Instructor's Solution Manual for Advanced Calculus

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Instructor’s Solution Manual for ADVANCED CALCULUS

Gerald B. Folland

Contents 1 Setting the Stage 1.1 Euclidean Spaces and Vectors 1.2 Subsets of Euclidean Space . . 1.3 Limits and Continuity . . . . . 1.4 Sequences . . . . . . . . . . . 1.5 Completeness . . . . . . . . . 1.6 Compactness . . . . . . . . . 1.7 Connectedness . . . . . . . . 1.8 Uniform Continuity . . . . . .

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1 1 1 2 3 4 5 5 7

2 Differential Calculus 2.1 Differentiability in One Variable . . . . . . . . . . . . . . 2.2 Differentiability in Several Variables . . . . . . . . . . . . 2.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . 2.5 Functional Relations and Implicit Functions: A First Look 2.6 Higher-Order Partial Derivatives . . . . . . . . . . . . . . 2.7 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . 2.8 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Extreme Value Problems . . . . . . . . . . . . . . . . . . 2.10 Vector-Valued Functions and Their Derivatives . . . . . .

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8 8 9 10 10 10 11 12 14 15 17

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3 The Implicit Function Theorem and its Applications 3.1 The Implicit Function Theorem . . . . . . . . . . 3.2 Curves in the Plane . . . . . . . . . . . . . . . . 3.3 Surfaces and Curves in Space . . . . . . . . . . . 3.4 Transformations and Coordinate Systems . . . . 3.5 Functional Dependence . . . . . . . . . . . . . .

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19 19 20 21 22 24

4 Integral Calculus 4.1 Integration on the Line . . . . . . . . . . 4.2 Integration in Higher Dimensions . . . . 4.3 Multiple Integrals and Iterated Integrals . 4.4 Change of Variables for Multiple Integrals 4.5 Functions Defined by Integrals . . . . . . 4.6 Improper Integrals . . . . . . . . . . . . 4.7 Improper Multiple Integrals . . . . . . . .

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25 25 26 27 28 30 31 32

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Contents

iv 5 Line and Surface Integrals; Vector Analysis 5.1 Arc Length and Line Integrals . . . . . 5.2 Green’s Theorem . . . . . . . . . . . . 5.3 Surface Area and Surface Integrals . . . 5.4 Vector Derivatives . . . . . . . . . . . . 5.5 The Divergence Theorem . . . . . . . . 5.6 Some Applications to Physics . . . . . 5.7 Stokes’s Theorem . . . . . . . . . . . . 5.8 Integrating Vector Derivatives . . . . . 6 Infinite Series 6.1 Definitions and Examples . . . . . . . . 6.2 Series with Nonnegative Terms . . . . . 6.3 Absolute and Conditional Convergence 6.4 More Convergence Tests . . . . . . . . 6.5 Double Series; Products of Series . . . .

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7 Functions Defined by Series and Integrals 7.1 Sequences and Series of Functions . . . . . . . . 7.2 Integrals and Derivatives of Sequences and Series 7.3 Power Series . . . . . . . . . . . . . . . . . . . 7.4 The Complex Exponential and Trig Functions . . 7.5 Functions Defined by Improper Integrals . . . . . 7.6 The Gamma Function . . . . . . . . . . . . . . . 7.7 Stirling’s Formula . . . . . . . . . . . . . . . . .

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8 Fourier Series 8.1 Periodic Functions and Fourier Series . . . . . . . . 8.2 Convergence of Fourier Series . . . . . . . . . . . . 8.3 Derivatives, Integrals, and Uniform Convergence . . 8.4 Fourier Series on Intervals . . . . . . . . . . . . . . 8.5 Applications to Differential Equations . . . . . . . . 8.6 The Infinite-Dimensional Geometry of Fourier Series

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43 43 44 45 46 47

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49 49 50 51 53 54 56 57

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59 59 60 61 62 63 65

NOTE: Users of Advanced Calculus should be aware of the web site www.math.washington.edu/ folland/Homepage/index.html where a list of corrections to the book can be found. In particular, some errors in the exercises and in the answers in the back of the book were discovered in the course of preparing this solution manual. The solutions given here pertain to the corrected exercises.

v

Chapter 1

Setting the Stage 1.1 Euclidean Spaces and Vectors 1. 2.

