♦♥❣ t❤❡ ❛①❡s ✕ ♥❡❣❛t✐✈❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ • ❚✇♦ ❦✐♥❞s ♦❢ ❢r✐❝t✐♦♥ ❡①❡rt t✇♦ ❢♦r❝❡s ❛❧♦♥❣ t❤❡ ❛①❡s ✕ ♥❡❣❛t✐✈❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❧♦❝✐t②✳
✺✳✹✳
✸✶✻
❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
❚❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✇✐❧❧ ❜❡ t✇♦ ❧✐♥❡❛r ❖❉❊ ♦❢ ♦r❞❡r 2 ♦❢ t❤❡ s❛♠❡ ❦✐♥❞s ❛s ❜❡❢♦r❡✿ x′′ = ax′ + bx ❛♥❞ y ′′ = cy ′ + dy ,
✇❤❡r❡
a ≤ 0 ❛♥❞ b ≤ 0 ❛♥❞ c ≤ 0 ❛♥❞ d ≤ 0 .
■♥ t❤✐s ❝❛s❡✱ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❛♣♣❧✐❡s✱ s❡♣❛r❛t❡❧②✳ ❲❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ s❝❛❧❛r ❡q✉❛t✐♦♥s ✐♥t♦ ♦♥❡ ✈❡❝t♦r ❡q✉❛t✐♦♥✿
x′′ = ax′ +bx y ′′ = cy ′ +dy
❧❡❛❞✐♥❣ t♦
●❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ❖❉❊✿
x′′ y ′′
=
a 0 0 c
x′ y′
+
b 0 0 d
x y
X ′′ = AX ′ + BX .
❍❡r❡✱ X=
x y
,
′
X =
x′ y′
′′
,
X =
x′′ y ′′
❛r❡ t❤❡ ♣♦s✐t✐♦♥ ✈❡❝t♦r✱ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❆❧s♦✱ A ❛♥❞ B ❛r❡ t✇♦ 2 × 2 ♠❛tr✐❝❡s t❤❛t ❡①♣r❡ss t❤❡ ✭❧✐♥❡❛r✮ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❢♦r❝❡ ✭✐✳❡✳✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✮ ♦♥ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡② ❞♦♥✬t ❤❛✈❡ t♦ ❜❡ ❞✐❛❣♦♥❛❧ ❛♥②♠♦r❡✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❞✐s❝r❡t❡ ❖❉❊s✿ ∆X ∆2 X = A + BX . ∆t2 ∆t
❍❡r❡✱ ❛ s♦❧✉t✐♦♥ X ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ t0 , t1 , ... ✭✐✳❡✳✱ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠✮✱ s♦ t❤❛t ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿
∆2 x ∆2 X ∆t2 = 2 . ∆y ∆t2 ∆t2
∆x ∆X ∆t = , ∆y ∆t ∆t
x X = , y
■♥ ♦r❞❡r t♦ s✐♠♣❧✐❢② t❤❡ ♥♦t❛t✐♦♥✱ ✇❡ ❞❡✜♥❡ t❤r❡❡ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs ✐♥ R2 ✿ ∆2 X (tn ) ∆t2 ∆X (tn ) Xn′ = ∆t Xn′′ =
Xn = X( tn )
❚❤❡ ❖❉❊ t❤❡♥ ♣r♦❞✉❝❡s t❤❡s❡ t❤r❡❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤♦s❡ t❤r❡❡ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs✿ ′′ Xn+1 = AXn′ + BXn ′ ′′ Xn+1 = Xn′ + Xn+1 ∆t ′ Xn+1 = Xn + Xn+1 ∆t
❲❡ ❛♣♣❧② t❤❡♠ ❜❡❧♦✇✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ t❤❡ t✇♦ s♣r✐♥❣s✳ ❍❡r❡✱ t❤❡ ♠❛tr✐❝❡s ❛r❡ ❞✐❛❣♦♥❛❧ ✿ A=
✇✐t❤
ax 0 0 ay
,
B=
bx 0 0 by
ax , ay ≤ 0 ❛♥❞ bx , by < 0 .
,
✺✳✹✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
✸✶✼
❇❡❧♦✇✱ ✇❡ ✇✐❧❧ ❜❡ ❣r❛❞✉❛❧❧② ❛❞❞✐♥❣ ♥❡✇ ❢♦r❝❡s t♦ t❤❡ s❡t✉♣ ❜② ❛❞❞✐♥❣ ❛ ♥❡✇ ♥♦♥✲③❡r♦ ❡♥tr② t♦ t❤❡ ♠❛tr✐❝❡s✳ ❚❤❡ r❡s✉❧ts ❛r❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐s❝r❡t❡ ♠♦❞❡❧s ♣r♦❞✉❝❡❞ ❜② ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇✐❧❧ ❜❡ t❤❡ s❛♠❡✿
X0 =
1 0
,
X0′
=
1 1
.
x✲❛①✐s✿ −0.5 0 0 0 . , B= A= 0 0 0 0
❲❡ st❛rt ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t ❛ s♣r✐♥❣ ❛❧♦♥❣ t❤❡
❚❤❡ r❡s✉❧t ✐s ❛♥ ♦s❝✐❧❧❛t✐♦♥ ❛❧♦♥❣ t❤❡
x✲❛①✐s
❛♥❞ ✉♥✐❢♦r♠ ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡
y ✲❛①✐s✿ 0 0 , A= 0 0
y ✲❛①✐s✳
❲❡ ♥♦✇ ❛❞❞ ❛♥ ✐❞❡♥t✐❝❛❧ s♣r✐♥❣ ❢♦r t❤❡
❆s ❛ r❡s✉❧t✱ ❜♦t❤
x
❛♥❞
y
B=
−0.5 0 0 −0.5
.
♦s❝✐❧❧❛t❡ ❛t t❤❡ s❛♠❡ ♣❡r✐♦❞ ❛♥❞ ❝♦♠❡ ❜❛❝❦ s✐♠✉❧t❛♥❡♦✉s❧②❀ ✐t✬s ❛ ❝②❝❧❡✳ ❚❤❡ ♣❛t❤
❛♣♣❡❛rs t♦ ❜❡ ❛♥ ❡❧❧✐♣s❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐❣✐♥✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ ♠♦t✐♦♥ ✐s s❧♦✇s ❞♦✇♥ ❛t t❤❡ t✐♣s ♦❢ t❤✐s ❡❧❧✐♣s❡✳
❊①❡r❝✐s❡ ✺✳✹✳✶ ❲❤② ✐s♥✬t t❤❡ tr❛❥❡❝t♦r② ❛ ❝✐r❝❧❡❄
▲❡t✬s ♠❛❦❡ t❤❡ ❧❛tt❡r s♣r✐♥❣ ♠♦r❡ r✐❣✐❞ ✿
A=
0 0 0 0
,
B=
−0.5 0 0 −1
.
✺✳✹✳
✸✶✽
❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
❆s ❛ r❡s✉❧t✱ x ❛♥❞ y st✐❧❧ ♦s❝✐❧❧❛t❡ ❜✉t ❛t ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s✳ ❚❤❡② ♠❛② ❝♦♠❡ ❜❛❝❦ s✐♠✉❧t❛♥❡♦✉s❧② ❛❢t❡r s❡✈❡r❛❧ ♣❡r✐♦❞s✳ ❚❤❛t ✇✐❧❧ ❝r❡❛t❡ ❛ ❝②❝❧❡✳ ❲❡ ♥♦✇ ❛❞❞ ❢r✐❝t✐♦♥ ✐♥ t❤❡ x✲❞✐r❡❝t✐♦♥✿ A=
−0.3 0 0 0
,
B=
−0.5 0 0 −1.
.
❲❤✐❧❡ y st✐❧❧ ♦s❝✐❧❧❛t❡s✱ t❤❡ x✲♣♦s✐t✐♦♥✬s ♦s❝✐❧❧❛t✐♦♥ ❞✐♠✐♥✐s❤❡s ❛s ✐t ❛♣♣r♦❛❝❤❡s 0✳ ❲❡ ❛❧s♦ ❛❞❞ ❢r✐❝t✐♦♥ ✐♥ t❤❡ y ✲❞✐r❡❝t✐♦♥ ♥♦✇✿ A=
−0.3 0 0 −0.2
,
B=
−0.5 0 0 −1.0
.
❇♦t❤ x ♦r y ♦s❝✐❧❧❛t✐♦♥s ❞✐♠✐♥✐s❤ ❛♥❞ t❤❡ ♣♦✐♥t ❛♣♣r♦❛❝❤❡s 0✳ ❯♣ ✉♥t✐❧ ♥♦✇✱ x ❛♥❞ y ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ t✇♦ ❖❉❊s ❛r❡ s♦❧✈❛❜❧❡ s❡♣❛r❛t❡❧②✳ ❚❤✐s ✐s ✇❤② ❡✈❡r②t❤✐♥❣ ✇❡ s❡❡ ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ❜② t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❋r♦♠ ♥♦✇ ♦♥✱ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ❞♦❡s♥✬t ❛♣♣❧② ❛♥②♠♦r❡✳ ◆♦t❡ ❛❧s♦ t❤❛t t❤❡ ❝❧❡✈❡r tr✐❝❦ ♦❢ ✐♥tr♦❞✉❝✐♥❣ t❤❡ ✈❡❧♦❝✐t✐❡s ❛s ♥❡✇ ✈❛r✐❛❜❧❡s ✇✐❧❧ ♣r♦❞✉❝❡ 4 × 4 ♠❛tr✐❝❡s✱ t❤❡ ❛♥❛❧②s✐s ♦❢ ✇❤✐❝❤ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s t❡①t✳ ❚❤✐s ✐s ✇❤② ✇❡ ♣r♦❝❡❡❞ ✇✐t❤ ❞✐s❝r❡t❡ ♠♦❞❡❧s ♦♥❧②✳
✺✳✹✳
✸✶✾
❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
❲❡ ✐♥t❡r♠✐① t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ♥♦✇ ❜② ♠❛❦✐♥❣ t❤❡ ❢r✐❝t✐♦♥ ✈❛r② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦t❤❡r t❤❛♥ t❤❡ t✇♦ ❛①❡s ✭❢♦r ❡①❛♠♣❧❡✱ t❤❡ s♣r✐♥❣s ♠❛② ❜❡ ♣❧❛❝❡❞ ✉♥❞❡r ✈❛r✐♦✉s ❛♥❣❧❡s✮✿ A=
−0.3 −0.3 0.6 −0.2
,
B=
−0.5 0 0 −1
.
❚❤❡ ♣❛tt❡r♥ ♦❢ ❜❛❝❦✲❛♥❞✲❢♦rt❤ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ 0 r❡♠❛✐♥s✳ ❆s ❛ ✜♥❛❧ ❡①♣❡r✐♠❡♥t✱ ✇❡ r❡♠♦✈❡ t❤❡ y ✲❛①✐s s♣r✐♥❣✿ A=
−0.3 −0.3 0.6 −0.2
,
B=
−0.5 0 0 0
.
❚❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ t❤❡ y ✲❝♦♦r❞✐♥❛t❡ ♠♦✈❡s ❛✇❛② ❢r♦♠ 0✳ ▲❡t✬s ❧♦♦❦ ❛t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❡♥❡r❣② ✐♥ t❤✐s ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❇✉t ✜rst ❧❡t✬s s❡❡ ✇❤❛t t❤❡ ❞❛t❛ t❡❧❧s ❛❜♦✉t t❤❡ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥✳ ❋♦r s✐♠♣❧✐❝✐t② ✇❡ ❝♦♥t✐♥✉❡ t♦ ❛ss✉♠❡ t❤❛t t❤❡ ♠❛ss ✐s ❡q✉❛❧ t♦ 1✳ ❚❤❡♥ t❤❡ t❤✐s ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿
❦✐♥❡t✐❝ ❡♥❡r❣②
✐s ❦♥♦✇♥ t♦ ❜❡
1 K(t) = ||X ′ (t)||2 . 2
❊✉❧❡r✬s ♠❡t❤♦❞ s❛♠♣❧❡s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ❣✐✈❡s ✉s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✱ t❤❡ ❡♥❡r❣② ✿ 1 Kn = ||Xn′ ||2 , 2
✐♥ ❡❛❝❤ r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ◆❡①t✱ s✉♣♣♦s❡ t❤❛t t❤❡ ❜❡ ❢♦✉♥❞ ❛s t❤❡ ✇♦r❦ ♦❢ t❤✐s ❢♦r❝❡✿ Z t W (t) = −
0
s❛♠♣❧❡❞ ❦✐♥❡t✐❝
✳ ❚❤❡♥ t❤❡ ♣♦t❡♥t✐❛❧
❢♦r❝❡ ✐s ❝♦♥s❡r✈❛t✐✈❡
❡♥❡r❣②
❝❛♥
X ′′ · X ′ dt .
❲❤❛t ❞♦❡s t❤❡ ❞❛t❛ t❡❧❧ ✉s ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥❄ ❖♥❝❡ ❛❣❛✐♥✱ ❊✉❧❡r✬s ♠❡t❤♦❞ s❛♠♣❧❡s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ❣✐✈❡s ✉s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✱ t❤❡ s❛♠♣❧❡❞ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✳ ❖✈❡r ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡ ❢r♦♠ tk−1 t♦ tk ✭❧❡♥❣t❤ ∆t✮✱ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ❣r♦✇s ❜②✿ Wk = −Xk′′ · Xk′ ∆t .
✺✳✹✳
✸✷✵
❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣
❲❡ ❛❞❞ t❤❡s❡ ✐♥ ❛❧❧ r♦✇s ✉♣ t♦ t❤❡ nt❤ t♦ ✜♥❞ t❤❡ ❝✉rr❡♥t s❛♠♣❧❡❞ Pn =
n X
♣♦t❡♥t✐❛❧ ❡♥❡r❣②
✿
Wk .
k=0
❚❤❡ ♥❡①t ❝♦❧✉♠♥ ❝♦♥t❛✐♥s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦✱ t❤❡ t♦t❛❧
s❛♠♣❧❡❞ ❡♥❡r❣②
✿
n X 1 ′ 2 Xk′′ · Xk′ ∆t . En = Kn + Pn = ||Xn || − 2 k=0
❆s ②♦✉ ❝❛♥ s❡❡✱ ✐t✬s ♥♦t ❝♦♥s❡r✈❡❞✦
❚❤❡ r❡s✉❧t ✐s t♦ ❜❡ ❡①♣❡❝t❡❞ ❢r♦♠ s❛♠♣❧✐♥❣ ✭❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢✮ ❛ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥✳
❉❡✜♥✐t✐♦♥ ✺✳✹✳✷✿ ❞✐s❝r❡t❡ ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❆ ❞✐s❝r❡t❡
❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r
✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿
′′ Xn+1 = f (Xn , Xn′ ) ′ ′′ Xn+1 = Xn′ + Xn+1 ∆t ′ Xn+1 = Xn + Xn+1 ∆t
❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ❛ ♥♦♥✲③❡r♦ ♥✉♠❜❡r ∆t✳ ■ts ♦❢ ✈❡❝t♦rs t❤❛t s❛t✐s✜❡s t❤❡s❡ ❡q✉❛t✐♦♥s✳
s♦❧✉t✐♦♥
✐s t❤r❡❡ s❡q✉❡♥❝❡s
❚❤❡ ❞✐s❝r❡t❡ ❖❉❊ ❝♦♥s❡r✈❡s ❡♥❡r❣② ✐❢ ✇❡ ❥✉st ❧♦♦❦ ❛t t❤❡ ✇♦r❦ t❤❡ r✐❣❤t ✇❛②✳ ❲❡ r❡❝♦♠♣✉t❡ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ✇✐t❤ t❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② Xk′ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❝✉rr❡♥t ❛♥❞ t❤❡ ♣r❡✈✐♦✉s ✈❡❧♦❝✐t✐❡s✿ 1 ′ + Xk′ ) ∆t . Wk = Xk′′ · (Xk−1 2
❚❤❡ ❧❛st ❝♦❧✉♠♥ ❝♦♥✜r♠s t❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣②✳ ❚❤✐s ❢❛❝t ✐s ❛❧s♦ ❡❛s② t♦ ♣r♦✈❡ ❛❧❣❡❜r❛✐❝❛❧❧② ❢♦r ❛❧❧ ❧✐♥❡❛r ♦r ♥♦♥✲❧✐♥❡❛r ❞✐s❝r❡t❡ s②st❡♠s✳
❚❤❡♦r❡♠ ✺✳✹✳✸✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❊♥❡r❣② ❚❤❡
❡♥❡r❣②
✐s ❝♦♥st❛♥t✳
♦❢ ❛ s♦❧✉t✐♦♥ X ♦❢ ❛ ❞✐s❝r❡t❡ ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞❡✜♥❡❞ ❛s n X 1 1 ′ ′ 2 En = ||Xn || − + Xk′ ) ∆t , Xk′′ · (Xk−1 2 2 k=0
✺✳✺✳
❆ ♣❡♥❞✉❧✉♠
✸✷✶
Pr♦♦❢✳
1 1 1 ′ ′′ ′ En+1 − En = ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 · (Xn′ + Xn+1 ) ∆t 2 2 2 1 ′ 1 1 ′ ′ ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 − Xn′ · (Xn+1 + Xn′ ) ∆t = 2 ∆t 2 1 1 ′ ′ ′ = ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 · Xn+1 − Xn′ · Xn′ 2 2 = 0.
✺✳✺✳ ❆ ♣❡♥❞✉❧✉♠
❉❡s❝r✐♣t✐♦♥✿ ✏❛♥ ♦❜❥❡❝t ✐s ❛tt❛❝❤❡❞ t♦ ❛ ✜①❡❞ ♣♦✐♥t ✇✐t❤ ❛ str✐♥❣ ❛♥❞ s✉❜❥❡❝t❡❞ t♦ t❤❡ ❣r❛✈✐t②✑✳ ▲❡t
x
❜❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❞
z
t❤❡ ✈❡rt✐❝❛❧✳
❲❡ ❝♦♠♣✉t❡ ✇❤❛t✬s ♥❡❝❡ss❛r②✱ ✐✳❡✳✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❢♦r t❤❡ s♣r❡❛❞s❤❡❡t ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ str✐♥❣ ✐s
L✳
❋✐rst✱ t❤❡
❣r❛✈✐t② ✐s t❤❡ ❝♦♥st❛♥t ✈❡❝t♦r
G =< 0, −gm > .
❚❤❡♥ t❤❡
0
r❡s✐st❛♥❝❡ R ♦❢ t❤❡ str✐♥❣ ✐s ❢♦✉♥❞ ❛s t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛✈✐t② ♦♥ t❤❡ ❧✐♥❡ ❢r♦♠
t♦ t❤❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥
X =< x, z >✿
R=− ❚❤❡♥ ✇❡ ✜♥❞ t❤❡
G·X G·X X = − X. ||X||2 L2
t❛♥❣❡♥t✐❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❣r❛✈✐t② ✿
F = R + G. ❖r✱ ✐t ✐s s✐♠♣❧②✿
F = G sin θ , ✇❤❡r❡
θ
✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ str✐♥❣ ✇✐t❤ t❤❡ ✈❡rt✐❝❛❧✳ ◆♦t❡ t❤❛t t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢♦r❝❡ ✐s ③❡r♦
✇❤❡♥ t❤❡ str✐♥❣ ✐s ✈❡rt✐❝❛❧ ❛♥❞ ✇❤❡♥ ✐t ✐s ❤♦r✐③♦♥t❛❧ ❛♥❞ ✐t r❡✈❡rs❡s ✐ts ❞✐r❡❝t✐♦♥✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ✐s ❛❧✇❛②s ♥❡❣❛t✐✈❡✳ ❆♥❞ t❤❡ t❛♥❣❡♥t✐❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s✿
A = F/m .
✺✳✺✳
❆ ♣❡♥❞✉❧✉♠
✸✷✷
❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❛t ♦r❞❡r ❥✉st ❛s ❜❡❢♦r❡✳ ❚❤❡ ✉♣❞❛t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥s ❛r❡ ❛❧s♦ ❢♦✉♥❞ ❡①❝❡♣t z ✐s ❢♦✉♥❞ ❢r♦♠ x ❜②✿
x 2 + y 2 = L2 , ✇❤❡r❡ L ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ str✐♥❣✳ ❊q✉✐✈❛❧❡♥t❧②✱
x = L cos θ, y = L sin θ . ❲❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❡✈❛❧✉❛t❡ t❤❡s❡ ❝♦❧✉♠♥ ❜② ❝♦❧✉♠♥ ❛♥❞ t❤❡♥ ❢♦r ❡❛❝❤ ✐t❡r❛t✐♦♥ ♦❢ t✿
❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ s✉❣❣❡sts t❤❛t ✇❡ ✐♥❞❡❡❞ ❤❛✈❡ ❛ ♣❡♥❞✉❧✉♠✳ ❋♦r t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ ❣✐✈❡ t❤❡ ♣❡♥❞✉❧✉♠ ❛♥♦t❤❡r ✕ ❤♦r✐③♦♥t❛❧ ✕ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠✱ y ✳ ❚❤❡ ✈❡❝t♦r ❛♥❛❧②s✐s r❡♠❛✐♥s t❤❡ s❛♠❡ ✇✐t❤ G =< 0, 0, −gm > ❛♥❞ X =< x, y, z >✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ♦♥❧② r❡q✉✐r❡s ❛❞❞✐♥❣ ❝♦❧✉♠♥s ❢♦r y ✳ ❚❤❡ ✈❛❧✉❡s ♦❢ z ✐s ❢♦✉♥❞ ❢r♦♠✿
x 2 + y 2 + z 2 = L2 . ❇❡❧♦✇ ✇❡ ♣❧♦t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ < x, y > ✇✐t❤ ❛ ♥♦♥✲③❡r♦ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿
✺✳✺✳
✸✷✸
❆ ♣❡♥❞✉❧✉♠
❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ♣❡♥❞✉❧✉♠ s✇✐♥❣s ❛s ❜❡❢♦r❡ ❜✉t t❤❡ ♣❧❛♥❡ ♦❢ t❤❡ s✇✐♥❣s ✐s ❝♦♥t✐♥✉♦✉s❧② r♦t❛t✐♥❣✳ ❲❡ ♥♦✇ ❣♦ ❜❛❝❦ t♦ t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ❛♥❞ ❛❞❞r❡ss ✐t ❛♥❛❧②t✐❝❛❧❧②✳ ❲❡ ❛♣♣❧② ◆❡✇t♦♥✬s ❧❛✇ t♦ t❤❡ t❛♥❣❡♥t✐❛❧ ❛①✐s ♦♥❧②✿ F = −mg sin θ = ma =⇒ a = −g sin θ .
❍♦✇ ✐s t❤✐s ❧✐♥❡❛r ❛❝❝❡❧❡r❛t✐♦♥ a ❛❧♦♥❣ t❤❡ t❛♥❣❡♥t r❡❧❛t❡❞ t♦ t❤❡ ❝❤❛♥❣❡ ✐♥ ❛♥❣❧❡ θ❄ ▲❡t s ❜❡ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❙✐♥❝❡ t❤❡ ❝✉r✈❡ ✐s ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s L✱ ✇❡ ❤❛✈❡✿ s = Lθ .
◆♦✇✱ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✱ ✐✳❡✳✱ s = s(t)✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t✇✐❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t✿ s = Lθ =⇒ v = s′ = Lθ′ =⇒ a = x′′ = Lθ′′ .
❚❤❡r❡❢♦r❡✱ ♦r
Lθ′′ = −g sin θ , θ′′ +
❚❤✐s ✐s ❛ ♥♦♥✲❧✐♥❡❛r ❖❉❊ ✇✐t❤ r❡s♣❡❝t t♦ θ✳
g sin θ = 0. L
❚♦ ❝♦♥✜r♠ ♦✉r s✐♠✉❧❛t✐♦♥s✱ ✇❡ ♣❧♦t ❛ ❢❡✇ s♦❧✉t✐♦♥s ✉s✐♥❣ ❊✉❧❡r✬s ♠❡t❤♦❞ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✿
❍❡r❡ θ ✐s ♣❧♦tt❡❞ ❛❣❛✐♥st α = θ′ ❀ ✐✳❡✳✱ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ (
θ′ = α, g α′ = − sin θ. L
❚❤❡ ❊✉❧❡r✬s s♦❧✉t✐♦♥s ❛r❡♥✬t ♣❡r✐♦❞✐❝ ❜✉t r❛t❤❡r ❛♣♣❡❛r t♦ ❜❡ s♣✐r❛❧s✦ ▲❡t✬s ❧✐♥❡❛r✐③❡ ✦ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❡q✉✐❧✐❜r✐✉♠ θ = α = 0✳ ◆♦✇✱ d sin θ (0) = 1 . dθ
✺✳✻✳
P❧❛♥❡t❛r② ♠♦t✐♦♥
✸✷✹
❚❤❡r❡❢♦r❡✱ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠s ✐s✿
(
θ′ = α, g α′ = − θ. L
❲❡ ❝❛♥ ❧♦♦❦ ❛t t❤❡ ❡✐❣❡♥✈❛❧✉❡s✿
A=
"
0 1 g − 0 L
❆❝❝♦r❞✐♥❣ t♦ t❤❡
#
=⇒ χA (λ) = det(A − λI) = det
❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■■
"
−λ 1 g − −λ L
#
g = λ + = 0 =⇒ λ1,2 = ± L 2
✐♥ ❈❤❛♣t❡r ✷✱ t❤❡ s②st❡♠ ❤❛s ❛
❝❡♥t❡r
❛t
0✦
r
g i. L
❖❢ ❝♦✉rs❡✱
✇❡ ❝❛♥ ❥✉st s♦❧✈❡ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠✿
g θ = α = − θ =⇒ θ = θ0 cos L ′′
′
r
g t L
,
✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿
θ(0) = θ0
❛♥❞
θ′ (0) = 0 .
❙♦✱ ♦✉r ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ✇❤❡♥ s♠❛❧❧ t❤❡ s✇✐♥❣s ❛r❡ ❝❧♦s❡ t♦ ♣❡r✐♦❞✐❝ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡r✐♦❞✿
2π ❚❤❡ ♣❡r✐♦❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❛♠♣❧✐t✉❞❡
r
g . L
θ0 ✦
✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥
❆ ❢❛♠✐❧✐❛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 ✶✳ ♣r♦♣♦rt✐♦♥❛❧ t♦ ❡✐t❤❡r ♦❢ t❤❡✐r ♠❛ss❡s✱ ✷✳ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ❝❡♥t❡rs✱ ❛♥❞ ✸✳ ❞✐r❡❝t❡❞ ❛❧♦♥❣ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡♠✳
✺✳✻✳
✸✷✺
P❧❛♥❡t❛r② ♠♦t✐♦♥
■♥ ♦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 ♦❢ t❤❡ ♠❛ss❡s✳
▲❡t✬s ❝♦♥♥❡❝t t❤✐s ❢❛♠✐❧✐❛r ♣❤②s✐❝s ❧❛✇ t♦ ❛♥♦t❤❡r✳
❑❡♣❧❡r✬s ▲❛✇s ♦❢ P❧❛♥❡t❛r② ▼♦t✐♦♥✿ ❚❤❡ ♠♦t✐♦♥ ♦❢ ♣❧❛♥❡ts ❛r♦✉♥❞ t❤❡ ❙✉♥ ❢♦❧❧♦✇s t❤❡s❡ ❧❛✇s✿ ✶✳ ❚❤❡ ♦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✳ ❲❡ ❦♥♦✇ t❤❡ ✈❡❝t♦r ❢♦r♠ ♦❢ ◆❡✇t♦♥✬s ❧❛✇ ✭✇✐t❤ t❤❡ ✜rst ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✮✿ X . ||X||3
F = −GmM
❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❢♦r t❤❡ ❧♦❝❛t✐♦♥ X ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✿ X ′′ = −GM
❲❡ ✇✐❧❧ ❝❛❧❧ ✐t
X . ||X||3
✳ ■t ✐s ♥♦♥✲❧✐♥❡❛r✳
◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥
❆s ❜❡❢♦r❡✱ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ Xn′′ t❤❡ ✈❡❧♦❝✐t② Xn′ ❛♥❞ t❤❡♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t② t❤❡ ♣♦s✐t✐♦♥ Xn ❛r❡ ❝♦♠♣✉t❡❞ ❜② ✇❤❛t ❛♠♦✉♥ts t♦ ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❚❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿ Xn ||Xn ||3 ′ ′′ = Xn + Xn+1 ∆t ′ = Xn + Xn+1 ∆t
′′ Xn+1 = −GM ′ Xn+1 Xn+1
❚❤❡ r❡s✉❧t ✐s ❛s ♣r❡❞✐❝t❡❞✿ t❤❡ ♣♦✐♥ts ❢♦r♠ ✇❤❛t ❧♦♦❦s ❧✐❦❡ ❛♥ ❡❧❧✐♣s❡ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ❢♦❝✐ ❛t t❤❡ ♦r✐❣✐♥ ✭❋✐rst ❑❡♣❧❡r✬s ▲❛✇✮✳
✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥
✸✷✻
■♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤✐s ✐❞❡❛✱ ✇❡ ✜rst ❞r❛✇ ❛ ❧✐♥❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✭t❤❡ ✜rst ❢♦❝✉s✮ t♦ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡ ✇❤❡r❡ ✐t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ t❛♥❣❡♥t✳ ❲❡ t❤❡♥ ❝♦♥t✐♥✉❡ t❤✐s ❧✐♥❡ t♦ t❤❡ ♦t❤❡r ❡♥❞ ♦❢ t❤❡ ❝✉r✈❡ ❛♥❞ ❞✐s❝♦✈❡r t❤❛t ✐♥❞❡❡❞ t❤❡ t❛♥❣❡♥t ✐s ❛❣❛✐♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤✐s ❝❤♦r❞✳ ❲❡ t❤❡♥ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❢♦❝✉s t♦ t❤❡ ✜rst ♣♦✐♥t ❛♥❞ ♣❧♦t t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ s❡❝♦♥❞ ♣♦✐♥t ❛❧♦♥❣ t❤❡ ❝❤♦r❞✳ ❚❤✐s ❣✐✈❡s ✉s t❤❡ s❡❝♦♥❞ ❢♦❝✉s ✭❧❡❢t✮✳
❲❡ ❢✉rt❤❡r t❡st ♦✉r ❝♦♥❥❡❝t✉r❡ ❜② ♣❧♦tt✐♥❣ t❤❡ ❧✐♥❡s r❡♣r❡s❡♥t✐♥❣ r❛②s ♦❢ ❧✐❣❤t t❤❛t st❛rt ❛t ♦♥❡ ❢♦❝✉s ❛♥❞ t❤❡♥ ❜♦✉♥❝❡ ♦✛ t❤❡ ❝✉r✈❡ t♦ t❤❡ ♦t❤❡r ❢♦❝✉s ✭r✐❣❤t✮✳ ❊①❡r❝✐s❡ ✺✳✻✳✶
■s t❤✐s r❡❛❧❧② ❛♥ ❡❧❧✐♣s❡❄ ❇② ❝❤❛♥❣✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇❡ ❝❛♥ ♣r♦❞✉❝❡ ♦t❤❡r ♣❛tt❡r♥s ♦❢ ❜❡❤❛✈✐♦r s✉❝❤ ❛s ✇❤❛t ❧♦♦❦s ❧✐❦❡ ❛ ❤②♣❡r❜♦❧❛✿
◆♦✇ ✇❤❛t ❛❜♦✉t t❤❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇❄
✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥
✸✷✼
❲❡ ❝❛♥ s❡❡ ✐♥ ♦✉r s✐♠✉❧❛t✐♦♥ t❤❛t t❤❡ ♣♦✐♥ts ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✭t❤❡ ✜rst ♦❜❥❡❝t✮ ❛r❡ ❝❧♦s❡r s♣❛❝❡❞ t❤❛♥ t❤❡ ♦♥❡s ❝❧♦s❡r t♦ t❤❡ ♦r✐❣✐♥✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♠♦t✐♦♥ ✐s ❢❛st❡r ✇❤❡♥ t❤❡② ❛r❡ ❝❧♦s❡r✳ ❚❤❛t ✐s ✇❤② t❤❡ ❧♦♥❣❡r tr✐❛♥❣❧❡s ❛r❡ t❤✐♥♥❡r✳
❲❡ ❝♦♥✜r♠ t❤❛t t❤❡ ❛r❡❛s ♦❢ t❤❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ s❛♠❡ ❜② ❝♦♠♣✉t✐♥❣ t❤❡♠ ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✈✐❛ t❤❡ P❛r❛❧❧❡❧♦❣r❛♠ ❋♦r♠✉❧❛ ✿ An+1
1 = det [Xn+1 Xn ] , 2
✇❤❡r❡ Xn+1 ❛♥❞ Xn ❛r❡ ❣✐✈❡♥ ❜② t❤❡✐r ❝♦❧✉♠♥ ✈❡❝t♦rs✳
❲❡ ✇✐❧❧ ❞❡r✐✈❡ t❤❡ ❧❛✇ ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣ t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✳ ❋✐rst ✇❡ ♥♦t❡ t❤❛t ✐t ♠✐❣❤t ❛♣♣❧② t♦ ❛❧❧ ❞✐s❝r❡t❡ tr❛❥❡❝t♦r✐❡s ♥♦t ❥✉st ❡❧❧✐♣s❡s✳ ❊①❡r❝✐s❡ ✺✳✻✳✷
Pr♦✈❡ t❤❛t ❛♥ ♦❜❥❡❝t ♠♦✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ s❛t✐s✜❡s t❤❡ ❛r❡❛ ♣r♦♣❡rt② ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② ✜①❡❞ ♣♦✐♥t✳ ❍✐♥t✿ ✉s❡ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr②✳
✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥
✸✷✽
❙❡❝♦♥❞✱ ✇❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ❧❛✇ ❛♣♣❧✐❡s t♦ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r❝❡s t❤❛♥ ❥✉st t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ❣r❛✈✐t② ✐s ❛ ❝❡♥tr❛❧ ❢♦r❝❡✱ ✐✳❡✳✱ ♦♥❡ t❤❛t ✐s ❞✐r❡❝t❡❞ ❛❧♦♥❣ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ♦r✐❣✐♥✳ ❚❤❡♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r s✉❝❤ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧s ✐s✿ ′′ Xn+1 = L(||Xn ||)Xn ,
❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ L✳
❚❤❡♦r❡♠ ✺✳✻✳✸✿ ❉✐s❝r❡t❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇
■❢ X ✐s ❛ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇✐t❤ ❝❡♥tr❛❧ ❢♦r❝❡✱ t❤❡ s❡❣♠❡♥t ❢r♦♠ t❤❡ ❢♦❝✉s t♦ t❤❡ ❧♦❝❛t✐♦♥ s✇❡❡♣s ❛♥ ❡q✉❛❧ ❛r❡❛ ✐♥ ❛♥ ❡q✉❛❧ t✐♠❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ t❤✐s ❛r❡❛ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❜② ❡❛❝❤ tr❛❥❡❝t♦r② ♦❢ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ♦r❞❡r t✇♦✱ t❤❡ ♠♦❞❡❧ ✐s ♣r♦❞✉❝❡❞ ❜② ❛ ❝❡♥tr❛❧ ❢♦r❝❡✳ Pr♦♦❢✳
❲❡ ❝♦♥s✐❞❡r t❤❡ ♠♦❞❡❧✿
′′ Xn+1 = f (Xn ) ,
✇❤❡r❡ f ✐s s♦♠❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥✱ ′ Xn+1 = Xn′ + Xn′′ ∆t = Xn′ + f (Xn )∆t .
❲❡ ✇✐❧❧ ✉s❡ t✇✐❝❡ t❤❡ P❛r❛❧❧❡❧♦❣r❛♠ ❋♦r♠✉❧❛ ❢♦r t❤❡ ♦r✐❡♥t❡❞ ❛r❡❛ A ♦❢ t❤❡ tr✐❛♥❣❧❡ s♣❛♥♥❡❞ ❜② ✈❡❝t♦rs M ❛♥❞ N ✿ A=
1 det [M N ] . 2
❍♦✇❡✈❡r✱ t❤❡ ✈❡❝t♦rs ❝❤♦s❡♥ ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤♦s❡ ✉s❡❞ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❋✐rst✱ s✉♣♣♦s❡ t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ❛r❡❛ An ✐s ❢♦r♠❡❞ ❜② Xn ❛♥❞ Xn′ ∆t❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❜② t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t✿ ∆t det [Xn Xn′ ] . 2 ′ ❙❡❝♦♥❞✱ s✉♣♣♦s❡ t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ❛r❡❛ An+1 ✐s ❢♦r♠❡❞ ❜② Xn ❛♥❞ Xn+1 ∆t❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ An =
❢♦❧❧♦✇✐♥❣ ❜② t❤❡ ❧✐♥❡❛r✐t② ❛♥❞ t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t✿
∆t ′ det Xn Xn+1 2 ∆t det [Xn (Xn′ + f (Xn )∆t)] = 2 ∆t = det [Xn Xn′ ] + ∆t det [Xn f (Xn )] . 2
An+1 =
❚❤❡r❡❢♦r❡✱ ❋✐♥❛❧❧②✱ An+1
∆t2 An+1 − An = det [Xn f (Xn )] . 2 = An ✐❢ ❛♥❞ ♦♥❧② ✐❢ Xn ❛♥❞ f (Xn ) ❛r❡ ♣❛r❛❧❧❡❧✳
❊①❡r❝✐s❡ ✺✳✻✳✹
❉❡r✐✈❡ ❑❡♣❧❡r✬s ❙❡❝♦♥❞ ▲❛✇ ❢r♦♠ t❤✐s t❤❡♦r❡♠✳ ❍✐♥t✿ ✉s❡ ✇❤❛t ②♦✉ ❦♥♦✇ ❛❜♦✉t t❤❡ ❛❝❝✉r❛❝② ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❲❡ t❤✉s ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡ ❡❧❧✐♣s❡ ✭♦r t❤❡ ♣❛r❛❜♦❧❛✱ ♦r t❤❡ ❤②♣❡r❜♦❧❛ s❡❡♥ ❜❡❧♦✇✮ ❞♦❡s ♥♦t ♠❛t❝❤ ✐ts st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥✱ x = a sin ωt, y = b sin ωt✳ ❚❤❡ ❢♦r♠✉❧❛s ❢✉❧❧② ❛♣♣❧② t♦ t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ s✐t✉❛t✐♦♥✳ P❧♦tt✐♥❣ t❤❡ ♦r❜✐t ❛❧♦♥❣ t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿
✺✳✻✳
P❧❛♥❡t❛r② ♠♦t✐♦♥
✸✷✾
❊①❡r❝✐s❡ ✺✳✻✳✺
■♠♣❧❡♠❡♥t ❛ s✐♠✉❧❛t✐♦♥ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐♥ t❤❡
3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳
❉❡♠♦♥str❛t❡ t❤❛t t❤❡ ♠♦t✐♦♥
✐s ♣❧❛♥❛r✳
❲❡ st❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✿ ❚❤❡♦r❡♠ ✺✳✻✳✻✿ P❧❛♥❡t❛r② ▼♦t✐♦♥ ✐♥ P♦❧❛r ❈♦♦r❞✐♥❛t❡s ❆♥② tr❛ ❥❡❝t♦r② ♦❢ t❤❡ ◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿
r= ✇❤✐❝❤ ✐s
• • • •
e = 1✱ 0 < e < 1✱ ♣❛r❛❜♦❧❛ ❢♦r e = 1✱ ❤②♣❡r❜♦❧❛ ❢♦r e > 1✱
❛ ❝✐r❝❧❡ ❢♦r
❛♥ ❡❧❧✐♣s❡ ❢♦r ❛ ❛
✇✐t❤ t❤❡ ❢♦❝✉s ❛t t❤❡ ♦r✐❣✐♥✳
❚❤❡ ❣r❛♣❤s ❛r❡ ♣❧♦tt❡❞ ❢♦r
e = 0, .5, 1, 2✿
p , 1 + e · cos θ
✺✳✼✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
✸✸✵
❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s t❤❡♦r❡♠ ❧✐❡s ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❋✐♥❛❧❧②✱ ❧❡t✬s ❣♦ ❜❛❝❦ t♦ t❤❡ q✉❡st✐♦♥ ♣♦s❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✿ ✇❤❛t ✐s t❤❡ ❝♦rr❡❝t tr❛❥❡❝t♦r②❄
▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡❞ ❛❜♦✉t t❤❡s❡ t✇♦ s❡tt✐♥❣s ✐♥ ❈❤❛♣t❡r ✹❍❉✲✷✳
•
❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ✏❧❛r❣❡✑ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ tr❛❥❡❝t♦r②✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ♣❛r❛❧❧❡❧ ✐♥ ❛❧❧ ❧♦❝❛t✐♦♥s✳ ❚❤❡♥ t❤❡ tr❛❥❡❝t♦r② ✐s ❛ ♣❛r❛❜♦❧❛ ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡
•
x✲❛①✐s
❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤ ❛♥❞
y ✲❛①✐s
✇✐t❤ t❤❡ ❢♦r❝❡✳
❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ❡✐t❤❡r ✏s♠❛❧❧✑ ♦r ❛t ❧❡❛st ♣❡r❢❡❝t❧② s♣❤❡r✐❝❛❧✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❣♦ r❛❞✐❛❧❧② t♦✇❛r❞ ✐ts ❝❡♥t❡r✳ ❚❤❡♥ t❤❡ tr❛❥❡❝t♦r② ✐s ❛♥ ❡❧❧✐♣s❡ ✭♦r ❛ ❤②♣❡r❜♦❧❛ ♦r ❛ ♣❛r❛❜♦❧❛✮ ✇✐t❤ ✐ts ❢♦❝✉s ❧♦❝❛t❡❞ ❛t t❤❡ ❝❡♥t❡r✳
❲❤❡♥ t❤❡ s✐③❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❊❛rt❤ ♠❛tt❡r✱ t❤✐♥❣s ❣❡t ❝♦♠♣❧✐❝❛t❡❞✳✳✳
✺✳✼✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
❲❡ ❤❛✈❡ ✐❣♥♦r❡❞ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❊❛rt❤✬s ❣r❛✈✐t② ♦♥ t❤❡ ❙✉♥✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ♠❛ss s✐❣♥✐✜❝❛♥t❧② ❧❛r❣❡r t❤❛♥ t❤❡ ♠❛ss
q
p
♦❢ t❤❡ ❙✉♥ ✐s
♦❢ t❤❡ ❊❛rt❤✿
p >> q . ❚❤✐s ✐s t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧♦✇s ✉s t♦ ♣❧❛❝❡ t❤❡ ❙✉♥ ❛t t❤❡ ♦r✐❣✐♥ ❛s ❛ st❛t✐♦♥❛r② ♦❜❥❡❝t✳ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ♣❧❛♥❡ts ♦❢ ❝♦♠♣❛r❛❜❧❡ s✐③❡s ❄
✺✳✼✳
✸✸✶
❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
❙✉♣♣♦s❡ ♥♦✇ ✇❡ ❤❛✈❡ t✇♦ ♣❧❛♥❡ts ✇✐t❤ ♠❛ss❡s p, q ✇✐t❤ ❧♦❝❛t❡❞ ❛t U, V r❡s♣❡❝t✐✈❡❧②✳ ❊✐t❤❡r ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ♦t❤❡r✳ ❚❤❡♥ t❤❡ t✇♦ ✐♥t❡r❛❝t✐♦♥s ❛♣♣❡❛r ✐♥ t❤❡ t✇♦ ❞❡♣❡♥❞❡♥t ❖❉❊s✿ U −V ||U − V ||3 V −U V ′′ = −Gp ||V − U ||3 U ′′ = −Gq
❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ t✇♦✲❜♦❞②
✳
♣r♦❜❧❡♠
▲❡t✬s ❛♣♣❧② ❊✉❧❡r✬s ♠❡t❤♦❞ t♦ s❡❡ ✇❤❛t ❝❛♥ ❤❛♣♣❡♥✳ ❲❡ st❛rt ✇✐t❤ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts✳ ❚❤❡② s❡❡♠ t♦ ❝✐r❝❧❡ ❡❛❝❤ ♦t❤❡r✿
✳✳✳✉♥t✐❧ ✇❡ r❡❛❧✐③❡ t❤❛t ❡✐t❤❡r ♦♥❡ ❝✐r❝❧❡s ❛ ❝❡rt❛✐♥ ♣♦✐♥t✱ t❤❡ s❛♠❡ ♣♦✐♥t ❢♦r ❜♦t❤✳ ❚❤✐s ♣♦✐♥t ❧✐❡s ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ t❤❡♥✱ 1 C = (U + V ) . 2
❚❤✐s ❢❛❝t ✐s ❝♦♥✜r♠❡❞ ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳
▲❡t✬s ❞♦✉❜❧❡ t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♣❧❛♥❡t✳ ❲✐t❤ ❡✈❡r②t❤✐♥❣ ❡❧s❡ r❡♠❛✐♥✐♥❣ t❤❡ s❛♠❡✱ t❤❡ t✇♦ ♣❧❛♥❡ts s❡❡♠ t♦ ❞❛♥❝❡ ❛✇❛② ✇❤✐❧❡ st✐❧❧ ❝✐r❝❧✐♥❣ ❡❛❝❤ ♦t❤❡r✳ ❚❤✐s t✐♠❡✱ t❤❡ ♠✐❞✲♣♦✐♥t ✐s ❛❧s♦ ❝✐r❝❧✐♥❣✳
■♥ ❢❛❝t✱ ✐t ❞♦❡s♥✬t s❡❡♠ t♦ r❡✈❡❛❧ ❛♥②t❤✐♥❣ ❛❜♦✉t ✇❤❛t ✐s ❣♦✐♥❣ ♦♥✳ ❲❤❛t ✐❢ ✇❡ ❧♦♦❦ ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ t✇♦✱ M=
p q U+ V, p+q p+q
✐♥st❡❛❞❄ ■t s❡❡♠s t♦ ❜❡ ♠♦✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡✦
♦❢ ♠❛ss
✺✳✼✳
❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
✸✸✷
❋✉rt❤❡r♠♦r❡✱ ❧❡t✬s ♣❧♦t t❤❡ t✇♦ tr❛❥❡❝t♦r✐❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss✱ ✐✳❡✳✱
P =U −M
❛♥❞
Q=V −M.
❚❤❡② s❡❡♠ t♦ tr❛❝❡ ❡❧❧✐♣s❡s✳ ▲❡t✬s st❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡s❡ ❝♦♥❥❡❝t✉r❡s✳
❚❤❡♦r❡♠ ✺✳✼✳✶✿ ❈❡♥t❡r ♦❢ ▼❛ss ❚✇♦✲❜♦❞② ❙②st❡♠
❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss M ♦❢ t❤❡ t✇♦✲❜♦❞② s②st❡♠ s❛t✐s✜❡s t❤❡ ✈❡❝t♦r ❖❉❊✿ M ′′ = 0 ,
❛♥❞✱ t❤❡r❡❢♦r❡✱ ♠♦✈❡s ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✳ Pr♦♦❢✳ ❲❡ s✐♠♣❧② s✉❜st✐t✉t❡✿
M
′′
′′ q p U+ V = p+q p+q p q = U ′′ + V ′′ p+q p+q q U −V V −U p + −Gq −Gp = p+q ||U − V ||3 p+q ||V − U ||3 V −U U −V pq + = −G 3 p + q ||U − V || ||U − V ||3
= 0. ❚❤❡♦r❡♠ ✺✳✼✳✷✿ ▲♦❝❛t✐♦♥s ✐♥ ❚✇♦✲❜♦❞② ❙②st❡♠
■❢ U ❛♥❞ V ❛r❡ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ t✇♦ ♣❧❛♥❡ts ✐♥ t❤❡ t✇♦✲❜♦❞② s②st❡♠ ❛♥❞ M ✐s ✐ts ❝❡♥t❡r ♦❢ ♠❛ss M ✱ t❤❡♥ P = U − M ❛♥❞ Q = V − M s❛t✐s❢② t❤❡ ◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ❛♥❞✱ t❤❡r❡❢♦r❡✱ tr❛❝❡ ❡❧❧✐♣s❡s✱ ♣❛r❛❜♦❧❛s✱ ♦r ❤②♣❡r❜♦❧❛s✳
✺✳✼✳
✸✸✸
❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
Pr♦♦❢✳
❲❡ ✜rst ♦❜s❡r✈❡ t❤❛t M ❧♦❝❛t❡❞ ♦♥ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ U ❛♥❞ V ✐♥ ♣r♦♣♦rt✐♦♥ t♦ t❤❡✐r ♠❛ss❡s✳ ❚❤❡r❡❢♦r❡✱ U − V ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ U − M ✿ M=
p q p+q U+ V =⇒ U − V = (U − M ) . p+q p+q q
◆♦✇✱ ✇❡ s✐♠♣❧② s✉❜st✐t✉t❡✿ P ′′ = (U − M )′′ = U ′′ − M ′′ = −Gq = −Gq
U −V −0 ||U − V ||3 p+q (U − M ) q
|| p+q (U − M )||3 q
U −M 2 p+q ||(U − M )||3 q 2 U −M q = −Gq p+q ||(U − M )||3 q3 P = −G . 2 (p + q) ||P ||3 = −Gq
■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❝❛s❡ ♦❢ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts ❛♥❞ s✐♠♣❧② ✐♥❝r❡❛s❡ t❤❡ y ✲❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ s❡❝♦♥❞ ♣❧❛♥❡t ❢r♦♠ 1 t♦ 2✳ ❲✐t❤ ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ ❝✐r❝❧✐♥❣✱ t❤❡② st❛rt t♦ ♠♦✈❡ ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳
❊✈❡♥t✉❛❧❧②✱ t❤❡② ❛r❡ ✢②✐♥❣ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✦ ❏✉st ❛s ✐♥ t❤❡ ❧❛st ❝❛s❡✱ ♣❧♦tt✐♥❣ t❤❡ tr❛❥❡❝t♦r✐❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❤❡❧♣ t♦ r❡✈❡❛❧ t❤❡ ♣❛tt❡r♥✳ ❚❤✉s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ t❤❡s❡ t❤r❡❡ s❡❡♠ t♦ ❜❡ t❤❡ ♠♦st ✐♥t❡r❡st✐♥❣ ♦✉t❝♦♠❡s✳ ■♥ t❤❡ ✜rst✱ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts ❞❛♥❝❡ ❛r♦✉♥❞ ❡❛❝❤ ♦t❤❡r ✭✐♥ r❡❛❧✐t②✱ ❛r♦✉♥❞ t❤❡ ❝♦♠❜✐♥❡❞ ❝❡♥t❡r ♦❢ ♠❛ss✮✳ ■♥ t❤❡ s❡❝♦♥❞✱ t❤❡② ❞❛♥❝❡ ❛✇❛② t♦❣❡t❤❡r ✭st✐❧❧ ❛r♦✉♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss✱ ✇❤✐❝❤ ✐s ♠♦✈✐♥❣✮✳ ❋✐♥❛❧❧②✱ t❤❡ t✇♦ r✉♥ ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ ❜❡②♦♥❞ s♦♠❡ ❞✐st❛♥❝❡ t❤❡② ❢❡❡❧ ❛❧♠♦st ♥♦ ♣✉❧❧✳✳✳
✺✳✼✳
✸✸✹
❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
❊①❡r❝✐s❡ ✺✳✼✳✸
❲❤② ❞♦♥✬t ❛♥② ♦❢ t❤❡ ♣❧❛♥❡ts ✐♥ t❤❡ s♦❧❛r s②st❡♠ ❜❡❤❛✈❡ t❤✐s ✇❛②❄ ❲❤❛t ♦t❤❡r ♣♦ss✐❜✐❧✐t✐❡s ❝❛♥ ②♦✉ t❤✐♥❦ ♦❢❄ ❲❤❛t ❛❜♦✉t t❤r❡❡
♣❧❛♥❡ts
❄
❋✐rst ❧❡t✬s r❡✈✐❡✇ ❛ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✷✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ ❊❛rt❤ ♦r❜✐ts t❤❡ ❙✉♥ ❛♥❞ t❤❡ ▼♦♦♥ ♦r❜✐ts ❛r♦✉♥❞ t❤❡ ❊❛rt❤ ♠❛② ❜❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ♣✐❝t✉r❡ ♦♥ ❧❡❢t ✇❤✐❧❡ t❤❡ ❛❝t✉❛❧ ❞❛t❛ ❛❜♦✉t t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ♦r❜✐ts ❛♥❞ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ▼♦♦♥ ♣r♦❞✉❝❡ t❤❡ ♣✐❝t✉r❡ ♦♥ r✐❣❤t✿
❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡ t❤r❡❡ ♣❧❛♥❡ts ❤❛✈❡ ♠❛ss❡s p, q, m✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t p >> q >> m ,
s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❙✉♥✲❊❛rt❤✲▼♦♦♥✱ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ s❡❝♦♥❞ ♦♥ t❤❡ ✜rst ❛♥❞ t❤❡ t❤✐r❞ ♦♥ t❤❡ s❡❝♦♥❞ ✐s ♥❡❣❧✐❣✐❜❧❡✱ ❥✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❙✉♣♣♦s❡ • U = 0 ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❙✉♥ ✜①❡❞ ❛t t❤❡ ♦r✐❣✐♥✱
• V ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❊❛rt❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❙✉♥✱ ❛♥❞ • X ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ▼♦♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❊❛rt❤✳
❚❤❡♥ t❤❡ t✇♦ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ ✇❡ ❤❛✈❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ◆❡✇t♦♥✬s ❖❉❊s ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✿ V ||V ||3 X X ′′ = −Gq ||X||3 V ′′ = −Gp
❇♦t❤ s♦❧✉t✐♦♥s ❢♦❧❧♦✇ t❤❡✐r r❡s♣❡❝t✐✈❡ ❡❧❧✐♣s❡s✳ ❲❡ t❤❡♥ ❛❞❞ t❤❡♠ ❛s ✈❡❝t♦rs✱ W =V +X,
t♦ ♣r♦❞✉❝❡ t❤❡ ♣❛t❤ ♦❢ t❤❡ t❤✐r❞ ♣❧❛♥❡t ❛r♦✉♥❞ t❤❡ ✜rst✳ ●❡♥❡r✐❝❛❧❧②✱ ✐t ❧♦♦❦s ❧✐❦❡ t❤✐s✿
✺✳✼✳
❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s
✸✸✺
❲❡ ♥❡①t ♠❛❦❡ ♦♥❡ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♣❧❛♥❡t ✐s ♥♦t ♥❡❣❧✐❣✐❜❧② s♠❛❧❧ r❡❧❛t✐✈❡ t♦ t❤❛t ♦❢ t❤❡ ✜rst ❛♥②♠♦r❡✳ ❲❡✱ t❤❡♥✱ ❝❛♥✬t ✜① t❤❡ ✜rst ❛t t❤❡ ♦r✐❣✐♥ ❛♥②♠♦r❡✳ ❲❡ ❤❛✈❡ ❛ t✇♦✲❜♦❞② s②st❡♠ ❥✉st ❛s ❛❜♦✈❡✿ U −V ||U − V ||3 V −U V ′′ = −Gp ||V − U ||3 W −U W −V W ′′ = −Gp −q 3 ||W − U || ||W − V ||3
U ′′ = −Gq
■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ t❤✐r❞ ♣❧❛♥❡t ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ✜rst t✇♦✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ r❡str✐❝t❡❞
t❤r❡❡
❜♦❞② ♣r♦❜❧❡♠
❲❤✐❧❡ t❤❡ t✇♦ ✜rst ♣❧❛♥❡ts ❞♦ t❤❡ s❛♠❡✱ t❤❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ♦❢ t❤❡ t❤✐r❞ ✐s ❡♥♦r♠♦✉s ❡✈❡♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿
❚❤❡r❡ ❛r❡ ♠❛♥② ♣❡r✐♦❞✐❝ tr❛❥❡❝t♦r✐❡s ❜✉t ✇❡ ✇✐❧❧ ♣♦✐♥t ♦✉t ♦♥❧② ♦♥❡✳ ❇❡❧♦✇ ✇❡ ❝♦♥✜r♠ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♣❧❛❝❡ ❛ s❛t❡❧❧✐t❡ ♦♥ t❤❡ ▼♦♦♥✬s ♦r❜✐t✱ 60 ❞❡❣r❡❡s ❛❤❡❛❞✱ t❤❛t ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ r❡✈♦❧✈❡ ✐♥ t❤✐s ❢❛s❤✐♦♥ ✐♥❞❡✜♥✐t❡❧②✳
✺✳✽✳
❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳
✸✸✻
◆♦✇✱ ✐♥ ❣❡♥❡r❛❧✳ ❙✉♣♣♦s❡ ♥♦✇ ✇❡ ❤❛✈❡ t❤r❡❡ ♣❧❛♥❡ts ✇✐t❤ ♠❛ss❡s
m1 , m2 , m3 ❧♦❝❛t❡❞ ❛t V1 , V2 , V3 r❡s♣❡❝t✐✈❡❧②✳
❊❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♦❜❥❡❝ts ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ ❡✐t❤❡r ♦❢ t❤❡ ♦t❤❡r t✇♦✳ ❚❤❡♥ t❤❡s❡ t❤r❡❡ ✐♥t❡r❛❝t✐♦♥ ❛♣♣❡❛r ✐♥ t❤❡ t❤r❡❡ ❞❡♣❡♥❞❡♥t ✈❡❝t♦r ❖❉❊s✿
V1 − V3 V1 − V2 − Gm3 3 ||V1 − V2 || ||V1 − V3 ||3 V2 − V1 V2 − V3 V2′′ = −Gm1 − Gm3 3 ||V2 − V1 || ||V2 − V3 ||3 V3 − V1 V3 − V2 V3′′ = −Gm1 − Gm2 3 ||V3 − V1 || ||V3 − V2 ||3
V1′′ = −Gm2
❚❤❡ ❣❡♥❡r❛❧ t❤r❡❡ ❜♦❞② ♣r♦❜❧❡♠ ❤❛s ♥♦ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥✳
✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ❲❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❤✐st♦r② ♦❢ ❜❛❧❧✐st✐❝s ✐s t❤❛t t❤❡ ♣❛r❛❜♦❧✐❝ tr❛❥❡❝t♦r✐❡s ✇❡r❡ ❦♥♦✇♥ s✐♥❝❡ ●❛❧✐❧❡♦ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ◆❡✇t♦♥ ✇♦r❦❡❞ ♦✉t t❤❡ ♣❤②s✐❝s ❜❡❤✐♥❞ ❛♥❞ t❤❡ ♠❛t❤❡♠❛t✐❝s ✭❝❛❧❝✉❧✉s✮ ❜❡❤✐♥❞ t❤✐s ✐❞❡❛✳ ◆♦♥❡t❤❡❧❡ss✱ ❛s r❡❝❡♥t❧② ❛s ♠✐❞✲✶✾t❤ ❝❡♥t✉r② t❤❡② ❛✐♠❡❞ ❝❛♥♥♦♥s ❜❛s❡❞ ❡♥t✐r❡❧② ♦♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ ❡①♣❡r✐♠❡♥t❛t✐♦♥✱ ✇✐t❤♦✉t ♠❛t❤❡♠❛t✐❝s✳ ❍❡r❡ ✐s ❛♥ ❡①❝❡r♣t ❢r♦♠
❚❤❡ ❊♠♣❡r♦r ◆❛♣♦❧❡♦♥✬s ◆❡✇ ❙②st❡♠ ♦❢ ❋✐❡❧❞ ❆rt✐❧❧❡r②
❇❡❧♦✇ ✐s t❤❡ ✏r❛♥❣❡ t❛❜❧❡✑ ❢r♦♠ t❤❡
❚❤❡ ❈♦♥❢❡❞❡r❛t❡ ❖r❞♥❛♥❝❡ ▼❛♥✉❛❧
✭✶✽✺✹✮✿
❞✉r✐♥❣ t❤❡ ❆♠❡r✐❝❛♥ ❈✐✈✐❧ ❲❛r
✭✶✽✻✵s✮✳ ❈❧❡❛r❧②✱ t❤❡ ♥✉♠❜❡rs ❝♦♠❡ ❢r♦♠ s❤♦♦t✐♥❣ ❛♥❞ t❤❡♥ ♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡✿
✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳
✸✸✼
❲✐t❤ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ ♣r♦❥❡❝t✐❧❡s✱ ✇❡ ❝❛♥ tr② t♦ r❡♣r♦❞✉❝❡ t❤❡s❡ r❡s✉❧ts✳ ❲❡ st❛rt ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿
x′′ = 0 , y ′′ = −32
x′ (0) = s0 cos α , y ′ (0) = s0 sin α
x(0) = 0 , y(0) = y0
✇❤❡r❡ s0 ✐s t❤❡ ✐♥✐t✐❛❧ s♣❡❡❞ ❛♥❞ α ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧✳ ❊①❛♠♣❧❡ ✺✳✽✳✶✿ ❞❛t❛
❚❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❤❛s ❛❧r❡❛❞② ❜❡❡♥ s♦❧✈❡❞✿
x = s0 cos α · t, y = y0 +s0 sin α · t −16t2 .
❲❡ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥ x t❤❛t y = 0✳ ❲❡ ✜♥❞ t❤❡ t✐♠❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✜rst ✭❝❤♦♦s✐♥❣ t❤❡ ♣♦s✐t✐✈❡ ✈❛❧✉❡✮✿ t=
−s0 sin α ±
p p (s0 sin α)2 − 4(−16)y0 1 = s0 sin α + (s0 sin α)2 + 64y0 . 2(−16) 32
❲❡ ❝♦♥s✐❞❡r t❤❡ ✶✷✲♣♦✉♥❞ ✜❡❧❞ ❤♦✇✐t③❡r✳ ❙✉♣♣♦s❡ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥✱ s0 = 1054 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❲❡ ❛❧s♦ ❡st✐♠❛t❡ t❤❡ ✐♥✐t✐❛❧ ❡❧❡✈❛t✐♦♥ ✭t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❝❛♥♥♦♥✮ ❛t y0 = 5 ❢❡❡t✳ ❚❤❡♥ t❤❡ ❞✐st❛♥❝❡s s❤♦✉❧❞ ❜❡✿ ❡❧❡✈❛t✐♦♥ ✐♥ ❞❡❣r❡❡s t✐♠❡ ✐♥ s❡❝♦♥❞s✱ t ❞✐st❛♥❝❡ ✐♥ ❢❡❡t✱ x ❞✐st❛♥❝❡ ✐♥ ②❛r❞s t❡st ❞✐st❛♥❝❡ 0 1 2 3 4 5
0.56 1.38 2.43 3.54 4.66 5.80
589 1451 2557 3722 4902 6085
❲❡ s❡❡ ❛ ♠✐s♠❛t❝❤ ✇✐t❤ t❤❡ ❞❛t❛ t❤❛t ❣r♦✇s ❢❛st ✇✐t❤ t❤❡ ❛♥❣❧❡✳
196 484 852 1241 1634 2028
195 539 640 847 975 1072
✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳
✸✸✽
❊①❡r❝✐s❡ ✺✳✽✳✷
❊①♣❧❛✐♥ ✇❤② t❤❡ r❡❞ ❝✉r✈❡ ❧♦♦❦s str❛✐❣❤t✳ ❚❤❡ r❡❛s♦♥ ✇❤② t❤❡ ◆❡✇t♦♥✐❛♥ ♠❡❝❤❛♥✐❝s ♦✈❡r❡st✐♠❛t❡s t❤❡ ❞✐st❛♥❝❡ ✐s ❦♥♦✇♥❀ ✐t ✐s t❤❡ ❛✐r✲r❡s✐st❛♥❝❡✳ ■♥ ❢❛❝t✱ ◆❡✇t♦♥ ❤✐♠s❡❧❢ ✇♦r❦❡❞ ♦✉t ❛❧❧ ♥❡❝❡ss❛r② ❞❡t❛✐❧s ♦❢ ❜❛❧❧✐st✐❝s ✐♥❝❧✉❞✐♥❣ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❛✐r r❡s✐st❛♥❝❡✳ ❍❡ ❛ss✉♠❡❞ ❤♦✇❡✈❡r t❤❛t t❤❡ r❡s✐st❛♥❝❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s♣❡❡❞✳ ❍❛❧❢ ❛ ❝❡♥t✉r② ❧❛t❡r ❊✉❧❡r t❤♦✉❣❤t✱ ❢♦r ❜❛❧❧✐st✐❝s✱ ✐t ✐s ❜❡tt❡r t♦ t❛❦❡ t❤❡ r❡s✐st❛♥❝❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ s♣❡❡❞✳ ❚❤❛t ✇❛s ✶✵✵ ②❡❛rs ❜❡❢♦r❡ t❤♦s❡ t❡sts✦ ❚❤❡ ❞r❛❣ ❡q✉❛t✐♦♥ ❣✐✈❡s t❤❡ ❢♦r❝❡ FD ♦❢ ❞r❛❣ ❡①♣❡r✐❡♥❝❡❞ ❜② ❛♥ ♦❜❥❡❝t ❞✉❡ t♦ ✐ts ♠♦✈❡♠❡♥t t❤r♦✉❣❤ t❤❡ ❛✐r✿
||FD || = 21 ρCA · s2 , ✇❤❡r❡
• ρ ✐s t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❛✐r✱
• A ✐s t❤❡ ❝r♦ss s❡❝t✐♦♥❛❧ ❛r❡❛ ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✱
• C ✐s t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t ✕ ❛ ❞✐♠❡♥s✐♦♥❧❡ss ❝♦❡✣❝✐❡♥t t❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ♣r♦❥❡❝t✐❧❡✬s ❣❡♦♠❡tr②✱ ❛♥❞ • s ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✳
❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ❢♦r❝❡ FD ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❛t ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♦♣♣♦s✐t❡ t♦ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② X ′ ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢♦r❝❡ ✐s
− 21 ρCA
1 sX ′ , M
✇❤❡r❡ M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✳ ❚❤❡♥ ♦✉r ■❱P ❜❡❝♦♠❡s✿ x′′ =
y ′′
1 − 12 ρCA sx′ M , 1 ′ 1 = −32 − 2 ρCA sy M
x′ (0) = s0 cos α y ′ (0) = s0 sin α
,
x(0) = 0
.
y(0) = y0
❲✐t❤ t❤✐s ✉♣❞❛t❡❞ ❞②♥❛♠✐❝s ♦❢ ♣r♦❥❡❝t✐❧❡s✱ ❧❡t✬s tr② ❛❣❛✐♥ t♦ r❡♣r♦❞✉❝❡ t❤❡ t❡st r❡s✉❧ts✳ ❊①❛♠♣❧❡ ✺✳✽✳✸✿ ❞❛t❛
❲❡ ♥❡❡❞ s♦♠❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ❞r❛❣ ❡q✉❛t✐♦♥✿ • t❤❡ ✇❡✐❣❤t M = 8.9 ♣♦✉♥❞s • t❤❡ ❛r❡❛ A = πr2 ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧✱ ✇❤❡r❡ t❤❡ r❛❞✐✉s ✐s r = 4.62/2 = 2.31 ✐♥❝❤❡s • t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❛✐r ρ = 0.074887 ♣♦✉♥❞s ♣❡r ❝✉❜✐❝ ❢♦♦t • t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t ♦❢ ❛ s♣❤❡r❡ C = 0.47 ❲❡ ✉s❡ ❊✉❧❡r✬s ♠❡t❤♦❞✿
✺✳✾✳
❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s
✸✸✾
❲❡ r❡♣❡❛t ✐t ❢♦r ❡❛❝❤ ❡❧❡✈❛t✐♦♥✱ s✐① t✐♠❡s✿ ❡❧❡✈❛t✐♦♥ ✐♥ ❞❡❣r❡❡s
❞✐st❛♥❝❡ ✐♥ ②❛r❞s
t❡st ❞✐st❛♥❝❡
0 1 2 3 4 5
187 405 629 822 984 1124
195 539 640 847 975 1072
▼✉❝❤ ❜❡tt❡r✦
✺✳✾✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s
❊①❛♠♣❧❡ ✺✳✾✳✶✿ s❛♠♣❧❡ ❇❱P
❙❡tt✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛s✐❞❡ ❢♦r ❛ ♠♦♠❡♥t✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ ♣r♦❜❧❡♠✿ ◮ ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = ax2 +bx+c✱ ✜♥❞ t❤❡ ♦♥❡s ✇✐t❤ f (0) = 20,
f ′ (0) =
−3✳
❍❡r❡✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡✱
x = 0✳ [0, ∞]✳
❚❤❡s❡ ❛r❡ ❝❛❧❧❡❞
✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳
❚❤❡ ✇♦r❞ ✏✐♥✐t✐❛❧✑ r❡❢❡rs t♦ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✱
0✱
♦❢ t❤❡ r❛②
▲❡t✬s ❝❤❛♥❣❡ t❤❡ ♣r♦❜❧❡♠✿
◮ ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = ax2 +bx+c✱ ✜♥❞ ♦♥❡s ✇✐t❤ f (0) = 20, f (5) = 30✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♣r♦✈✐❞❡❞ ❢♦r t✇♦ ✈❛❧✉❡s ♦❢ x ❛♥❞ ♥♦ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ♣r♦✈✐❞❡❞✳ ❚❤❡s❡ ❛r❡ ❝❛❧❧❡❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ✇♦r❞ ✏❜♦✉♥❞❛r②✑ r❡❢❡rs t♦ t❤❡ ❡♥❞✲♣♦✐♥ts✱ 0 ❛♥❞ 5✱ ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❜♦✉♥❞❛r② ♣♦✐♥ts✱ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [0, 5]✳
✺✳✾✳
❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s
✸✹✵
❊①❡r❝✐s❡ ✺✳✾✳✷
❋✐♥❞ ❛❧❧ t❤❡s❡ ♣♦❧②♥♦♠✐❛❧s✳ ❊①❛♠♣❧❡ ✺✳✾✳✸✿ ✉♥✐t❛r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s
▲❡t✬s ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ a = 1✿ • ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = x2 + bx + c✱ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ f (0) = 20, f ′ (0) = −3✳ • ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = x2 + bx + c✱ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ f (0) = 20, f (5) = 30✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ❝❡rt❛✐♥ ✉♥✐q✉❡♥❡ss✱ ✐✳❡✳✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥❀ t❤❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ✐s✱ ♦❢ ❝♦✉rs❡✱ c = 20, b = −3✳ ❋♦r t❤❡ ❧❛tt❡r ❝❛s❡ ✇❡ ♥❡❡❞ ♠♦r❡ ❛❧❣❡❜r❛✿ c = 20 =⇒ 52 + b · 5 + 20 = 30 =⇒ b = (30 − 52 − 20)/5 = 3 .
❊①❛♠♣❧❡ ✺✳✾✳✹✿ ❝❛♥♥♦♥
▲❡t✬s r❡✈✐❡✇ ❛ ❢❛♠✐❧✐❛r ♣r♦❜❧❡♠ ❢r♦♠ ❱♦❧✉♠❡s ✶ ❛♥❞ ✷✳ ❋r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄
❲❡ ❦♥♦✇ t❤✐s ❛s ❛♥ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ■♥❞❡❡❞✱ t❤❡ ❖❉❊ ❤❛s ❜❡❡♥ s♦❧✈❡❞✿
x = 200t, y = 200 −16t2 .
✺✳✾✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s
✸✹✶
◆♦✇ ✇❡ ❥✉st ♥❡❡❞ t♦ ✜♥❞ t❤❡ s♦❧✉t✐♦♥ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✿ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ X(0) = (x(0), y(0)) = (0, 200)❀ • t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ X ′ (0) = (x′ (0), y ′ (0)) = (200, 0)✳
❲❡ s❝r♦❧❧ ❞♦✇♥ t❤❡ s♣r❡❛❞s❤❡❡t t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ y ❝❧♦s❡ t♦ 0✿ ❛r♦✉♥❞ t1 = 3.55 s❡❝♦♥❞s✱ ✇✐t❤ t❤❡ ✈❛❧✉❡ ♦❢ x ❛t t❤❡ t✐♠❡ ❝❧♦s❡ t♦ 710 ❢❡❡t✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡♥ s✉❜st✐t✉t❡✿ r √ √ 200 5 2 5 2 2 y(t1 ) = 200 − 16t1 = 0 =⇒ t1 = = =⇒ x1 = x(t1 ) = 200t1 = 200 ≈ 707 . 16 2 2 ◆♦✇ ✇❤❛t ❝♦✉❧❞ ❜❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤✐s s❡tt✐♥❣❄ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ ♦♥❡✱ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ♠❛② ❜❡ t❤❡ ✜♥❛❧ ❧♦❝❛t✐♦♥ ✇❤❡♥✱ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❛r❡ tr②✐♥❣ t♦ ❤✐t ❛ t❛r❣❡t ✕ ❛t ❛ ♣❛rt✐❝✉❧❛r ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❙✉♣♣♦s❡ ✇❡ ✇❛♥t t❤❡ ❝❛♥♥♦♥ ❜❛❧❧ t♦ ❜❡ ❛t (200, 500) ❛❢t❡r 2 s❡❝♦♥❞s✳ ■♥ t❤❛t ❝❛s❡ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❛♥♥♦♥ ✐s ✉♥s♣❡❝✐✜❡❞ ❛♥❞ ✐t ✐s ✇❤❛t ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✜♥❞✳ ❲❡ t❤✉s ♥❡❡❞ t♦ ✜♥❞ ❛ s♦❧✉t✐♦♥ ✭t❤❡r❡ ❝♦✉❧❞ ❜❡ ♦♥❡✱ ♠❛♥②✱ ♦r ♥♦♥❡✮ t❤❛t s❛t✐s✜❡s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✿ • t❤❡ ✜rst ❜♦✉♥❞❛r② ❧♦❝❛t✐♦♥✿ X(0) = (x(0), y(0)) = (0, 200)❀ • t❤❡ s❡❝♦♥❞ ❜♦✉♥❞❛r② ❧♦❝❛t✐♦♥✿ X(2) = (x(2), y(2)) = (500, 100)✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ❛ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ▲❡t✬s s♦❧✈❡ ✐t✳
❲❤❡r❡ ❞♦ t❤❡ s♦❧✉t✐♦♥s ❝♦♠❡ ❢r♦♠❄ ❚❤❡ ❖❉❊ ❤❛s ❜❡❡♥ s♦❧✈❡❞ ❛♥❞ t❤❡ ✜rst ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❣✐✈❡s ✉s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ x = x0 +ut x =0 +ut =⇒ . y = y0 +vt −16t2 y = 200 +vt −16t2 ❲❡ ♥♦✇ ✜♥❞ u, v ❢r♦♠ t❤❡ s❡❝♦♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❖♥❡ ❞✐♠❡♥s✐♦♥ ❛t ❛ t✐♠❡✿ ❢♦r x✱ ❛♥❞ ❢♦r y ✱
x(2) = v · 2 = 500 =⇒ u = 250;
y(2) = 200 + v · 2 − 16 · 22 = 100 =⇒ 2v = 100 − (200 − 16 · 22 ) =⇒ v = (−100 + 16 · 4)/2 = −18 .
◆♦t❡ t❤❛t ❛ ♠♦r❡ ♣r❛❝t✐❝❛❧ s✐t✉❛t✐♦♥ ✐s ✇❤❡♥ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t②✱ ♠❛t❤❡♠❛t✐❝❛❧❧② t❤❡ s♣❡❡❞✱ r❡♠❛✐♥s t❤❡ s❛♠❡ ❛♥❞ ✐t ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧ t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞✳
✺✳✾✳
❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s
✸✹✷
❊①❛♠♣❧❡ ✺✳✾✳✺✿ s♣r✐♥❣
❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊ ❢♦r t❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣✿ y ′′ (x) + y(x) = 0 .
❇✉t t❤✐s t✐♠❡ ✇❡ ❛r❡♥✬t tr②✐♥❣ t♦ ♣r❡❞✐❝t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♦❜❥❡❝t ✇✐t❤ ❦♥♦✇♥ ♣♦s✐t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t②✳ ❲❡ ❛s❦ ♦✉rs❡❧✈❡s ✐❢ t❤❡ s②st❡♠ ❝❛♥ ❜r✐♥❣ t❤❡ ♦❜❥❡❝t ❢r♦♠ ❛ ♣❛rt✐❝✉❧❛r ♣♦s✐t✐♦♥ t♦ ❛♥♦t❤❡r ✐♥ ❛ s♣❡❝✐✜❝ ❛♠♦✉♥t ♦❢ t✐♠❡✱ ❢♦r ❡①❛♠♣❧❡✿ y(0) = 0, y(π/2) = 2 .
❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ t♦ ♦✉r ❖❉❊ ✐s y(x) = A sin x + B cos x .
◆♦✇✱ ❢r♦♠ t❤❡ ✜rst ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✇❡ ♦❜t❛✐♥✿ 0 = A · 0 + B · 1 =⇒ B = 0 .
❋r♦♠ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✇❡ ♦❜t❛✐♥✿ 2 = A · 1 =⇒ A = 2 .
❚❤✉s ✐♠♣♦s✐♥❣ t❤❡s❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♣r♦❞✉❝❡s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✿ y(x) = 2 sin x.
❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t✇♦ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✱ r❡❣✐♦♥s ❛r❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❛♥❞ s♦ ❛r❡ t❤❡✐r ❜♦✉♥❞❛r✐❡s✳ ❙✉❝❤ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r❡ ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳
❈❤❛♣t❡r ✻✿ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s
❈♦♥t❡♥ts
✻✳✶ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✻✳✹ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ❚❤❡ ❤❡❛t P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s ✻✳✶✵ ❚❤❡ ✇❛✈❡ P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✶ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡ ✳ ✳ ✳
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✸✹✸ ✸✺✸ ✸✻✵ ✸✻✺ ✸✼✵ ✸✼✺ ✸✽✸ ✸✾✶ ✸✾✸ ✹✵✵ ✹✵✺
✻✳✶✳ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜ ❥❡❝ts
▼♦t✐♦♥ ❤❛♣♣❡♥s ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✳ ❍♦✇❡✈❡r✱ ②♦✉ ❝❛♥✬t ❜❡ ❛t t✇♦ ♣❧❛❝❡s ❛t t❤❡ s❛♠❡ t✐♠❡✦ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ ♣r♦❝❡ss❡s t❤❛t ❤❛♣♣❡♥ ✕ s❡♣❛r❛t❡❧② ❜✉t ♥♦t ✐♥❞❡♣❡♥❞❡♥t❧② ✕ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ♦❢ ❛ r❡❣✐♦♥✳ ❆ ❝✉♣ ♠❛② ❜❡ ❣r❛❞✉❛❧❧② ❝♦♦❧✐♥❣ ❢r♦♠ t❤❡ ♦✉ts✐❞❡✿
❚❤❡ t❡♠♣❡r❛t✉r❡ ✈❛r✐❡s ✇✐t❤ t✐♠❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ❜✉t ❛❧s♦ ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❖r ✐t ♠❛② ❜❡ ✇❛r♠✐♥❣ ✉♣ ♦♥ ❛ st♦✈❡✿
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✹✹
❚❤❡ ♣❛tt❡r♥s ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❡♠♣❡r❛t✉r❡ ✐♥ t❤❡ ❝✉♣ ❞✐✛❡r ✐♥ t❤❡s❡ t✇♦ s❝❡♥❛r✐♦s✳ ❲❡ st❛rt ✇✐t❤ ❛ r❡✈✐❡✇ ❛♥❞ t❤❡♥ ♣r♦❝❡❡❞ t♦ t❤❡
1✲❞✐♠❡♥s✐♦♥❛❧
❝❛s❡✳
▲❡t✬s r❡❝❛❧❧ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣ ✭❈❤❛♣t❡r ✶✮✿
◮ ✏❚❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✑✳ ❆♥ ❡①❛♠♣❧❡ ✐s ❝♦♦❧✐♥❣ ♦❢ ❛ ❝✉♣ ♦❢ ❝♦✛❡❡ ♦r ✇❛r♠✐♥❣ ✉♣ ❛ ❝❛♥ ♦❢ s♦❞❛✳ ❲❡ ✐♥tr♦❞✉❝❡ ✈❛r✐❛❜❧❡s✿ ✶✳
t
✷✳
u
✐s t❤❡ t✐♠❡✳ ✐s t❤❡ t❡♠♣❡r❛t✉r❡✳
❲❡ t❤❡♥ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿
∆u = k(r − u) , ∆t ✇❤❡r❡
r
✐s t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ❢♦r s♦♠❡ ❝♦♥st❛♥t
k > 0✳
❲❡ ❝❛♥ s❡❡ t❤❛t ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❤✐❣❤❡r
t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ❞❡❝r❡❛s❡s ❛♥❞ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❧♦✇❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ✐♥❝r❡❛s❡s✳ ❚❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✭❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✮✿
u(tn+1 ) = u(tn ) + K(r − u(tn )) · ∆t , ❝♦♠❜✐♥❡❞ ✇✐t❤
tn+1 = tn + ∆t . ❚❤❡s❡ ❢♦r♠✉❧❛s ❛r❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥s✳ ❲❡ ❝❛♥ s❡❡ ❜❡❧♦✇ s❡✈❡r❛❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❝✉♣ ♦❢ ❝♦✛❡❡ ✐s ❝♦♦❧✐♥❣ ❞♦✇♥ ❛♥❞ ❛ ❝❛♥ ♦❢ s♦❞❛ ✐s ✇❛r♠✐♥❣ ✉♣✿
❇♦t❤ t❡♠♣❡r❛t✉r❡s ❛r❡ ❛♣♣r♦❛❝❤✐♥❣ t❤❛t ♦❢ t❤❡ r♦♦♠ ❛s t❤❡② ❡①❝❤❛♥❣❡ t❤❡ ❤❡❛t ✇✐t❤ t❤❡ s✉rr♦✉♥❞✐♥❣ ❛✐r ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ✐♥ ❈❤❛♣t❡r ✶✳ ❚❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ r❡♠❛✐♥s ❝♦♥st❛♥t ❜❡❝❛✉s❡ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐s t♦♦ ❧❛r❣❡ ✐♥ ❝♦♠♣❛r✐s♦♥✳ ❲❡ ♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✳
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
❲❡ ✜rst t❛❦❡ t❤✐s ♠♦❞❡❧ ❛s t❤❛t ♦❢
✸✹✺
t✇♦ ♦❜❥❡❝ts ✐♥ t❤❡ r♦♦♠ ❛t t❤❡ s❛♠❡ t✐♠❡✿ s♦❞❛
❛✐r
❝♦✛❡❡
❲❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s✿
u
✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ s♦❞❛ ❛♥❞
v
✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❝♦✛❡❡✳
❚❤❡♥✱ ✇❡ ❤❛✈❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✿
u(tn+1 ) = u(tn ) + K(r − u(tn )) · ∆t ◆❡①t st❡♣✿ ❲❤❛t ✐❢ t❤❡ t✇♦ ♦❜❥❡❝ts ❛r❡
❛❞❥❛❝❡♥t
❛♥❞
v(tn+1 ) = v(tn ) + K(r − v(tn )) · ∆t .
t♦ ❡❛❝❤ ♦t❤❡r❄ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ t✇♦ t♦✉❝❤ ❡❛❝❤ ♦t❤❡r
✇❤✐❧❡ ✐♥s✉❧❛t❡❞ ❡❧s❡✇❤❡r❡ ✭♥♦ ❛✐r✮✿ s♦❞❛
❝♦✛❡❡
❚❤❡♥ t❤❡ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts ❡①❝❤❛♥❣❡s t❤❡ ❤❡❛t ✇✐t❤ t❤❡ ♦t❤❡r ❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✳ ❚❤❡ ❧❛✇ st✐❧❧ ❛♣♣❧✐❡s✿
◮
✏❚❤❡ r❛t❡ ♦❢ ❤❡❛t ❡①❝❤❛♥❣❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡ ❛♥❞
t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ♦❜❥❡❝t✑✳
❆t ❡❛❝❤ st❡♣✱ t❤❡r❡❢♦r❡✱ t❤❡ ❞②♥❛♠✐❝s r❡♠❛✐♥s t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡ ❜✉t t❤❡ r❡❢❡r❡♥❝❡ ✐s t♦ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ♦t❤❡r ♦❜❥❡❝t r❛t❤❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✳ ❲❡ t❛❦❡ t❤❡ t✇♦ ❢♦r♠✉❧❛s ❢♦r♠✉❧❛s ❛❜♦✈❡ ❛♥❞ ❥✉st r❡♣❧❛❝❡
•
❢♦r s♦❞❛
•
❢♦r ❝♦✛❡❡
u✿ r
r❡♣❧❛❝❡❞ ✇✐t❤ ❝♦✛❡❡
v✿ r
r❡♣❧❛❝❡❞ ✇✐t❤ s♦❞❛
r
✭❛✐r✮ ✇✐t❤ t❤❡ ♦t❤❡r ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡✿
v✱ u✳
❚❤✐s ✐s t❤❡ r❡s✉❧t✿
u(tn+1 ) = u(tn ) + k v(tn ) − u(tn ) · ∆t
❛♥❞
v(tn+1 ) = v(tn ) + k u(tn ) − v(tn ) · ∆t .
❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❤❡❛t ❛❞❞❡❞ t♦ ♦♥❡ ✐s t❤❡ ❤❡❛t t❛❦❡♥ ❢r♦♠ t❤❡ ♦t❤❡r❀ t❤❛t
❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❊♥❡r❣② ✦
❊①❛♠♣❧❡ ✻✳✶✳✶✿ s♦❞❛✲❝♦✛❡❡ ▲❡t✬s ❝❛rr② ♦✉t t❤✐s ♣❧❛♥ ✇✐t❤ ❛
s♣r❡❛❞s❤❡❡t✳
❲❡ ❝❤♦♦s❡ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡s✿ s♦❞❛
k = .1
40 ❛♥❞ ❝♦✛❡❡ 100 ❞❡❣r❡❡s❀ t❤❡② ❛r❡ ✐♥ t❤❡ ✜rst r♦✇✳
❲❡ ❝❤♦s❡
❢♦r ❛ s❧♦✇ ❡①❝❤❛♥❣❡✳ ❚❤❡ t✇♦ ❢♦r♠✉❧❛s ❛r❡ s✐♠✐❧❛r✿
❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✶❪✲❘❬✲✶❪❈✮✯❘✷❈
❛♥❞
❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✲✶❪✲❘❬✲✶❪❈✮✯❘✷❈
❇❡❧♦✇ ✐s t❤❡ ♣r❡❝❡❞❡♥❝❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡s❡ ❢♦r♠✉❧❛s✿
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✹✻
❆s ❡①♣❡❝t❡❞✱ t❤❡ ❞②♥❛♠✐❝s ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧✿
❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ t❡♠♣❡r❛t✉r❡s ❝♦♥✈❡r❣❡ t♦ t❤❡ ♠✐❞❞❧❡ ✐♥st❡❛❞ t♦ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✿ 40 + 100 = 70 . 2
❊①❡r❝✐s❡ ✻✳✶✳✷ Pr♦✈❡ t❤❡ ❧✐♠✐t✳ ❖♥❡ st❡♣ ❢✉rt❤❡r✳ ❲❤❛t ✐❢ t❤❡r❡ ❛❧s♦ ❛
❝✉♣ ♦❢ t❡❛ ❄
❲❡ ❛ss✉♠❡ ❛❣❛✐♥ t❤❛t t❤❡ t❤r❡❡ t♦✉❝❤ ❡❛❝❤ ♦t❤❡r ✇❤✐❧❡ ✐♥s✉❧❛t❡❞ ❡❧s❡✇❤❡r❡✿ s♦❞❛ ❝♦✛❡❡ t❡❛ ❚❤❡ ❢✉♥❝t✐♦♥s ❛r❡ r❡s♣❡❝t✐✈❡❧②✿ u v
w
❚❤❡♥ t❤❡ s♦❞❛ ❡①❝❤❛♥❣❡s ❤❡❛t ✇✐t❤ t❤❡ ❝♦✛❡❡ ❥✉st ❛s ❜❡❢♦r❡ ❛♥❞ t❤❡ t❡❛ ❛❧s♦ ❡①❝❤❛♥❣❡s ❤❡❛t ✇✐t❤ t❤❡ ❝♦✛❡❡ ❥✉st ❛s ❜❡❢♦r❡✳ ❊✐t❤❡r ❤❛s ♦♥❧② ♦♥❡ ♥❡✐❣❤❜♦r✦ ❲❡ ✇r✐t❡ ❢♦r t❤❡s❡ t✇♦✿ u(tn+1 ) = u(tn ) + k v(tn ) − u(tn ) · ∆t
w(tn+1 ) = w(tn ) + k v(tn ) − w(tn ) · ∆t .
❚❤❡ ❝♦✛❡❡ ✐s ♥♦✇ ✐♥ ❛ ♥❡✇ ♣♦s✐t✐♦♥✳ ■t ❤❛s t✇♦ ♥❡✐❣❤❜♦rs✦ ❙✐♥❝❡ ✇❡ ❡①♣❡❝t t❤❡ ❤❡❛t ❛❞❞❡❞ t♦ t❤❡ ❝♦✛❡❡ ✐s t❤❡ ❤❡❛t t❛❦❡♥ ❢r♦♠ t❤❡ t✇♦ ♦t❤❡rs✱ ✇❡ s✐♠♣❧② ❝♦♣② t❤❡ t❡r♠s ❢r♦♠ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ v(tn+1 ) = v(tn ) + k u(tn ) − v(tn ) · ∆t + k w(tn ) − v(tn ) · ∆t .
❊①❛♠♣❧❡ ✻✳✶✳✸✿ s♦❞❛✲❝♦✛❡❡✲t❡❛
❲❡ ❝❤♦♦s❡ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡s✿ s♦❞❛ 40✱ ❝♦✛❡❡ 100✱ ❛♥❞ t❡❛ 85 ❞❡❣r❡❡s✳ ❚❤❡ t✇♦ ❢♦r♠✉❧❛s t❤❡ s♦❞❛ ❛♥❞ t❤❡ t❡❛ ❛r❡ t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✿ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✶❪✲❘❬✲✶❪❈✮✯❘✷❈ ❛♥❞ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✲✶❪✲❘❬✲✶❪❈✮✯❘✷❈
❚❤❡② t❛❦❡ ❢r♦♠ t❤❡✐r r✐❣❤t ❛♥❞ ❧❡❢t ♥❡✐❣❤❜♦rs r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❢♦r♠✉❧❛s t❤❡ ❝♦✛❡❡ ✐s ♥❡✇✿ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✰✵✳✶✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✹✼
■t t❛❦❡s ❢r♦♠ t❤❡✐r ❧❡❢t ❛♥❞ r✐❣❤t ♥❡✐❣❤❜♦rs ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜rst r♦✇s✿
❚❤❡ ❞②♥❛♠✐❝s ✐s s✐♠✐❧❛r t♦ t❤❡ ❧❛st✿
❚❤❡ s♦❞❛ ✐s ✇❛r♠✐♥❣ ✉♣✱ t❤❡ ❝♦✛❡❡ ✐s ❝♦♦❧✐♥❣ ❞♦✇♥✱ ✇❤✐❧❡ t❤❡ t❡❛ ✜rst ✐s ✇❛r♠✐♥❣ ✉♣ ✭❜❡❝❛✉s❡ t❤❡ ❛❞❥❛❝❡♥t ❝♦✛❡❡ ✐s ❤♦tt❡r✮ ❛♥❞ t❤❡♥ ❝♦♦❧✐♥❣ ❞♦✇♥ ✭❛s t❤❡ ❝♦✛❡❡ ✐s✮✳ ❚❤❡② ❛r❡ ❛❧❧ ❝♦♥✈❡r❣✐♥❣ ♦♥ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡✿ 40 + 100 + 85 = 75 . 3
◆❡①t st❡♣✿ ❲❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡r❡ ✐s ❛ s❡r✐❡s ♦❢ ♦❜❥❡❝ts t❤❛t ❡①❝❤❛♥❣❡ t❤❡✐r ❤❡❛t ✇✐t❤ t❤❡✐r ♥❡✐❣❤❜♦rs✳ ❊①❛♠♣❧❡ ✻✳✶✳✹✿ t❡♥ ❝❛♥s
❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ 10 ♦❜❥❡❝ts ✭♦r ❝♦♥t❛✐♥❡rs✮ ✇✐t❤ ❝♦❧❞ ❡♥❞s ❛♥❞ ❤♦t ❝❡♥t❡r✳ ❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s r❡♣❡❛t❡❞ ✭❡✐❣❤t t✐♠❡s✮ ✐♥ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ 10 ❝♦❧✉♠♥s✳ ❲❡ ❝♦❧♦r t❤❡ ❝❡❧❧s t♦ s❡❡ t❤❡ ❞②♥❛♠✐❝s✿
❚❤❡ ❞②♥❛♠✐❝s ✐s ❛s ❡①♣❡❝t❡❞✿ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡✳ ❲❡ ❛❧s♦ r❡❛❧✐③❡ t❤❛t t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✦ ■t ❝❛♥ ❛❧s♦ ❜❡ ✈✐s✉❛❧✐③❡❞ ❜② ✐ts ❣r❛♣❤✿
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✹✽
❆♥♦t❤❡r ❡①♣❡r✐♠❡♥t✱ ✇✐t❤ ❤♦t ❛♥❞ ❝♦❧❞ ❡♥❞s✿
❖♥❝❡ ❛❣❛✐♥✱ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡✦ ❊①❡r❝✐s❡ ✻✳✶✳✺
■♠♣❧❡♠❡♥t ❤❡❛t tr❛♥s❢❡r ✐♥ ❛ ❝✐r❝✉❧❛r ❛rr❛♥❣❡♠❡♥t✳ ❚❤❡ ♠♦❞❡❧ ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ♠❡t❛❧
r♦❞ ✿
❚❤✐s ✐s ❛♥ ✐♥s✉❧❛t❡❞ r♦❞ ❛♥❞ ♠✐❣❤t ❜❡ ❤❡❛t❡❞ ❛t ♦♥❡ ❡♥❞ ❛s ❛❜♦✈❡✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ t❤❡♥ ✈❛r✐❡s ❛❧♦♥❣ ✐ts ❧❡♥❣t❤ ❡✈❡♥ ✇❤❡♥ t❤❡ t✐♠❡ ✐s ✜①❡❞✳ ❲❡ ♥❡❡❞ ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✦ ❲❡ r❡✲✐♥tr♦❞✉❝❡ t❤❡ ✈❛r✐❛❜❧❡s✿ ✶✳ t ✐s t❤❡ t✐♠❡✳ ✷✳ x ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ✸✳ u ✐s t❤❡ t❡♠♣❡r❛t✉r❡✳ ❆❧❧ t❤r❡❡ ❛r❡ ❥✉st ♥✉♠❜❡rs✳ ❍♦✇ ❞♦ t❤❡② ❞❡♣❡♥❞ ♦♥ ❡❛❝❤ ♦t❤❡r❄ ❚❤❡r❡ ✐s ♥♦ ♠♦t✐♦♥✦ ❚❤❡r❡❢♦r❡✱ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ t ♦♥❡ ❝❛♥ ❜❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ x ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤✐s ♠❡❛♥s t❤❛t t ❛♥❞ x ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✳ ◆❡①t✱ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ t ❛♥❞ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ x ✇❡ ❝❛♥ ♠❡❛s✉r❡ t❤❡ t❡♠♣❡r❛t✉r❡ u✳ ❚❤✐s ♠❡❛♥s t❤❛t u ❞❡♣❡♥❞s ♦♥ t ❛♥❞ x✳ ❙♦✱ ✇❡ ❤❛✈❡ t❤❡
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✹✾
❢♦❧❧♦✇✐♥❣ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠✿
t ց x ❲❡ s✐♠♣❧② ❤❛✈❡ ❛
u
ր
❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✿ u = f (t, x) .
❇❡❢♦r❡ ❞❡✈❡❧♦♣✐♥❣ ❛ ♠♦❞❡❧ ❛♥❞ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✱ ✇❤❛t ✐s ❛ r❡❛s♦♥❛❜❧❡ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛♥ ❡♥✈✐s✐♦♥❄ ▲❡t✬s s❡t ✉♣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❙✉♣♣♦s❡ t❤❡ r♦❞ ✐s ❝♦♦❧ ✕ ♥❡❣❛t✐✈❡ ✕ ❛t ♦♥❡ ❡♥❞ ❛♥❞ ❤♦t ✕ ♣♦s✐t✐✈❡ ✕ ❛t t❤❡ ♦t❤❡r✳ ❋✐rst ✇❡ ✐♠❛❣✐♥❡ t❤❛t ✇❡ s✐t ♥❡①t t♦ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ♦♥ t❤❡ r♦❞ ❛♥❞ ♦❜s❡r✈❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❡♠♣❡r❛t✉r❡ ♦✈❡r
t✐♠❡✳
❏✉❞❣✐♥❣ ❜② ◆❡✇t♦♥✬s ▲❛✇✱ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ ❣r❛❞✉❛❧❧② ❝♦♥✈❡r❣❡ t♦ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦✈❡r
t✐♠❡✳ ❙✉♣♣♦s❡ t❤✐s ✈❛❧✉❡ ✐s
0✿
◆♦✇ ✇❡ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ✏❢r❡❡③❡✑ t✐♠❡ ❛♥❞ r✉♥ ❛❧♦♥❣ t❤❡ r♦❞ ♦❜s❡r✈✐♥❣ ❤♦✇ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❝❤❛♥❣✐♥❣ ♦✈❡r
s♣❛❝❡✳
❙✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡ ✈❛r✐❡s ❧✐♥❡❛r❧② ❢r♦♠ ♦♥❡ ❡♥❞ t♦ t❤❡ ♦t❤❡r✳ ❏✉❞❣✐♥❣ ❜② t❤❡ ❈♦♥s❡r✈❛t✐♦♥ ♦❢
❊♥❡r❣② ▲❛✇✱ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ ❣r❛❞✉❛❧❧② ❛✈❡r❛❣❡ ✐ts❡❧❢ ✇✐t❤ t✐♠❡✳ ■t ✐s ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ r❡♠❛✐♥ ❧✐♥❡❛r✿
❲❡ ❝❛♥ tr② t♦ ✐♠❛❣✐♥❡ ✇❤❛t t❤❡ ❢✉♥❝t✐♦♥
f
✐s ❧✐❦❡✳ ❚❤❡ ❖❉❊ ❤❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② s♦❧✉t✐♦♥s✿
u = r + Ce−kt . ❚♦ ❣❡t t❤❡ ❣❡♥❡r❛❧ s❤❛♣❡✱ ✇❡ ❝❛♥ tr②✿
u = xe−t
♦r
u = x/t .
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts
✸✺✵
▲❡t✬s ❝♦❧❧❡❝t t❤❡ ❞❛t❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❚❤❡ ✐♥♣✉ts ❜❡✐♥❣ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r ❢♦r♠ ❛♥
❛rr❛②✱ s❛② x ✐♥ r♦✇s ❛♥❞ t ✐♥ ❝♦❧✉♠♥s✿
t\x 0.0 1.5 2.0 2.5 0 1 2 ■♥ ♦t❤❡r ✇♦r❞s✱ ♦✉r ❢✉♥❝t✐♦♥
f
❤❛s t❤❡
(t, x)✲♣❧❛♥❡
♦r ✐ts s✉❜s❡t ❢♦r ❛ ❞♦♠❛✐♥✳
◆♦✇ t❤❡ ♦✉t♣✉ts✳ ❲❡ ❥✉st ♣❧❛❝❡ t❤❡♠ ❛t ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s ♦❢ t❤❡ ❛❜♦✈❡ t❛❜❧❡✿
t\x 0 1 2
0 1 3 0
1.5 2 4 0
2 2.5 0 0 −1
P❧♦tt✐♥❣ t❤❡ s♦❧✉t✐♦♥s t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♠❛t❝❤❡s t❤✐s ❞❡s❝r✐♣t✐♦♥✿
❚❤✐s t✐♠❡✱ ❛❧❧ ❜✉t t❤❡ ♦❜❥❡❝ts ❛t t❤❡ ❡♥❞s ❢♦❧❧♦✇ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ t❤❛t r❡❢❡rs t♦ ✐ts
t✇♦ ♥❡✐❣❤❜♦rs✿
v(ti+1 ) = v(ti ) + k u(ti ) − v(ti ) · ∆t + k w(ti ) − v(ti ) · ∆t .
❲❡ r❡✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❙♦✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢
t
❛♥❞
x✳
❚❤❡ t✐♠❡ ✐s ❛s ❜❡❢♦r❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ♥♦❞❡s ♦❢ ♦✉r ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡
t✲❛①✐s✿
..., ti−1 , ti , ti+1 , ..., ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥t✿
∆t = ti+1 − ti . ❚❤❡ s♣❛❝❡ ✐s s✐♠✐❧❛r✳ ❚❤❡s❡ ❛r❡ t❤❡ ♥♦❞❡s ♦❢ ♦✉r ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡
x✲❛①✐s
✭t❤❡ r♦❞✮✿
..., xn−1 , xn , xn+1 , ... ❲❡ ❛r❡ ♠❛❦✐♥❣ ❛ s✇✐t❝❤ ❢r♦♠ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡s ♦♥❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ t♦ t❤❡ ♣r❡✈✐♦✉s ♥♦t❛t✐♦♥✿
u(·) v(·) w(·) ... xn−1 xn xn+1 ... ... u(·, xn−1 ) u(·, xn ) u(·, xn+1 ) ... ... −•− −•− −•− ... ❚❤❡ ❤❡❛t ✐s ❡①❝❤❛♥❣❡❞ ❜❡t✇❡❡♥ t❤❡s❡ ✏❝♦♥t❛✐♥❡rs✑ t❤r♦✉❣❤ t❤❡s❡ ✏♣✐♣❡s✑✳
✻✳✶✳ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ❚❤❡♥ ♦✉r ❢♦r♠✉❧❛✱ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿
✸✺✶
v(ti+1 ) = v(ti ) + k u(ti ) − v(ti ) · ∆t + k w(ti ) − v(ti ) · ∆t
u(ti+1 , xn ) = u(ti , xn ) + k u(ti , xn−1 ) − u(ti , xn ) · ∆t + k u(ti , xn+1 ) − u(ti , xn ) · ∆t .
❚❤❡ ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛❄ ❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t✿
∆t u(ti , xn ) = u(ti+1 , xn ) − u(ti , xn ) .
❲❡ ❛❞❞ ❛ s✉❜s❝r✐♣t t t♦ ✐♥❞✐❝❛t❡ t❤❡ ✈❛r✐❛❜❧❡ ❥✉st ❛s ✇✐t❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✦ ❚❤❡♥✱ ♦✉r ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿ ∆t u(ti , xn ) = k u(ti , xn−1 ) − u(ti , xn ) · ∆t + k u(tn , xi+1 ) − u(ti , xn ) · ∆t .
❚❤❡ t✇♦ t❡r♠s ❛r❡ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ❡①❝❤❛♥❣❡❞ ✇✐t❤ t❤❡ t✇♦ ♥❡✐❣❤❜♦rs✳ ❉✐✈✐❞✐♥❣ ❜② ∆t s✐♠♣❧✐✜❡s t❤✐♥❣s✿
∆t u(ti , xn ) = k u(ti , xn−1 ) − u(ti , xn ) + u(ti , xn+1 ) − u(ti , xn ) . ∆t
❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ t❡r♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡❄ ■s t❤✐s t❤❡ s✉♠ ♦❢ t✇♦ ❞✐✛❡r❡♥❝❡s❄✦ ◆♦✳ ❋✐rst✱ t❤❡s❡ ❛r❡ ❞✐✛❡r❡♥❝❡s ❜✉t ♥♦t ❞✐✛❡r❡♥❝❡s ∆✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ ❛t t❤❡ ❧♦❝❛t✐♦♥ ✐s s✉❜tr❛❝t❡❞ ❢r♦♠ t❤❛t ♦❢ ❛ ♥❡✐❣❤❜♦r✱ ♥♦ ♠❛tt❡r ❧❡❢t ♦r r✐❣❤t✱ ❛ s♠❛❧❧❡r ♦r ❧❛r❣❡r ♣♦s✐t✐♦♥ ✇✐t❤✐♥ t❤❡ x✲❛①✐s✳ ❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ x✿ ∆x u(ti , xn ) = u(ti , xn+1 ) − u(ti , xn ) .
❘✐❣❤t ♠✐♥✉s ❧❡❢t✦ ▲❡t✬s ✜♥❞ t❤❡ ❞✐✛❡r❡♥❝❡s ✇✐t❤ r❡s♣❡❝t t♦ s♣❛❝❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭· st❛♥❞s ❢♦r ti ✮✿ u ·, xn−1 ) − u(·, xn + u ·, xn+1 ) − u(·, xn = − u(·, xn − u(·, xn−1 )] + u(·, xn+1 ) − u(·, xn ) = − ∆x u(·, xn ) + ∆x u(·, xn+1 ) .
❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡✦
■t ♠❛❦❡s s❡♥s❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤r❡❡ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✇✐t❤ t❤❡s❡ t❡♠♣❡r❛t✉r❡s✿ 4
7
9
❚❤❡ ❤❡❛t ✢♦✇s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ 4 ← ← 7 ← ← 9
• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡ ❧❡❢t ❝❡❧❧❄ ❯♣✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ♥❡✐❣❤❜♦r ✐s ✇❛r♠❡r✳
• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡ r✐❣❤t ❝❡❧❧❄ ❉♦✇♥✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ♥❡✐❣❤❜♦r ✐s ❝♦♦❧❡r✳
♠✐❞❞❧❡ ❝❡❧❧❄ ❉♦✇♥✦ ❲❤②❄ ❚❤❡r❡ ❛r❡ t✇♦ ♥❡✐❣❤❜♦rs✱ ♦♥❡ ❝♦♦❧❡r ❛♥❞ t❤❡ ♦t❤❡r ✇❛r♠❡r✳✳✳ ❚❤❡② ❤❛✈❡ ♦♣♣♦s✐t❡ ❡✛❡❝ts ♦♥ t❤✐s ❝❡❧❧✦ ❙♦✱ ✇❤② ❞♦✇♥❄
• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡
❚❤✐s ✐s ✇❤②✿
↑ 4 ↑ ↓ 7 ↓ ↓ 9 ↓
✻✳✶✳
❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ◮
❇❡❝❛✉s❡
7
✐s ❝❧♦s❡r t♦
9
t❤❛♥ t♦
✸✺✷
4✳
❚❤❡ ❤❡❛t ✢♦✇s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ❜✉t t❤❡ ♠✐❞❞❧❡ ♦♥❡ ❧♦s❡s ♠♦r❡ t❤❛♥ ✐t ❣✐✈❡s✿
4 ⇔ ⇔ 7 ← ← 9 ▲❡t✬s t❛❦❡ t❤✐s ❛♣❛rt✳ ✏❈❧♦s❡✑ ✐s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s✳ ✏❈❧♦s❡r✑ ✐s✱ t❤❡r❡❢♦r❡✱ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s✦ ❚❤✐s ✐❞❡❛ ✐s♥✬t ♥❡✇❀ ✐t✬s ❛❧❧ ❛❜♦✉t
❝♦♥❝❛✈✐t②
✭❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✱ ❈❤❛♣t❡r ✷❉❈✲✹✮ ♦❢ t❤❡
❢✉♥❝t✐♦♥✿
▲❡t✬s ✜♥❞ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ s♣❛❝❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ♦✉r ❡q✉❛t✐♦♥✿
−∆x u(·, xn ) + ∆x u(·, xn+1 ) = ∆x ∆x u(·, xn ) . ❲❡ ❤❛✈❡ t❤❡r❡❢♦r❡✿
∆t u(ti , xn ) = k∆x ∆x u(ti , xn ) . ∆t ❲❛r♥✐♥❣✦
❚❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ ♦♥❧② ✇♦r❦s ✇❤❡♥ k ✐s ❝♦♥st❛♥t✳ ❙♦✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❝♦♥❝❛✈✐t② ❞❡t❡r♠✐♥❡s ❜② ❤♦✇ ♠✉❝❤ t❤❡ ✈❛❧✉❡ ✐s ♣✉s❤❡❞ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿
❚❤❡ r❡s✉❧t ✐s ❛✈❡r❛❣✐♥❣✳ ❚❤✐s ✐s t❤❡ s❤♦rt❡♥❡❞ ✈❡rs✐♦♥✿
∆t u = k∆2x u ∆t ■t ✐s ❝❛❧❧❡❞ t❤❡
❤❡❛t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✳
❚❤❡ r❡❛❧✐t② ✐s ♠♦r❡ ❝♦♠♣❧❡①✿
◮
❚❤❡ tr❛♥s❢❡r r❛t❡ ♠✐❣❤t ❜❡ ❞✐✛❡r❡♥t ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✳
✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✸
✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
❲❡ st❛rt ✐♥ t❤✐s s❡❝t✐♦♥ ✇✐t❤ ❤❡❛t tr❛♥s❢❡r ✇✐t❤✐♥ ❛ ♠❡t❛❧ r♦❞✳ ■t✬s ❤❡❛t✲tr❛♥s❢❡rr✐♥❣ ♣r♦♣❡rt✐❡s ♠✐❣❤t ❜❡ ♥♦♥✲✉♥✐❢♦r♠✿
❚❤❡ r♦❞ ✐s s❡❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤②♣♦t❤❡t✐❝❛❧ ✇❛②✿ ❛s ✐❢ ✐t✬s s♣❧✐t ✐♥t♦ ♣✐❡❝❡s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❤❡❛t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ str✐♥❣ ♦❢ ❝♦♥t❛✐♥❡rs ❛♥❞ ❡❛❝❤ ❝♦♥t❛✐♥❡r ❡①❝❤❛♥❣❡s t❤❡ ❤❡❛t ✇✐t❤ ✐ts t✇♦ ♥❡✐❣❤❜♦rs t❤r♦✉❣❤ t❤❡ ♣✐♣❡s✿
❍❡r❡ ✇❡ ❤❛✈❡✿ • x, y, z ❛r❡ s♦♠❡ ♦❢ t❤❡ r♦♦♠s✳ • a, b ❛r❡ s♦♠❡ ♦❢ t❤❡ ♣✐♣❡s✳
❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ ❝♦♥t❛✐♥❡rs ✇✐t❤ ♥♦❞❡s ❛♥❞ ♣✐♣❡s ✇✐t❤ ❡❞❣❡s✳ ❆ ❝❛r❡❢✉❧ ❧♦♦❦ r❡✈❡❛❧s t❤❛t t♦ ♠♦❞❡❧ ❤❡❛t tr❛♥s❢❡r✱ ✇❡ ♥❡❡❞ t♦ s❡♣❛r❛t❡❧② r❡❝♦r❞ t❤❡ ❡①❝❤❛♥❣❡ ♦❢ ❤❡❛t ♦❢ ❡❛❝❤ ❝♦♥t❛✐♥❡r ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs✳ ❚❤❡ ♣r♦❝❡ss ✇❡ st✉❞② ♦❜❡②s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧✐❛r ❧❛✇ ♦❢ ♣❤②s✐❝s✳ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✿ ❚❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ✐ts t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ❛♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝t✳ ❖❢ ❝♦✉rs❡✱ ❝♦♦❧✐♥❣ ♠❡❛♥s ❤❡❛t✐♥❣ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ♦t❤❡r ♦❜❥❡❝t ✐s ❤✐❣❤❡r✳ ❖✉r ✐♥✐t✐❛❧ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ❛❧❧ ❝♦♥t❛✐♥❡rs ❤❛✈❡ ❡q✉❛❧ s✐③❡✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ ❝♦♥t❛✐♥❡rs ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡✐r r❡s♣❡❝t✐✈❡ t❡♠♣❡r❛t✉r❡s✳ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❡ ❧❛✇ ♦❢ ❝♦♦❧✐♥❣ ❛s ❢♦❧❧♦✇s✿ ◮ ❚❤❡ r❛t❡ ♦❢ ❤❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❡✈❡r② t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡
❜❡t✇❡❡♥ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ✐♥ t❤❡♠✳
✻✳✷✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✹
❚❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② ✐♥ ❝♦♥t❛✐♥❡r x ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ❤❡❛t ✐♥ ❝♦♥t❛✐♥❡rs x ♦✈❡r t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ❢r♦♠ ti t♦ ti+1 ✐s ❡q✉❛❧ t♦ u(ti+1 , x) − u(ti , x) = −
s✉♠ ♦❢ t❤❡ ♦✉t✢♦✇ g t❤r♦✉❣❤ t❤❡ ♣✐♣❡s ♦❢ x .
❚❤❡ ♦✉t✢♦✇ ❣✐✈❡s t❤❡ ❛♠♦✉♥t ♦❢ ✢♦✇ ❛❝r♦ss ❛♥ ❡❞❣❡ ✭❢r♦♠ t❤❡ ❝♦♥t❛✐♥❡r t♦ ✐ts ♥❡✐❣❤❜♦r✮ ♣❡r ✉♥✐t ♦❢ t✐♠❡✳ ❙♣❡❝✐✜❝❛❧❧②✱ t❤❡ ✢♦✇ ✐s ♣♦s✐t✐✈❡ ❛t x ✐❢ ✐t ✐s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ❢♦r y ❀ t❤❡♥✿ u(ti+1 , x) − u(ti , x) = − g(ti , a) − g(ti , b) = g(ti , a) − g(ti , b) . ❲❛r♥✐♥❣✦ ❲❤❛t ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s r❛t❡❄
■t ❝❛♥♥♦t
❜❡ s♦ ❢❛st t❤❛t t❤❡ ✉♣❞❛t❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❤❡❛t ♦❢ t❤❡ ❝♦♥t❛✐♥❡r s✉r♣❛ss❡s t❤❛t ♦❢ t❤❡ ♦t❤❡r✦
■♥ ♦r✲
❞❡r t♦ ❛✈♦✐❞ t❤✐s ✏♦✈❡rs❤♦♦t✑✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❤❡❛t s❤♦✉❧❞♥✬t ❜❡ ♠♦r❡ t❤❛t ❤❛❧❢✲❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ ❤❡❛t ♦❢ t❤❡ ♦t❤❡r ❝♦♥t❛✐♥❡r✳
◆♦✇✱ ✇❡ ♥❡❡❞ t♦ ❡①♣r❡ss g ✐♥ t❡r♠s ♦❢ u✳ ❚❤❡ ✢♦✇ g(t, a) t❤r♦✉❣❤ ♣✐♣❡ a ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ✐♥ a ❛♥❞ t❤❡ ♦t❤❡r ❝♦♥t❛✐♥❡r ❛❞❥❛❝❡♥t t♦ a✳ ❙♦✱ ♣✐♣❡ a : g(ti , a) = −K(a) u(ti , x) − u(ti , y) ∆t ♣✐♣❡ b : g(ti , b) = −K(b) u(ti , z) − u(ti , x) ∆t
❍❡r❡✱ K(a) ≥ 0 r❡♣r❡s❡♥ts t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ t❤❡ ♣✐♣❡ a ❛t ❛ ❣✐✈❡♥ t✐♠❡ ♦✈❡r t❤❡ s❛♠❡ t✐♠❡ ♣❡r✐♦❞✳
❚❤❡s❡ ♥✉♠❜❡rs K(a) ❛r❡ s✐♠♣❧② t❤❡ ♣r♦♣♦rt✐♦♥❛❧✐t② ❝♦❡✣❝✐❡♥ts✳ ❚❤❡②✱ t❤❡r❡❢♦r❡✱ ♣r♦❞✉❝❡ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳ ❚❤❡ ❜❡♥❡✜t ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✇❡ ✇♦♥✬t ♥❡❡❞ t♦ ❤❛✈❡ s❡♣❛r❛t❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞✦ ❊①❡r❝✐s❡ ✻✳✷✳✶
❈❛♥ K ❛❧s♦ ❞❡♣❡♥❞ ♦♥ t❄ ❚❤❡ r❡s✉❧t ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ h h i i u(ti+1 , x) = u(ti , x) + − K(a) u(ti , x) − u(ti , y) ∆t − − K(b) u(ti , z) − u(ti , x) ∆t
▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣ ❢♦r ❞✐s❝r❡t❡ ❢♦r♠s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✳
❲❡ ❤❛✈❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❞❡❝♦♠♣♦s❡ ✐t ✐♥t♦ n ❡❞❣❡s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b .
❚❤❡s❡ ❛r❡ t❤❡ ❡❞❣❡s✿ c1 = [x0 , x1 ], c2 = [x1 , x2 ], ..., cn = [xn−1 , xn ] .
❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ✐s ❛ 0✲❢♦r♠ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ ♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛ 1✲❢♦r♠✿
✻✳✷✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✺
❚❤❡ 0✲❢♦r♠s ❤❛✈❡ ♥♦❞❡s ❛s ✐♥♣✉ts ❛♥❞ t❤❡ 1✲❢♦r♠s ❤❛✈❡ ❡❞❣❡s ❛s ✐♥♣✉ts ❜✉t t❤❡ ♦✉t♣✉ts ❛r❡ r❡❛❧ ♥✉♠❜❡rs✿
❲❤❛t ♠❛❦❡s t❤✐s ❞✐✛❡r❡♥t ❢r♦♠ ❖❉❊s ✐s t❤❛t t❤❡ ❢✉♥❝t✐♦♥s ✇✐❧❧ ❤❛✈❡ t✇♦ ✈❛r✐❛❜❧❡s ✕ ♦♥❡ ❢♦r ❧♦❝❛t✐♦♥ ❛♥❞ ♦♥❡ ❢♦r t✐♠❡✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t u = u(t, x) ✐s s✐♠♣❧② ❛ ♥✉♠❜❡r ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ❝♦♥t❛✐♥❡r x✳ ❚❤✐s ✐s ✶✳ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ ❧♦❝❛t✐♦♥ x ❛♥❞ ✷✳ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ t✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ s❡t✉♣ ❢♦r ❞✐s❝r❡t❡ ❢♦r♠s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❲❡ st❛rt ✇✐t❤ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ t❤❡ x✲ ❛♥❞ t❤❡ t✲❛①❡s✿
❚❤❡② ❞❡✈❡❧♦♣ ✐♥ ♣❛r❛❧❧❡❧✿
• ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✐♥ t❤❡ x✲❛①✐s ❝♦♥s✐sts ♦❢ n + 1 ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b , ❛♥❞ n ❡❞❣❡s✿
s1 , s2 , ..., sn−1 , sn , ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥ts✿
∆xk = xk − xk−1 , k = 1, 2, ..., n . • ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡ ✐♥t❡r✈❛❧ [c, d] ✐♥ t❤❡ t✲❛①✐s ❝♦♥s✐sts ♦❢ m + 1 ♥♦❞❡s✿ c = t0 , t1 , t2 , ..., tm−1 , tm = d , ❛♥❞ m ❡❞❣❡s✿
q1 , q2 , ..., qm−1 , qm , ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥ts✿
∆ti = ti − ti−1 , i = 1, 2, ..., m . ❲❡ ✉s❡ t❤❡s❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s t♦ ❝♦♥str✉❝t ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐♥ t❤❡ xt✲♣❧❛♥❡✳ ❚❤❡ ❧✐♥❡s x = xk ❛♥❞ t = ti ❝✉t t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d] ✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s [xk , xk+1 ] × [ti , ti+1 ]✳
✻✳✷✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✻
❲❡ t❛❦❡ ♦✉r ❡q✉❛t✐♦♥✿
h
i
h
i
u(ti+1 , x) − u(ti , x) = − K(a) u(ti , x) − u(ti , y) ∆t − − K(b) u(ti , z) − u(ti , x) ∆t , ❛♥❞ r❡✇r✐t❡ ✐t✿
h i h i u(ti+1 , xk ) − u(ti , xk ) = − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t − − K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t . ❲❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♥❡①t t❡♠♣❡r❛t✉r❡ ❛t ❛ ❧♦❝❛t✐♦♥ t❤❛t ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡s ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ❛❞❥❛❝❡♥t ❧♦❝❛t✐♦♥s✿
u(ti+1 , xk ) = u(ti , xk ) − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t + K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t ■t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥s✦ ❙✉♣♣♦s❡ ♦✉r r♦❞ ✐s s♣❧✐t ✐♥t♦ t❡♥ ♣✐❡❝❡s ❛s ✐❢ t❤❡② ❛r❡ s❡♣❛r❛t❡ ❝♦♥t❛✐♥❡rs ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✳ ❊❛❝❤ ♦❢ t❤❡♠ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ✭✇✐❞❡r✮ ❝❡❧❧ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❚❤✐s ✐s ✇❤❛t ②♦✉ s❡❡ ❛t t❤❡ t♦♣ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❜❡❧♦✇✿
❚❤❡ ♣✐♣❡s ❛r❡ s❤♦✇♥ ❛s ✭♥❛rr♦✇❡r✮ ❝❡❧❧s✳ ❚❤❡ t✐♠❡ ❛①✐s ✐s ✈❡rt✐❝❛❧✳ ◆♦✇✱ ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝♦♥t❛✐♥❡rs ❝♦♥t❛✐♥s ❛ ♥✉♠❜❡r ✇❤✐❝❤ ✐s t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t t❤❛t ❧♦❝❛t✐♦♥ ❛t t❤❛t t✐♠❡✳
u
❛t
◆❡①t✱ ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣✐♣❡s ❝♦♥t❛✐♥s ❛ ♥✉♠❜❡r ✇❤✐❝❤ ✐s t❤❡
✻✳✷✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
♣❡r♠❡❛❜✐❧✐t②
k
✸✺✼
❛t t❤❛t ❧♦❝❛t✐♦♥ ❛t t❤❛t t✐♠❡✳ ❚❤❡ ❤❡❛t ✢♦✇ t❤r♦✉❣❤ t❤❡ ♣✐♣❡s ✐s ❝♦♠♣✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡
❢♦r♠✉❧❛ ❞✐s❝✉ss❡❞ ❛❜♦✈❡ ❛♥❞ t❤❡ ♥❡✇ ✈❛❧✉❡ ♦❢
u
✐s ❝♦♠♣✉t❡❞ ❡✈❡r② t✐♠❡ ✇❡ ♠♦✈❡ t♦ t❤❡ ♥❡①t r♦✇✳
❚❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ✐s ❛s ❢♦❧❧♦✇s✿
❂❘❬✲✶❪❈✲❘❈❬✲✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✷❪✮✰❘❈❬✶❪✯✭❘❬✲✶❪❈❬✷❪✲❘❬✲✶❪❈✮ ❚❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ❢♦r♠✉❧❛✿
❊①❛♠♣❧❡ ✻✳✷✳✷✿ ✐♥s✉❧❛t❡❞ r♦❞
❙✉♣♣♦s❡ ✇❡ ♣✉t ✐♥s✉❧❛t♦rs t♦ t❤❡ ❡♥❞s ♦❢ ♦✉r r♦❞ ❛♥❞ s✉♣♣♦s❡ t❤❡② ❤❛✈❡ ✐♥✐t✐❛❧❧② ❛ ❧♦✇❡r t❡♠♣❡r❛t✉r❡ t❤❛♥ t❤❛t ♦❢ t❤❡ r♦❞✳ ❚❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ ❡♥❞s q✉✐❝❦❧② ❝♦♦❧s ❞♦✇♥✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❍❡❛t tr❛♥s❢❡r ❝♦♥t✐♥✉❡s ✇✐t❤✐♥ t❤❡ ❜♦❞② ♦❢ t❤❡ r♦❞ ✇✐t❤ ✈✐rt✉❛❧❧② ♥♦ tr❛♥s❢❡r t❤r♦✉❣❤ t❤❡ ✐♥s✉❧❛t♦rs✳ ❲❡ s❡t t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢
u
t♦ ❣♦ ❢r♦♠
0
t♦
1
❛♥❞ t❤❡♥ ❜❛❝❦ t♦
0
❛t t❤❡ ♦t❤❡r ❡♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡
♠❛② ❤❛✈❡✿
u(0, x) = 10 sin(πx) . ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✐s ③❡r♦ ❛t t❤❡ ❡♥❞s ❛♥❞ ♥♦♥✲③❡r♦ t❤r♦✉❣❤♦✉t t❤❡ r❡st ♦❢ t❤❡ r♦❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠❛② ❤❛✈❡✿
K(0) = K(1) = 0
❛♥❞
K(x) = .45
❢♦r
0 < x < 1.
❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❡❛❝❤
t
✭t❤❡ ❣r❛♣❤ ♦❢
u
✐s ♦❢ ❝♦✉rs❡ ❛ s✉r❢❛❝❡✮✿
u(t, ·)
♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❢♦r
❆❜♦✈❡ ✇❡ s❡❡ ❡✐t❤❡r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤❡❛t t❤r♦✉❣❤♦✉t t❤❡ r♦❞ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ ♦r t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t✦ ◆♦t❡ t❤❛t t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ r♦❞ r❡♠❛✐♥s t❤❡ s❛♠❡ ✭❣✐✈❡♥ ✉♥❞❡r ✏t♦t❛❧✑ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✮✳ ❆s ❛ ✇❛r♥✐♥❣✱ ✐❢ ✇❡ ❝❤♦♦s❡ ♦✉r ❝♦❡✣❝✐❡♥t ♦❢ ♣❡r♠❡❛❜✐❧✐t② ❧❛r❣❡r t❤❛♥
1/2✱
t❤❡ s✐♠✉❧❛t✐♦♥ ❢❛✐❧s q✉✐❝❦❧②✿
✻✳✷✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✽
❊①❛♠♣❧❡ ✻✳✷✳✸✿ ❤❡❛t❡❞ r♦❞ ❙✉♣♣♦s❡ ♦✉r r♦❞ ✐s ✐♥s❡rt❡❞ t♦ ✜① ❛ ❣❛♣ ✐♥ ❛ ❧♦♥❣ ♣✐♣❡ ✇✐t❤ ✇❛r♠❡r t❡♠♣❡r❛t✉r❡✳ ❖♥❝❡ ✐♥✱ t❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞ st❛rt t♦ ❡①❝❤❛♥❣❡ ❤❡❛t ✇✐t❤ t❤❡ ♣✐♣❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❡✛❡❝t ♦♥ t❤❡ ♣✐♣❡✬s t❡♠♣❡r❛t✉r❡ ✐s ♥❡❣❧✐❣✐❜❧❡✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❲❡ s❡t t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ u t♦ ❜❡ 0 t❤r♦✉❣❤♦✉t t❤❡ r♦❞ ✇❤✐❧❡ t❤❡ ♦✉ts✐❞❡ t❡♠♣❡r❛t✉r❡ ✐s 5✳ ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✐s .45 t❤r♦✉❣❤♦✉t t❤❡ r♦❞✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥✿
❚❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t✦ ■❢ ✇❡ ❝❤♦♦s❡ ♦✉r ❝♦❡✣❝✐❡♥t ♦❢ ♣❡r♠❡❛❜✐❧✐t② ❧❛r❣❡r t❤❛♥ 1/2✱ t❤❡ s✐♠✉❧❛t✐♦♥ ❢❛✐❧s q✉✐❝❦❧②✿
❊①❡r❝✐s❡ ✻✳✷✳✹ ❲❤❛t ❦✐♥❞ ♦❢ ❝✉r✈❡ ❞♦ ✇❡ s❡❡ ❛t t❤❡ ❡❞❣❡ ✭x = 0✮ ♦❢ t❤✐s s✉r❢❛❝❡❄
❊①❛♠♣❧❡ ✻✳✷✳✺✿ s✐♠♣❧✐✜❡❞ ❆ s✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥ ♠❛❦❡s k ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✐♠❡ ❛♥❞ ♣✉ts ✐t ❛t t❤❡ t♦♣ r♦✇✳ ❚❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝❡✿
✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②
✸✺✾
❚❤❡ r❡s✉❧t ✐s ♦♥ t❤❡ ❧❡❢t✿
❊①❡r❝✐s❡ ✻✳✷✳✻
❲❤❛t ❤❛♣♣❡♥❡❞ ✐♥ t❤❡ s❡❝♦♥❞ s♣r❡❛❞s❤❡❡t❄ ❊①❛♠♣❧❡ ✻✳✷✳✼✿ ❤❡❛t❡❞ r♦❞
❲❤❛t ✇✐❧❧ ❤❛♣♣❡♥ t♦ t❤❡ r♦❞ ❤❡❛t❡❞ ❛t t❤❡ ❡♥❞s ✇❤❡♥ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✈❛r✐❡s ❄ ❚❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ✐s st✐❧❧ 0 ✇❤✐❧❡ t❤❡ ♦✉ts✐❞❡ t❡♠♣❡r❛t✉r❡ ✐s 5✳ ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✇✐❧❧ r❡♠❛✐♥ .45 t❤r♦✉❣❤♦✉t t❤❡ ❤❛❧❢ ♦❢ t❤❡ r♦❞ ❛♥❞ .1 ♦♥ t❤❡ ❧❡❢t✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥✿
❚❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t ❛❣❛✐♥ ❜✉t✱ ♣r❡❞✐❝t❛❜❧②✱ t❤❡ ❧❡❢t ❡♥❞ ✐s ❢❛❧❧✐♥❣ ❜❡❤✐♥❞✦ ❲❛r♥✐♥❣✦ ❚❤❡ r♦❞ ✐s ✐ts ♦✇♥ ✉♥✐✈❡rs❡✳
❲❡ ✇✐❧❧ ♥♦✇ ✐♥t❡r♣r❡t t❤❡ ❞✐✛❡r❡♥❝❡s ✇❡ s❡❡ ✐♥ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ u(ti+1 , xk ) = u(ti , xk ) − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t + K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t .
■❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ❜②✿ ∆f (ck ) = f (xk+1 ) − f (xk )
❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳
✻✳✸✳
❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s
❚❤❡ ♣❛rt✐❛❧
✸✻✵
❞✐✛❡r❡♥❝❡s ♦❢ u ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ t ❛r❡ ❞❡✜♥❡❞ ❛t t❤❡s❡ ❡❞❣❡s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆x u (ti , sk ) = u(ti , xk+1 ) − u(ti , xk )
❛♥❞
∆t u (qi , xk ) = u(ti+1 , xk ) − u(ti , xk )
❲❡ t❛❦❡ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❛♥❞ r❡✇r✐t❡ ✐t ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿
∆u (qi , xk ) = K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡❄
✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❊①❛♠♣❧❡ ✻✳✸✳✶✿ ❧❛r❣❡ t❡❛ ❝✉♣
❲❤❛t ✐❢ t❤❡ t❡❛ ❝✉♣ ✐t
3
t✐♠❡s ❧❛r❣❡r t❤❛♥ t❤❡ ♦t❤❡r t✇♦❄ s♦❞❛ ❝♦✛❡❡
❲❡ ❛❞❥✉st t❤❡ ❢♦r♠✉❧❛ ❜② ❞✐✈✐❞✐♥❣ ❜②
3
t❡❛
t❤❡ ❛♠♦✉♥t t❤❛t ✐s ❛❞❞❡❞ t♦
w
✭t❡❛✮✿
❂❘❬✲✶❪❈✲✵✳✶✯❘✷❈✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✲✵✳✶✯❘✷❈❬✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶✴✸ ❚❤❡ ✐❞❡❛ ✐s t❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡s ♦♥ t❤❡ t❡❛ ✐s ♦♥❡ t❤✐r❞✿
❚❤❡ ♣❛tt❡r♥ ♦❢ ❛✈❡r❛❣✐♥❣ ❝♦♥t✐♥✉❡s ❜✉t t❤❡ ❛✈❡r❛❣❡ ❤❛s ❝❤❛♥❣❡❞✿
40 + 100 + 85 · 3 = 79 . 5 ❆ ✏q✉✐❝❦✲❛♥❞✲❞✐rt②✑ ✜① ❤❛s ♣r♦❞✉❝❡❞ t❤❡ ❡①♣❡❝t❡❞ r❡s✉❧t✦
❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♠♦❞❡❧ ❢♦r ❤❡❛t tr❛♥s❢❡r ✐♥ ❛ r♦❞✳ ❚❤✐s t✐♠❡✱ t❤❡ ❝♦♥t❛✐♥❡rs✬ s✐③❡s ♠❛② ✈❛r② ❛♥❞ s♦ ❞♦ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✳ ❲❤❛t ♠❛tt❡rs ♥♦✇ ✐s t❤❡ ❥✉①t❛♣♦s✐t✐♦♥✿
•
❤❡❛t ✈s✳ t❡♠♣❡r❛t✉r❡✱
•
✇❡✐❣❤t ✈s✳ ❞❡♥s✐t②✳
❚❤❡ ❧❛tt❡r ✐s t❤❡ ❛✈❡r❛❣❡ ❛♠♦✉♥t ♦❢ t❤❡ ❢♦r♠❡r✿
t❡♠♣❡r❛t✉r❡
=
❤❡❛t s✐③❡
✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❛♥❞
❞❡♥s✐t② =
✸✻✶ ✇❡✐❣❤t . s✐③❡
❚❤❡ s✐③❡s ❛r❡ ♥♦t❤✐♥❣ ❜✉t ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ♥♦❞❡ xk ✳ ▲❡t✬s ❝❛❧❧ t❤❡♠ ∆sk ✳ ✭❚❤❡ ❧❛tt❡r ❢♦r♠✉❧❛ ♠❛t❝❤❡s t❤❡ ♦♥❡ ✐♥ ❈❤❛♣t❡r ✸■❈✲✸✳✮ ■❢ u ✐s t❤❡ t❡♠♣❡r❛t✉r❡✱ t❤❡♥ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ r♦♦♠ x ✐s✿ u(·, xk ) · ∆sk .
■t ✐s ❤❡❛t t❤❛t ✐s ❜❡✐♥❣ ❡①❝❤❛♥❣❡❞✳ ❇✉t ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡s t❤❛t ❞r✐✈❡s t❤❡ ❡①❝❤❛♥❣❡✳ ❖✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿ u(ti+1 , xk ) · ∆sk = u(ti , xk ) · ∆sk + K(sk+1 )∆x u(ti , sk+1 )∆t − K(sk )∆x u(ti , sk )∆t . ❲❛r♥✐♥❣✦ ❚❤❡ s✐③❡
∆sk
✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✳
❚❤❡ ♥❡✇ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ t❡♠♣❡r❛t✉r❡ ✐s✿ u(ti+1 , xk ) = u(ti , xk ) + K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t/∆sk
❲❡ ✉s❡ ✐t ❢♦r s✐♠✉❧❛t✐♦♥s✳ ❊①❛♠♣❧❡ ✻✳✸✳✷✿ r♦❞
❲❡ ♣❧❛❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ ∆s ❛t t❤❡ t♦♣ ♦❢ ❡❛❝❤ ❝♦❧✉♠♥✿
❚❤❡ ❢♦r♠✉❧❛s ✐s ❛s ❢♦❧❧♦✇s✿ ❂❘❬✲✶❪❈✲✵✳✶✯❘✷❈✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✴❘✸❈✲✵✳✶✯❘✷❈❬✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶✴❘✸❈
❚❤✐s ✐s t❤❡ ♥❡✇ ❞❡♣❡♥❞❡♥❝❡✿
✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s
✸✻✷
❲❡ ❛❧s♦ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ❤❡❛t ❛♥❞ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡❀ t❤❡ ❧❛st t✇♦ ❝♦❧✉♠♥s✳ ❚❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② ✐s ❝♦♥✜r♠❡❞✦ ❚❤❡ ❛✈❡r❛❣✐♥❣ ❧♦♦❦s ❛ ❜✐t ❞✐✛❡r❡♥t✿
❊①❛♠♣❧❡ ✻✳✸✳✸✿ ❡①❝❤❛♥❣❡ ✇✐t❤ ♦✉ts✐❞❡
◆♦✇✱ s✉♣♣♦s❡ t❤❡ r♦❞ ✐s ♥♦t ✐ts ♦✇♥ ✉♥✐✈❡rs❡ t❤✐s t✐♠❡ ❛♥❞ ✐♥t❡r❛❝ts ✇✐t❤ t❤❡ ♦✉ts✐❞❡✳ ❚❤❡♥✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ✐ts ❝❡❧❧s✱ t❤❡r❡ ✐s ❡①❝❤❛♥❣❡ ✇✐t❤ t❤❡ ❛✐r ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡✳ ❚❤❡ ❧❛tt❡r ✐s ❡①❡❝✉t❡❞ ❥✉st ❛s ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ◆❡✇t♦♥✬s ▲❛✇ ♠♦❞❡❧ ❜✉t ❢♦r ❡❛❝❤ ❝♦♥t❛✐♥❡r ✐♥❞❡♣❡♥❞❡♥t❧②✦ ❲❡ ❥✉st ❛❞❞ ❛♥ ❡①tr❛ t❡r♠ t♦ t❤❡ ❢♦r♠✉❧❛ ✭0 r♦♦♠ t❡♠♣❡r❛t✉r❡✮✿ ✲✵✳✶✯❘❬✲✶❪❈
❖❢ ❝♦✉rs❡✱ t❤❡r❡ ✐s ♥♦ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② t❤✐s t✐♠❡✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ t❡♠♣❡r❛t✉r❡ st❛❜✐❧✐③❡s t♦✇❛r❞ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ✭t❤❡ st❡❛❞② st❛t❡✮✿
❚❤✐s ♦❜s❡r✈❛t✐♦♥ ❝♦♥✜r♠s t❤❡ ♦r✐❣✐♥❛❧ ◆❡✇t♦♥✬s ▲❛✇ ♠♦❞❡❧✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ t❤❡ r♦❞ ❛♥❞ t❤❡ ❤❡❛t ❡①❝❤❛♥❣❡ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡ r♦❞ ✐s s❡❡♥ ❛s s♣❧✐t ✐♥t♦ ♣✐❡❝❡s ❛s ✐❢ t❤❡② ❛r❡ s❡♣❛r❛t❡ ♣✐❡❝❡s✿ t❤❡ ❤❡❛t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ r♦✇ ♦❢ r♦♦♠s ❛♥❞ ❡❛❝❤ r♦♦♠ ❡①❝❤❛♥❣❡s t❤❡ ♠❛t❡r✐❛❧ ✇✐t❤ ✐ts t✇♦ ♥❡✐❣❤❜♦rs t❤r♦✉❣❤ ✐ts ✇❛❧❧s✿
✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s
✸✻✸
❇❡❧♦✇ ✇❡ ❤❛✈❡✿ • a = AB ✐s ♦♥❡ ♦❢ t❤❡ r♦♦♠s✳
• b, c ❛r❡ t❤❡ t✇♦ ❛❞❥❛❝❡♥t r♦♦♠s✱ ❧❡❢t ❛♥❞ r✐❣❤t✳ • A, B ❛r❡ t❤❡ ✇❛❧❧s ♦❢ a✱ ❧❡❢t ❛♥❞ r✐❣❤t✳
❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ r♦♦♠s ✇✐t❤ ❡❞❣❡s ❛♥❞ ✇❛❧❧s ✇✐t❤ ♥♦❞❡s✿
❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ❛❜♦✈❡❀ ✐♥st❡❛❞ ♦❢ r♦♦♠s ❛♥❞ ✇❛❧❧s ❜❡t✇❡❡♥ t❤❡♠✱ ✇❡ s♣♦❦❡ ♦❢ ❝♦♥t❛✐♥❡rs ❛♥❞ ♣✐♣❡s ❜❡t✇❡❡♥ t❤❡♠✳ ❘❡❝❛❧❧✿ • a ✐s ♦♥❡ ♦❢ t❤❡ ❝♦♥t❛✐♥❡rs✳
• b, c ❛r❡ t❤❡ t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs✱ ❧❡❢t ❛♥❞ r✐❣❤t✳ • A, B ❛r❡ t❤❡ ♣✐♣❡s t❤❛t st❛rt ❛t a✱ ❧❡❢t ❛♥❞ r✐❣❤t✳
❚❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿
❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ ❝♦♥t❛✐♥❡rs ✇✐t❤ ♥♦❞❡s ❛♥❞ t❤❡ ♣✐♣❡s ✇✐t❤ ❡❞❣❡s✿
✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❚❤✐s ✐s ❞✉❛❧✐t② ✿ ❚✇♦ ✐❞❡♥t✐❝❛❧ st❡♣s ❛♥❞ ✇❡ ❛r❡ ❜❛❝❦ ✇❤❡r❡ ✇❡ st❛rt❡❞✳ ▲❡t✬s r❡✈✐❡✇✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❝❡❧❧s✿
❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❝❡❧❧s ❛r❡ ✐♥❞✐❝❛t❡❞ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇✿ ∆xk = xk+1 − xk ❛♥❞ ∆sk = sk+1 − sk .
❚❤❡ ❢♦r♠❡r ♦♥❡s ❤❛✈❡♥✬t ❛♣♣❡❛r❡❞ ②❡t✳ ❯♥❞❡r t❤❡ st❛r ♦♣❡r❛t♦r✱ t❤❡ 0✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r ❝♦rr❡s♣♦♥❞ t♦ t❤❡ 1✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ♦✉r r❡❝t❛♥❣❧❡ ❞✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿
❲❡ t✉r♥ t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿
♦r✱ ♦r✱
u(ti+1 , xk ) = u(ti , xk ) + K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t/∆sk , ∆u K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) (qi , xk ) = , ∆t ∆sk ∆ K∆x u ∆u (qi , xk ) = (ti , xk ) . ∆t ∆sk
❲❤❡♥ K ✐s ❝♦♥st❛♥t✱ ✇❡ ❤❛✈❡ ❛♥ ❛❜❜r❡✈✐❛t❡❞ ❢♦r♠✿ ∆(∆x u) ∆u (qi , xk ) = K (ti , xk ) ∆t ∆sk
❚❤✐s ✐s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ❢♦r♠s ✐♥ t❤❡ ❧❡❢t✲ ❛♥❞ r✐❣❤t✲❤❛♥❞ t❡r♠s ♦❢ t❤❡ ❡q✉❛t✐♦♥ r❡s✐❞❡✱ r❡s♣❡❝t✐✈❡❧②✿
✸✻✹
✻✳✹✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
✸✻✺
✻✳✹✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
❲❡ st❛rt ❢r♦♠ s❝r❛t❝❤✳ ❚❤✐s ✐s ✇❤❛t ♦✉r s②st❡♠ ♦❢ ❝♦♥t❛✐♥❡rs ❛♥❞ ♣✐♣❡s ❧♦♦❦s ❧✐❦❡✿
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ♣✉rs✉❡ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♣❡r♠❡❛❜✐❧✐t② K ✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣✳ ❲❡ ❞❡❝♦♠♣♦s❡ t❤❡ s❡❣♠❡♥t [a, b] ✐♥t♦ n − 1 ✐♥t❡r✈❛❧s✿
a = x1 , x2 , x3 , ..., xn−1 , xn = b . ❚❤❡♥ t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡✿
∆xk = xk+1 − xk . ❙✉♣♣♦s❡ t❤❡ ❛♠♦✉♥t ♦❢ ✢♦✇ ✕ ❧❡❢t t♦ r✐❣❤t ✕ ♦❢ ❤❡❛t ✭♦r ♠❛t❡r✐❛❧✮ ❛❧♦♥❣ t❤❡ ♣✐♣❡
sk = [xk−1 , xk ] ❞✉r✐♥❣ ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡
qj = [tj−1 , tj ] ✐s ❞❡♥♦t❡❞ ❜②✿
g = g(qj , sk ) . ❚❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② ✭♦r ♠❛t❡r✐❛❧✮ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿
◮ ❚❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ❤❡❛t ✐♥ ❝♦♥t❛✐♥❡r x ♦✈❡r ❛ ❣✐✈❡♥ ✐♥t❡r✈❛❧ ♦❢ t✐♠❡ ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ✕ ❛❝❝♦✉♥t✐♥❣ ❢♦r t❤❡ ❞✐r❡❝t✐♦♥s ✕ ♦❢ t❤❡ ✢♦✇ ❛❧♦♥❣ t❤❡ t✇♦ ♣✐♣❡s t❤❛t ❧❡❛✈❡ x✳ ❚❤❡ st❛t❡♠❡♥t t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ ❢♦r♠✿
∆t u (qi , xk ) = g(qi , sk−1 ) − g(qi , sk ) ◆❡①t✱ ✇❡ s❡♣❛r❛t❡❧② r❡❝♦r❞ t❤❡ ❡①❝❤❛♥❣❡ ♦❢ ❤❡❛t ❜❡t✇❡❡♥ ❡❛❝❤ ♣❛✐r ♦❢ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✿ ❚❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ✐ts t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ♦❜❥❡❝t✳
✻✳✹✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
✸✻✻
■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✭♦r ♠❛t❡r✐❛❧✮✱ ✇❡ ✐♥❝❧✉❞❡❞ t❤❡ t❡♠♣❡r❛t✉r❡ ✭♦r t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ♠❛t❡r✐❛❧✱ r❡s♣❡❝t✐✈❡❧②✮ ✐♥ ♦✉r ❛♥❛❧②s✐s✳ ❚❤❡ ❤❡❛t r❡s✐❞❡s ❛♥❞ ♠♦✈❡s ✇✐t❤✐♥ t❤❡ ❞✉❛❧
❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ x✲❛①✐s✿
❲❡ ❤❛✈❡ ✉t✐❧✐③❡❞ t❤❡ ✐❞❡❛ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ xk = [sk−1 , sk ] ✐♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ❝♦♥t❛✐♥❡r xk ✿ ∆sk = sk − sk−1 .
❚❤❡ ♥❡✇ ✐❞❡❛ ✐s t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ sk = [xk−1 , xk ] ✐♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ♣✐♣❡ sk ✿ ∆xk = xk − xk−1 .
❙♦✱ t❤❡ ❧❛✇ ♦❢ ❝♦♦❧✐♥❣ t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ◮ ✶✳ ❚❤❡ ✢♦✇ ♦❢ ❤❡❛t ✭❞✉r✐♥❣ ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡✮ ❛❧♦♥❣ ❛ ♣✐♣❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡
♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥ t❤❡ t✇♦ ❝♦♥t❛✐♥❡rs ❛t t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ ♣✐♣❡ ✭✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡✮✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ ❛ss✉♠❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ✷✳ ❚❤❡ ✢♦✇ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡✳
■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❛❧s♦ ♠❛❞❡ ♦♥❡ ♠♦r❡ ❛ss✉♠♣t✐♦♥✿ ◮ ✸✳ ❚❤❡ ❡✛❡❝t ♦❢ t❤❡ ✢♦✇ ♦♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡
❝♦♠♣❛rt♠❡♥t✳ ❲❡ ♠❛❦❡ ❛ ♥❡✇
❛ss✉♠♣t✐♦♥ ✿
◮ ✹✳ ❚❤❡ ✢♦✇ ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣✐♣❡✳
❲❡ ❞❡❛❧ ✇✐t❤ t❤❡s❡ ❢♦✉r q✉❛♥t✐t✐❡s✿ ✶✳ u(ti , xk ) − u(ti , xk−1 ) = ∆x u (ti , sk )
✷✳ ∆ti
1 ∆sk 1 ✹✳ ∆xk
✸✳
■♥ ♦r❞❡r t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♥❡✇ ✐t❡♠ ✭★✹✮✱ s✐♠♣❧② r❡✐♥t❡r♣r❡t t❤❡ ♣❡r♠❡❛❜✐❧✐t②✿ K(sk ) = m(sk )
✇❤❡r❡ m(sk ) r❡✢❡❝ts t❤❡ ❝❛rr②✐♥❣ ❝❛♣❛❝✐t② ♦❢ t❤❡ ♣✐♣❡✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ s②st❡♠ ♦❢ ♣✐♣❡s ♥♦✇ ❝♦♠❡s ✐♥t♦ ♣❧❛②✳
1 , ∆xk
✻✳✹✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
✸✻✼
❊①❛♠♣❧❡ ✻✳✹✳✶
❚❤❡ ❧❡♥❣t❤s ♦❢ ♣✐♣❡s ❛♥❞ ♦❢ t❤❡ ❝♦♠♣❛rt♠❡♥ts ✐♥t❡r❛❝t t♦ ♣r♦❞✉❝❡ ❛ r✐❣✐❞ str✉❝t✉r❡✿
❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ✐♥t♦ ♦✉r ❢♦r♠✉❧❛ ❛♥❞ t❤❡ r❡s✉❧t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ m(sk+1 ) ∆x1k+1 ∆x u(ti , sk+1 ) − m(sk ) ∆x1 k ∆x u(ti , sk ) ∆u (qi , xk ) = . ∆t ∆sk
❲❡ ❝❛♥ ✉s❡ ✐t ❢♦r s✐♠✉❧❛t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✹✳✷✿ r♦❞
❲❡ ♣❧❛❝❡ t❤❡ ✈❛❧✉❡ ♦❢ ∆x ❛t t❤❡ t♦♣ ♦❢ ❡❛❝❤ ❝♦❧✉♠♥✿
❲❡ s✐♠♣❧② ❝❤❛♥❣❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ♣❡r♠❡❛❜✐❧✐t② ❛s ❢♦❧❧♦✇s✿ ❂✶✴❘❬✲✶❪❈
❚❤✐s ✐s t❤❡ ♥❡✇ ❞❡♣❡♥❞❡♥❝❡✿
❚❤✐s ✇❛②✱ t❤❡ r❡st ♦❢ ❢♦r♠✉❧❛s r❡♠❛✐♥ t❤❡ s❛♠❡✳ ❚❤❡ ❛✈❡r❛❣✐♥❣ ❧♦♦❦s ❞✐✛❡r❡♥t ✐♥ t❤❡ t✇♦ ❤❛❧✈❡s ♦❢ t❤❡ r♦❞✿
✻✳✹✳
❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
✸✻✽
❚❤❡ ♦♥❡ ✇✐t❤ t❤❡ s❤♦rt❡r ♣✐♣❡s st❛❜✐❧✐③❡s ❡❛r❧✐❡r✳ ❊①❡r❝✐s❡ ✻✳✹✳✸
❲❤❛t ✐❢ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✈❛r✐❡s ✇✐t❤ t✐♠❡❄ ❊①❡r❝✐s❡ ✻✳✹✳✹
❲❤❛t ✐❢ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ ❡❛❝❤ ✇❛❧❧ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs❄ ❊①❛♠♣❧❡ ✻✳✹✳✺✿ r✐♥❣
❘♦♦♠s ❛♥❞ ✇❛❧❧s ♦♥ t❤❡ ❧❡❢t ❛♥❞ ❝♦♥t❛✐♥❡rs ❛♥❞ ♣✐♣❡s ♦♥ t❤❡ ♠✐❞❞❧❡✿
■t✬s t❤❡ s❛♠❡ ❝✐r❝❧❡✦ ❚❤❛t ✐s ✇❤② t❤❡ ♠❡❛s✉r❡♠❡♥ts ❛r❡ t❤❡ s❛♠❡ ❢♦r ❜♦t❤ ❡✈❡♥ t❤♦✉❣❤ s✉❝❤ ❛ ❝♦♥✲ str✉❝t✐♦♥ ✇♦✉❧❞♥✬t ✜t ✐♥ t❤❡ ♣❧❛♥❡✳ ❊①❛♠♣❧❡ ✻✳✹✳✻✿ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s
◆♦✇✱ s✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡s ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞ ❛r❡ ♠❛✐♥t❛✐♥❡❞ ❜✉t ❞✐✛❡r❡♥t✳ ■t✬s ❛s ✐❢ t❤❡ r♦❞ ♠❡❡ts t❤❡ ♦✉ts✐❞❡ ❜② ♣r♦tr✉❞✐♥❣ ✐♥t♦ t✇♦ s❡♣❛r❛t❡ r♦♦♠s✳ ❲❡ ✉s❡ t❤❡ ✜rst ❛♥❞ ❧❛st ❝❡❧❧s ❛s t❤♦s❡ r♦♦♠s ❛♥❞ ♠❛❦❡ t❤❡✐r ✈❛❧✉❡s ❝♦♥st❛♥t✿
❚❤❡ t❡♠♣❡r❛t✉r❡ st❛❜✐❧✐③❡s t♦✇❛r❞ ❛ ❧✐♥❡❛r ✭t❤❡ st❡❛❞② st❛t❡✮✳ ❚❤❡r❡ ✐s ♥♦ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣②✳
✻✳✹✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②
✸✻✾
❲❡ ♥♦✇ ✐♥t❡r♣r❡t ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐♥ t❡r♠s ♦❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❲❡ ❞❡❝♦♠♣♦s❡ t❤❡ s❡❣♠❡♥t ✐♥t♦ n − 1 ✐♥t❡r✈❛❧s ❜② ❣✐✈✐♥❣ ♥♦❞❡s t♦ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ ♥❛♠❡s✿ p = s1 , s2 , s3 , ..., sn−1 , sn = q .
❚❤❡♥ t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡✿ ∆sk = sk+1 − sk .
◆♦✇✱ ✇❤❛t ❛r❡ t❤❡ ♥♦❞❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡❞❣❡s ♦❢ t❤✐s ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥❄ ❚❤❡ ♥♦❞❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❝♦✉rs❡✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿ x1 ✐♥ [s1 , s2 ], x2 ✐♥ [s2 , s3 ], ..., xn−1 ✐♥ [sn−1 , sn ] .
❲❡ ❛♣♣❧② t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥s t♦ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ g = g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿
∆f ∆x
✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❢✉♥❝t✐♦♥ ♦❢
∆g(xk ) = g(sk+1 ) − g(sk )
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ♦❢ g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿ g(sk+1 ) − g(sk ) ∆g (xk ) = ∆x sk+1 − sk
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳
❚❤❡ ❝♦♥tsr✉❝t✐♦♥ ✐s ♦✉t❧✐♥❡s ✐♥ t❤❡ ♠✐❞❞❧❡ r♦✇ ❜❡❧♦✇✿
❚❤❡ r❡st ✐s ✐♥ t❤❡ ❜♦tt♦♠ r♦✇✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆f (sk+1 ) − ∆f (sk )
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳
✻✳✺✳ ❚❤❡ ❤❡❛t P❉❊
✸✼✵
❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆x2
− ∆f (s ) ∆x k sk+1 − sk
∆f (s ) ∆x k+1
❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳
❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s r❡❛❞② t♦ ❜❡ ❞✐✛❡r✲ ❡♥t✐❛t❡❞ ❛❣❛✐♥✦ ❇❛❝❦ t♦ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✳ ❲❤❡♥ m ✐s ❝♦♥st❛♥t✱ ✇❡ ❤❛✈❡✿ ❖r✱ ✐♥ t❤❡ s✐♠♣❧✐✜❡❞ ❢♦r♠✿
∆ u(t,x)
∆x x∆x ∆t u(t, x) =m ∆t ∆x
.
∆t u ∆2 u = m x2 ∆t ∆x
❲❡ s❤♦✉❧❞ ❦❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❞❡✜♥❡❞ ❛t t❤❡ ✈❡rt✐❝❛❧ ❡❞❣❡s ❛♥❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳
❊①❡r❝✐s❡ ✻✳✹✳✼
Pr♦✈❡ t❤❛t ✐❢ t❤❡ t❡♠♣❡r❛t✉r❡s ❝❤❛♥❣❡ ❧✐♥❡❛r❧② ❛❧♦♥❣ t❤❡ r♦❞ ✭∆x = ∆x = 1✮✱ t❤❡♥ t❤❡② ❞♦♥✬t ❝❤❛♥❣❡ ✇✐t❤ t✐♠❡ ✭st❡❛❞② st❛t❡✮✳
✻✳✺✳ ❚❤❡ ❤❡❛t P❉❊
❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ♦❢ ❤❡❛t tr❛♥s❢❡r✳ ❙✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡ ❢✉♥❝t✐♦♥ u ✐s ❞❡✜♥❡❞ ❢♦r ❛❧❧ x ❛♥❞ t ✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ s✉❜s❡t U ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ ✐t ✐s s❛♠♣❧❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡ r❡❝t❛♥❣❧❡ [a, b] × [0, ∞) ❝♦♥t❛✐♥❡❞ ✐♥ t❤❛t s✉❜s❡t✳ ❲❡ r❡✜♥❡ t❤✐s ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❛♥❞ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐s❝r❡t❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ❝♦♥st❛♥t ♣❡r♠❡❛❜✐❧✐t② k✱ ∆u ∆2 u (ti , xk ) , (qi , xk ) = K ∆t ∆x2
❛s ∆x → 0 ❛♥❞ ∆t → 0✳ ❲❡ ❤❛✈❡ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✿
∂u ∂ 2u =K 2 ∂t ∂x
✻✳✺✳
❚❤❡ ❤❡❛t P❉❊
✸✼✶
❆ s♦❧✉t✐♦♥ u ♦❢ t❤✐s ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭P❉❊✮ ✐s ❛ ❢✉♥❝t✐♦♥✿ • ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ r❡❝t❛♥❣❧❡✱
• ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ✐♥s✐❞❡ ♦❢ ✐t✱ ❛♥❞
• t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ x ✐♥s✐❞❡ t❤❡ r❡❝t❛♥❣❧❡✱ s✉❝❤ t❤❛t
• t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ ♣❛✐r (x, t) s✉❝❤ t❤❛t x ✐s ✐♥ (a, b) ❛♥❞ t ✐s ✐♥ (0, ∞)✳
❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ P❉❊ ✐s ❛s ❢♦❧❧♦✇s✿
• ❆ ♣♦s✐t✐✈❡ ❝♦♥❝❛✈✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ❣♦❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ♣♦s✐t✐✈❡ s❧♦♣❡ ✇✐t❤ r❡s♣❡❝t t♦ t✳
• ❆ ♥❡❣❛t✐✈❡ ❝♦♥❝❛✈✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ❣♦❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ♥❡❣❛t✐✈❡ s❧♦♣❡ ✇✐t❤ r❡s♣❡❝t t♦ t✳
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ♣❛tt❡r♥s ❢♦r u ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s s❤♦✇♥ ❜❡❧♦✇✿
❇❡❝❛✉s❡ t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ ❛ s♦❧✉t✐♦♥ ✐s ♥♦t ♠❛❞❡ s♣❡❝✐✜❝ ❜② ❛ s✐♥❣❧❡✲♥✉♠❜❡r ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛s ✇❡ ❦♥♦✇✳ ❲❡ ✐♠♣♦s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢✿ • t❤❡
✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱ ♣r♦✈✐❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ u ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣✿ u(0, x) = i(x) ,
✇❤❡♥ a ≤ x ≤ b,
✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ i ✐s ❣✐✈❡♥✱ ❛♥❞ • t❤❡
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✱ ♣r♦✈✐❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ u ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞✿ u(t, a) = α(t) ❛♥❞ u(t, b) = β(t) ,
✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥s α ❛♥❞ β ❛r❡ ❣✐✈❡♥✳
❊①❡r❝✐s❡ ✻✳✺✳✶
❯♥❞❡r t❤❡ ❛❜♦✈❡ r❡str✐❝t✐♦♥s ✇❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t i(x)❄
✇❤❡♥ t ≥ 0 ,
✻✳✺✳
❚❤❡ ❤❡❛t P❉❊
✸✼✷
❊①❡r❝✐s❡ ✻✳✺✳✷
❉❡s❝r✐❜❡ t❤❡ ♠♦❞❡❧ s❡t✉♣s ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥s ✇✐t❤ ②♦✉r ❝❤♦✐❝❡s ♦❢ i✱
α✱
❛♥❞
β✳
❊①❛♠♣❧❡ ✻✳✺✳✸✿ st❡❛❞② st❛t❡
❆ st❡❛❞② st❛t❡ s♦❧✉t✐♦♥ ✐s ♦♥❡ t❤❛t ❞♦❡s♥✬t ❝❤❛♥❣❡ ✇✐t❤ t✐♠❡✿
u(t, x) = s(x) , ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥
s✳
■♥ ♦t❤❡r ✇♦r❞s✱
ut = 0 . ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ P❉❊ t❤❛t
uxx = 0 . ❚❤❡r❡❢♦r❡✱
u(t, x) = s(x)
✐s ❧✐♥❡❛r ♦♥
x✳
■❢ t❤❡ ✐♥✐t✐❛❧ st❛t❡
i
✐s ❧✐♥❡❛r✱ t❤❡ s②st❡♠ ✇✐❧❧ r❡♠❛✐♥ ✐♥ ✐t✳
❊①❛♠♣❧❡ ✻✳✺✳✹✿ ♠❛✐♥t❛✐♥❡❞ t❡♠♣❡r❛t✉r❡
❙✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡s ❛t t❤❡ ❡♥❞s ❛r❡ ❝♦♥st❛♥t ❛♥❞ ❡q✉❛❧✿
α = β = 0. ❚❤❛t✬s t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✦ ❆s ♠♦❞❡❧✐♥❣ ❤❛s s❤♦✇♥✱ t❤❡ t❡♠♣❡r❛t✉r❡s ✐♥ t❤❡ r♦❞ ✇✐❧❧ ❛♣♣r♦❛❝❤ t❤✐s ♥✉♠❜❡r✿
u(t, x) → 0
❛s
t → ∞.