✁✂☎✄✝✆ ✞✠✟☛✡✌☞✎✍✑✏✓✒✔✟☛✡✕☞✎✍✑✏✓✒✔✟☛✡✖✏✗✟✘✄✚✙✜✛ ✞ ✢✂✣✄✝✆ ☞✎✍✤✙✜✒✔✟☛✡✥✙✠✟☛✡✕✏✗✟✦✄✚✞ ✁★✧✩✢✪✄✚✞✫☞✎✍✤✙✜✒✬✡✌☞✎✍✑✏✓✒✎✙✭✡ ☞✎✍✑✏✓✒✮✏✂✡✕✏✯✧✱✰✲✄✳✍✤✴ ✄✕✶✠✷✹✸✗✸✻✺✜✼✩☞✎✍✤✴✜✽✠✞✘✧✩✙✜✛ , ✞✾✒✿✄✕✶✠✷✹✸✗✸✻✺✜✼✩☞✎✍❀✛ ✞☛✽✠✙✜✒❁✄✕❂✠❃❄✽✠❅ , ,✵ . ✟ ✟ ✟ ✁❇❆❈✢✂ ✄❉☞❊✁❋❆✖✢●✒✂✧❍☞❊✁❇❆❈✢■✒✲✄❏ ✁❑ ❆✖✙▲✁✪✧▼✢◆✡❖ ✢✂

. Taking the plus sign gives (a); adding these

✁■P●✡❈✧✗✧✗✧▲✡◗✁❙❘✫ ✟ ✄✌❚ ❯✮❘ ❱ P ✁ ❯ ✟ ✡❲✙❁❚ P❨❳❬❩❪❭ ❯ ❳ ❘ ✁❫❩❴✧✗✁ ❯ . The Pythagorean theorem follows immediately. ✟ ☞❊❛❜✒❝✄❞ ❡❢✍◗❛✎❣❤ ☞❊❛❜✒ With ❵ proof, ❡❢✄✕❛✎❣ as in the❛❝✐❦ ❥ equality holds precisely when the minimum value of ❵ ❡ is 0, for some . Thus equality holds in Cauchy’s inequality precisely when and that ❣ is, when are linearly dependent. ❡❧✧✗❣◆✄❖ ❡✬✗ ❣❤ ❡ ❣ The triangle inequality is an equality precisely❣ when , that is, when the angle from to ❡ is 0, or when is a positive scalar multiple of or vice versa. ❡✬✜✄♠♥☞♦❡♣✍✪❣■✒❫✡✥❣❤✣qr ❡s✍✪❣✯✗✡✌ ❣❤ ❡✬▲✍t ❣✯✣q✚ ❡✉✍✪❣❤ ❣❤▼✍t ❡✬✈q✚ ❡♣✍✇❣❤ , so . Likewise, . ❡s✧✓❣◆✄✕✰ ❡♣①✖❣ ❡★②❦❣❤▼✄♠ ❡●✗ ❣❤ ❡❦②★❣◆✄✌③ ❡♣✄✕③ ❣◆✄✌③ then , so ; hence if also then or . (a) If ❡❢✧▼④❦✄✝❣✪✧✠④ ❡◆②◆④❦✄✝❣⑤②◆④ ☞♦❡❢✍✥❣■✒☛✧✠④❋✄✝✰ ☞♦❡⑥✍❲❣■✒❀②◆④❦✄⑦③ (b) then and , so by (a), either ❡s✍✪If❣✪✄✕③ ④⑧✄✕③ and or ; the latter possibility is excluded. ❡❋②★❡♣✄⑨③ ❡ ❣ ❡❦②★❣✪✄✕③ ❡★②★❣ . If then too. If not, then is a (c) We always have ❡ and ❡❦②⑩☞♦are ❡❦②★proportional, ❣✿✒✭✄✕ ❶ ③ identities with the plus and minus signs gives (b).

3. 4.

5. 6. 7.

nonzero vector perpendicular to , so

.