❋r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ◆❡✇t♦♥✬s ▲❛✇✱ ✇❡ ❦♥♦✇ t❤❛t t❤✐s ❝♦♥✈❡r❣❡♥❝❡ ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧✳ ❚❤❡r❡✲ ❢♦r❡✱ ✇❡ s❤♦✉❧❞ tr② ❢♦r
K = 1✿ u(t, x) = i(x)e−t .
❲❡ s✉❜st✐t✉t❡ t❤✐s ✐♥t♦ t❤❡ P❉❊✿
−i(x)e−t = i′′ (x)e−t . ❚❤❡♥✱
−i(x) = i′′ (x) . ❚❤❛t✬s ❛ ❢❛♠✐❧✐❛r s❡❝♦♥❞ ♦r❞❡r ❖❉❊ ❢r♦♠ ❈❤❛♣t❡r ✶✦ ❚❤❡ s♦❧✉t✐♦♥ ✐s✿
i(x) = A sin x + B cos x . ❙♦✱ t❤✐s ✐s ♣♦ss✐❜❧❡✦
◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✳ ❚❤❡ ✐❞❡❛ ✐s✿
◮
❲❤❛t ✐❢
u
✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✐♥✐t✐❛❧ st❛t❡
i
❜② ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ❢✉♥❝t✐♦♥ ♦❢
▲❡t✬s t❡st t❤✐s ✐❞❡❛ ❜② ❛ss✉♠✐♥❣ t❤❛t ✐t✬s tr✉❡✿
u(t, x) = i(x)g(t) . ❙✉❜st✐t✉t✐♥❣
u
✐♥t♦ t❤❡ P❉❊ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿
ig ′ = Ki′′ g . ▲❡t✬s r❡❛rr❛♥❣❡ t❤❡ t❡r♠s ❛♥❞ ✏s❡♣❛r❛t❡ t❤❡ ✈❛r✐❛❜❧❡s✑✿
i′′ g′ = . Kg i
t❄
✻✳✺✳
❚❤❡ ❤❡❛t P❉❊
✸✼✸
▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥✿
i′′ (x) g ′ (t) = . Kg(t) i(x)
❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t ✐s✱ t❤❡r❡❢♦r❡✱ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ x✳ ❈♦♥✈❡rs❡❧②✱ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ x ✐s✱ t❤❡r❡❢♦r❡✱ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ t✳ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s t❤✐s❄ ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛ r♦♦❢ t♦ ❜❡ ❜✉✐❧t ❢♦r♠ ✈❡rt✐❝❛❧ r♦❞s✳ ❱❛r✐♦✉s s❤❛♣❡s ❛r❡ ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ r♦❞s ❛r❡ ♣♦✐♥t❡❞ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ✭❧❡❢t ❛♥❞ ♠✐❞❞❧❡✮✿
❚❤❡ ♦♥❧② ✇❛② t♦ ❤❛✈❡ ❛ ❝r♦ss ♣❛tt❡r♥ ✐s ❛ ✢❛t r♦♦❢ ✭r✐❣❤t✮✳ ❚❤❡r❡❢♦r❡✱ t❤✐s q✉❛♥t✐t② ♠✉st ❜❡ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❜♦t❤ t ❛♥❞ x✦ ❇♦t❤ s✐❞❡s ❛r❡ ❡q✉❛❧ t♦ s♦♠❡ ❝♦♥st❛♥t ♥✉♠❜❡r✱ s❛② −λ✳ ❲❡ ❤❛✈❡✿ g ′ (t) = −λ K g(t) ❛♥❞ i′′ (x) = −λ i(x) .
❲❡ ❝❛♥ s♦❧✈❡ t❤❡s❡ ❢❛♠✐❧✐❛r ❖❉❊s✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊ ✭❈❤❛♣t❡r ✶✮✳ ■ts s♦❧✉t✐♦♥ ✐s✿ g(t) = Ce−λKt .
❙✐♥❝❡ ✇❡ ❡①♣❡❝t ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♦♥❧② t❤❡ ❝❛s❡✿ λ > 0.
❲✐t❤ t❤✐s ✐♥ ♠✐♥❞✱ ✇❡ s♦❧✈❡ t❤❡ s❡❝♦♥❞ ❖❉❊✳ ❚❤❡ s♦❧✉t✐♦♥ ✐s✿ √ √ i(x) = A sin( λ x) + B cos( λ x) ,
❢♦r s♦♠❡ ❝♦♥st❛♥t A, B ✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✻✳✺✳✺✿ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊ ❚❤❡ ❢✉♥❝t✐♦♥
u(t, x) = e
−λKt
√
A sin( λ x) + Be
−λKt
√
cos( λ x)
✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✳
❋♦r ❢✉rt❤❡r ❛♥❛❧②s✐s✱ ✇❡ ♠❛❦❡ t✇♦ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥s✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ t✐♠❡ st❛rts ❛t 0✿ a = 0,
❙❡❝♦♥❞✱ ✇❡ ❛ss✉♠❡ t❤❡ ③❡r♦
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✿ α(t) = β(t) = 0 .
❚❤❡♥ t❤❡ ③❡r♦ ❢✉♥❝t✐♦♥ ✐s ❛❧✇❛②s ❛ s♦❧✉t✐♦♥✳
✻✳✺✳
❚❤❡ ❤❡❛t P❉❊
✸✼✹
❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♣r♦❞✉❝❡✿
√ √ u(0, t) = i(0)g(t) = 0 =⇒ i(0) = 0 =⇒ A sin( λ 0) + B cos( λ 0) = 0 =⇒ B = 0 , ❛♥❞
√ √ √ u(t, b) = i(b)g(t) = 0 =⇒ i(b) = 0 =⇒ A sin( λ b) + 0 cos( λ b) = 0 =⇒ λ b = πn ,
❢♦r s♦♠❡ ✐♥t❡❣❡r
n✳
❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿
❚❤❡♦r❡♠ ✻✳✺✳✻✿ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊ ❲✐t❤ ❩❡r♦ ❇♦✉♥❞❛r② ❈♦♥❞✐✲ t✐♦♥ ■❢
❢♦r s♦♠❡ ✐♥t❡❣❡r
√ n✱
λ = πn/b
t❤❡♥ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ③❡r♦
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿
√ un (t, x) = Be−λKt sin( λ x)
❋♦r ❡❛❝❤
t✱
t❤✐s ✐s ❛ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞✴s❤r✉♥❦ s✐♥✉s♦✐❞✳ ❆s
t
❣r♦✇s✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ✐ts
❛♠♣❧✐t✉❞❡ ❞✐♠✐♥✐s❤❡s ❜✉t t❤❡ ♣❡r✐♦❞ r❡♠❛✐♥s t❤❡ s❛♠❡✳ ❚❤❡ ❢✉♥❝t✐♦♥ ❛❜♦✈❡ ✐s ♣❧♦tt❡❞ ❜❡❧♦✇✿
❆♥ ✐♠♣♦rt❛♥t ♦❜s❡r✈❛t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿
❈♦r♦❧❧❛r② ✻✳✺✳✼✿ ❙✉♠ ♦❢ ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❍❡❛t P❉❊ ❚❤❡ s✉♠ ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥❀ t❤❡r❡❢♦r❡✱
u(t, x) =
N X n=1
Bn e−λn Kt sin(
p
λn x)
✇❤❡r❡
p λn = πn/b, n = 1, 2, ..., N ,
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ③❡r♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳
❊①❡r❝✐s❡ ✻✳✺✳✽ Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳
❲❡ r❡❝♦❣♥✐③❡ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s✳ ❲❤❡♥ ❝♦♥✈❡r❣❡♥t✱ ✐ts s✉♠ ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✳
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
✸✼✺
✻✳✻✳ ❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
❲❡ ✉s❡❞ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✐♥t❡r✈❛❧s✱ ❛s ✇❡❧❧ ❛s t❤♦s❡ ♦❢ t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡✱ ✐♥ ♦r❞❡r t♦ st✉❞②
❝❤❛♥❣❡
✳ ❚❤✐s t✐♠❡✱ ✇❡ ♥❡❡❞ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡
n✲❞✐♠❡♥s✐♦♥❛❧
✐♥❝r❡♠❡♥t❛❧
❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳ ❚❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s
✇✐❧❧ ❝♦♠❡ ❢r♦♠ ❝❡❧❧ ❞❡❝♦♠♣♦s✐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 ✲♣❧❛♥❡✿
[c, d] = {y : c ≤ y ≤ d} ,
[a, b] × [c, d] = {(x, y) : a ≤ x ≤ b, c ≤ y ≤ d} .
❆
❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d]
✐s ♠❛❞❡ ♦❢ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②
❛s ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s
❲❡ st❛rt ✇✐t❤ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧
[a, b]
✐♥ t❤❡
[a, b]
x✲❛①✐s
✐♥ t❤❡
✐♥t♦
n
x✲❛①✐s
❛♥❞
[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✳
❲❡ ❞❡❝♦♠♣♦s❡ ❛♥ ✐♥t❡r✈❛❧
✐♥t❡r✈❛❧s✿
[y0 , y1 ], [y1 , y2 ], ..., [ym−1 , ym ] , ✇✐t❤
y0 = c, yn = d✳
[c, d]
✐♥ t❤❡
y ✲❛①✐s
✐♥t♦
m
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
❚❤❡ ❧✐♥❡s y = yj ❛♥❞ x = xi ❝r❡❛t❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ [xi , xi+1 ] × [yj , yj+1 ]✳ ❚❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ ❧✐♥❡s✱
✸✼✻
r❡❝t❛♥❣❧❡
[a, b] × [c, d]
✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s
Xij = (xi , yj ), i = 1, 2, ..., n, j = 1, 2, ..., m , ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡
♥♦❞❡s
♦❢ t❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐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✿
◆♦✇✱ ❛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♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
✸✼✼
❲❡ ❤❛✈❡ ❞✐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 ♦❢ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s✱ t❤❡ ❝❡❧❧s ❛r❡✿
• • •
0✲❝❡❧❧✱ ✐s {k} ✇✐t❤ k = ..., −2, −1, 0, 1, 2, 3, ... ❆♥ ❡❞❣❡✱ ♦r ❛ 1✲❝❡❧❧✱ ✐s [k, k + 1] ✇✐t❤ k = ..., −2, −1, 0, 1, 2, 3, ... ❆♥❞✱ 1✲❝❡❧❧s ❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ t❤❡s❡ 0✲❝❡❧❧s✳ ❆ ♥♦❞❡✱ ♦r ❛
❊①❛♠♣❧❡ ✻✳✻✳✷✿ ❞✐♠❡♥s✐♦♥ ✷
n = 2 ❣r✐❞✱ ✇❡ ❞❡✜♥❡ ❝❡❧❧s ❢♦r ❛❧❧ ✐♥t❡❣❡rs k, m ❛s 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]✳
❋♦r t❤❡ ❞✐♠❡♥s✐♦♥
• • •
♣r♦❞✉❝ts✿
❆ ♥♦❞❡✱ ♦r ❛
❲❡ ❛❧s♦ ❤❛✈❡✿
• •
❚❤❡
2✲❝❡❧❧s
❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ t❤❡s❡
❆♥❞✱ st✐❧❧✱ t❤❡
1✲❝❡❧❧s
1✲❝❡❧❧s✳
❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛❧♦♥❣ t❤❡
0✲❝❡❧❧s✳
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥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❡❞✿
❊①❛♠♣❧❡ ✻✳✻✳✸✿ ❞✐♠❡♥s✐♦♥ ✸
i, m, k ✱ ✇❡ ❤❛✈❡✿ ♦r ❛ 0✲❝❡❧❧✱ ✐s {i} × {m} × {k}✳
❋♦r ❛❧❧ ✐♥t❡❣❡rs
•
❆ ♥♦❞❡✱
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
• • •
✸✼✾
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 ❛ ❆ ❆
❚❤✉s✱ ♦✉r ❛♣♣r♦❛❝❤ t♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ s♣❛❝❡✱ ✐♥ ❛♥② ❞✐♠❡♥s✐♦♥✱ ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿
◮
n✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✐s ❝♦♠♣♦s❡❞ ❛❧♦♥❣ (k − 1)✲❝❡❧❧s✱ k = 1, 2, ..., n✳
❚❤❡
♦t❤❡r
❚❤❡ ❡①❛♠♣❧❡s s❤♦✇ ❤♦✇ t❤❡
n✲❞✐♠❡♥s✐♦♥❛❧
♦❢ ❝❡❧❧s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t
k ✲❝❡❧❧s
❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦
❛r❡ ❛tt❛❝❤❡❞ 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✐♦♥ ✐s ❡q✉❛❧ t♦ m✱ ❛♥❞ ✐t ✐s ❛❧s♦ ❝❛❧❧❡❞ ❛♥ m✲❝❡❧❧✱ ✇❤❡♥ t❤❡r❡ ❛r❡ m ❡❞❣❡s ❛♥❞ m − n ✈❡rt✐❝❡s ♦♥ t❤✐s ❧✐st✳ ❘❡♣❧❛❝✐♥❣ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s ✐♥ t❤❡ ♣r♦❞✉❝t ✇✐t❤ ♦♥❡ ♦❢ ✐ts ❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥ (n − 1)✲❝❡❧❧ ❝❛❧❧❡❞ ❛ ❜♦✉♥❞❛r② ❝❡❧❧ ♦❢ P ✳ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❛ ♣♦✐♥t
❉❡✜♥✐t✐♦♥ ✻✳✻✳✺✿ ❢❛❝❡ ❘❡♣❧❛❝✐♥❣ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s ✐♥ t❤❡ ♣r♦❞✉❝t ✇✐t❤ ♦♥❡ ♦❢ ✐ts ❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥
(n − 1)✲❝❡❧❧
❝❛❧❧❡❞ ❛
❡♥❞✲♣♦✐♥ts ❝r❡❛t❡s ❛♥
P ✳ ❘❡♣❧❛❝✐♥❣ s❡✈❡r❛❧ ❡❞❣❡s ✇✐t❤ k < n✱ ❝❛❧❧❡❞ ❛ ❜♦✉♥❞❛r② ❝❡❧❧ ♦❢ P ✳
❢❛❝❡
k ✲❝❡❧❧✱
♦❢
♦♥❡ ♦❢ t❤❡✐r
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
✸✽✵
❚❤✉s✱ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ ❛①❡s ✕ ✐♥t♦ ♥♦❞❡s ❛♥❞ ❡❞❣❡s ✕ ❝r❡❛t❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s♣❛❝❡ ✕ ✐♥t♦ ❝❡❧❧s ♦❢ ❛❧❧ ❞✐♠❡♥s✐♦♥s✳ ❊①❛♠♣❧❡ ✻✳✻✳✻✿ ✸❞ ❞❡❝♦♠♣♦s✐t✐♦♥
❇❡❧♦✇✱ ❛
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 ■♥ ❱♦❧✉♠❡s ✷ ❛♥❞ ✸✱ ✇❡ ❛ss✐❣♥❡❞ ♥✉♠❜❡rs t♦ t❤✐♥❣s ❛s ❧♦❝❛t✐♦♥ ✕ ♥♦❞❡s ♦r
0✲❝❡❧❧s
❛r❡ t❤❡ ❢❛❝❡s ♦❢ t❤❡
1✲❝❡❧❧s✱
❡t❝✳
♣♦✐♥ts ✇✐t❤✐♥ ❝❡❧❧s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ t♦ r❡♣r❡s❡♥t 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✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❇❡❧♦✇ ✇❡ s❡❡
m = 0, 1, 2✱
m
✐♥ ❛ ❝❡❧❧
r❡s♣❡❝t✐✈❡❧②✿
❋✐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 s❡✈❡r❛❧ ❛❞❥❛❝❡♥t ❝❡❧❧s✳ ❙❡❝♦♥❞❧②✱ t❤❡s❡ ♣♦✐♥ts ❛r❡ ✉s❡❞ ❢♦r ♠❡r❡ ❝♦r♥❡rs ♦r ♠✐❞✲♣♦✐♥ts ❡t❝✳
❜♦♦❦❦❡❡♣✐♥❣✳
❲❡ t❤❡♥ ❝❛♥ ❝❤♦♦s❡ t❤❡♠ t♦ ❜❡ t❤❡ ❡♥❞✲♣♦✐♥ts ♦r
■♥ 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 ♦❢ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢
t❤❡ ❧✐♥❡ t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉♥❝❤❛♥❣❡❞❀ ✐✳❡✳✱ ✐t✬s ❛ s✐♥❣❧❡ ♥✉♠❜❡r✳ 1 ♦✈❡r R ✿
1✲❢♦r♠s
❚❤✐s ✐s ❤♦✇ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢
0✲
❛♥❞
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
✸✽✶
❚❤❡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 ♦✈❡r Rn ✱ ♦r s✐♠♣❧② ❛ k ✲❢♦r♠✱ ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ k ✲❝❡❧❧s ♦❢ Rn ✳ ❆♥❞ t❤❡s❡ ❛r❡ 0✲✱ 1✲✱ ❛♥❞ 2✲❢♦r♠s ♦✈❡r R2 ✿
❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ♥❛t✉r❡ ♦❢ ❛ ❢♦r♠ ❛s ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛♥ ✉s❡ ❛rr♦✇s ✭R1 ✮✿
❍❡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✿
✻✳✻✳
❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s
❍❡r❡ ✇❡ ❤❛✈❡ t✇♦ ❢♦r♠s✿ • ❛ 0✲❢♦r♠ q ✇✐t❤ q(0) = 2, q(1) = 4, q(2) = 3, ...❀ ❛♥❞ • ❛ 1✲❢♦r♠ s ✇✐t❤ 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✐❣♥❡❞ t✇♦ s②♠❜♦❧s✿ • ♦♥❡ ✐s ✐ts ♥❛♠❡ ✭❛ ❧❛tt❡r✮ ❛♥❞
• t❤❡ ♦t❤❡r ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦r♠ ❛t t❤❛t ❧♦❝❛t✐♦♥ ✭❛ ♥✉♠❜❡r✮✳
❚❤✐s ✐❞❡❛ ✐s ✐❧❧✉str❛t❡❞ ❢♦r ❢♦r♠s ♦✈❡r R1 ❛♥❞ R2 r❡s♣❡❝t✐✈❡❧②✿
❲❡ ❤❛✈❡ ❛ 0✲❢♦r♠ q ❛♥❞ ❛ 1✲❢♦r♠ s ✐♥ 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 ✮✿
❚❤❡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❤❡ ♣✐♣❡✳
✸✽✷
✻✳✼✳
❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡
✸✽✸
• ❆ 2✲❢♦r♠✿ t❤❡ ✢♦✇ r❛t❡ ♦❢ t❤❡ ❧✐q✉✐❞ ❛❝r♦ss t❤❡ ♠❡♠❜r❛♥❡✳ • ❆ 3✲❢♦r♠✿ t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❧✐q✉✐❞ ✐♥s✐❞❡ t❤❡ ❜♦①✳
✻✳✼✳ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
2✿
❛ ♣❧❛t❡
❍❡❛t tr❛♥s❢❡r ❡①❤✐❜✐ts ❛ ❞✐s♣❡rs❛❧ ♣❛tt❡r♥ ✇✐t❤ ♥♦ ♣❛rt✐❝✉❧❛r ❞✐r❡❝t✐♦♥✿
❚❤❡ ❤❡❛t ✐s ❡①❝❤❛♥❣❡❞ ✕ ✇✐t❤ ❛❞❥❛❝❡♥t ❧♦❝❛t✐♦♥s✳ ❚❤❡ ♣❛tt❡r♥ ❝r❡❛t❡❞ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✳ ❚❤❡ ♣r♦❝❡ss ✐s ❛❧s♦ ❦♥♦✇♥ ❛s ❞✐✛✉s✐♦♥✳ ❚♦ ♠♦❞❡❧ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥✱ ✐♠❛❣✐♥❡ ❛ ❣r✐❞ ♦❢ sq✉❛r❡ r♦♦♠s ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ ❡❛❝❤ r♦♦♠ ❝❤❛♥❣❡s ❜② ❛ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❢♦✉r ❛❞❥❛❝❡♥t r♦♦♠s✳ ■ts s♣r❡❛❞s❤❡❡t s✐♠✉❧❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❢♦r♠✉❧❛✿
❂❘❈✲✳✷✺✯❦✯✭✭❘❈✲❘❈❬✲✶❪✮✰✭❘❈✲❘❈❬✶❪✮✰✭❘❈✲❘❬✲✶❪❈✮✰✭❘❈✲❘❬✶❪❈✮✮ ❖♥❧② ❛ ♣r♦♣♦rt✐♦♥✱ k ✱ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ♣r❡s✉♠❡❞ ❧❡♥❣t❤ ♦❢ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧✱ ♦❢ t❤❡ t♦t❛❧ ❛♠♦✉♥t ✐s ❡①❝❤❛♥❣❡❞✳
❊①❛♠♣❧❡ ✻✳✼✳✶✿ ❛ ❞r♦♣ ❞✐✛✉s❡❞ ❚❤❡ t✇♦ ✐♠❛❣❡s ❜❡❧♦✇ ❛r❡ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ✕ ❛ s✐♥❣❧❡ ✐♥✐t✐❛❧ ❤♦t s♣♦t ♦r ❛ ❞r♦♣ ♦❢ ❧✐q✉✐❞ ✕ ❛♥❞ t❤❡ r❡s✉❧ts ❛❢t❡r 1, 700 ✐t❡r❛t✐♦♥s ♦❢ s✉❝❤ ❛ s✐♠✉❧❛t✐♦♥ ❢♦r h = .01 ✭t❤❡ ✈❛❧✉❡ ❛t t❤❡ ❜♦r❞❡r ✜①❡❞ ❛t 0✮✿
❆s ❛ ✇❛r♥✐♥❣✱ ♦♥❡ ♥❡❡❞s t♦ ✉s❡ ❛♥ ❡①tr❛ s❤❡❡t ✭❛s ❛ ❜✉✛❡r✮❀ ♦t❤❡r✇✐s❡ s♣r❡❛❞s❤❡❡t✬s s❡q✉❡♥t✐❛❧ ✕ ❝❡❧❧✲❜②✲❝❡❧❧ ✐♥ ❡❛❝❤ r♦✇ ❛♥❞ t❤❡♥ r♦✇✲❜②✲r♦✇ ✕ ♠❛♥♥❡r ♦❢ ❡✈❛❧✉❛t✐♦♥ ✇✐❧❧ ❝❛✉s❡ ❛ s❦❡✇❡❞ ♣❛tt❡r♥✿
✻✳✼✳
❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
2✿
❛ ♣❧❛t❡
✸✽✹
❊①❛♠♣❧❡ ✻✳✼✳✷✿ ❛ s♦✉r❝❡ ♦❢ ❤❡❛t
❖♥ ❛ ❧❛r❣❡r s❝❛❧❡✱ t❤❡ s✐♠✉❧❛t✐♦♥ ♣r♦❞✉❝❡s ❛ r❡❛❧✐st✐❝ ❝✐r❝✉❧❛r ♣❛tt❡r♥ ❡✈❡♥ t❤♦✉❣❤ ✐t st❛rts str❛✐❣❤t✳ ❇❡❧♦✇ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡✲♣♦✐♥t ❜✉t ♣❡r♠❛♥❡♥t s♦✉r❝❡ ♦❢ ❤❡❛t s❤♦✇♥ ❛❢t❡r 3 ❛♥❞ t❤❡♥ ❛❢t❡r 3000 ✐t❡r❛t✐♦♥s✿
❊①❛♠♣❧❡ ✻✳✼✳✸✿ ❤❡❛t❡r
❋♦r ❡❛❝❤ t✱ ♦✉r ❢✉♥❝t✐♦♥ u(t, ·, ·) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛♥❞ ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞ ❜② ✐ts ❣r❛♣❤✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ sq✉❛r❡ ❤❡❛t❡r ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ❛ r♦♦♠ ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞②♥❛♠✐❝s ♦❢ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ♦✈❡r t❤❡ ✢♦♦r✿
❊①❛♠♣❧❡ ✻✳✼✳✹✿ ✇❛❧❧s
❚✇♦ ♦♣♣♦s✐t❡ ✇❛❧❧s ♦❢ ❛ r♦♦♠ ❛r❡ ✇❛r♠ ❛♥❞ t❤❡ ♦t❤❡r t✇♦ ❛r❡ ❝♦❧❞✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ ❞❡✈❡❧♦♣s ✐♥t♦ ❛ s❛❞❞❧❡✲❧✐❦❡ ♣❛tt❡r♥✿
✻✳✼✳
❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
2✿
✸✽✺
❛ ♣❧❛t❡
❊①❡r❝✐s❡ ✻✳✼✳✺
■♠♣❧❡♠❡♥t t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡s✳ ❚❤✉s✱ ✇❡ ❤❛✈❡
❢♦✉r
✇❛❧❧s ♦❢ t❤❡ r♦♦♠ ♦r✱ ❞✉❛❧❧②✱ ❢♦✉r ♣✐♣❡s ❧❡❛✈✐♥❣ t❤❡ r♦♦♠ ❛♥❞ t❤❛t ✐s ❤♦✇ t❤❡ ❤❡❛t ✭♦r
♠❛t❡r✐❛❧✮ ✐s ❡①❝❤❛♥❣❡❞ ✇✐t❤ t❤❡ ♥❡✐❣❤❜♦rs✳ ❚❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✳
❚❤❡ ❧❛tt❡r ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ♣r❡❢❡r❛❜❧❡ ❜❡❝❛✉s❡ ♦✉r t❡♠♣❡r❛t✉r❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥
w
✐s t❤❡♥ ❛
0✲❢♦r♠
✐♥
❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❛❜♦✈❡ s✐♠✉❧❛t✐♦♥s✱ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ✇✐❧❧ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ t❤❡ ✇❛❧❧s✴♣✐♣❡s✳ ❘❡❝❛❧❧ t❤❛t ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛
❜♦①
B
✐♥ t❤❡
txy ✲s♣❛❝❡
❝♦♠❡s ❢r♦♠ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✐ts t❤r❡❡
❡❞❣❡s ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✺✿
t0 q1 t1 q2 t2 q3 ... x0 s1 x1 s2 x2 s3 ... y0 p1 y1 p2 y2 p3 ... ❲❡ ♠❛❦❡ ❛ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧ ❝♦♥t❛✐♥❡rs ❤❛✈❡ ❡q✉❛❧ s✐③❡s✳
❚❤❡♥✱ ❢♦r ❝♦♥t❛✐♥❡r ❧♦❝❛t❡❞ ❛t
✻✳✼✳ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡ (xi , yj )✱ t❤✐s ✐s t❤❡
✸✽✻
t♦t❛❧ ✐♥✢♦✇ ✿
• −K(tk , si , yj−1 )∆y u (tk , si , pj−1 ) • −K(tk , si−1 , yj )∆x u (tk , si−1 , yj ) +K(tk , si , yj )∆x u (tk , si , yj ) • +K(tk , si , yj )∆y u (tk , si , pj ) •
❚❤❡ ❢♦✉r t❡r♠s ❛r❡ t❤❡ ✐♥✢♦✇s ❛❝r♦ss ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ✇❛❧❧s ♦❢ t❤❡ ❝♦♥t❛✐♥❡r ❛♥❞ t❤❡② ❛r❡ ❛rr❛♥❣❡❞ ❛❝❝♦r❞✐♥❣❧②✳ ❚❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ❤❡❛t ❡①❝❤❛♥❣❡❞ ❛❝r♦ss ❡❛❝❤ ✇❛❧❧ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ K ≥ 0✳ ■t ✐♥❝♦r♣♦r❛t❡s t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ t❤❡ ✇❛❧❧s✱ t❤❡✐r s✐③❡s✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧✱ ❛♥❞ s♦ ♦♥✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❤❡❛t✱ ♦r t❡♠♣❡r❛t✉r❡✱ u = u(t, x, y) ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆t u (qk , xi , yj ) = t♦t❛❧ ✐♥✢♦✇ ,
❛♥❞ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ t♦ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✐s s✐♠♣❧②✿ u(tk , xi , yj ) = u(tk−1 , xi , yj ) + t♦t❛❧ ✐♥✢♦✇ .
❚❤❡ s♣r❡❛❞s❤❡❡t ❝♦♥s✐sts ♦❢ s❡✈❡r❛❧ s❤❡❡ts ❝♦♠♣✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿ • t❤❡ ♣❡r♠❡❛❜✐❧✐t② ❢♦r ❡❛❝❤ ✇❛❧❧✱
• t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ❢♦r ❡❛❝❤ ❝♦♥t❛✐♥❡r✱ • t❤❡ ❜✉✛❡r ✭❝♦♣✐❡❞ ❝✉rr❡♥t ✈❛❧✉❡s✮✱
• t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ✢♦✇ ♦❢ t❡♠♣❡r❛t✉r❡ ❢♦r ❡❛❝❤ ✇❛❧❧✱ • t❤❡ t♦t❛❧ ✢♦✇ ❢♦r ❡❛❝❤ ❝♦♥t❛✐♥❡r✱ ❛♥❞ ✜♥❛❧❧② • t❤❡ ❝✉rr❡♥t ✈❛❧✉❡s ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡✳
❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s✿
t❤❡ ❝✉rr❡♥t ✈❛❧✉❡ ❂ t❤❡ ❜✉✛❡r ✈❛❧✉❡ ✰ t❤❡ t♦t❛❧ ✐♥✢♦✇ .
❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ sq✉❛r❡ ❝❡❧❧s r❡♣r❡s❡♥t t❤❡ ❝♦♥t❛✐♥❡rs ❛♥❞ t❤❡ ♥❛rr♦✇ ♦♥❡s t❤❡ ✇❛❧❧s✳
❊①❛♠♣❧❡ ✻✳✼✳✻✿ ❝♦✛❡❡ ❙✉♣♣♦s❡ ❝♦✛❡❡ ✐s ♣♦✉r❡❞ ✐♥t♦ ❛♥ ✐♥s✉❧❛t❡❞ ❝✉♣✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ s✐❞❡s ♦❢ t❤❡ ❝✉♣ q✉✐❝❦❧② ❝♦♦❧s ❞♦✇♥✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜rst t✇♦ s❤❡❡ts✿ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✭③❡r♦ ❛r♦✉♥❞ t❤❡ ❡❞❣❡ ♦❢ t❤❡ ❝✉♣✮ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ✭❤♦t ✐♥s✐❞❡✱ ❝♦❧❞ ♦✉ts✐❞❡✮✿
✻✳✼✳
❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
2✿
❛ ♣❧❛t❡
✸✽✼
❍❡❛t tr❛♥s❢❡r ❝♦♥t✐♥✉❡s ✇✐t❤✐♥ t❤❡ ❜♦❞② ♦❢ t❤❡ ❝♦✛❡❡ ✇✐t❤ ✈✐rt✉❛❧❧② ♥♦ tr❛♥s❢❡r t❤r♦✉❣❤ t❤❡ ✇❛❧❧s ♦❢ t❤❡ ❝✉♣✳ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts ❛❢t❡r 50 ✐t❡r❛t✐♦♥s✿
❖♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐s ♣r❡s❡r✈❡❞✳ ❊①❛♠♣❧❡ ✻✳✼✳✼✿ s♦❞❛
❙✉♣♣♦s❡ ❛ ❝❛♥ ♦❢ s♦❞❛ ✐s t❛❦❡♥ ♦✉t ♦❢ t❤❡ r❡❢r✐❣❡r❛t♦r✳ ❚❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ s✐❞❡s ♦❢ t❤❡ ❝❛♥ st❛rt t♦ ❡①❝❤❛♥❣❡ ❤❡❛t ✇✐t❤ t❤❡ ♦✉ts✐❞❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❡✛❡❝t ♦♥ t❤❡ ♦✉ts✐❞❡ t❡♠♣❡r❛t✉r❡ ✐s ♥❡❣❧✐❣✐❜❧❡✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜rst t✇♦ s❤❡❡ts✿ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✭❤✐❣❤ t❤r♦✉❣❤♦✉t✮ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ✭❝♦❧❞ ✐♥s✐❞❡✱ ❤♦t ♦✉ts✐❞❡✮✿
❚❤❡ t❡♠♣❡r❛t✉r❡ ♦✉ts✐❞❡ t❤❡ ❝❛♥ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞✦ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts ❛❢t❡r 50 ✐t❡r❛t✐♦♥s✿
✻✳✼✳ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡
✸✽✽
❊①❡r❝✐s❡ ✻✳✼✳✽
❲❤❛t ✐❢ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ ❛ ✇❛❧❧ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs❄ ❲❤❡♥ K ✐s ❝♦♥st❛♥t ❛s ❛♥♦t❤❡r s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ♦✉r ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s r❡❝♦❣✲ ♥✐③❛❜❧❡✿ • −∆y u (tk , si , pj−1 ) • +∆x u (tk , si , yj ) ∆t u (qk , xi , yj ) = K −∆x u (tk , si−1 , yj ) • +∆y u (tk , si , pj ) • = K ∆x u (tk , si , yj ) − ∆x u (tk , si−1 , yj ) + ∆y u (tk , si , pj ) − ∆y u (tk , si , pj−1 ) = K ∆x ∆x u (tk , xi , yj ) + ∆y ∆y u (tk , x , y ) i j = K ∆2x u (tk , xi , yj ) + ∆2y u (tk , xi , yj ) .
❚❤❡s❡ ❛r❡ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡s ✐♥tr♦❞✉❝❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✸✳
❆s t❤❡ tr❛♥s❢❡r ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧✱ ♦✉r ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆t u = K ∆2x u + ∆2y u ∆ .
❊①❡r❝✐s❡ ✻✳✼✳✾
❉❡r✐✈❡ ❛ ✈❡rs✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡ K ✳ ❊①❛♠♣❧❡ ✻✳✼✳✶✵✿ ♠✐❣r❛t✐♦♥
❚❤❡ ❞✐s❝r❡t❡ ❤❡❛t ❡q✉❛t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ ✉s❡❞ t♦ ♠♦❞❡❧ ♠♦✈✐♥❣ ♣♦♣✉❧❛t✐♦♥s✳ ■♥❞❡❡❞✱ ✐t ✐s ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t ♣❡♦♣❧❡ ♠✐❣r❛t❡ ❢r♦♠ t❤❡✐r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ t♦ ❛♥ ❛❞❥❛❝❡♥t ❛r❡❛ ✇✐t❤ ❛ ❧♦✇❡r ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②✳ ❇❡❧♦✇ ✇❡ ✐♠❛❣✐♥❡ t✇♦ ❛r❡❛s ✐♥ ❛ ❝♦✉♥tr②✿ ♦♥❡ ✐s ❞❡♥s❡❧② ❛♥❞ ✉♥✐❢♦r♠❧② ♣♦♣✉❧❛t❡❞ ✭u(0, x, y) = 1✮ ❛♥❞ t❤❡ ♦t❤❡r t❤✐♥❧② ❛♥❞ r❛♥❞♦♠❧② ✭u(0, x, y) ❝❧♦s❡ t♦ 0✮✳ ❚❤❡ t✇♦ ♣❛rts ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ r✐✈❡r ✭k = 0✮✳ ◆♦✇ ❛ ❜r✐❞❣❡ ✐s ❜✉✐❧t ♦✈❡r t❤❡ r✐✈❡r ❛s ✇❡❧❧ ❛ r♦❛❞ ✭k = 1✮ ❢r♦♠ t❤❡ ❜r✐❞❣❡ t♦ ❛ t♦✇♥ ♦♥ ❛ ❝✐t② ♦♥ t❤❡ ♦t❤❡r ❡♥❞ ♦❢ t❤❡ ❝♦✉♥tr②✳ ❚❤❡ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s s❤♦✇♥ ♦♥ t❤❡ ❧❡❢t ❛♥❞ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✐s s❤♦✇♥ ♦♥ t❤❡ r✐❣❤t✿
✻✳✼✳
❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
2✿
❛ ♣❧❛t❡
✸✽✾
❆t t❤✐s ♣♦✐♥t✱ ♣❡♦♣❧❡ st❛rt t♦ ❝r♦ss t❤❡ r✐✈❡r ❛♥❞ s♣r❡❛❞ ✐♥t♦ t❤❡ t❤✐♥❧② ♣♦♣✉❧❛t❡❞ ❛r❡❛ ✕ ❡s♣❡❝✐❛❧❧② ❛❧♦♥❣ t❤❡ r♦❛❞✳ ❲❡ ❛❧s♦ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❝✐t② ✐s ❛ ❝♦♥st❛♥t s♦✉r❝❡ ♦❢ ♥❡✇ s❡tt❧❡rs ❛rr✐✈✐♥❣ ❢r♦♠ t❤❡ ♦✉ts✐❞❡✱ u(t, x0 , y0 ) = 1✳
❊①❡r❝✐s❡ ✻✳✼✳✶✶ ✭❛✮ ■♥❝♦r♣♦r❛t❡ ✐♥t♦ t❤❡ ♠♦❞❡❧ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❣r♦✇✐♥❣ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t r❛t❡s✳ ✭❜✮ ■♥❝♦r♣♦r❛t❡ ✐♥t♦ t❤❡ ♠♦❞❡❧ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ s✉st❛✐♥❛❜✐❧✐t② ❧✐♠✐ts ✭❧♦❝❛t✐♦♥✲❞❡♣❡♥❞❡♥t✮ ♦♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✳ ❉❡r✐✈❡ t❤❡ P❉❊✳
❊①❛♠♣❧❡ ✻✳✼✳✶✷✿ ❞✐✛✉s✐♦♥ ✐♥ ❛ ✢♦✇ ■❢ ✇❡ ❞r♦♣ ❞②❡ ✐♥ ❛ ❧✐q✉✐❞✱ ✐t ✇✐❧❧ ❞✐✛✉s❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✳ ❲❤❛t ✐❢ t❤❡ ❧✐q✉✐❞ ✐s ✐♥ ♠♦t✐♦♥❄ ▲❡t✬s s❡❡ ✇❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❧❡t ❛ ❞r♦♣ ♦❢ ❞②❡ t♦ ❜❡ t❛❦❡♥ ❜② ❛ ✢♦✇✳ ❆s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡✱ ✇❡ ❢♦❧❧♦✇ t❤❡ r✉❧❡ t❤❛t✿ ◮ ❚❤❡ ❛♠♦✉♥t ♦❢ ❞②❡ ✐♥ ❡❛❝❤ ❝❡❧❧ ✐s s♣❧✐t ❛♥❞ ♣❛ss❡❞ t♦ t❤❡ ❛❞❥❛❝❡♥t ❝❡❧❧s ❛❧♦♥❣ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❛❧✉❡s ❛tt❛❝❤❡❞ t♦ t❤❡ ❡❞❣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ 1✬s ♦♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s ❛♥❞ 0s ♦♥ t❤❡ ✈❡rt✐❝❛❧ ❡❞❣❡s✳ ❚❤❡♥ t❤❡ ✢♦✇ ✇✐❧❧ ❜❡ ♣✉r❡❧② ❤♦r✐③♦♥t❛❧✳ ■❢ ✇❡ r❡✈❡rs❡ t❤❡ ✈❛❧✉❡s✱ ✐t ✇✐❧❧ ❜❡ ♣✉r❡❧② ✈❡rt✐❝❛❧✳ ◆♦✇ ✇❤❛t ✐❢ ✇❡ ❝❤♦♦s❡
✻✳✼✳ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥
1✬s
2✿
✸✾✵
❛ ♣❧❛t❡
♦♥ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❡❞❣❡s❄
1
→ − 1
→ −
1 y
1
• → − 1 y 1
• → − 1 y
1 y
1
• → − 1 y 1
• → − 1 y
1 y
1
• → − 1 y 1
• → − 1 y
■t ✐s s✐♠♣❧❡✿ t❤❡ ❛♠♦✉♥t ♦❢ ❞②❡ ✐♥ ❡❛❝❤ ❝❡❧❧ ✐s s♣❧✐t ✐♥ ❤❛❧❢ ❛♥❞ ♣❛ss❡❞ t♦ t❤❡ t✇♦ ❛❞❥❛❝❡♥t ❝❡❧❧s ❛❧♦♥❣ t❤❡ ✈❡❝t♦rs✳ ❚❤❡ ❧✐q✉✐❞ ✐s ✢♦✇✐♥❣ ❞♦✇♥ ❛♥❞ r✐❣❤t ❡q✉❛❧❧②✳ ❲❡ s❡❡ s♦♠❡ ❞✐s♣❡rs❛❧✱ ❜✉t t❤❡ ♣r❡❞♦♠✐♥❛♥t❧② ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s♣r❡❛❞✐♥❣ ♦❢ t❤❡ ❞②❡ ✐s ❛❧s♦ ❡✈✐❞❡♥t✿
❆ ♠♦r❡ ❣❡♥❡r❛❧ ❡①❛♠♣❧❡ ✐s t❤✐s✳ ■♠❛❣✐♥❡ t❤❛t ❛ ❧✐q✉✐❞ ♠♦✈❡s ❛❧♦♥❣ t❤❡ sq✉❛r❡ ❣r✐❞ ✇❤✐❝❤ ✐s t❤♦✉❣❤t ♦❢ ❛s ❛ s②st❡♠ ♦❢ ♣✐♣❡s ✿
■♥ t❤❡ ♣✐❝t✉r❡ ❛❜♦✈❡✱ t❤❡ ♥✉♠❜❡rs r❡♣r❡s❡♥t ✏✢♦✇s✑ t❤r♦✉❣❤ t❤❡s❡ ✏♣✐♣❡s✑✱ ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡
• •
x, y ✲❛①❡s✳
■♥ ♣❛rt✐❝✉❧❛r✱
t❤❡ ✏ 1✑ ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛s✿ ✏ 1 ❝✉❜✐❝ ❢♦♦t ♣❡r s❡❝♦♥❞ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✑✳ t❤❡ ✏ 2✑ ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛s✿ ✏ 2 ❝✉❜✐❝ ❢❡❡t ♣❡r s❡❝♦♥❞ ✉♣✇❛r❞✑✳
❍♦✇❡✈❡r✱ ❢♦r t❤❡ s❛❦❡ ♦❢ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ♠❛tt❡r✱ t❤❡s❡ ♥✉♠❜❡rs ❤❛✈❡ t♦ ❜❡ ♥♦r♠❛❧✐③❡❞✳ ❚❤❡♥ t❤❡ ❛♠♦✉♥t ❣♦❡s r✐❣❤t ❛♥❞
2/3
❣♦❡s ✉♣✳ ❖❢ ❝♦✉rs❡✱ ✇❡ t❤✐s ✐s ❛ ❞✐s❝r❡t❡
1/3
♦❢
1✲❢♦r♠✳
❊①❡r❝✐s❡ ✻✳✼✳✶✸
❈r❡❛t❡ ❛ s♣r❡❛❞s❤❡❡t ❛♥❞ ❝♦♥✜r♠ t❤❡ ♣❛tt❡r♥✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ✢♦✇ t❛❦❡s ❤♦r✐③♦♥t❛❧❧② t✇✐❝❡ ❛s ♠✉❝❤ ♠❛t❡r✐❛❧ ❛s ✈❡rt✐❝❛❧❧②❄
❋♦❧❧♦✇✐♥❣ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ❢♦r ❞✐♠❡♥s✐♦♥
1✱
✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❧❡♥❣t❤s ♦❢ ♣✐♣❡s ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡
❝❡❧❧s ✐♥t♦ t❤❡ ♠♦❞❡❧s✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✿
∆u =α ∆t
∆2 u ∆2 u + ∆x2 ∆y 2
.