8. This follows from the definitions by a simple calculation.

1.2 Subsets of Euclidean Space 1. (a)–(d): See the answers in the back of the text. ✄✌❼ ✄ ✄ ☞❊➁➃➂❜✰✾✒✯➄❬✍✑✏✑q✥➁❢q✌✏▲➅ ❷ ❷❦❿❋➀ (e) ❷■❸❺❹❜❻ and ❽❾❷ . ✄ ☞♦✰✣➂❜✰✾✒✹➅ ✄ ☞❊➇❄➂❨➁❬✒✯➄✈➇ ✟ ✡✥➁ ✟ q✚✏▲➅ , and (f) ❷ ❸❺❹❜❻ ✍✑❷❦ ✏✠➂❜➆❀✰▲➀ ➉■② ✰✈➅ , ❷ ➀ ➀ . segment ➈ ✄✌❼ ✄ ✄ ✰✣➂✱✏✮➉■② ✰✣➂✱✏✮➉ (g) ❷ ❸❺❹❜❻ and ❽❾❷ ❷ ➈ ➈ . 1

❽❴❷

is the union of the unit circle and the line

Chapter 1. Setting the Stage

2

✁✖✐

✄

☞✂✩✁ ➂❨✁✬✒

❷■❸❺❹❜❻ , there is a ball 2. If contained in ✁ ❷ . is open, so every point of is an interior point of and hence of ❷ , so in fact ☎✄⑨❷ ❸❺❹✹❻ and is an interior ✒ ❷ ❸❺❹❜❻ is open by ☞ ✒ point of ❷ ❸❺❹✹☞ ❻ . Thus Proposition 1.4a. Next, ❷ and ❽❴❷ are the complements of ❷✝✆ ❸❺❹✹❻ and ❷■❸❺❹❜❻■❿ ❷✞✆ ❸❺❹✹❻ , respectively, so they are closed by Proposition 1.4b. ✁✕✐

P

✁

P

✟ ❷ ❿❇❷ ✟ , some 3. We use Proposition If centered is contained in either P ✟ 1.4a. ✁ P ball ✁⑩✐ P✠at ✟ P ❷ ✟ or ❷ and ✟ ✟ point of ❷ ❿ ❷ . If ❷ ❷ , there are balls and centered hence ✁ in ❷ ❿ ❷ , so isP an interior ✟ , respectively; the smaller of these balls is contained in ❷ P ✟ ❷ ✟ , so again and contained in and at ❷ ❷ P✡✟ ✟ ✁ ❷ . is an interior point of ❷ P

4. The complements of ❷ ❿❦❷ ✟ and ❷ by Exercise 3 and Proposition 1.4b.

P

✟

❷ ✟

are

❷ P✆

✟

❷ ✟✆

and

5. This 1.4: ✄ remarks preceding✄ Proposition ☞ ✒ ☞ ✒ follows from the ❷✞✆ ❸❺❹✹❻❫❿⑥❽❴❷ . ❷✞✆ ❸❺❹❜❻ , whereas ❷ ❷■❸❺❹❜❻❫❿★❽❾❷ and ❷ ✆ 6. One example (in 7.

❥✔☛

and

❼

❥ P

❷ P✆ ❦ ❿ ❷ ✟ ✆ , respectively, which are both open ❥☞☛

is the disjoint union of

.

✁★✍✪❡●✏✕✖✁

then

✁❑✜✄♠♥☞❊✁❦✍✪❡❾✒❫✡❲❡✬✈q✖✤ ✁ ✡✌ ❡✬

1.3 Limits and Continuity

3. 4. 5. 6.

7.

. Thus, if ❷✗✄✘

☞✂✁▲➂❜❡➃✒

then

❷✗✄✘

☞✂✁✤✡✌ ❡✬ ➂❾③➃✒

.

☞♦✰✣➂❨➁❬✒❁✄✳✏ ➁✚✙ ✰ ☞♦✰✣➂❨➁❬✒❁✄✳✍✑✏ ➁✚✕ ✰ for and ❵ for . ➇✧✣ ✰ ☞❊➇✬➂❜✰✾✒❁✄❈➇✛✎✢✜✤✣✦✥ as . ☞❊❛✮➂❨❛❨✒✿✄ ✏✓✩✽ ▲★ ❛✫✪✬✣✦✥ ❛✭✣ ✰ as . P ✟ ✟ ➇❍➁❙✈q ✟ ☞❊➇ ✡◗➁ ✒ ☞❊➇✬➂❨➁✫✒✗✣q P ☞❊➇ ✟ ✡◗➁ ✟ ✒✔✣ ✰ ➇❄➂❨➁✮✣ ✰ ✪ (a) Since , we have ❵ as . ✞▲➇ ✪ ✍✇➁ ✪ ✣q⑤✞✫☞❊➇ ✪ ✡❲➁ ✪ ✒ ☞❊➇✬➂❨➁✫✒✗✫q⑤✞❬ ➇■✯✣ ✰ ➇❄➂❨➁✮✣ ✰ (b) Since , we have ❵ as . ☞❊➇❄➂❨➁❬✰✒ ✣❏➁ ✚ ➇ ✣ ✰ ☞♦✰✣➂❨➁❬✒❁✄❈➁ as , so take ❵ . ❵ ☞❊➇❄✲➂ ✱✫✒ ☞ ✱➃➂❨➁❬✒ ✳ ✱❇✄✖ ❶ ✰ ☞♦✰✣➂❜✰✾✒ ☞❊➇❄➂❜✰✾✒☛✄ ❵ ☞♦✰✣➂❨➁✫✒☛and ❵ are continuous for since ❵ is continuous except at . Moreover, ❵ ✄✖✰ ➇✬➂❨➁ for all , also continuous. ❵ ➁❢✄ ✰ ➁⑥✄✚➇ ✟ ➇t✄✳ ❶ ✰ The two formulas for ❵ agree along the curves except at ☞♦✰✣➂❜✰✾✒❁✄✖✰ and ☞❊➇✬➂ ✟P ➇ ✟ ,✒❁✄ ✟P ✣ ❶ , so✰ ❵ is➇✚continuous ✣ ✰ but ❵ as . the origin. It is discontinuous there since ❵ ☞❊➇❫✒✗✣q✚ ➇✿ ➇ ☞❊➇❫✔✒ ✣ ✰✲✄ ☞♦✰✾✒ ✧ ➇ ✣ ✰ ✱❇✄❈ ❶ ✰ ✱ Since for all , we have ❵ as .➇❴Suppose . If is irrational, then ☞✳✱✣✒☛✴ ✄ ❵ ✱❋✄✕ ❶ ✰ ➇ ❵ ✱ ❊ ☞ ✿ ✒ ✖ ✄ ✰ ✱ ☞✳✱✫✒✿✄✖ ✰ P , but there are points arbitrarily close to with . If is rational, then , ❵ ❵ ❵ ➇ ✱ ☞❊➇❴✒✗✏✙ ✟ ✱❴ ✱ but there are points arbitrarily close to with ❵ . In both cases ❵ is discontinuous at . ☞❊➇❫✒✗✫q ➇■ ➇ ✱✇✄✕ ❶ ✰ ☞ ✱✣✒✑✄✕ ✳ ❶ ✰ for all ✱ , so ❵ is☞❊➇❴continuous at 0. If is rational,✱ then , but there are Clearly➇ ❵ ❵ ✒✿✄✕✰ ✱ ❵ ❵ points arbitrarily close to with ; hence is discontinuous at . If is irrational and ✱ q✷✶ ☞❊➇❴✒✗✸✕✌✏✓✽✩✶ ➇✂✹ ✍ ✱❴✵ ✏✕is the rational number with denominator , then ❵ for ✺ distance from to the nearest ✵ ; ✱ q ✶ in any hence ❵ is continuous at . (There are only finitely many rational numbers with denominator

1. (a) ❵ (b) ❵ (c) ❵ 2.

, and

✄ ✰✣➂✬✏❝✍✍✌✏✎ P ➉ ✄ ✰✣➂✱✏✓✒ P ❯ ) is ❷ , for which ✑✓✒ ❷ ❯ ➈ ➈ .

8. The sets in Exercise 1a and 1f are both examples. 9. If

❷ ❸❺❹✹❻ , ❽❴❷

bounded interval.)