✻✳✽✳
❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2
✸✾✶
❚❤❡✐r s✉♠ ✐s ❝❛❧❧❡❞ t❤❡ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r✳
✻✳✽✳ ❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥
2
❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s
❝❛s❡ ♦❢ ❤❡❛t tr❛♥s❢❡r✳ ❙✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡ ❢✉♥❝t✐♦♥ u ✐s ❞❡✜♥❡❞ ❢♦r ❛❧❧ x✱ y ✱ ❛♥❞ t ✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ s✉❜s❡t U
♦❢ t❤❡ s♣❛❝❡ ❛♥❞ ✐t ✐s s❛♠♣❧❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡ ❜♦① [a, b] × [c, d] × [0, ∞) ❝♦♥t❛✐♥❡❞ ✐♥ t❤❛t s✉❜s❡t✳ ❲❡ r❡✜♥❡ t❤✐s ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❛♥❞ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐s❝r❡t❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ❝♦♥st❛♥t ♣❡r♠❡❛❜✐❧✐t② K ✱ ∆u (qi , xk , yj ) = K ∆t
❛s ❲❡ ❤❛✈❡ t❤❡ ❤❡❛t
❡q✉❛t✐♦♥ ✿
∆2 u ∆2 u (ti , xk , yj ) + (ti , xk , yj ) , ∆x2 ∆y 2
∆x → 0, ∆y → 0, ❛♥❞ ∆t → 0 . ∂u =α ∂t
❚❤❡ t❡r♠ ✐♥ ♣❛r❡♥t❤❡s❡s ✐s ❝❛❧❧❡❞ t❤❡ ▲❛♣❧❛❝❡
∂ 2u ∂ 2u + ∂x2 ∂y 2
♦♣❡r❛t♦r ♦❢ u✳
❆ s♦❧✉t✐♦♥ u ♦❢ t❤✐s ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭P❉❊✮ ✐s ❛ ❢✉♥❝t✐♦♥ • ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ ❜♦①✱
• ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ✐♥s✐❞❡ ♦❢ ✐t✱ ❛♥❞
• t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ✐♥s✐❞❡ t❤❡ ❜♦①✱ s✉❝❤ t❤❛t
• t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ (t, x, y) ✳
❆ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✿
❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ P❉❊ ✐s t❤❛t✿ • ❆ ♣♦s✐t✐✈❡ ❝♦♥❝❛✈✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ♦r y ❣♦❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ♣♦s✐t✐✈❡ s❧♦♣❡ ✇✐t❤ r❡s♣❡❝t t♦ t✳
• ❆ ♥❡❣❛t✐✈❡ ❝♦♥❝❛✈✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ♦r y ❣♦❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ♥❡❣❛t✐✈❡ s❧♦♣❡ ✇✐t❤ r❡s♣❡❝t t♦ t✳
❲❡ ✐♠♣♦s❡ t❤❡s❡ ❝♦♥❞✐t✐♦♥s t♦ ♠❛❦❡ t❤❡ s♦❧✉t✐♦♥ s♣❡❝✐✜❝✿
✻✳✽✳
❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2
• t❤❡
✸✾✷
✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱ ♣r♦✈✐❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ u ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣✿ u(0, x, y) = i(x, y),
✇❤❡♥ a ≤ x ≤ b, c ≤ y ≤ d
✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ i ✐s ❣✐✈❡♥✱ ❛♥❞ • t❤❡
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✱ ♣r♦✈✐❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ u ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❢♦r ❛❧❧ t ≥ 0✿ u(t, a, y) = α(t, y), u(t, b, y) = β(t, y)
✇❤❡♥ c ≤ y ≤ d ,
u(t, x, c) = γ(t, y), u(t, x, d) = δ(t, x)
✇❤❡♥ a ≤ x ≤ b ,
❛♥❞ ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥s α✱ β ✱ γ ✱ ❛♥❞ δ ❛r❡ ❣✐✈❡♥✳ ❊①❡r❝✐s❡ ✻✳✽✳✶
❯♥❞❡r t❤❡ ❛❜♦✈❡ r❡str✐❝t✐♦♥s ✇❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t i(x)❄ ❊①❡r❝✐s❡ ✻✳✽✳✷
❉❡s❝r✐❜❡ t❤❡ ♠♦❞❡❧ s❡t✉♣s ❣✐✈❡♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇✐t❤ ②♦✉r ❝❤♦✐❝❡s ♦❢ i✱ α✱ ❡t❝✳ ❚❤❡ ✐❞❡❛ ❤♦✇ t♦ ❛♣♣r♦❛❝❤ ✜♥❞✐♥❣ s♦❧✉t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿ ◮ ❲❤❛t ✐❢ u ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✐♥✐t✐❛❧ st❛t❡ i ❜② ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ❢✉♥❝t✐♦♥ ♦❢ t❄
▲❡t✬s t❡st t❤✐s ✐❞❡❛ ❜② ❛ss✉♠✐♥❣ t❤❛t ✐t✬s tr✉❡✿ u(t, x, y) = i(x, y) · g(t) .
❙✉❜st✐t✉t✐♥❣ u ✐♥t♦ t❤❡ P❉❊ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ig ′ = K(ixx + iyy )g .
▲❡t✬s r❡❛rr❛♥❣❡ t❤❡ t❡r♠s ❛♥❞ s❡♣❛r❛t❡ ✈❛r✐❛❜❧❡s✿ g′ i′′ = . Kg i
▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥✿
g ′ (t) ixx (x, y) + iyy (x, y) = . Kg(t) i(x, y)
❲❡ r❡♣❡❛t t❤❡ ❛r❣✉♠❡♥t ❢r♦♠ t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿ ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t ✐s✱ t❤❡r❡❢♦r❡✱ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ x, y ✱ ✇❤✐❧❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❞❡♣❡♥❞s ♦♥❧② ♦♥ x, y ✐s✱ t❤❡r❡❢♦r❡✱ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ t✳ ❚❤❡r❡❢♦r❡✱ t❤✐s q✉❛♥t✐t② ♠✉st ❜❡ ❝♦♥st❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❜♦t❤ t ❛♥❞ x, y ✦ ❇♦t❤ s✐❞❡s ❛r❡ ❡q✉❛❧ t♦ s♦♠❡ ❝♦♥st❛♥t ♥✉♠❜❡r✱ s❛② −λ✳ ❲❡ ❤❛✈❡✿ g ′ (t) = −λ K g(t) ❛♥❞ ixx (x, y) + iyy (x, y) = −λ i(x, y) .
❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊ ✭❈❤❛♣t❡r ✶✮✳ ■ts s♦❧✉t✐♦♥ ✐s✿ g(t) = Ce−λKt .
❙✐♥❝❡ ✇❡ ❡①♣❡❝t ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♦♥❧② t❤❡ ❝❛s❡✿ λ > 0.
◆♦✇ t❤❡ P❉❊✿ ixx (x, y) + iyy (x, y) = −λ i(x, y) .
✻✳✾✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
❲❡ ❛r❡ ❣✉❡ss✐♥❣ ❛❣❛✐♥✿ ❲❤❛t ✐❢
u
1✿
✸✾✸
s♣r✐♥❣s ❛♥❞ str✐♥❣s
✐s t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ❞❡♣❡♥❞ ♦♥❧② ♦♥
x
❛♥❞
y
r❡s♣❡❝t✐✈❡❧②❄
❲❡ tr②✿
u(x, y) = X(x) + Y (y) . ❲❡ s✉❜st✐t✉t❡✿
(X + Y )xx + (X + Y )yy = −λ (X + Y ) . ❚❤❡♥✿
X ′′ + Y ′′ = −λ (X + Y ) . ❲❡ s❡♣❛r❛t❡ t❤❡ ✈❛r✐❛❜❧❡s ❛❣❛✐♥✿
X ′′ + λ X = −(Y ′′ + λ Y ) . ❚❤❡ ❛r❣✉♠❡♥t ❛❜♦✈❡ ❛♣♣❧✐❡s ❛❣❛✐♥ ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤✐s ✐s ❛ ❝♦♥st❛♥t✱ s❛②
X ′′ + λ X = p
❛♥❞
p✳
❲❡ ❤❛✈❡✿
Y ′′ + λ Y = −p .
❚❤❡ s♦❧✉t✐♦♥ ✐s ❦♥♦✇♥ ❢r♦♠ ❈❤❛♣t❡r ✶✿
√ √ p X(x) = A sin( λ x) + B cos( λ x) + , λ ❛♥❞
√ √ p Y (y) = D sin( λ y) + E cos( λ y) − , λ
❢♦r s♦♠❡ ❝♦♥st❛♥t ❲❡ ❛❧s♦ s❡t
p=0
A, B, D, E ✳ ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✻✳✽✳✸✿ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊ ❉✐♠ ❚❤❡ ❢✉♥❝t✐♦♥
2
u(t, x, y) =
√ √ √ √ e−λKt A sin( λ x) + B cos( λ x) + D sin( λ y) + E cos( λ y) ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❢♦r ❛♥② ♥✉♠❜❡rs
✻✳✾✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
1✿
A, B, D, E ✳
s♣r✐♥❣s ❛♥❞ str✐♥❣s
Pr❡✈✐♦✉s❧②✱ ✇❡ st✉❞✐❡❞ t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ❛tt❛❝❤❡❞ t♦ ❛ ✇❛❧❧ ❜② ❛ ✭♠❛ss✲❧❡ss✮ s♣r✐♥❣✳ ■♠❛❣✐♥❡ t❤✐s t✐♠❡ ❛
str✐♥❣ ♦❢ ♦❜❥❡❝ts
❝♦♥♥❡❝t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✇✐t❤ s♣r✐♥❣s✿
✻✳✾✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s
✸✾✹
▲❡t u = u(t, x) ♠❡❛s✉r❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ ♦❜❥❡❝t ❛ss♦❝✐❛t❡❞ ✇✐t❤ ♣♦s✐t✐♦♥ x ❛t t✐♠❡ t✳ ❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ❢♦r x ❛♥❞ t ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✳ ❋✐rst✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡✱ x✳ ❊❛❝❤ ♦❜❥❡❝t ✐s ❧♦❝❛t❡❞ ❛t ❛ ♣r✐♠❛r② ♥♦❞❡ ♦❢ t❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛ ✭♣♦ss✐❜❧② ✈❛r✐❛❜❧❡✮ ❞✐st❛♥❝❡ h = ∆x t♦ ✐ts ♥❡✐❣❤❜♦r ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐s ∆s✳ ❆❝❝♦r❞✐♥❣ t♦ ❍♦♦❦❡✬s ❧❛✇✱ t❤❡ ❢♦r❝❡ ❡①❡rt❡❞ ❜② t❤❡ s♣r✐♥❣ ✐s✿ H = −kS ,
✇❤❡r❡ S ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s♣r✐♥❣ ❢r♦♠ ✐ts ❡q✉✐❧✐❜r✐✉♠ st❛t❡ ❛♥❞ t❤❡ ❝♦♥st❛♥t✱ st✐✛♥❡ss✱ k r❡✢❡❝ts t❤❡ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s♣r✐♥❣✳ ❚❤❡♥✱ ✐❢ t❤✐s ✐s t❤❡ s♣r✐♥❣ t❤❛t ❝♦♥♥❡❝ts ❧♦❝❛t✐♦♥s xp−1 ❛♥❞ xp ✱ ✐ts ❝♦♠♣r❡ss✐♦♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ S = u(tj , xp−1 ) − u(tj , xp ) .
❚❤❡r❡❢♦r❡✱ t❤❡ ❢♦r❝❡ ♦❢ t❤✐s s♣r✐♥❣ ✐s Hp = K(tj , sp ) [u(tj , xp−1 ) − u(tj , xp )] = −K(tj , sp )∆x u (tj , sp ) = −(k∆x u)(tj , sp ) ,
✇❤❡r❡ k ✐s t❤❡ ❧♦❝❛t✐♦♥✲ ❛♥❞ t✐♠❡✲❞❡♣❡♥❞❡♥t st✐✛♥❡ss ♦❢ t❤❡ s♣r✐♥❣s✳ ❚❤✐s ❢♦r♠✉❧❛✱ Hp = −(k∆x u)(tj , sp ) ,
❡①♣r❡ss❡s t❤❡ ❢♦r❝❡ ❛✛❡❝t✐♥❣ ❛♥ ♦❜❥❡❝t ❛t s♦♠❡ ❧♦❝❛t✐♦♥ ✐♥ t❡r♠s ♦❢ s♦♠❡ q✉❛♥t✐t② t❤❛t ♠❛② ❞❡♣❡♥❞ ♦♥ ❜♦t❤ ❧♦❝❛t✐♦♥ ❛♥❞ t✐♠❡✳ ■t ❝❛♥ ❤❛✈❡ ❞✐✛❡r❡♥t ✐♥t❡r♣r❡t❛t✐♦♥s ❢♦r ❞✐✛❡r❡♥t ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ u✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛♥ ♦s❝✐❧❧❛t✐♥❣ str✐♥❣ ✿
❍❡r❡✱ t❤❡ ♣✐❡❝❡s ♦❢ t❤❡ str✐♥❣ ❛r❡ ✈❡rt✐❝❛❧❧② ❞✐s♣❧❛❝❡❞ ✭t❤❛t✬s ∆x u✮ ✇❤✐❧❡ t❤❡ ✇❛✈❡s ♣r♦♣❛❣❛t❡ ❤♦r✐③♦♥t❛❧❧②✳ ❆t ✐ts s✐♠♣❧❡st✱ ✇❡ ❤❛✈❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✇❡✐❣❤ts t❤❛t ❝❛♥ ♠♦✈❡ ♦♥❧② ✈❡rt✐❝❛❧❧② ❝♦♥♥❡❝t❡❞ ❜② ✈❡rt✐❝❛❧ s♣r✐♥❣s✿
✻✳✾✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
1✿
s♣r✐♥❣s ❛♥❞ str✐♥❣s
✸✾✺
❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♦❜❥❡❝ts✱ ∆x✱ r❡♠❛✐♥ ✉♥❝❤❛♥❣❡❞ ✇❤✐❧❡ t❤❡ ❤❡✐❣❤t✱ u(t, x)✱ ✈❛r✐❡s ✇✐t❤ t✐♠❡✳ ❚❤❡ ✈❡rt✐❝❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✐s ∆x u✳ ❆♣♣❧②✐♥❣ ❍♦♦❦❡✬s ❧❛✇ ❛❣❛✐♥✱ ✇❡ ✜♥❞ t❤❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦r❝❡ ❡①❡rt❡❞ ❜② t❤❡ s♣r✐♥❣✿ H = −k∆x u .
❆ ♠♦r❡ ❝♦♠♣❧❡① ♠♦❞❡❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❲❡ st✐❧❧ ✐♠❛❣✐♥❡ t❤❛t t❤❡ str✐♥❣ ✐s ♠❛❞❡ ♦❢ s♣r✐♥❣s ✇✐t❤ ✇❡✐❣❤ts ❜✉t t❤❡ s♣r✐♥❣ ❝❛♥ ❣♦ ❞✐❛❣♦♥❛❧❧②✿
❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✇❡✐❣❤ts st✐❧❧ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞❀ t❤❛t✬s ∆x✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ✉s❡ ❍♦♦❦✬s ▲❛✇ ✿ ❚❤❡ ❢♦r❝❡ ❡①❡rt❡❞ ❜② t❤❡ s♣r✐♥❣ ✐s F = −kS ✱ ✇❤❡r❡ S ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s♣r✐♥❣ ❢r♦♠ ✐ts ♥♦♥✲str❡t❝❤❡❞ st❛t❡ ❛♥❞ k ✐s t❤❡ st✐✛♥❡ss ♦❢ t❤✐s s♣r✐♥❣✳ ❚❤❡s❡ ❛r❡ t❤❡ q✉❛♥t✐t✐❡s ✇❡ ✇✐❧❧ ❤❛✈❡ t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t✿
❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ ♦❢ ❡❛❝❤ s♣r✐♥❣ ✐s 0✳ ❚❤❡♥ S ✐s s✐♠♣❧② t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✇❡✐❣❤ts✳ ❙✉♣♣♦s❡ H ✐s t❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢♦r❝❡ F ✳ ❊①❛♠✐♥✐♥❣ s✐♠✐❧❛r tr✐❛♥❣❧❡s r❡✈❡❛❧s✿ S F = . H ∆x u
❚❤❡r❡❢♦r❡✱
−kS S = , H ∆x u
t❤❡♥ H = −k∆x u .
❚❤❡ ❡q✉❛t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡ ❢♦r t❤❡ ♣r❡✈✐♦✉s ❛♣♣r♦❛❝❤✦ ❋r♦♠ t❤✐s ❡q✉❛t✐♦♥ ✇❡ ❞❡r✐✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ❣♦✈❡r♥s t❤❡s❡ ✇❛✈❡s ♠♦❞❡❧❡❞ ❜② ❛♥② ♦❢ t❤❡s❡ t❤r❡❡ ❛♣♣r♦❛❝❤❡s✳ ▲❡t F (xi , tj ) ❜❡ t❤❡ ❢♦r❝❡ t❤❛t ❛❝t❡❞ ♦♥ t❤❡ ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t xi ❛t t✐♠❡ tj ✳ ❚❤❡r❡ ❛r❡ t✇♦ ❍♦♦❦❡✬s ❢♦r❝❡s ❛❝t✐♥❣ ♦♥ t❤✐s ♦❜❥❡❝t ❢r♦♠ t❤❡ t✇♦ ❛❞❥❛❝❡♥t s♣r✐♥❣s✿ Hi−1 ❛♥❞ Hi ✱ ♣✉❧❧✐♥❣ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ F (xi , tj ) = Hi−1 −Hi = −(k∆x u)(tj , si−1 ) +(k∆x u)(tj , si ) = ∆x (k∆x u)(tj , xi ) .
◆♦t❡ t❤❛t t❤✐s ✐s t❤❡ s❛♠❡ ❡①♣r❡ss✐♦♥ ❛s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✦
✻✳✾✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s
✸✾✻
❙❡❝♦♥❞✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ t❡♠♣♦r❛❧ ✈❛r✐❛❜❧❡✱ t✳ ❊❛❝❤ ✐♥❝r❡♠❡♥t ∆t ♦❢ t✐♠❡ ♠❛② ❤❛✈❡ ❛ ❞✐✛❡r❡♥t ❞✉r❛t✐♦♥✳ ◆♦✇ s✉♣♣♦s❡ t❤❛t t❤❡ ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t xi ❤❛s ♠❛ss m(xi )✳ ❚❤❡♥✱ ❜② t❤❡ ❙❡❝♦♥❞ ◆❡✇t♦♥✬s ▲❛✇✱ t❤❡ t♦t❛❧ ❢♦r❝❡ ✐s F (tj , xi ) = m(xi )a(tj , xi ) ,
✇❤❡r❡ a(tj , xi ) ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ♦❜❥❡❝t xi ❛t t✐♠❡ tj ✳ ❆s ✇❡ ❦♥♦✇✱ t❤✐s ✐s t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✱ a(tj , xi ) =
∆2 u (tj , xi ) . ∆t2
❲❡ ❤❛✈❡ ♥♦✇ t❤❡ ❞✐✛❡r❡♥❝❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ✿ ∆2 u 1 (tj , xi ) = ∆x (k∆x u)(tj , xi ) 2 ∆t m(xi )
◆♦✇ ✇❡ ✇✐❧❧ ❞❡r✐✈❡ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✐♥ ♦r❞❡r t♦ s✐♠✉❧❛t❡ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❲❡ ♠❛❦❡ s❡✈❡r❛❧ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥s✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥ts ❛r❡ ❡q✉❛❧✿ ∆t = 1✳ ❚❤❡♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ ♦✉r ❡q✉❛t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ m∆2t u = m u(t + 1, x) − 2u(t, x) + u(t − 1, x) .
❋♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ♦r✐❣✐♥❛❧ ❡①♣r❡ss✐♦♥✿
h i h i ∆x (k∆x u) = k u(t, x − 1) − u(t, x) + k u(t, x + 1) − u(t, x) .
❙❡❝♦♥❞✱ ✇❡ ❛ss✉♠❡ t❤❛t k ❛♥❞ m ❛r❡ ❝♦♥st❛♥t✳ ❚❤❡♥ ❥✉st s♦❧✈❡ ❢♦r u(x, t + 1)✿
✇❤❡r❡
u(t + 1, x) = 2u(t, x) − u(t − 1, x) + α u(t, x + 1) − 2u(t, x) + u(t, x − 1) , α=
k . m
❚♦ ✈✐s✉❛❧✐③❡ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ❛rr❛♥❣❡ t❤❡ t❡r♠s ✐♥ ❛ t❛❜❧❡ t♦ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❛s ❛ s♣r❡❛❞s❤❡❡t✿ x−1
x x+1 t−1 −u(t − 1, x) = αu(t, x − 1) +2(1 − α)u(t, x) +αu(t, x + 1) t u(t + 1, x) t+1
❊✈❡♥ t❤♦✉❣❤ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s t❤❡ s❛♠❡✱ t❤❡ t❛❜❧❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❛t ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ ♠❛❦❡s ✐t ♥❡❝❡ss❛r② t♦ ❧♦♦❦ t✇♦ st❡♣s ❜❛❝❦✱ ♥♦t ❥✉st ♦♥❡✳ ❚❤❛t✬s ✇❤② ✇❡ ❛❧s♦ ❤❛✈❡ t✇♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❲❡ s✉♣♣♦s❡✱ ❢♦r s✐♠♣❧✐❝✐t②✱ t❤❛t α = 1✳ ❊①❛♠♣❧❡ ✻✳✾✳✶✿ ❛❧❣❡❜r❛
❈❤♦♦s✐♥❣ t❤❡ s✐♠♣❧✐✜❡❞ s❡tt✐♥❣s ❛❧❧♦✇s ✉s t♦ ❡❛s✐❧② s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❜② ❤❛♥❞✿ ∆2t u = ∆2x u; 1 0 u(t, x) = 1 0
✐❢ t = 0, x = 1; ✐❢ t = 0, x 6= 1; ✐❢ t = 1, x = 2; ✐❢ t = 1, x 6= 2.
■♥✐t✐❛❧❧②✱ t❤❡ ✇❛✈❡ ❤❛s ❛ s✐♥❣❧❡ ❜✉♠♣ ❛♥❞ t❤❡♥ t❤❡ ❜✉♠♣ ♠♦✈❡s ♦♥❡ st❡♣ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✳ ❚❤❡
✻✳✾✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
1✿
s♣r✐♥❣s ❛♥❞ str✐♥❣s
✸✾✼
♥❡❣❛t✐✈❡ ✈❛❧✉❡s ♦❢ x ❛r❡ ✐❣♥♦r❡❞✳ ◆♦✇✱ s❡tt✐♥❣ k = 1 ♠❛❦❡s t❤❡ ♠✐❞❞❧❡ t❡r♠ ✐♥ t❤❡ t❛❜❧❡ ❞✐s❛♣♣❡❛r✳ ❚❤❡♥ ❡✈❡r② ♥❡✇ t❡r♠ ✐s ❝♦♠♣✉t❡❞ ❜② t❛❦✐♥❣ ❛♥ ❛❧t❡r♥❛t✐♥❣ s✉♠ ♦❢ t❤❡ t❤r❡❡ t❡r♠s ❛❜♦✈❡✱ ❛s s❤♦✇♥ ❜❡❧♦✇✿ t\x 0 1 2 3 4 5 ..
1 1 0 0 0 0 0 ..
2 0 1 0 0 0 0 ..
3 0 0 1 0 0 0 ..
4 5 6 7 .. 0 0 [0] 0 .. 0 [0] 0 [0] .. 0 0 (0) 0 .. 1 0 0 0 .. 0 1 0 0 .. 0 0 1 0 .. .. .. .. .. ..
❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✇❛✈❡ ✐s ❛ s✐♥❣❧❡ ❜✉♠♣ r✉♥♥✐♥❣ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛t s♣❡❡❞ 1✿
❊①❡r❝✐s❡ ✻✳✾✳✷
❙❡t ✉♣ ❛♥❞ s♦❧✈❡ ❛♥ ■❱P ✇✐t❤ 2 ❜✉♠♣s✱ n ❜✉♠♣s✳ ❚❤❡ s✐♠♣❧❡st ✇❛② t♦ ✐♠♣❧❡♠❡♥t t❤✐s ❞②♥❛♠✐❝s ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ✐s t♦ ✉s❡ t❤❡ ✜rst t✇♦ r♦✇s ❢♦r t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡♥ ❛❞❞ ♦♥❡ r♦✇ ❢♦r ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡ ❊①❝❡❧ ❢♦r♠✉❧❛ ✐s✿ ❂❘✶❈✺✯❘❬✲✶❪❈❬✲✶❪✰✷✯✭✶✲❘✶❈✺✮✯❘❬✲✶❪❈✰❘✶❈✺✯❘❬✲✶❪❈❬✶❪✲❘❬✲✷❪❈
❍❡r❡ ❝❡❧❧ ❘✶❈✺ ❝♦♥t❛✐♥s t❤❡ ✈❛❧✉❡ ♦❢ α✳ ❊①❛♠♣❧❡ ✻✳✾✳✸✿ ❜✉♠♣s
❚❤❡ s✐♠♣❧❡st ♣r♦♣❛❣❛t✐♦♥ ♣❛tt❡r♥ ✐s ❣✐✈❡♥ ❜② α = 1✳ ❇❡❧♦✇ ✇❡ s❤♦✇ t❤❡ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ❜✉♠♣✱ ❛ t✇♦✲❝❡❧❧ ❜✉♠♣✱ ❛♥❞ t✇♦ ❜✉♠♣s✿
✻✳✾✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
1✿
s♣r✐♥❣s ❛♥❞ str✐♥❣s
✸✾✽
■♥ t❤❡ s❡❝♦♥❞ r♦✇✱ t❤❡ s✇✐♥❣ ♦❢ t❤❡ ✇❛✈❡ ✐s ✈✐s✉❛❧✐③❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❊①❡r❝✐s❡ ✻✳✾✳✹
▼♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t t♦ ✐♥tr♦❞✉❝❡ ✇❛❧❧s ✭♦♥❡ ❛♥❞ t❤❡♥ t✇♦✮ ✐♥t♦ t❤❡ ♣✐❝t✉r❡✿
❊①❡r❝✐s❡ ✻✳✾✳✺
▼♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t t♦ ❛❝❝♦♠♠♦❞❛t❡ ♥♦♥✲❝♦♥st❛♥t ❞❛t❛ ❜② ❛❞❞✐♥❣ ✈❛r✐❛❜✐❧✐t② t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✭❛✮ t❤❡ st✐✛♥❡ss k ✭s❤♦✇♥ ❜❡❧♦✇✮✱ ✭❜✮ t❤❡ ✇❡✐❣❤ts m✱ ✭❝✮ t❤❡ s♣❛❝❡ ✐♥t❡r✈❛❧s ∆x✱ ✭❞✮ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧s ∆t✳
❊①❡r❝✐s❡ ✻✳✾✳✻
■♠♣❧❡♠❡♥t ❛ s♣r❡❛❞s❤❡❡t s✐♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❝❛s❡ ♦❢ ♥♦♥✲❝♦♥st❛♥t m✳ ❍✐♥t✿ ❨♦✉ ✇✐❧❧ ♥❡❡❞ t✇♦ ❜✉✛❡rs✳
✻✳✾✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s
❇❡❧♦✇ ✇❡ ❝♦♥s✐❞❡r ❛
r♦♣❡ ✇✐t❤ ✜①❡❞ ❡♥❞s✳
❚❤✐s ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿
u(t, 0) = u(t, n) = 0, ❲❡ ❛❣❛✐♥ ❝❤♦♦s❡
✸✾✾
❢♦r ❛❧❧
t.
α = 1✳
❊①❛♠♣❧❡ ✻✳✾✳✼✿ ✇❤✐♣
u(0, x)✳ ■t ✐s ❛ s♠❛❧❧ u(1, x) = u(0, x − 1)✱ ♦❢ t❤❡ ✜rst✳ ❚❤❡
❚❤❡ s✐♠✉❧❛t✐♦♥ st❛rts ✇✐t❤ ❛♥ ❛r❜✐tr❛r② s❤❛♣❡ ❣✐✈❡♥ ❜② t❤❡ ✜rst ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ♣✐❡❝❡ ♦❢ ❛ s✐♥✉s♦✐❞✳ ❚❤❡ s❡❝♦♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ❥✉st ❛ s❤✐❢t✱
s✐♠✉❧❛t✐♦♥ ❝♦♥t✐♥✉❡s t♦ ♣r♦❞✉❝❡ t❤✐s s❤✐❢t ❛t ❡✈❡r② ✐t❡r❛t✐♦♥❀ ✐♥ ❢❛❝t✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ s❤✐❢t ✐s t❤❡ s♣❡❡❞ ♦❢ ♣r♦♣❛❣❛t✐♦♥ ♦❢ t❤❡ s❤❛♣❡✳ ❚❤❡ r♦♣❡ ❜❡❤❛✈❡s ❧✐❦❡ ❛ ✇❤✐♣✿
❚❤✐s ❡✛❡❝t ✐s ❦♥♦✇♥ ❛s ❛ ✏tr❛✈❡❧✐♥❣ ✇❛✈❡✑✳ ❋✉rt❤❡r♠♦r❡✱ ❜❡❝❛✉s❡ t❤❡ ❡♥❞ ✐s ❛tt❛❝❤❡❞ t♦ t❤❡ ✇❛❧❧✱ t❤❡ ✇❛✈❡ ❜♦✉♥❝❡s ♦✛ t❤❡ ❡♥❞✳ ❲❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥t
α
✐s
1.01
✐♥st❡❛❞ ♦❢
1✱
t❤❡ ♠♦❞❡❧ ❜r❡❛❦s ❞♦✇♥ ✈❡r② q✉✐❝❦❧②✿
❊①❛♠♣❧❡ ✻✳✾✳✽✿ ✈✐❜r❛t✐♥❣ str✐♥❣
❚❤❡ s✐♠✉❧❛t✐♦♥ st❛rts ✇✐t❤ ❛ s✐♥✉s♦✐❞ ❣✐✈❡♥ ❜② t❤❡ ✜rst ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ s❡❝♦♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ❛♥ ❡①❛❝t ❝♦♣②✱
u(1, x) = u(0, x)✱
u(x, 0) = sin(πx/n)✳
❚❤❡
♦❢ t❤❡ ✜rst✳ ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✱ t❤❡♥✱ ✐s
③❡r♦ ✭t❤❡r❡ ✐s ♦❢ ❝♦✉rs❡ ❛❝❝❡❧❡r❛t✐♦♥✮✳ ❚❤❡ s✐♠✉❧❛t✐♦♥ ♣r♦❞✉❝❡s ♠♦r❡ s✐♥✉s♦✐❞s ❛♥❞ ❛ ♣❡r❢❡❝t ✈✐❜r❛t✐♦♥✿
✻✳✶✵✳ ❚❤❡ ✇❛✈❡ P❉❊
✹✵✵
❲❡ ❝❛♥ ❛❧s♦ ✈✐s✉❛❧✐③❡ u ❜② ✐ts ❣r❛♣❤ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿
❊①❡r❝✐s❡ ✻✳✾✳✾
❲❤❛t ✐❢ ❤❛❧❢ ♦❢ t❤❡ str✐♥❣ ✐s ♠❛❞❡ ♦❢ ❛ st✐✛❡r ♠❛t❡r✐❛❧❄
✻✳✶✵✳ ❚❤❡ ✇❛✈❡ P❉❊
❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ♦❢ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✳ ❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥ u ✐s ❞❡✜♥❡❞ ❢♦r ❛❧❧ x ❛♥❞ t ✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ s✉❜s❡t U ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ ✐t ✐s s❛♠♣❧❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ r❡❝t❛♥❣❧❡ [0, b] × [0, d] ❝♦♥t❛✐♥❡❞ ✐♥ t❤❛t s✉❜s❡t✳ ❲❡ ♥♦✇ ✉t✐❧✐③❡ t❤❡ r❡s✉❧ts ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ✇❡✐❣❤ts ❛♥❞ s♣r✐♥❣s✿
❚❤❡s❡ ❛r❡ ♦✉r ❛ss✉♠♣t✐♦♥s✿ • ❚❤❡ ❛rr❛② ♦❢ N ✐❞❡♥t✐❝❛❧ ✇❡✐❣❤ts❀ m ✐s ❝♦♥st❛♥t✳
• ❚❤❡ ✇❡✐❣❤ts ❛r❡ ❞✐str✐❜✉t❡❞ ❡✈❡♥❧② ♦✈❡r t❤❡ ❧❡♥❣t❤ L = N ∆x ♦❢ t❤❡ str✐♥❣✳
✻✳✶✵✳
❚❤❡ ✇❛✈❡ P❉❊
✹✵✶
• ❚❤❡ t♦t❛❧ ♠❛ss ✐s M = N m✳
• ❚❤❡ s♣r✐♥❣s ❛r❡ ✐❞❡♥t✐❝❛❧❀ k ✐s ❝♦♥st❛♥t✳
• ❚❤❡
t♦t❛❧ s♣r✐♥❣ ❝♦♥st❛♥t ♦❢ t❤❡ ❛rr❛② ✐s K = k/N ✳
• ❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ str✐♥❣ ✐s ✉♥✐❢♦r♠✿ ∆x = ∆s = h✳
• ❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✐s ❛❧s♦ ✉♥✐❢♦r♠✿ ∆t = ∆q ✳
❲❡ ✇✐❧❧ ♥❡❡❞ ❛❣❛✐♥ ❛ ❞✉❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧✿
❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❢♦r ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ ❛s ❢♦❧❧♦✇s✿ m
❋✉rt❤❡r♠♦r❡✿
u(t + ∆t, x) − 2u(t, x) + u(t − ∆t, x) (∆t)2 . = k u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x)
u(t + ∆t, x) − 2u(t, x) + u(t − ∆t, x) (∆t)2 K(∆x)2 u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x) = m (∆x)2 2 k/N (N ∆x) u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x) = Nm (∆x)2 2 KL u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x) = . M (∆x)2
❙♦❧✈✐♥❣ ❢♦r u(t + ∆t, x)✱ ✇❡ ♦❜t❛✐♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ u(t + ∆t, x) = 2u(t, x) − u(t − ∆t, x) + α
✇❤❡r❡
α=
∆t ∆x
2
u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x) ,
KL2 . M
❊①❡r❝✐s❡ ✻✳✶✵✳✶
❙❡t ✉♣ ❛♥❞ s♦❧✈❡ t❤❡ ■❱P ❢♦r t❤❡ ✜♥✐t❡ str✐♥❣✱ ❢♦r t❤❡ s✐♠♣❧✐✜❡❞ s❡tt✐♥❣s✳ ❍✐♥t✿ ♠✐♥❞ t❤❡ ❡♥❞s✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❢♦✉♥❞ t❤❡ ❞✐s❝r❡t❡
✇❛✈❡ ❡q✉❛t✐♦♥ ✿ ∆2 u ∆2 u = α . ∆t2 ∆x2
■t ✐s ❛ P❉❊ ✇✐t❤ r❡s♣❡❝t t♦ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ❲❡ r❡✜♥❡ ♦✉r ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❛♥❞ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛s ∆x → 0 ❛♥❞ ∆t → 0✳ ❲❡ ❤❛✈❡ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ✿ ∂ 2u ∂ 2u = α ∂t2 ∂x2
■t ✐s ❛ P❉❊ ✇✐t❤ r❡s♣❡❝t t♦ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✳ ❆ s♦❧✉t✐♦♥ u ♦❢ t❤✐s P❉❊ ✐s ❛ ❢✉♥❝t✐♦♥✿
✻✳✶✵✳
❚❤❡ ✇❛✈❡ P❉❊
✹✵✷
• ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ r❡❝t❛♥❣❧❡✱
• t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ✐♥s✐❞❡ ♦❢ ✐t✱ ❛♥❞
• t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ x ✐♥s✐❞❡ t❤❡ r❡❝t❛♥❣❧❡✱ s✉❝❤ t❤❛t
• t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ ♣❛✐r (x, t) s✉❝❤ t❤❛t x ✐s ✐♥ (0, b) ❛♥❞ t ✐s ✐♥ (0, d)✳
■♥ ❝♦♥tr❛st t♦ t❤❡ ❤❡❛t tr❛♥s❢❡r P❉❊✱ ✇❡ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ✜♥❞ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ■♥ ♦r❞❡r t♦ s♦❧✈❡ t❤✐s P❉❊✱ ✇❡ ✐♥tr♦❞✉❝❡ ♥❡✇ ✈❛r✐❛❜❧❡s✿ p = x + αt ❛♥❞ q = x − αt ,
❚❤❡♥✱
1 1 (p − q) . x = (p + q) ❛♥❞ t = 2 2α
❊①❡r❝✐s❡ ✻✳✶✵✳✷
❲❤❛t ✐s t❤❡ ♠❛tr✐① ♦❢ t❤✐s ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥❄ ❚❤❡ ♥❡✇
✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥✿ v(p, q) = u(t, x) = u
1 1 (p + q), (p − q) 2 2α
,
❛♥❞ u(t, x) = v(x + αt, x − αt) . ❚❤❡♦r❡♠ ✻✳✶✵✳✸✿ ▼✐①❡❞ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ♦❢ ❲❛✈❡
❙✉♣♣♦s❡ u ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ s❛t✐s✜❡s t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✳ ❚❤❡♥ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ v ✐s ③❡r♦✳ Pr♦♦❢✳
▲❡t✬s ✜rst ❧✐st t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ♦❧❞ ✈❛r✐❛❜❧❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥❡✇ ♦♥❡s✿ 1 ∂x 1 ∂t 1 ∂t 1 ∂x = , = ❛♥❞ = , =− . ∂p 2 ∂q 2 ∂p 2α ∂q 2α
◆♦✇ ❧❡t✬s ❞✐✛❡r❡♥t✐❛t❡ v ✳ ❲❡ ✉s❡ t❤❡ ❈❤❛✐♥
❘✉❧❡ ✭❈❤❛♣t❡r ✹❍❉✲✸✮ s❡✈❡r❛❧ t✐♠❡s✳ ❋✐rst✿
∂v ∂ ∂u ∂x ∂u ∂t ∂u 1 ∂u 1 (p, q) = u(t, x) = + = + . ∂p ∂p ∂x ∂p ∂t ∂p ∂x 2 ∂t 2α
❙❡❝♦♥❞✿ ∂ ∂v ∂ ∂u 1 ∂u 1 ∂ 2v ❙✉❜st✐t✉t❡✳ (p, q) = (p, q) = + ∂q∂p ∂q ∂p ∂q ∂x 2 ∂t 2α 2 2 ∂ 2 u ∂t 1 ∂ u ∂x ∂ u ∂x ∂ 2 u ∂t 1 = + + + 2 2 ∂q ∂t∂x∂q 2 ∂t∂x ∂q ∂t ∂q 2α ∂x 2 2 2 ∂ u1 1 1 ∂ u 1 1 ∂ u 1 ∂ 2u = − ❇② ❈❧❛✐r❛✉t✬s t❤❡♦r❡♠✳✳✳ + + + 2 − 2 ∂x 2 ∂x∂t 2α 2 ∂t∂x 2 ∂t 2α 2α ∂ 2u 1 1 ∂ 2u 1 1 = − 2 , ✳✳✳ ❛♥❞ ❜❡❝❛✉s❡ u s❛t✐s✜❡s t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✳ ∂x2 2 2 ∂t 2α 2α = 0.