1.4. Sequences

3

❡✇✐✇❥ ☛ ✙⑨✰ ☞ ➂✄✂☎☞♦❡➃✒❨✒ ✄ ✁✥➄✆☎✂ ☞❊✁✬✒✘✐ ✁ ➅ ✁ ✄ ✁ ☎ ➀ 8. Given ❡t✐ and , let . Then is open, and hence so is . ✙✝✰ ☞ ➂❜❡➃✒ ✂☎☞❊✁✬✒❤✍✞✂✈☞♦❡➃✒✗✰✕ such❡ that We have ✁s✍❋☎ ❡✬, so ✵ ✄✝☎ . But this says that ✕ there✂ exists ✵ ✵ , so is continuous at . One can replace “open” by “closed” in the hypothesis by whenever the reasoning of the second paragraph of the proof of Theorem 1.13. ✂

✁

points✒❧of and the points of ☎ has 9. The fact that since is a one-to-one correspondence between the ✂✈☞ ✁ ✄ ✂✈☞ ✒ ✁ P ➆✠✟ ☎✳➆ ✟ . (ii) If ✦✄✡☎ , the consequences , ✁✪➄☛following ✂✈☞❊✁✬✒❤✐ ➅✭✄☞✂ ✎ ☞ ✒ that we shall use: (i) If ✟ ✄ ➀ . ❣ ✌ ✐ ✂☎☞ ✒ ✙ ✰ ☞ ➂❜❣■✒ ✂ P P ❴ ❽ ❷ ✍☎ . Since ✄ Suppose , and let be small enough so that is continuous, ✂ ✎ ☞ ☞ ➂❜❣✿✒❨✒ ✂ ✎ ☞♦❣✿✒ is a neighborhood of by Theorem☞ 1.13 it. Hence it ➂❜❣✿✒ and the remarks following ✂✈☞ ✒ ❷ ❷ ❷ and points not in , and therefore contains points in and points not in contains points in ✂☎☞ ✒ ❣❲✐ ✎☞ ✂☎☞ ✒❨✒ ❷ . It follows that ❽ ❷ . P ❣❲✐ ✎☞ ✂☎☞ ✒❨✒ ❡ ✏ ✄ ✂ ✎ ☞♦❣✿✒ ✙ ✰ ☞ ➂❜❡➃✒ ✁ P P P Conversely, suppose ❽ ☞ ❷ ➂❜❡➃✒❨,✒☛and be small enough so that ✄ . ✂ ✎ ✂☎☞ ✄ ☞✎let ✂ ✎ ✒ ✎ ☞ ☞ ➂❜❡➃✒❨; ✒ let ❣ Since is continuous, ✂☎☞ ✒ is a neighborhood of by Theorem 1.13 again. ✂☎☞ ✒ ☞ ➂❜❡➃✒ not in ❷ , and so contains points in ❷ and points Hence it contains points❡★ in✐ ❷ and points❣◗ ✐✑✂✈☞ ✒ not in ❷ . It follows that ❽❴❷ and hence ❽❾❷ .

1.4 Sequences ✛

✶

➇❾❘✘✄

✛ ✤ ✙ ✡✖✶ ✎ P ✙❝✖ ✡ ✶ ✎ ✓P ✒ ✟

✣

✛ ✙ ✙ .

1. (a) Divide top and bottom by to get ✔✼✄✔✖✕ ✶❬✩✽ ✶❙✫q✌✏✓✩✽ ✶ ✣ ✰ . P (b) P ➇❴❘ ✰ ✛ ✞ ✍ ✛ ✞ ✶ ✟ ✟ (c) Diverges since is , , and for infinitely many each. ➇➃❘✦✍✪✞❬✜✄✳✏✱✴✜✽✫ ✶✉✍⑩❂❬✏✕ ✶✧✙⑤❂❝✡✕✏✱✴ ✎ P 2. whenever . ✏ ✙ ✶ ✍✥✏ ✏ ✄ ✣ ✰ ➇➃❘✘✄✳✏✯✧ ✧ ✧✗✧✗✧ ✙ ✞ ✶ ✶ 3. .

➇➃❘ ✣ ✱ ➁✜❘ ✣✘✗ ☞❊➇❾❘☎➂❨➁✜❘▲✒ ✣ ☞✳✱➃➂✙✗✻✒ and , then . By continuity of addition and multiplication (Theorem 4. If 1.10) and the sequential characterization of continuity (Theorem 1.15), the result follows. ✂✈☞❊✁✬✰ ✒ ✣✛✚ ✁ ✣ ❡ ✙ ✰ ✙ ✰ ✂✈☞❊✁✬✒✗✍✜✎✚ ✸✕ ✰ ✕✚ ✁❑✍✭❡✬ ✕ ✵. 5. If ✁❄❘ ✣ ❡ as , for any there such that✶✗✙ whenever ✁✬❘ exists ✍❲❡✬ ✕ ✵ ✂☎☞❊✁✬❘▼✒❑✍✣✚✎✡✕ , there exists such that whenever . On If ✢ ✵ ✢ , and hence ✂✈☞❊✁✬✒s✣✤ ❶ ✚ ✁ ✣ ❡ ✖ ✙⑨✰ ✙✚✰ ✁ ✵ the other hand, if as , there exists such that for every there is an ✰ ✕✚ ✁❦✍⑩❡✬ ✕ ✮ ✂✈☞❊✁✬✒■✥ ✍ ❨✚ ✙ ✁❄❘ ✄✳✏✓✽✩✶ ✁✬❘ ✣ ❡ ✂☎☞❊✁❄❘✠✒✦✣✦ ❶ with ✚ ✵ but . Let be such a point for ✵ . Then but . ✁❙❘✉✐ ✁❙❘❋✄r ❶ ❡ ✁❙❘ ✣ ❡ ✁❄❘☎➅ 6. If ❷ ,✙ ✰ , and , then the sequence ➀ ☞ must ➂❜❡➃✒ assume ❡ infinitely many distinct values, and for ,❡ all but finitely many of them are in ; thus is ✶ an accumulation ☞♦❡❙➂✱✏✓✽✩point ❬ ✶ ✒ of ❷ . point of ❷ , for each positive integer the ball contains Conversely, if is an accumulation ❡ ✁●❘ be one. points of ❷ other than ; let ❡ ❡✇✐ ❡⑤✐ ✽ ❡ 7. If is an accumulation point of ❷ , then 1.14 and Exercise 6. If ❷ by Theorem ❷ and is not ❡ an accumulation point of ❷ , there is a neighborhood of that contains only finitely many points of ❡ ❡ ❷ . If is less than the minimum distance from to any of these points (which do not coincide with since ❡◗✐ ✽ ☞ ➂❜❡➃✒ ❡ ❡◗✐ ✽ ❷ ), ❷ . is a neighborhood of that is disjoint from ❷ , and hence

Chapter 1. Setting the Stage

4

1.5 Completeness

✄ ☞✎✍ ✏✠➂❄✍✑✏✓✽✜✛ ✙✜✒

☞✎✏✓✽▼✛ ✙✈➂❄✏✓✒

✍✑✏

✏

1. (a) ❷ , so the inf and sup are and . ❿ (b) The supremum is the 0th element of the sequence; the infimum is the limit of the odd-numbered subsequence. ✄ ❃❄✁✽ ✫➂ ✥ ✒ ❃❄✽✁ ✥ ➈ (c) ❷ , so the inf and sup are and .

➇❴❘✦✄✖✙✼ ✔ ✕❙☞ ✶✣❃❄✽✠✞✜✒ (Exercise 1c in ✂ 1.4). ✱✪✄ ✰☎✄ ✱ P ✱ ✟ ✱ ✜ ✄✆✄✆❑✄ ✐r☞♦✰✣➂✱✏✓✒ ❾➇ ❘★✄ ✰☎✄ ✱ P ✱ ✟ ✄✆✄✆✄ ✱☎❘ ✏✗✰ ❘ , let , considered as a fraction with denominator If ➇❾❘✾➅ ✱ ✱⑥✄✳✰ ➇●❘❧✄⑦✏✓✽✩✶ . ➀ Then is a subsequence of the given sequence that converges to . For , take ; for ✱s✄✳✏ ➇❾❘✘✄ ☞ ✶❧✍ ✏✓✒❨✩✽ ✶ , take . ✔✟✞❲➇❾❘✘✡ ✄ ✠ ✠❙✡ ✄ ✠✟ ✠❫✄✕✰ (a) If ✝ , then and hence or 1. ✱❴ ✕✌✏ ✱ ✄✌❆❧✏ ✱❫✏✙✌✏ (b) The limit is zero if , 1 if , and nonexistent (or infinite) if . ➇❄P✂✄ ✛ ✮ ✙ ✕❈✙ ➇➃❘ ✕✖✙ ➇❾✆❘ ☛ P ✄ ✛ ✙✤✡◗➇❾❘ ✕ ✛ ✙❝✡✥✙⑧✄✌✙ ➇❴❘ ✕t✙ We have .➇❫If☞❘ ☛ P ✄ ,✙✤then . ➇➃By induction, for all ✶ ✡ ➃ ➇ ❘ ✙ ➃ ➇ ✯ ❘ ✡ ➃ ➇ ✲ ❘ ✄ ▲ ✙ ❾ ➇ ❘ ✮ ✙ ➃ ➇ ❀ ❘ ✱ ✧ ❘ ⑨ ✄ ➃ ➇ ❘ ✛ ✛ ✛ ✛ .➇➃❘☎This being the case, . Thus the sequence ➅ ✠ ✠✤✄ ✛ ✙✤✌ ✡ ✠ ➀ is increasing and✠❙bounded above , hence ✠ ✟ ✄✡✠❬✡✥✙ ✄✕✙ ✠❄✄✳ ✍✑✏ by 2, so it converges to a limit . We have ➇✬❘ ✙ ✰ ✶