✻✳✶✵✳
❚❤❡ ✇❛✈❡ P❉❊
✹✵✸
❙♦❧✈✐♥❣ t❤❡ r❡s✉❧t✐♥❣ P❉❊✱
∂ 2v = 0, ∂p∂q ✐s ❡❛s②✳ ■♥❞❡❡❞✱ ❛♥② ❢✉♥❝t✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥ ♦♥❧② ♦♥ p s❛t✐s✜❡s ✐t✿ v(p, q) = f (p) .
❆♥❞ s♦ ❞♦❡s ❛♥② ❢✉♥❝t✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥❧② ♦♥ q ✿ v(p, q) = g(p) .
❲❡ ❤❛✈❡ t❤❡♥ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ P❉❊✿ u(t, x) = f (x + αt) ❛♥❞ u(t, x) = g(x − αt) ,
❢♦r ❛♥② ❝❤♦✐❝❡ ♦❢ f ❛♥❞ g ✳ ❲❤❛t ❛r❡ t❤❡s❡ s♦❧✉t✐♦♥s❄ ❚❤❡② ❛r❡ t❤❡ tr❛✈❡❧✐♥❣ ✇❛✈❡s t❤❛t ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ t❤❡ s❤❛♣❡ r❡♠❛✐♥s t❤❡ s❛♠❡ ❛s ✐t ✐s ♠♦✈✐♥❣ ❛❧♦♥❣ t❤❡ r♦♣❡ ❛t t❤❡ s♣❡❡❞ α✳ ❚❤❡ t✇♦ ❛❜♦✈❡ ❛r❡ ❛ ❧❡❢t tr❛✈❡❧✐♥❣ ✇❛✈❡ ❛♥❞ ❛ r✐❣❤t tr❛✈❡❧✐♥❣ ✇❛✈❡ r❡s♣❡❝t✐✈❡❧②✿
❚❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤✐s t✐♠❡ t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s✉♠ ♦❢ t✇♦ ❧❡❢t✴r✐❣❤t tr❛✈❡❧✐♥❣ ✇❛✈❡s ✐s ❛ ❧❡❢t✴r✐❣❤t tr❛✈❡❧✐♥❣ ✇❛✈❡✳ ▼♦r❡♦✈❡r✱ t❤❡ s✉♠ ♦❢ ❛ ❧❡❢t tr❛✈❡❧✐♥❣ ✇❛✈❡ ❛♥❞ ❛ r✐❣❤t tr❛✈❡❧✐♥❣ ✇❛✈❡ ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✦ ■♥❞❡❡❞✱ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ v(p, q) = f (p) + g(q) ,
❢♦r ❛♥② ♣❛✐r ♦❢ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥s f ❛♥❞ g ✱ ✐s ③❡r♦✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✻✳✶✵✳✹✿ ❙♦❧✉t✐♦♥ ♦❢ ❲❛✈❡ ❊q✉❛t✐♦♥
❋♦r ❛♥② t✇♦ t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✱ t❤❡ ❢✉♥❝t✐♦♥ u(t, x) = f (x + αt) + g(x − αt)
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✳ ❊①❡r❝✐s❡ ✻✳✶✵✳✺
❍❛✈❡ ✇❡ ❢♦✉♥❞ ❛❧❧ s♦❧✉t✐♦♥s❄ ❊①❛♠♣❧❡ ✻✳✶✵✳✻✿ st❛♥❞✐♥❣ ✇❛✈❡s
❲❤❛t ✐❢ t❤❡ t✇♦ tr❛✈❡❧✐♥❣ ✇❛✈❡s ❛r❡ ✐❞❡♥t✐❝❛❧❄ ❲❤❛t ✐❢ f ❛♥❞ g ❛r❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❄ ▲❡t✬s ❝❤♦♦s❡✿ f (p) = g(p) = sin(p) .
❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ tr❛✈❡❧✐♥❣ ✇❛✈❡s✱ t❤❡ r✐❣❤t✱ u1 (t, x) = sin(x − αt) ,
✻✳✶✵✳
❚❤❡ ✇❛✈❡ P❉❊
✹✵✹
❛♥❞ ❧❡❢t✱
u2 (t, x) = sin(x + αt) . ❲❤❛t ✐s t❤❡✐r s✉♠ u❄
❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ tr❛✈❡❧✐♥❣✦ ▲❡t✬s ❝♦♥✜r♠ t❤✐s ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❲❡ ❤❛✈❡✿
u(t, x) = u1 (t, x) + u2 (t, x) = sin(x − αt) + sin(x + αt) = 2 sin x cos(αt) , ❜② t❤❡ s✉♠✲t♦✲♣r♦❞✉❝t tr✐❣♦♥♦♠❡tr✐❝ ✐❞❡♥t✐t②✱
sin a + sin b = 2 sin
a+b 2
cos
a−b 2
.
❍❡r❡ t❤❡ s✐♥✉s♦✐❞ ♦❢ u = 2 sin x ✐s str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧② ❜② ❛ t✐♠❡✲❞❡♣❡♥❞❡♥t ♠✉❧t✐♣❧❡✱ cos(αt)✳ ❚❤✐s ❞❡s❝r✐❜❡s ❛ ✇❛✈❡ t❤❛t ♦s❝✐❧❧❛t❡s ✕ ✉♣ ❛♥❞ ❞♦✇♥ ✕ ❜✉t ❞♦❡s♥✬t ♠♦✈❡ ❤♦r✐③♦♥t❛❧❧②✳ ❆♥ ♦s❝✐❧❧❛t✐♥❣ s♣r✐♥❣ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ t❤✐s t❤❛t ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ◆❡①t✱ ✇❡ ✐♠♣♦s❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ♦♥ t❤✐s ❢✉♥❝t✐♦♥✿ ♦♥❡ ❢♦r t❤❡ ✭✈❡rt✐❝❛❧✮ ❧♦❝❛t✐♦♥ ♦❢ ❡❛❝❤ ✇❡✐❣❤t ♦♥ t❤❡ str✐♥❣ ❛♥❞ ♦♥❡ ❢♦r t❤❡ ✭✈❡rt✐❝❛❧✮ ✈❡❧♦❝✐t②✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✻✳✶✵✳✼✿ ❉✬❆❧❡♠❜❡rt✬s ❋♦r♠✉❧❛
❚❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✱
❤❛s ❛ s♦❧✉t✐♦♥✿
2 ∂ 2u ∂ u = α ∂t2 ∂x2 u(0, x) = h0 (x) ∂u (0, x) = h1 (x) ∂t
h0 (x − αt) + h0 (x + αt) 1 + u(t, x) = 2 2α
✇❤❡♥
• h0 ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❛♥❞ • h1 ✐s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✳
Z
x+αt
h1 (s) ds x−αt
✻✳✶✶✳
❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
2✿
✹✵✺
❛ ♠❡♠❜r❛♥❡
❊①❡r❝✐s❡ ✻✳✶✵✳✽
❈♦♥✜r♠ t❤❡ ❢♦r♠✉❧❛✳ ❍✐♥t✿ ❙♣❧✐t t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦✳
✻✳✶✶✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
2✿
❛ ♠❡♠❜r❛♥❡
❲❡ ♠♦❞❡❧ ❛ ♠❡♠❜r❛♥❡ ♦❢ ❛ ❞r✉♠ ❛s ✇❡❧❧ ❛s t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❧✐q✉✐❞✿
❍❡r❡✱ t❤❡ ♣✐❡❝❡s ♦❢ t❤❡ ♠❡♠❜r❛♥❡ ❛r❡ ✈❡rt✐❝❛❧❧② ❞✐s♣❧❛❝❡❞ ❛♥❞ t❤❡ ✇❛✈❡s ♣r♦♣❛❣❛t❡ ❤♦r✐③♦♥t❛❧❧②✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ r❡♣r❡s❡♥t t❤✐s ♠❡♠❜r❛♥❡ ❜② ✇❡✐❣❤ts ❝♦♥♥❡❝t❡❞ ❜② s♣r✐♥❣s ❜✉t✱ t❤✐s t✐♠❡ ✐t ✐s ♥♦t ❛ str✐♥❣ ❜✉t ❛♥ ❛rr❛② ✿
❖♥ t❤❡ ❧❡❢t✱ ✇❡ s❡❡ t❤❡ ✈✐❡✇ ❢r♦♠ ❛❜♦✈❡ ❛♥❞ ♦♥ t❤❡ r✐❣❤t ❢r♦♠ ❛s✐❞❡✳ ❲❡ ✉s❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤✐s r❡❝t❛♥❣❧❡ ❛♥❞ t❤✐s ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ u✱ ✇❤✐❝❤ ✐s ❛ 0✲❢♦r♠✳
❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❡❞❣❡s✿ ∆x ❛♥❞ ∆y ✳
✻✳✶✶✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡
✹✵✻
❘❡❝❛❧❧ t❤❛t ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❜♦① B ✐♥ t❤❡ xyt✲s♣❛❝❡ ❝♦♠❡s ❢r♦♠ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✐ts t❤r❡❡ ❡❞❣❡s ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✺ ❛♥❞ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✿ t0 q1 t1 q2 t2 q3 ... x0 s1 x1 s2 x2 s3 ... y0 p1 y1 p2 y2 p3 ...
❲❡ ♠❛❦❡ ❛ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡q✉❛❧✳ ❲❡ ✉s❡ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿ ✐❢ H ✐s t❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢♦r❝❡ F ♦❢ t❤❡ s♣r✐♥❣✱ t❤❡♥ H = −K∆x u ♦r H = −K∆y u ,
❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r t❤✐s s♣r✐♥❣ ✐s ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ x✲ ♦r t❤❡ y ✲❛①✐s✳
❚❤❡ ❢♦r❝❡s ❡①❡rt❡❞ ♦♥ t❤❡ ♦❜❥❡❝t ❛t ❧♦❝❛t✐♦♥ x ❛r❡ t❤❡ ❢♦✉r ❢♦r❝❡s ♦❢ t❤❡ ❢♦✉r s♣r✐♥❣s ❛tt❛❝❤❡❞ t♦ ✐t✳ ❊❛❝❤ t❡r♠ ✐s t❤❡ ❞✐✛❡r❡♥❝❡✿ t✇♦ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ t✇♦ ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡ ❛❧❣❡❜r❛ t❤❛t ❢♦❧❧♦✇s ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ✇❡ ✉s❡❞ ❢♦r t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✳ ❋♦r t❤❡ ✇❡✐❣❤t ❧♦❝❛t❡❞ ❛t (xi , yj )✱ t❤✐s ✐s t❤❡ t♦t❛❧ ❢♦r❝❡ ❛t t✐♠❡ tk ✿ F (tk , xi , yj ) =
• −K(tk , si , yj−1 )∆y u (tk , si , pj−1 ) • −K(tk , si−1 , yj )∆x u (tk , si−1 , yj ) +K(tk , si , yj )∆x u (tk , si , yj ) • +K(tk , si , yj )∆y u (tk , si , pj ) •
❚❤❡ ❢♦✉r t❡r♠s ❛r❡ t❤❡ ❢♦r❝❡s ♦❢ t❤❡ ❢♦✉r s♣r✐♥❣s ❛♥❞ t❤❡② ❛r❡ ❛rr❛♥❣❡❞ ❛❝❝♦r❞✐♥❣❧②✳
❚❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ✭✈❡rt✐❝❛❧✮ ❞✐s♣❧❛❝❡♠❡♥ts ♦❢ t❤❡ ✇❡✐❣❤ts u = u(t, x, y) ❝♦♠❡s ❢r♦♠ t❤❡ ❙❡❝♦♥❞ ◆❡✇t♦♥✬s ▲❛✇ ✭♠❛ss t✐♠❡s ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❢♦r❝❡✮✿ m(xi , yj )
∆2 u (tk , xi , yj ) = F (tk , xi , yj ) . ∆t
❙✐♥❝❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♥♦✇ ❦♥♦✇♥✿ a(tk , xi , yj ) =
1 F (tk , xi , yj ) , m(xi , yj )
✇❡ ❤❛✈❡ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❞❡r✐✈❡❞ ❥✉st ❛s ❜❡❢♦r❡✿ v(qk , xi , yj ) = v(qk−1 , xi , yj ) +a(tk , xi , yj )∆tk , u(tk , xi , yj ) = u(tk−1 , xi , yj ) +v(qk , xi , yj )∆tk .
❚❤❡ s♣r❡❛❞s❤❡❡t ❝♦♥s✐sts ♦❢ s❡✈❡r❛❧ s❤❡❡ts ❝♦♠♣✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿ • t❤❡ st✐✛♥❡ss ❢♦r ❡❛❝❤ s♣r✐♥❣✱ • t❤❡ ♠❛ss ♦❢ ❡❛❝❤ ✇❡✐❣❤t✱
• t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ❢♦r ❡❛❝❤ ✇❡✐❣❤t✱
• t❤❡ ♥❡①t ❧♦❝❛t✐♦♥ ❢♦r ❡❛❝❤ ✇❡✐❣❤t ✭✐✳❡✳✱ t❤❡ ✈❡❧♦❝✐t②✮✱ • t❤❡ ✜rst ❜✉✛❡r ✭❝♦♣✐❡❞ ❢r♦♠ t❤❡ s❡❝♦♥❞ ❜✉✛❡r✮✱
• t❤❡ s❡❝♦♥❞ ❜✉✛❡r ✭❝♦♣✐❡❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t ✈❛❧✉❡s✮✱
• t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ ❡♥❞s ❛♥❞ t❤❡ ❍♦♦❦✬s ❢♦r❝❡ ❢♦r ❡❛❝❤ s♣r✐♥❣✱ • t❤❡ t♦t❛❧ ❢♦r❝❡ ❢♦r ❡❛❝❤ ✇❡✐❣❤t✱
• t❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t②✱ ❛♥❞ ✜♥❛❧❧②
• t❤❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❢♦r ❡❛❝❤ ✇❡✐❣❤t✳
❚✇♦ ❡①❛♠♣❧❡s ❛r❡ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ❞♦♠❛✐♥ ✐s ❝❤♦s❡♥ t♦ ❜❡ t❤❡ sq✉❛r❡ [1, 40] × [1, 40] ✇✐t❤ ∆x = ∆y = 1✳
✻✳✶✶✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
2✿
✹✵✼
❛ ♠❡♠❜r❛♥❡
❊①❛♠♣❧❡ ✻✳✶✶✳✶✿ ❞r✉♠
❆ ❞r✉♠ ✐s ❛ ❝✐r❝✉❧❛r ♠❡♠❜r❛♥❡ t❤❡ ❡❞❣❡ ♦❢ ✇❤✐❝❤ ✐s ❛tt❛❝❤❡❞ t♦ ❛ r✐♥❣✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s
u(t, x, y) = 0
✇❤❡♥
x2 + y 2 > 202 .
❲❡ ❝❤♦♦s❡ t❤❡ ✐♥✐t✐❛❧ s❤❛♣❡ ♦❢ t❤❡ ❞r✉♠ t♦ ❜❡ ❛ r♦t❛t❡❞ s✐♥✉s♦✐❞✿
u(0, x, y) = cos ❛♥❞ ③❡r♦ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②
p
(x −
20)2
+ (y −
20)2
π , 40
u(1, x, y) = u(0, x, y) . ❚❤❡ s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❛r❡ ❜❡❧♦✇✿
❚❤❡ ❞r✉♠ s❦✐♥ s❤r✐♥❦s t♦ ✢❛t ❛♥❞ t❤❡♥ ❝r❡❛t❡s t❤❡ s❛♠❡ s❤❛♣❡ ♦♥ t❤❡ ♦t❤❡r s✐❞❡✳
❊①❛♠♣❧❡ ✻✳✶✶✳✷✿ ❜r❡❛❦✇❛t❡r
❙❡❡♠✐♥❣❧② ❤❛✈✐♥❣ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ✐t✱ ❝❛♥ ♦✉r ♠♦❞❡❧ r❡❛s♦♥❛❜❧② r❡♣r♦❞✉❝❡ ✇❛✈❡s ♦❢ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❧✐q✉✐❞❄
❇❡❧♦✇ ✇❡ s❤♦✇ ❛ s✐♠✉❧❛t✐♦♥ ♦❢ ❛ ❜r❡❛❦✇❛t❡r ♣r♦t❡❝t✐♥❣ ❛ ❤❛r❜♦r✳
✏♣✉❧s❡✑ ✭ts✉♥❛♠✐❄✮ ♦✉ts✐❞❡ t❤❡ ❤❛r❜♦r✱
u(0, 20, 3) = 1, u(0, x, y) = 0
❢♦r t❤❡ r❡st ♦❢
❛♥❞
u(1, x, y) = 0
❢♦r ❛❧❧
(x, y) .
❚❤❡ ❜r❡❛❦✇❛t❡r ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ r♦✇ ♦❢ ✜①❡❞ ✈❛❧✉❡s✿
u(t, x, 10) = 0
❢♦r ❛❧❧
✇✐t❤ ❛ s✐♥❣❧❡ ❣❛♣✳ ❚❤❡ s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❛r❡ ❜❡❧♦✇✿
x 6= 20 ,
(x, y) ,
■t st❛rts ✇✐t❤ ❛ s✐♥❣❧❡
✻✳✶✶✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡
✹✵✽
❚❤❡ ❧❛r❣❡ ✇❛✈❡s ♦✉ts✐❞❡ ♣r♦❞✉❝❡ ✈❡r② ♠✐❧❞✱ ❝✐r❝✉❧❛r ✇❛✈❡s ✐♥s✐❞❡✳ ❲❡ r❡♣❡❛t t❤❡ ❛❧❣❡❜r❛ ✇❡ ✉s❡❞ ✐♥ ♦✉r ❛♥❛❧②s✐s ♦❢ ❤❡❛t tr❛♥s❢❡r ❢♦r ❞✐♠❡♥s✐♦♥ 2✳ ❲❤❡♥ K ✐s ❝♦♥st❛♥t✱ ❛s ❛♥♦t❤❡r s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥✱ t❤❡ t♦t❛❧ ❢♦r❝❡ ✐s s✐♠♣❧✐✜❡❞✿ F (xi , yj , tk ) = • −∆y u (tk , si , pj−1 ) • +∆x u (tk , si , yj ) = K ∆2x u (tk , xi , yj ) + ∆2y u (tk , xi , yj ) . = K −∆x u (tk , si−1 , yj ) • +∆y u (tk , si , pj ) •
❚❤❡s❡ ❛r❡ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡s✳ ❖✉r ♣❛rt✐❛❧ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿
✇❤❡r❡ α > 0 ✐s ❛ ❝♦♥st❛♥t✳
∆2 u K 2 2 ∆ u + ∆ u , = x y ∆t2 m
❊①❡r❝✐s❡ ✻✳✶✶✳✸
❉❡r✐✈❡ ❛ ✈❡rs✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡ K ✳ ❏✉st ❛s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ ❝❛♥ ❛r❣✉❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♣♦✐♥ts✳ ❋✐rst✱ ❛ ❧♦♥❣❡r s♣r✐♥❣ ✐s ❧❡ss st✐✛ t❤❛♥ ❛ s❤♦rt❡r ♦♥❡ ♠❛❞❡ ♦❢ t❤❡ s❛♠❡ ♠❛t❡r✐❛❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ K ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ ∆x = ∆y ✳ ❙❡❝♦♥❞✱ t❤❡ ✇❡✐❣❤ts ❛r❡✱ ✐♥ ❢❛❝t✱ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ s♣r✐♥❣s ❛♥❞✱ t❤❡r❡❢♦r❡✱ m ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ∆x✳ ❚❤❡♥ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿ ∆2 u α 2 2 ∆ u + ∆ u =α = x y ∆t2 ∆x2
∆2 u ∆2 u + ∆x2 ∆y 2
.
❋✉rt❤❡r♠♦r❡✱ ✇❤❡♥ u ✐s ❞❡✜♥❡❞ t❤r♦✉❣❤♦✉t t❤❡ r❡❣✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ❢♦r ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✿ 2 2 2 ∂ u =α ∂t2
∂ u ∂ u + ∂x2 ∂y 2
✇✐t❤ t❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s✱ ❛❣❛✐♥✱ t❤❡ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r ♦❢ u✳
,
✻✳✶✶✳ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥
❊①❡r❝✐s❡ ✻✳✶✶✳✹ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ t❤❡ ❧❛st✳
2✿
❛ ♠❡♠❜r❛♥❡
✹✵✾
❊①❡r❝✐s❡s
❈♦♥t❡♥ts
✶ ❊①❡r❝✐s❡s✿ ❇❛s✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊①❡r❝✐s❡s✿ ❆♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✸ ❊①❡r❝✐s❡s✿ ❊✉❧❡r✬s ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①❡r❝✐s❡s✿ ●❡♥❡r❛❧✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ❛♥❞ s❡tt✐♥❣ ✉♣ ❖❉❊s ✻ ❊①❡r❝✐s❡s✿ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✼ ❊①❡r❝✐s❡s✿ ❙②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❊①❡r❝✐s❡s✿ ❙❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❊①❡r❝✐s❡s✿ ❆❞✈❛♥❝❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❊①❡r❝✐s❡s✿ P❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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✹✶✵ ✹✶✷ ✹✶✸ ✹✶✹ ✹✶✺ ✹✶✻ ✹✶✼ ✹✶✽ ✹✶✾ ✹✷✵ ✹✷✶
✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝s
❊①❡r❝✐s❡ ✶✳✶
❊①❡r❝✐s❡ ✶✳✸
❋✐♥❞ ❛❧❧ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ x−2 ❞❡✜♥❡❞ ♦♥ (−∞, 0)∪ (0, +∞)✳
❱❡r✐❢② t❤❛t t❤❡ ❢✉♥❝t✐♦♥ y = cx2 ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ xy ′ = 2y .
❆r❡ t❤❡r❡ ❛♥② ♦t❤❡rs❄
❊①❡r❝✐s❡ ✶✳✷
❋✐♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐①✿ F =
2 1 1 2
.
❊①❡r❝✐s❡ ✶✳✹
❋✐♥❞ ❛❧❧ ❝✉r✈❡s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s✿ xy 2 = C .
✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝s
✹✶✶
❊①❡r❝✐s❡ ✶✳✺
❋✐♥❞ ❛❧❧ ❝✉r✈❡s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s✿ x2 y = C .
❊①❡r❝✐s❡ ✶✳✻
❋✐♥❞ ❛❧❧ ❝✉r✈❡s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s✿ xy 2 = C .
❊①❡r❝✐s❡ ✶✳✼
❱❡r✐❢② t❤❛t −2x2 y + y 2 = 1 ✐s ❛♥ ✐♠♣❧✐❝✐t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ 2xy + (x2 − y)y ′ = 0✳ ❋✐♥❞ ♦♥❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥✳ ❊①❡r❝✐s❡ ✶✳✽
■♥ t❤❡ sq✉❛r❡ [−3, 3] × [−3, 3]✱ ♣❧♦t t❤❡ ❞✐r❡❝✲ t✐♦♥ ✜❡❧❞ ✭s❧♦♣❡ ✜❡❧❞✮ ❢♦r t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ dy y = −x✳ ❙❦❡t❝❤ t❤❡ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ❛♥❞ 3 dx ♦t❤❡r s♦❧✉t✐♦♥ ❝✉r✈❡s✳ ❊①❡r❝✐s❡ ✶✳✾
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ y ′ + 2xy 2 = 0✳ ❊①❡r❝✐s❡ ✶✳✶✵
❙♦❧✈❡ t❤❡ ■❱P✿ (4y + 2x − 5)dx + (6y + 4x − 1)dy, y(−1) = 2 . ❊①❡r❝✐s❡ ✶✳✶✶
❨♦✉r ❧♦❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❋✐♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 y 2 5 10 5 0 0 3 7
✷✳ ❊①❡r❝✐s❡s✿ ❆♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s
✹✶✷
✷✳ ❊①❡r❝✐s❡s✿ ❆♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s
❊①❡r❝✐s❡ ✷✳✶
❊①❡r❝✐s❡ ✷✳✶✵
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❜② ✉s✐♥❣ ❛♥ ❛♣♣r♦✲ ♣r✐❛t❡ s✉❜st✐t✉t✐♦♥✿ y ′ = 1 + ey−x+5 ✳
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ xyy ′ = x2 + 3y 2 .
❊①❡r❝✐s❡ ✷✳✷
❋♦r t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y ′ = 1+e3y+5 ✱ ❡①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥ u = 3y + 5✳
❊①❡r❝✐s❡ ✷✳✶✶
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ (2x2 y + 3x2 ) dx + (2x2 y + 4y 3 ) dy .
❊①❡r❝✐s❡ ✷✳✸
❙♦❧✈❡ ❜② s❡♣❛r❛t✐♥❣ ✈❛r✐❛❜❧❡s✿ y ′ = xy .
❊①❡r❝✐s❡ ✷✳✶✷
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ y ′ = −6xy .
❊①❡r❝✐s❡ ✷✳✹
❙♦❧✈❡ ❜② t❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦r✿ y ′ = y/x . ❊①❡r❝✐s❡ ✷✳✺
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ y ′ + 2xy 2 = 0 . ❊①❡r❝✐s❡ ✷✳✻
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ x
dy − y = x2 sin x . dx
❊①❡r❝✐s❡ ✷✳✼
❙♦❧✈❡ t❤❡ ■❱P✿ (4y + 2x − 5) dx + (6y + 4x − 1) dy = 0, y(−1) = 2 . ❊①❡r❝✐s❡ ✷✳✽
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❜② ✉s✐♥❣ ❛♥ ❛♣♣r♦✲ ♣r✐❛t❡ s✉❜st✐t✉t✐♦♥✿ dy = 1 + ey−x+5 . dx ❊①❡r❝✐s❡ ✷✳✾
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❜② ✉s✐♥❣ ❛♥ ❛♣♣r♦✲ ♣r✐❛t❡ s✉❜st✐t✉t✐♦♥✿ p dy = 2 + y − 2x + 3 . dx
✸✳ ❊①❡r❝✐s❡s✿ ❊✉❧❡r✬s ♠❡t❤♦❞
✸✳ ❊①❡r❝✐s❡s✿ ❊✉❧❡r✬s ♠❡t❤♦❞
❊①❡r❝✐s❡ ✸✳✶
❈❛rr② ♦✉t n = 4 st❡♣s ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐t❤ h = .5 ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿
y ′ = y − x, y(0) = 2 . ❊①❡r❝✐s❡ ✸✳✷
❙✉♣♣♦s❡ y ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜✲ ❧❡♠ y ′ = x2 + y 2 , y(0) = 0 . ❋✐♥❞ y(1) ❜② ♠❡❛♥s ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐t❤ st❡♣ h = .2✳ ❊①❡r❝✐s❡ ✸✳✸
❯s❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐t❤ n = 4 st❡♣s t♦ ❡st✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♣r♦❜❧❡♠✿
y ′ = 2x − y, y(0) = 1, ♦♥ t❤❡ ✐♥t❡r✈❛❧ [0, 1]✳ ❊①❡r❝✐s❡ ✸✳✹
❯s❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐t❤ 4 st❡♣s t♦ ❡st✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♣r♦❜❧❡♠✿
y ′ = 2x − y, y(0) = 1, ♦♥ t❤❡ ✐♥t❡r✈❛❧ [0, 1]✳ ❊①❡r❝✐s❡ ✸✳✺
▲✐♥❡❛r✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❖❉❊ ❛t x = 2✿ √ y′ = x − 1 − 1 . ❉♦ ♥♦t s♦❧✈❡✳ ❊①❡r❝✐s❡ ✸✳✻
❙❡t ✉♣ ✭❞♦♥✬t s♦❧✈❡✮ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♣r♦❜❧❡♠s ✕ ❢♦r x ❛♥❞ y ✕ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿ ❆♥ ♦❜❥❡❝t ✐s t❤r♦✇♥ ✉♣ ❢r♦♠ ❛ ❜✉✐❧❞✐♥❣ ♦❢ ❤❡✐❣❤t h ❛t 45 ❞❡❣r❡❡s ✇✐t❤ s♣❡❡❞ s✳ ❊①❡r❝✐s❡ ✸✳✼
❙♦❧✈❡ ❜② s❡♣❛r❛t✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s✿
y ′ = xy .
✹✶✸
✹✳ ❊①❡r❝✐s❡s✿ ●❡♥❡r❛❧✐t✐❡s
✹✶✹
✹✳ ❊①❡r❝✐s❡s✿ ●❡♥❡r❛❧✐t✐❡s
❊①❡r❝✐s❡ ✹✳✶
Pr♦✈✐❞❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛♥❞ s❦❡t❝❤ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ s♦❧✉t✐♦♥ s❡t ✇✐t❤♦✉t ✐t✳ ❊①❡r❝✐s❡ ✹✳✷
■♥❞✐❝❛t❡ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ tr✉❡ ♦r ❢❛❧s❡✳ ✶✳ y ′ = y 2 ✐s ❛ s❡❝♦♥❞ ♦r❞❡r ❖❉❊✳ ✷✳ y(t) = |t| ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ■❱P y ′ = 1, y(0) = 0✳ ✸✳ y(t) = −t ✐s ❛ s♦❧✉t✐♦♥ ♦❢ ❛ ❉❊ ✇✐t❤ t❤❡ s❧♦♣❡ ✜❡❧❞ ❜❡❧♦✇✳ 1
✹✳ ❚❤❡ ■❱P y ′ = 2 , y(0) = 1 ❤❛s ♠♦r❡ t +1 t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥✳ ✺✳ ❚❤❡ ❖❉❊ y ′ = yt2 sin t ❝❛♥ ❜❡ s♦❧✈❡❞ ❜② s❡♣✲ ❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s✳ ❊①❡r❝✐s❡ ✹✳✸
❱❡r✐❢② t❤❛t −2x2 y + y 2 = 1 ✐s ❛♥ ✐♠♣❧✐❝✐t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ 2xy + (x2 − y)
dy = 0. dx
❋✐♥❞ ♦♥❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥✳ ❊①❡r❝✐s❡ ✹✳✹
✭❛✮ ❙t❛t❡ t❤❡ ❊①✐st❡♥❝❡✲❯♥✐q✉❡♥❡ss ❚❤❡♦r❡♠ ❢♦r ❛ s②st❡♠ ♦❢ t✇♦ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ✭❜✮ ❋♦r ✇❤❛t ✈❛❧✉❡s t0 , a, b ❞♦❡s t❤❡ t❤❡♦r❡♠ ❣✉❛r❛♥t❡❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ ■❱P x′ = y/t, y ′ = x, x(t0 ) = a, y(t0 ) = b . ❊①❡r❝✐s❡ ✹✳✺
❋✐♥❞ t❤❡ ✈❛❧✉❡s x0 ❛♥❞ a ❢♦r ✇❤✐❝❤ t❤❡ ❊①✐st❡♥❝❡✲ ❯♥✐q✉❡♥❡ss ❚❤❡♦r❡♠ ❣✉❛r❛♥t❡❡s ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ❢♦r t❤❡ ■❱P✿ y ′ = y|x|, y(x0 ) = a .
✺✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ❛♥❞ s❡tt✐♥❣ ✉♣ ❖❉❊s
✹✶✺
✺✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ❛♥❞ s❡tt✐♥❣ ✉♣ ❖❉❊s
❊①❡r❝✐s❡ ✺✳✶
❲❤❛t ✐s t❤❡ ❖❉❊ ♦❢ ❛♥ ♦❜❥❡❝t ♠♦✈✐♥❣ ❤♦r✐③♦♥t❛❧❧② t❤r♦✉❣❤ ❛ ♠❡❞✐✉♠ ✇❤♦s❡ r❡s✐st❛♥❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♦❜❥❡❝t✬s ✈❡❧♦❝✐t②❄ ❉❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♦❜❥❡❝t✳
♦❜❥❡❝t✬s ✈❡❧♦❝✐t②✿ v ′ = −ky, k > 0 .