2. One example is 3.

4. 5.

, and hence

6.

7.

8.

or

. The latter alternative is impossible since

for all .

❾➇ ❘ ✶ ✁▼❘ ✄⑨➇➃✆ ❘ ☛ P ✽✩➇➃❘ ➇❾❘✆☛ ✟ ✄⑨➇➃❘✆☛ P ✡❲➇❾❘ (a) Let ✁✩❘✆☛ be✄ the✏❤✡✚th☞✎✏✓term of the Fibonacci sequence, so . Since , we ✽ ✁✓❘✜✒✤✄ ☞✂✓✁ ❘❀✡⑨✏✓✒❨✽ ✁✓❘ ➇❴❘✆☛ P ✶ ✶⑧✡⑨✏ P by dividing through by . Replacing by we get obtain ✁✓❘✆☛ ✟ ✄ ✂☞ ✁✓❘✆☛ P ✡✖✏✓✒❨✽ ✁✓❘✆☛ P , and substituting in ✁▲❘✆☛ P ✄ ☞✂✁✓❘✯P ✡✖✏✓✒❨✽ ✁✓❘ gives ✁✩❘✆☛ ✟ ✄ ☞ ✙ ✁✓❘❝✡❈✏✓✒❨✽✣☞✂✁✓❘✤✡✖✏✓✒ . ☞❊➇❴✒❁✄♠☞ ✙▲➇❁✡❦✏✓✒❨✽✣☞❊➇☛✡★✏✓✒❤✄✖✙❄✍ ☞❊➇☛✡❦✏✓✒ ✎ ➇ ☞✎✍■✒❁✄✏✍ is ✁✩an increasing ✁✓function of☞✂✁✓❘▲,✒✹and (b) The function ❵ ❵ ☞✎✍■✒✦✄✒✍ . ✁✩❘ ✕✑✍ ✁✩❘✆☛ ✄ ✂✁✓❘✠✒✹✕ ☞ ✎ ☞ ● ✍ ✦ ✒ ✒ ✄ ✍ ✍✙✑✍ ❘ ✆ ❘ ☛ ✄ ✙ ✟ ✟ ❵ ❵ ❵ Hence,✁✜P⑧ if ✄ ✧ , ✁▲and if ✍ then . ✏ ✕✓✍ then ✁ ✟ ✄ ✙ ❵ ✙✓✍ ❘ ✕✓ ✶ ✁✩❘ ✙✓✍ ✶ for odd and for even. Next, Since ✁✓✆ ❘ ☛ ✟ ✍ ✁✓❘✘✄ ✂☞ ✁✓❘▲✒❄✍ and✁✓❘✦✄♠☞✎✏■✡ ✁✓❘✯✍ , ✁ it❘✟ ✒❨follows ✽✣☞✂✁✩❘❑✡t✏✓that ✒ ✶ ❵ , which by the hint is positive for odd and negative ✶ for even. ✁ P✗➅ ✁ ➅ (c) By (b), ➀ ✟ ❯ ✎ is an increasing sequence and ➀ ✟ ❯ is a decreasing ✍ ✠✔P ✠✟ ✔☞ ✠ ✒❁✡ ✄ ✠ ❯ sequence, bounded above and ✍ , and hence both are equal to . below, respectively, by . Their limits and both satisfy ❵ ❯ ✁❄✖❘ ▼✕ ➅ ❡ ✙❈✰ ☞ ➂❜❡➃✒ ✁●✗❘ ✕ ✌ If ➀ converges , then ✁■❘ contains ✁❄❘✙for all ☞✎sufficiently large .✌⑧Conversely, ❡ to , and ✘❝✐ ✏✠➂❜❡➃✒ ✄✌✙✈➂✹✞✈➂✚✫➂✆✄✆✄✆if✄ P contains infinitely many , we can pick , and then for , every ball about ✶ ✙✷✶ ❯ ✎ P so that ✁❙✗❘ ✕✑✐ ☞ ✌ ✎ ➂❜❡➃✒ ; then ✁❄✖❘ ✕✬✣ ❡ . we can pick ❯ ✁●❘☎➅ be a sequence of distinct points If ❷ is bounded and infinite, let ➀ ❡ of ❷ . By Theorem 1.19, this set ✂ 1.4 its limit is an accumulation point of ❷ . (At has a convergent subsequence, and by Exercise 6 in ✁❄❘ ❡ most one