❉❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♦❜❥❡❝t✳ ❲❤❛t ✐❢ k ❣r♦✇s ✇✐t❤ t✐♠❡❄
❊①❡r❝✐s❡ ✺✳✷
❊①❡r❝✐s❡ ✺✳✽
✭❛✮ ■♥ t❤❡ t❤❡♦r② ♦❢ ❧❡❛r♥✐♥❣✱ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ❛ s✉❜❥❡❝t ✐s ♠❡♠♦r✐③❡❞ ✐s ❛ss✉♠❡❞ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❛♠♦✉♥t t❤❛t ✐s ❧❡❢t t♦ ❜❡ ♠❡♠♦r✐③❡❞✳ ❙✉♣♣♦s❡ M ❞❡♥♦t❡s t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ s✉❜❥❡❝t t♦ ❜❡ ♠❡♠✲ ♦r✐③❡❞ ❛♥❞ A(t) ✐s t❤❡ ❛♠♦✉♥t ♠❡♠♦r✐③❡❞ ❛t t✐♠❡ t✳ ❙❡t ✉♣ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r A(t)✳ ✭❜✮ ❙✉♣♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ♠❛t❡r✐❛❧ ✐s ❢♦r✲ ❣♦tt❡♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ A(t)✳ ❙❡t ✉♣ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r A(t)✳
❉❡r✐✈❡ t❤❡ ❡q✉❛t✐♦♥s ❞❡s❝r✐❜✐♥❣ t❤❡ ♣r❡❞❛t♦r✲♣r❡② s②st❡♠ ❛♥❞ ♣❧♦t t❤❡ ♣❤❛s❡ ♣♦rtr❛✐t✳ ❊①❡r❝✐s❡ ✺✳✾
❙❡t ✉♣ ❛♥❞ s♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ❞❡✲ s❝r✐❜❡s t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦❢ ♠❛ss M s✉s✲ ♣❡♥❞❡❞ ✈❡rt✐❝❛❧❧② ❜② ❛ s♣r✐♥❣ ✇✐t❤ ❍♦♦❦❡ ❝♦♥st❛♥t k✳ ❊①❡r❝✐s❡ ✺✳✶✵
❊①❡r❝✐s❡ ✺✳✸
❙✉♣♣♦s❡ ♣♦✐♥t T ❣♦❡s ❛❧♦♥❣ t❤❡ ❧✐♥❡ x = 1 ✇❤✐❧❡ ❞r❛❣❣✐♥❣ ♣♦✐♥t P ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❜② ❛ str✐♥❣ P T ♦❢ ❧❡♥❣t❤ 1✳ ❙✉♣♣♦s❡ T st❛rts ❛t (1, 0) ❛♥❞ P ❛t (2, 0)✳ ❋✐♥❞ t❤❡ ♣❛t❤ ♦❢ P ✳
❙❡t ✉♣ ✭❞♦♥✬t s♦❧✈❡✮ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♣r♦❜❧❡♠s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s✐t✉❛t✐♦♥s✿ ✭❛✮ ❆♥ ♦❜❥❡❝t ✐s t❤r♦✇♥ ✉♣ ❢r♦♠ ❛ ❜✉✐❧❞✐♥❣ ♦❢ ❤❡✐❣❤t h ❛t 45 ❞❡❣r❡❡s ✇✐t❤ s♣❡❡❞ s❀ ✭❜✮ ❆♥ ♦❜❥❡❝t t❤r♦✇♥ tr❛✈❡❧s ❢♦r 2 s❡❝♦♥❞s ❛♥❞ t❤❡♥ ❤✐ts t❤❡ ❣r♦✉♥❞ ❛t 4✺ ❞❡❣r❡❡s ❛♥❞ s♣❡❡❞ s✳
❊①❡r❝✐s❡ ✺✳✹
❙❡t ✉♣ ✭❞♦♥✬t s♦❧✈❡✮ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♣r♦❜❧❡♠s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s✐t✉❛t✐♦♥s✿ ✭❛✮ ❆♥ ♦❜❥❡❝t ✐s t❤r♦✇♥ ✉♣ ❢r♦♠ ❛ ❜✉✐❧❞✐♥❣ ♦❢ ❤❡✐❣❤t h ❛t 45 ❞❡❣r❡❡s ✇✐t❤ s♣❡❡❞ s❀ ✭❜✮ ❆♥ ♦❜❥❡❝t t❤r♦✇♥ tr❛✈❡❧s ❢♦r ✷ s❡❝♦♥❞s ❛♥❞ t❤❡♥ ❤✐ts t❤❡ ❣r♦✉♥❞ ❛t 45 ❞❡❣r❡❡s ❛♥❞ s♣❡❡❞ s✳
❊①❡r❝✐s❡ ✺✳✶✶
❙❡t ✉♣ ❛♥❞ s♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ❞❡✲ s❝r✐❜❡s t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦❢ ♠❛ss M ♣❧❛❝❡❞ ♦♥ t♦♣ ♦❢ ❛ s♣r✐♥❣ ✇✐t❤ ❍♦♦❦❡ ❝♦♥st❛♥t k st❛♥❞✐♥❣ ✈❡rt✐❝❛❧❧② ♦♥ t❤❡ ❣r♦✉♥❞✳ ❊①❡r❝✐s❡ ✺✳✶✷
❊①❡r❝✐s❡ ✺✳✺
❙❡t ✉♣ ❛♥❞ s♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ❞❡✲ s❝r✐❜❡s t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦❢ ♠❛ss M ♣❧❛❝❡❞ ♦♥ t♦♣ ♦❢ ❛ s♣r✐♥❣ ✇✐t❤ ❍♦♦❦❡ ❝♦♥st❛♥t k st❛♥❞✐♥❣ ✈❡rt✐❝❛❧❧② ♦♥ t❤❡ ❣r♦✉♥❞✳ ❊①❡r❝✐s❡ ✺✳✻
✭❛✮ ❉❡s❝r✐❜❡ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧✳ ✭❜✮ ❙❡t ✉♣ ❛ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ♠♦❞❡❧ ❛♥❞ ✜♥❞ ✐ts ❡q✉✐❧✐❜r✐❛✳ ❊①❡r❝✐s❡ ✺✳✼
❙✉♣♣♦s❡ ❛♥ ♦❜❥❡❝t ✐s ♠♦✈✐♥❣ ❤♦r✐③♦♥t❛❧❧② t❤r♦✉❣❤ ❛ ♠❡❞✐✉♠ ✇❤♦s❡ r❡s✐st❛♥❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡
✭❛✮ ❉❡s❝r✐❜❡ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧✳ ✭❜✮ ❙❡t ✉♣ ❛ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ♠♦❞❡❧ ❛♥❞ ✜♥❞ ✐ts ❡q✉✐❧✐❜r✐❛✳ ❊①❡r❝✐s❡ ✺✳✶✸
✭❛✮ ■♥ t❤❡ t❤❡♦r② ♦❢ ❧❡❛r♥✐♥❣✱ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ❛ s✉❜❥❡❝t ✐s ♠❡♠♦r✐③❡❞ ✐s ❛ss✉♠❡❞ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❛♠♦✉♥t t❤❛t ✐s ❧❡❢t t♦ ❜❡ ♠❡♠♦r✐③❡❞✳ ❙✉♣♣♦s❡ M ❞❡♥♦t❡s t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ s✉❜❥❡❝t t♦ ❜❡ ♠❡♠✲ ♦r✐③❡❞ ❛♥❞ A(t) ✐s t❤❡ ❛♠♦✉♥t ♠❡♠♦r✐③❡❞ ❛t t✐♠❡ t✳ ❙❡t ✉♣ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r A(t)✳ ✭❜✮ ❙✉♣♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ♠❛t❡r✐❛❧ ✐s ❢♦r✲ ❣♦tt❡♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ A(t)✳ ❙❡t ✉♣ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r A(t)✳
✻✳ ❊①❡r❝✐s❡s✿ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s
✹✶✻
✻✳ ❊①❡r❝✐s❡s✿ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s
❊①❡r❝✐s❡ ✻✳✶
❆ s❦❡t❝❤ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❛♥ ❖❉❊ y = f (t, y) ✐s s❤♦✇♥ ❜❡❧♦✇✳ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ s✐❣♥ ♦❢ f ❄ ′
❊①❡r❝✐s❡ ✻✳✷
❙✉♣♣♦s❡ y = 0 ✐s ❛ st❛❜❧❡ ✭✉♥st❛❜❧❡✮ ❡q✉✐❧✐❜r✐✉♠ ♦❢ y ′ = f (x, y)✳ ❙✉♣♣♦s❡ g(x, y) = f (x, y) ✐❢ |y| < 1/x✳ ❲❤❛t ❝❛♥ ②♦✉ t❡❧❧ ❛❜♦✉t y ′ = g(x, y)❄ ❊①❡r❝✐s❡ ✻✳✸
■♥ t❤❡ sq✉❛r❡ [−3, 3] × [−3, 3]✱ ♣❧♦t t❤❡ ❞✐r❡❝t✐♦♥ ✜❡❧❞ ❢♦r t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y
dy = −x . dx
❙❦❡t❝❤ t❤❡ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ❛♥❞ t❤r❡❡ ♦t❤❡r s♦✲ ❧✉t✐♦♥ ❝✉r✈❡s✳ ❊①❡r❝✐s❡ ✻✳✹
❙❦❡t❝❤ t❤❡ ❞✐r❡❝t✐♦♥ ✜❡❧❞ ♦❢ t❤❡ s②st❡♠✿ x′ = 2x − y, y ′ = x − 3y
❛♥❞ s❡✈❡r❛❧ ♦❢ ✐ts tr❛❥❡❝t♦r✐❡s✱ ✐❞❡♥t✐❢② t❤❡ t②♣❡ ♦❢ ❡q✉✐❧✐❜r✐✉♠✳ ❊①❡r❝✐s❡ ✻✳✺
❆❝❝♦r❞✐♥❣ t♦ ◆❡✇t♦♥✬s ❧❛✇ ♦❢ ❝♦♦❧✐♥❣✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ T ♦❢ ❛ ❜♦❞② ✐♠♠❡rs❡❞ ✐♥ ❛ ♠❡❞✐✉♠ ♦❢ ❝♦♥st❛♥t t❡♠♣❡r❛t✉r❡ A ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❞✐❢✲ ❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ x✿ dT = k(A − T ), k > 0 . dx
❙♦❧✈❡ ✐t✳ ■♥t❡r♣r❡t ②♦✉r s♦❧✉t✐♦♥ t♦ ❡①♣❧❛✐♥ ✇❤② t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❜♦❞② ✇✐❧❧ ❜❡❝♦♠❡ ❡q✉❛❧ t♦ A✱
❡✈❡♥t✉❛❧❧②✳
✼✳ ❊①❡r❝✐s❡s✿ ❙②st❡♠s
✹✶✼
✼✳ ❊①❡r❝✐s❡s✿ ❙②st❡♠s
❊①❡r❝✐s❡ ✼✳✶
❊①❡r❝✐s❡ ✼✳✽
❙❦❡t❝❤ t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❖❉❊s X ′ = F X ✐❢ t❤❡ ♠❛tr✐① F ❤❛s t❤❡s❡ ♣❛✐rs ♦❢ ❡✐❣❡♥✲ ✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs✿
❙✉♣♣♦s❡ λ1 , λ2 ❛r❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❙❦❡t❝❤ t❤❡ ♣❤❛s❡ ♣♦r✲ tr❛✐t ♦❢ t❤❡ s②st❡♠ ✐❢ ✭❛✮ λ1 , λ2 ❛r❡ ❝♦♠♣❧❡① ✇✐t❤ Reλ1 < 0❀ ✭❜✮ λ1 = λ2 ✇✐t❤ ❛ s✐♥❣❧❡ ❧✐♥❡❛r❧② ✐♥❞❡✲ ♣❡♥❞❡♥t ❡✐❣❡♥✈❡❝t♦r❀ ✭❝✮ λ1 , λ2 ❛r❡ r❡❛❧ ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s✳
λ1 = 2, V1 =
1 0
❛♥❞ λ2 = −1, V2 =
1 1
.
❊①❡r❝✐s❡ ✼✳✷
❲r✐t❡ t❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ♦❢ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❖❉❊s X ′ = F X ✐❢ t❤❡ ♠❛tr✐① F ❤❛s t❤❡s❡ ♣❛✐rs ♦❢ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs✿ λ1 = i, V1 =
1 i
❛♥❞ λ2 = −i, V2 =
1 −i
,
❛♥❞ ♣r♦✈✐❞❡ ♦♥❡ ♥♦♥✲tr✐✈✐❛❧ r❡❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②s✲ t❡♠✳ ❊①❡r❝✐s❡ ✼✳✸
❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ x′ = 2x − y, y ′ = x . ❊①❡r❝✐s❡ ✼✳✹
❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ x′ = 4x − 5y, y ′ = 5x − 4y . ❊①❡r❝✐s❡ ✼✳✺
❙✉♣♣♦s❡ λ1 , λ2 ❛r❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❙❦❡t❝❤ t❤❡ ♣❤❛s❡ ♣♦rtr❛✐t ♦❢ t❤❡ s②st❡♠ ✐❢ ✭❛✮ λ1 , λ2 ❛r❡ ❝♦♠✲ ♣❧❡① ✇✐t❤ Re λ1 < 0❀ ✭❜✮ λ1 = λ2 ✇✐t❤ ❛ s✐♥❣❧❡ ❧✐♥✲ ❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ❡✐❣❡♥✈❡❝t♦r❀ ✭❝✮ λ1 , λ2 ❛r❡ r❡❛❧ ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s✳ ❊①❡r❝✐s❡ ✼✳✻
❙♦❧✈❡ t❤❡ s②st❡♠ x′ = −3x − 4y, y ′ = 2x + y . ❊①❡r❝✐s❡ ✼✳✼
❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ x′ = 2x − y, y ′ = x✳
✽✳ ❊①❡r❝✐s❡s✿ ❙❡❝♦♥❞ ♦r❞❡r
✹✶✽
✽✳ ❊①❡r❝✐s❡s✿ ❙❡❝♦♥❞ ♦r❞❡r
❊①❡r❝✐s❡ ✽✳✶
❊①❡r❝✐s❡ ✽✳✽
❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ y = Cert t♦ s♦❧✈❡ t❤✐s ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ y ′′ − 3y ′ + 2y = 0✳
❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠
❊①❡r❝✐s❡ ✽✳✷
❋✐♥❞ t❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ✭✶✮ y ′′ − 6y ′ + 9y = 0❀ ✭✷✮ 2y ′′ −5y ′ −3y = 0❀ ✭✸✮ y ′′ +4y ′ +7y = 0✳
(x − 1)y ′′ − xy + y = 0, y(0) = −2, y ′ (0) = 6,
✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✳ ✭❖♥❡ ❡①tr❛ ♣♦✐♥t ❢♦r r❡♣r❡s❡♥t✐♥❣ t❤❡ s♦❧✉t✐♦♥ ✐♥ ❛ ♠♦r❡ ❝♦♠♣❛❝t ❢♦r♠✳✮ ❊①❡r❝✐s❡ ✽✳✾
❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿
❊①❡r❝✐s❡ ✽✳✸
❋✐♥❞ t❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts✿ ✶✳ y − 6y + 9y = 0✱ ′′
y ′′ + 2y ′ + 4y = 0 . ❊①❡r❝✐s❡ ✽✳✶✵
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥
✷✳ 2y ′′ − 5y ′ − 3y = 0✱
y ′′ + y ′ x = 0 .
✸✳ y ′′ + 4y ′ + 7y = 0✳ ❊①❡r❝✐s❡ ✽✳✶✶
❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿
❊①❡r❝✐s❡ ✽✳✹
❚❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ xy ′′ + y ′ = 0 ❤❛s ❛ ❦♥♦✇♥ s♦❧✉t✐♦♥ y1 = ln x✳ ❋✐♥❞ ❛♥♦t❤❡r s♦❧✉t✐♦♥ y2 ❧✐♥✲ ❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ ✜rst✳
y ′′ + 2y = 0, y(0) = 0, y ′ (0) = 1 . ❊①❡r❝✐s❡ ✽✳✶✷
❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿
❊①❡r❝✐s❡ ✽✳✺
❚❤❡ ❢✉♥❝t✐♦♥ y1 = ex ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤♦♠♦❣❡✲ ♥❡♦✉s ❡q✉❛t✐♦♥ y ′ ‘ − 3y ′ + 2y = 0 .
❙♦❧✈❡ t❤❡ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ y ′′ − 3y + 2y = 5e3x . ❊①❡r❝✐s❡ ✽✳✻
●✐✈❡♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y ′′ − 2y + 2y = 0✱ s♦❧✈❡ ✭❛✮ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ y(π) = 1, y ′ (π) = 1,
❛♥❞ ✭❜✮ t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠ y(0) = 1, y(π) = 1 . ❊①❡r❝✐s❡ ✽✳✼
❙♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ x2 y ′′ + (y ′ )2 = 0 .
y ′′ + 2y ′ + 4y = 0 .
✾✳ ❊①❡r❝✐s❡s✿ ❆❞✈❛♥❝❡❞
✹✶✾
✾✳ ❊①❡r❝✐s❡s✿ ❆❞✈❛♥❝❡❞
❊①❡r❝✐s❡ ✾✳✶
❊①❡r❝✐s❡ ✾✳✽
▲✐♥❡❛r✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❖❉❊ ❛t x = 0✿
❇② t❤❡ ♣♦✇❡r s❡r✐❡s ♠❡t❤♦❞✱ s♦❧✈❡ t❤❡ ■❱P
y ′ = cos x − 1 .
❉♦ ♥♦t s♦❧✈❡✳
y ′′ + y ′ − 2y = 0, y(0) = 1, y ′ (0) = −2 . ❊①❡r❝✐s❡ ✾✳✾
❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥✿
❊①❡r❝✐s❡ ✾✳✷
❙♦❧✈❡ t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✿
2xy
y ′′ − 4y = 0, y(0) = 1, y(1) = 2 .
dy = 4x2 + 3y 2 . dx
❊①❡r❝✐s❡ ✾✳✶✵ ❊①❡r❝✐s❡ ✾✳✸
❱❡r✐❢② t❤❛t t❤❡ ♣♦✇❡r s❡r✐❡s y=
∞ X
n=1
(−1)n+1 n x n
✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥
❙♦❧✈❡ t❤✐s ❡①❛❝t ❡q✉❛t✐♦♥ y 3 dx + 3y 2 x dy = 0 . ❊①❡r❝✐s❡ ✾✳✶✶
❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❝♦♠✲ ♣❧❡① ❞♦♠❛✐♥✿
(x + 1)y ′′ + y ′ = 0 . ❊①❡r❝✐s❡ ✾✳✹
●✐✈❡♥ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥
y ′′ + y ′ + y = 0 . ❊①❡r❝✐s❡ ✾✳✶✷
Pr♦✈✐❞❡ t❤❡ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ❢♦r
′′
(cos x)y + y = 0,
✜♥❞ t✇♦ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠ ♦❢ ♣♦✇❡r s❡r✐❡s✳ Pr♦✈✐❞❡ ❛❧❧ t❡r♠s ✉♣ t♦ x2 ✳ ❘❡❝❛❧❧ t❤❛t 1 1 cos x = 1 − x2 + x4 − .. . 2 4!
❊①❡r❝✐s❡ ✾✳✺
Pr♦✈✐❞❡ t❤❡ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ❢♦r y′ + y = 1 . ❊①❡r❝✐s❡ ✾✳✻
❙✉♣♣♦s❡ y1 , y2 ❛r❡ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ′′
′
y′ + y = 1 . ❊①❡r❝✐s❡ ✾✳✶✸
❱❡r✐❢② t❤❛t t❤❡ ♣♦✇❡r s❡r✐❡s y =
∞ X (−1)n+1
n=1
n
xn ✐s ❛
s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ (x = 1)y ′′ +y ′ = 0✳ ❊①❡r❝✐s❡ ✾✳✶✹
●✐✈❡♥ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ (cos x)y ′′ + y = 0✱ ✜♥❞ t✇♦ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠ ♦❢ ♣♦✇❡r s❡r✐❡s✳ Pr♦✈✐❞❡ ❛❧❧ t❡r♠s ✉♣ t♦ x2 ✳ ❘❡❝❛❧❧ 1 1 t❤❛t cos x = 1 − x2 + x4 − ...✳ 2
4!
y + p(x)y + q(x) = 0
❛♥❞ c1 , c2 ❛r❡ ❝♦♥st❛♥ts✳ ❙❤♦✇ t❤❛t y = c1 y1 + c2 y2 ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥✳ ❊①❡r❝✐s❡ ✾✳✼
❋✐♥❞ ❛ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ ❛ ♣❛rt✐❝✉❧❛r s♦❧✉t✐♦♥ ♦❢ y (3) + y ′′′ = 3ex + 4x2 .
❊①❡r❝✐s❡ ✾✳✶✺
❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ (x−1)y ′′ −xy ′ +y = 0✱ y(0) = −2✱ y ′ (0) = 6✱ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ♣♦✇❡r s❡✲ r✐❡s✳ ✭❖♥❡ ❡①tr❛ ♣♦✐♥t ❢♦r r❡♣r❡s❡♥t✐♥❣ t❤❡ s♦❧✉t✐♦♥ ✐♥ ❛ ♠♦r❡ ❝♦♠♣❛❝t ❢♦r♠✳✮
✶✵✳ ❊①❡r❝✐s❡s✿ P❉❊s
✹✷✵
✶✵✳ ❊①❡r❝✐s❡s✿ P❉❊s
❊①❡r❝✐s❡ ✶✵✳✶
❉❡♠♦♥str❛t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ u(t, x) = sin t · sin x ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ P❉❊✿ utt = uxx .
❙✉❣❣❡st ❛♥♦t❤❡r s♦❧✉t✐♦♥✳ ❊①❡r❝✐s❡ ✶✵✳✷
❈♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ y|x 1 2 3 4 5 6 0 2 0 −1 1 2 −1 2 1 −2 1 1 −1 1 2 2 2 −3 1 0 −1 2 2 0 2 −1 −1 3 4 2 1 0 0 −2 −1 5 2 0 0 1 −3 −1 ❊①❡r❝✐s❡ ✶✵✳✸
❙❡t ✉♣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❢♦r ❤❡❛t ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ t✇♦ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ❛♥❞ ♥♦ ❡①❝❤❛♥❣❡ ✇✐t❤ t❤❡ ♦✉ts✐❞❡✿ xn+1 = ❴❴❴❴❴❴❴, yn+1 = ❴❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✵✳✹
❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ u(t, x, y) ✐s ✉s❡❞ t♦ r❡♣r❡s❡♥t t❤❡ t❡♠♣❡r❛t✉r❡ ❛t ♣♦✐♥t (x, y) ♦♥ t❤❡ ♠❡t❛❧ ♣❧❛t❡ [0, 1]×[0, 1] ❛t t✐♠❡ t✳ ✭❛✮ ❊①♣r❡ss ✐♥ t❡r♠s ♦❢ u t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❡❞❣❡ ♦❢ t❤❡ ♣❧❛t❡ ✐s ♠❛✐♥t❛✐♥❡❞ ❛t 0 ❛t ❛❧❧ t✐♠❡s✳ ✭❜✮ ❊①♣r❡ss ✐♥ t❡r♠s ♦❢ u t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ t❤❡ t❡♠♣❡r❛t✉r❡ ❡①♣♦♥❡♥t✐❛❧❧② ❞❡❝❛②s✳ ❊①❡r❝✐s❡ ✶✵✳✺
❈❛rr② ♦✉t n = 4 st❡♣s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✇✐t❤ t❤r❡❡ ❝❡❧❧s✱ ③❡r♦ ✈❛❧✉❡s ♦✉ts✐❞❡✱ ③❡r♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ ❛♥❞ ②♦✉r ♦✇♥ ❝❤♦✐❝❡ ♦❢ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t✿ xk+1,m = xk,m +(xk,m−1 +xk,m+1 +1)∆t, m = 1, 2, 3 .
✶✶✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣
✹✷✶
✶✶✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣
❈♦♠♣✉t✐♥❣ ❛ss✐❣♥♠❡♥t ★✶✿ ❋r❡❡ ❢❛❧❧✳ ❈❛rr② ♦✉t✱ ♦r ❈♦♠♣✉t✐♥❣ ❛ss✐❣♥♠❡♥t ★✸✿ ❊✉❧❡r✬s ♠❡t❤♦❞ ♦♥ ✜♥✐s❤✱ ♦r r❡♣r♦❞✉❝❡ ♦♥❡ ♦❢ t❤❡s❡ ♣r♦❥❡❝ts ❢r♦♠ ❝❛❧❝✉❧✉s ✶ t❤❡ ♣❧❛♥❡✳ ❋♦r ♦♥❡ ♦❢ t❤❡ s②st❡♠s ♦❢ ❖❉❊s ❜❡❧♦✇✱ ♣❡r✲ ✭t❤❡ ♦✉t♣✉t s❤♦✉❧❞ ❜❡ ❛ s❤♦rt ♣r❡s❡♥t❛t✐♦♥ ❞❡♠♦♥str❛t✐♥❣ ❢♦r♠ t❤❡ s❛♠❡ t❛s❦s ❛s ✐♥ t❤❡ ❧❛st ❛ss✐❣♥♠❡♥t✿ ❛♥ ❊①❝❡❧ s♣r❡❛❞s❤❡❡t ❛❝❝♦♠♣❛♥✐❡❞ ❜② ❛♥ ❡①♣❧❛♥❛t✐♦♥✮✿ ✶✳ x′ = sin y, y ′ = tan(y 2 ) ✶✳ ❍♦✇ ❞♦ ■ t❤r♦✇ ❛ ❜❛❧❧ ❞♦✇♥ ❛ st❛✐r❝❛s❡ s♦ t❤❛t ✐t ✷✳ x′ = |y|, y ′ = cos(x + y) ❜♦✉♥❝❡s ♦✛ ❡❛❝❤ st❡♣❄ p ✸✳ x′ = x + y, y ′ = cos y · |y| ✷✳ ❍♦✇ s❤♦✉❧❞ ②♦✉ t❤r♦✇ ❛ ❜❛❧❧ ❢r♦♠ t❤❡ t♦♣ ♦❢ ❛ 100 √ st♦r② ❜✉✐❧❞✐♥❣ s♦ t❤❛t ✐t ❤✐ts t❤❡ ❣r♦✉♥❞ ❛t 100 ❢❡❡t ✹✳ x′ = y, y ′ = sin y · y ♣❡r s❡❝♦♥❞❄ ✺✳ x′ = x − y, y ′ = [x + y] ✭t❤❡ ❋▲❖❖❘ ❢✉♥❝t✐♦♥ ✐♥ ✸✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ✉s❡ ❛ ❝❛♥♥♦♥ ✇✐t❤ ❛ ♠✉③③❧❡ ✈❡❧♦❝✐t② ❊①❝❡❧✮ ♦❢ 100 ❢❡❡t ♣❡r s❡❝♦♥❞ t♦ ❜♦♠❜❛r❞ t❤❡ ✐♥s✐❞❡ ♦❢ ❛ 1 ❢♦rt✐✜❝❛t✐♦♥ 300 ❢❡❡t ❛✇❛② ✇✐t❤ ✇❛❧❧s 20 ❢❡❡t ❤✐❣❤✳ ✻✳ x′ = cos(x + y), y ′ = ✹✳ ■ ❤❛✈❡ ❛ t♦② ❝❛♥♥♦♥ ❛♥❞ ■ ✇❛♥t t♦ s❤♦♦t ✐t ❢r♦♠ ❛ t❛❜❧❡ ❛♥❞ ❤✐t ❛ s♣♦t ♦♥ t❤❡ ✢♦♦r 10 ❢❡❡t ❛✇❛② ❢r♦♠ t❤❡ t❛❜❧❡✳
x−y
❈♦♠♣✉t✐♥❣ ❛ss✐❣♥♠❡♥t ★✹✿ ❱❛♥ ❞❡r P♦❧ ♦s❝✐❧❧❛✲ t♦r✳ ❆♣♣❧② t❤❡ q✉❛❧✐t❛t✐✈❡ ♠❡t❤♦❞s ❛♥❞ ❊✉❧❡r✬s ♠❡t❤♦❞
✺✳ ❍♦✇ ❤❛r❞ ❞♦ ■ ❤❛✈❡ t♦ ♣✉s❤ ❛ t♦② tr✉❝❦ ❢r♦♠ t❤❡ t♦ ❛♥❛❧②③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡❝♦♥❞ ♦r❞❡r ❡q✉❛t✐♦♥✿ ✢♦♦r ✉♣ ❛ 30 ❞❡❣r❡❡ ✐♥❝❧✐♥❡ t♦ ♠❛❦❡ ✐t r❡❛❝❤ t❤❡ x′′ = m(1 − x2 )x′ − x . t♦♣ ♦❢ t❤❡ t❛❜❧❡ ❛t ③❡r♦ s♣❡❡❞❄ ✻✳ ❍♦✇ ❢❛st ❞♦❡s t❤❡ s❤❛❞♦✇ ♦❢ ❛ ❢❛❧❧✐♥❣ ❜❛❧❧ ♦♥ ❛ s❧✐❞✐♥❣ ❧❛❞❞❡r ♠♦✈❡❄ ✼✳ ❍♦✇ ❢❛st ❞♦ ■ ❤❛✈❡ t♦ ♠♦✈❡ ♠② ❤❛♥❞ ✇❤✐❧❡ s♣✐♥♥✐♥❣ ❛ s❧✐♥❣ ✐♥ ♦r❞❡r t♦ t❤r♦✇ t❤❡ r♦❝❦ 100 ❢❡❡t ❛✇❛②❄
❈♦♠♣✉t✐♥❣ ❛ss✐❣♥♠❡♥t ★✷✿ ❊✉❧❡r✬s ♠❡t❤♦❞✳
❋♦r ♦♥❡ ♦❢ t❤❡ ❖❉❊s ❜❡❧♦✇✱ s✉❜❥❡❝t ✐t t♦ ❊✉❧❡r✬s ♠❡t❤♦❞ ❛♥❛❧✲ ②s✐s ✇✐t❤ ❊①❝❡❧✿ ✶✳ ❆♥❛❧②③❡ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ❛♣♣❧② t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss t❤❡♦r❡♠s✳ ✷✳ P❧♦t s✉✣❝✐❡♥t❧② ♠❛♥② s♦❧✉t✐♦♥s✳ ✸✳ ❋✐♥❞ ♣❛tt❡r♥s ♦❢ t❤❡ s❡t ♦❢ s♦❧✉t✐♦♥s✱ s✉❝❤ ❛s✿ ♣❡r✐✲ ♦❞✐❝✐t②✱ ♠♦♥♦t♦♥✐❝✐t②✱ ❛s②♠♣t♦t❡s✱ s②♠♠❡tr②✱ ❛♥❞ ❛♥② ♦t❤❡r✳ ❖❉❊s✿ ✶✳ y ′ = tan(y 2 ) ✷✳ y ′ = cos(x + y) ✸✳ y ′ = cos y ·
p |y| √ ✹✳ y ′ = sin y · y
✺✳ y ′ = [x + y] ✭t❤❡ ❋▲❖❖❘ ❢✉♥❝t✐♦♥ ✐♥ ❊①❝❡❧✮ ✻✳ y ′ =
1 x−y
■♥❞❡①
❛❞❞✐t✐♦♥ ✐s ♣r❡s❡r✈❡❞✱ ✶✹✹ ❆❧❣❡❜r❛ ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s✱ ✷✹✹ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs✱ ✷✶✹ ❆❧❣❡❜r❛ ♦❢ ❈♦♥t✐♥✉✐t②✱ ✷✸✸ ❆❧❣❡❜r❛ ♦❢ ❉❡r✐✈❛t✐✈❡s✱ ✷✹✹ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✷✸✷ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✷✸✷ ❛r❣✉♠❡♥t ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r✱ ✷✷✺ ❜❛s✐s✱ ✶✽✺ ❇❛s✐s ♦♥ t❤❡ P❧❛♥❡✱ ✶✽✼ ❇✐❥❡❝t✐♦♥s ❛♥❞ ❉❡t❡r♠✐♥❛♥ts✱ ✶✻✼ ❝❡❧❧✱ ✸✼✾ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✷✾ ❝❡❧❧ ❢✉♥❝t✐♦♥✱ ✸✻ ❝❡❧❧s✱ ✷✾ ❝❡♥t❡r✱ ✸✵✸ ❈❡♥t❡r ♦❢ ▼❛ss ❚✇♦✲❜♦❞② ❙②st❡♠✱ ✸✸✷ ❈❤❛✐♥ ❘✉❧❡✱ ✸✼ ❈❤❛r❛❝t❡r✐st✐❝ P♦❧②♥♦♠✐❛❧✱ ✶✼✾ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✱ ✶✼✻ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✱ ✷✾✸ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs✱ ✶✼✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✕ ❘❡❛❧ ❊✐❣❡♥✈❛❧✉❡s✱ ✶✾✹ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ ❈♦♠♣❧❡① ❊✐❣❡♥✈❛❧✉❡s✱ ✷✹✵ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■✱ ✷✾✽ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■■✱ ✸✵✸ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■✱ ✷✶✼ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■■✱ ✷✶✽ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ❊✐❣❡♥✈❛❧✉❡s✱ ✷✸✻ ❈♦❧✉♠♥s ❛r❡ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs✱ ✶✺✽ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡✱ ✷✶✹ ❝♦♠♣❧❡① ♥✉♠❜❡r✱ ✷✶✸ ❈♦♠♣♦♥❡♥t✲✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡r✐❡s✱ ✷✹✺ ❈♦♠♣♦♥❡♥t✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s✱ ✷✷✷ ❝♦♠♣♦s✐t✐♦♥✱ ✷✵✾ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs✱ ✷✵✾ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❊♥❡r❣②✱ ✸✷✵ ❈♦♥st❛♥t ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s✱ ✾✼ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s✱ ✷✷✸ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s✱ ✷✹✻ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P✱ ✻✵ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P✱ ✺✺ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊✱ ✺✻ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s✱ ✷✸✸ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ ✷✸✸ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡✱ ✷✷✷ ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②✱ ✷✹✺ ❈♦♥✈❡rs✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs✱ ✷✷✻ ❉✬❆❧❡♠❜❡rt✬s ❋♦r♠✉❧❛✱ ✹✵✹
❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✱ ✷✹✷ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧✱ ✷✹✹ ❞❡t❡r♠✐♥❛♥t✱ ✶✻✸✱ ✷✸✺ ❉❡t❡r♠✐♥❛♥t ■s ■♥tr✐♥s✐❝✱ ✶✻✽ ❉✐✛ ❂❃ ❈♦♥t✱ ✷✹✸ ❞✐✛❡r❡♥❝❡✱ ✸✷ ❞✐✛❡r❡♥t✐❛❧ ✵✲❢♦r♠✱ ✹✶ ❞✐✛❡r❡♥t✐❛❧ ✶✲❢♦r♠✱ ✹✵ ❞✐s❝r❡t❡ ❢♦r♠✱ ✸✽✶ ❞✐s❝r❡t❡ ❢♦r♠s✱ ✸✵ ❞✐s❝r❡t❡ ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r✱ ✸✷✵ ❉✐s❝r❡t❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇✱ ✸✷✽ ❞✉❛❧ ❝❡❧❧s✱ ✶✷✵ ❞✉❛❧ ❢♦r♠s✱ ✶✷✶ ❡✐❣❡♥s♣❛❝❡✱ ✶✼✷✱ ✷✸✺ ❊✐❣❡♥s♣❛❝❡ ❙♦❧✉t✐♦♥s✱ ✷✾✷ ❡✐❣❡♥✈❛❧✉❡✱ ✶✼✵✱ ✷✸✺ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ ❊✐❣❡♥✈❡❝t♦rs✱ ✶✼✻ ❊✐❣❡♥✈❛❧✉❡s ❛s ❘♦♦ts✱ ✶✼✻ ❡✐❣❡♥✈❡❝t♦r✱ ✶✼✵ ❡✐❣❡♥✈❡❝t♦rs✱ ✶✼✶ ❊q✉✐❧✐❜r✐✉♠ ♦❢ ▲✐♥❡❛r ❙②st❡♠✱ ✷✽✾ ❊rr♦r ❇♦✉♥❞✱ ✽✻✱ ✷✺✸ ❊✉❧❡r s♦❧✉t✐♦♥✱ ✼✻✱ ✷✽✷✱ ✷✽✾ ❊①✐st❡♥❝❡✱ ✺✾✱ ✷✼✾ ❡①✐st❡♥❝❡✱ ✺✷ ❊①✐st❡♥❝❡ ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊✱ ✺✷ ❡①✐st❡♥❝❡ ♣r♦♣❡rt②✱ ✷✼✼ ❢❛❝❡✱ ✸✼✾ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ✹✷ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■✱ ✸✹ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■■✱ ✸✺ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ ✷✹✻ ❣❧✉✐♥❣✱ ✸✼✼ ❤❡❛t ❡q✉❛t✐♦♥✱ ✸✽✸ ✐❞❡♥t✐t② ♦♣❡r❛t♦r✱ ✶✹✾ ■♠❛❣❡s ♦❢ ▲✐♥❡s✱ ✶✺✵ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✱ ✷✶✶ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✱ ✺✷✱ ✷✼✼✱ ✷✽✾ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛✱ ✷✹✸ ✐♥t❡❣r❛❧ ♦❢ ✶✲❢♦r♠✱ ✹✷ ✐♥✈❡rs❡ ♠❛tr✐①✱ ✷✵✾ ■♥✈❡rs❡ ♦❢ ▼❛tr✐① ♦❢ ❉✐♠❡♥s✐♦♥ ✷✱ ✷✵✾ ■❱P✱ ✷✼✼ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ✷✸✷ ▲✐♥❡ ❈♦❧❧❛♣s❡s✱ ✶✻✼ ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s✱ ✶✵✵ ▲✐♥❡❛r ❖❉❊✱ ✻✸
✹✷✸ ❧✐♥❡❛r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✱ ✶✶✹ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✶✹✼✕✶✹✾✱ ✶✺✽✕✶✻✵ ▲✐♥❡❛r ❖♣❡r❛t♦r ❛t ✵✱ ✶✹✽ ▲✐♥❡❛r ❖♣❡r❛t♦r ✐♥ ❚❡r♠s ♦❢ ❇❛s✐s✱ ✶✾✵ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❛♥❞ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s✱ ✶✹✼ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✈s✳ ▼❛tr✐❝❡s✱ ✶✹✼ ▲♦❝❛❧✐t②✱ ✷✸✷ ▲♦❝❛t✐♦♥s ✐♥ ❚✇♦✲❜♦❞② ❙②st❡♠✱ ✸✸✷ ▼❛tr✐① ♦❢ ❈♦♠♣♦s✐t✐♦♥✱ ✷✵✾ ▼❛tr✐① ♦❢ ❘♦t❛t✐♦♥✱ ✶✻✵ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥✱ ✻✵ ▼✐①❡❞ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ♦❢ ❲❛✈❡✱ ✹✵✷ ♠♦❞✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r✱ ✷✷✺ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙♦❧✉t✐♦♥s✱ ✽✾ ▼✉❧t✐♣❧❡s ♦❢ ❊✐❣❡♥✈❡❝t♦rs✱ ✶✼✶ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs✱ ✷✷✼ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✱ ✸✺✸✱ ✸✻✺ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ▲✐♥❡❛r ❖❉❊✱ ✻✼ ◆♦♥✲♣♦s✐t✐✈❡ ❉✐s❝r✐♠✐♥❛♥t✱ ✷✹✵✱ ✷✹✶ ◆♦♥✲③❡r♦ ❉❡t❡r♠✐♥❛♥t✱ ✷✸✺ ◆♦♥✲❩❡r♦ ❙♦❧✉t✐♦♥s✱ ✶✻✺✱ ✶✻✼ ❖❉❊✱ ✹✼ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✱ ✶✶✹ ❖♥❡✲s✐❞❡❞ ❊rr♦r ❇♦✉♥❞✱ ✽✽ ❖♥❡✲t♦✲♦♥❡ ▲✐♥❡❛r ❖♣❡r❛t♦r✱ ✶✻✷ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥✱ ✷✺✼ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥✱ ✷✻✶ P❧❛♥❡t❛r② ▼♦t✐♦♥ ✐♥ P♦❧❛r ❈♦♦r❞✐♥❛t❡s✱ ✸✷✾ ♣♦✇❡r s❡r✐❡s✱ ✷✹✽ Pr❡✐♠❛❣❡s ♦❢ ❩❡r♦✱ ✶✽✵ Pr❡s❡r✈✐♥❣ ❆❞❞✐t✐♦♥✱ ✶✹✹ Pr❡s❡r✈✐♥❣ ❙❝❛❧❛r ▼✉❧t✐♣❧✐❝❛t✐♦♥✱ ✶✹✻ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ❞♦♠❛✐♥s✱ ✶✶✾ ♣r♦❞✉❝t ♦❢ ♠❛tr✐① ❛♥❞ ✈❡❝t♦r✱ ✶✸✵ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s✱ ✷✷✹ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s✱ ✷✷✺ ❘❛❞✐✉s ♦❢ ❈♦♥✈❡r❣❡♥❝❡✱ ✷✹✾ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✱ ✷✹✾ ❘❡♣r❡s❡♥t❛t✐♦♥✱ ✶✾✶ ❘❡♣r❡s❡♥t❛t✐♦♥ ■♥ ❚❡r♠s ♦❢ ❊✐❣❡♥s♦❧✉t✐♦♥s✱ ✷✾✸ s❛❞❞❧❡✱ ✷✾✽ s❛♠♣❧✐♥❣✱ ✹✷
s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡r✈❡❞✱ ✶✹✻ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡✱ ✶✷✶ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✶✷✷ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s✱ ✷✹✺ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✷✷ ❙❤✐❢t❡❞ ❙♦❧✉t✐♦♥s✱ ✷✼✻ s✐♥❣✉❧❛r ♠❛tr✐①✱ ✶✻✸ ❙✐♥❣✉❧❛r ▼❛tr✐① ❛♥❞ ❉❡t❡r♠✐♥❛♥t✱ ✶✻✹ s♦❧✉t✐♦♥ ♦❢ ❖❉❊✱ ✹✽ s♦❧✉t✐♦♥ ♦❢ s②st❡♠ ♦❢ ❖❉❊s✱ ✷✼✻✱ ✷✽✾ ❙♦❧✉t✐♦♥ ♦❢ ❲❛✈❡ ❊q✉❛t✐♦♥✱ ✹✵✸ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊✱ ✸✼✸ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊ ❉✐♠❡♥s✐♦♥ ✷✱ ✸✾✸ ❙♦❧✉t✐♦♥s ♦❢ ❍❡❛t P❉❊ ❲✐t❤ ❩❡r♦ ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥✱ ✸✼✹ ❙♦❧✉t✐♦♥s ♦❢ ❙❡♣❛r❛❜❧❡ ❖❉❊s✱ ✻✷ s♣r❡❛❞s❤❡❡t✱ ✸✾✻ st❛❜❧❡ ❛♥❞ ✉♥st❛❜❧❡ ❢♦❝✉s✱ ✸✵✷ st❛❜❧❡ ♥♦❞❡✱ ✷✾✼ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r✱ ✷✶✹ ❙t❛t✐♦♥❛r② ❙♦❧✉t✐♦♥s✱ ✽✾ s✉♠✱ ✸✸ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ ❙❡r✐❡s✱ ✷✹✼ s✉♠ ♦❢ s❡r✐❡s✱ ✷✹✺ ❙✉♠ ♦❢ ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❍❡❛t P❉❊✱ ✸✼✹ ❙✉♠ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s✱ ✷✷✸ ❙✉♠ ❘✉❧❡ ❢♦r ❙❡r✐❡s✱ ✷✹✻ ❚❡r♠✲❜②✲❚❡r♠ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ■♥t❡❣r❛t✐♦♥✱ ✷✺✵ ❚✐♠❡ ■♥❞❡♣❡♥❞❡♥t ❖❉❊s✱ ✽✽ ❚r❛❝❡ ❛♥❞ ❉✐s❝r✐♠✐♥❛♥t✱ ✶✾✼ tr❛❝❡ ♦❢ ♠❛tr✐①✱ ✶✼✽ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✱ ✶✸✸ ❯♥✐q✉❡♥❡ss✱ ✻✵✱ ✷✼✾ ✉♥✐q✉❡♥❡ss✱ ✺✸ ❯♥✐q✉❡♥❡ss ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊✱ ✺✹ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t✱ ✷✷✷ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s✱ ✷✺✵ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♠✱ ✷✹✺ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt②✱ ✷✼✼ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs ❆r❡ ❈♦❧✉♠♥s✱ ✶✺✾ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠✱ ✸✵✻ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❖❉❊✱ ✹✽ ❲❡✐❡rstr❛ss ▼✲❚❡st✱ ✷✹✽ ③❡r♦ ♦♣❡r❛t♦r✱ ✶✹✽