can be equal to ; throw it out if necessary.)

➇❙❘

✶

✘✍

✙✘✱

✼✜✛✣✢ ➇❾❘✲➄

◆➅

✶✥✤✧✦

✙✘✱

✘✍

✦

➀ ➇❙❘ ✙✘✱❍✡ 9. If✔✟✞ there are many for which , then and hence ✼★✛✣❤ ✢ ➇➃✩ ❘ infinitely ✤✘✱❫✍ ✶ ✼★✛✣✢ ➇❾for ❘ ➄ all ✶✪✤✧◆ ✦ ➅✲q✘✱❍✡ . If there are only finitely many for which , then ✝ ➀ ✦ ✔✟✞⑤✼✜✛✣✢❤➇❫❘⑧q✘✱❁✡ ✱❢q ✔✟✞⑤✼✜✛✣✢✂➇❫❘✲✘ q ✱ ✝ for sufficiently large, . Since is arbitrary, we have ✱❧✄ ✔✟✞ ✼✜✛✫✢✂➇❴and ❘ hence ✝ and hence . ✝ ➇❫❘✖✕▼➅

➀ 10. We define subsequence ➇❾❘✗✕ a✙⑦ ★✼ ✛✣✢ ➇➃❘◆➄✔✶✬✤ so that❘✖✕✜✰ ✘ ☛ P ✍❈☞✎✏✓✽ ➀ ✌☎✒✬✕t➇➃❘✗✕ ✭ have ✔✟✞✣✔✖✣ ✕ ✱ ✝ .

✶ ✕

❍P ✄ ✏

✏

P

recursively. We take , and for , we choose ✔✟✞ ❯ ✼✜✛✣✢ ❯ ✎ ✂ P r ✡ ▲ ✏ ⑧ ➅ ✌ ✍ ✎ ☞ ✓ ✏ ✽ ✌✈✒ ❯✎ . Then,✔✟◗ with ✭✯✮ as in the of ✝ , we ❘✗✕✚✰ ✘ ☛ P ✞ ➇❴❘✖✕✘✄ ✔✟✞ ✄ definition ✔✟⑤ ✞ ✼✜✛✣✢✂➇➃❘ ✭ ✝ ✭✯✮ ✝ . It follows that ✝ . Similarly for ✶

✌✖✙

✶

✙

✶

1.6. Compactness

✚✣ ✱

✙

✱

✕

✕ ✱

5

✶

➇❾❘✖✕ ⑦✰ ♣✍ ❖➇❴❘ ❧✡ 11. If✔✟✞ ★✼ ✛✣❤ , then for any we have for infinitely many . It follows ✄✖ that ✢ ➇➃✩ ❘ ✤ ✑✍ ✔✟✞✣✔✖✕✣✱☎➇❾❘✲q ✭✡ ✰ and ✝ ; since can be arbitrarily small, the same is true with . ✝ ➁ ✔✟✞ ✼✜✛✣✢ ✔✟✞ ✔ ✕✣✱ ➇❙❘✘✍ ❴➃q ✤ and➁ ✝ ✤ ❧✍ , the assertion that for ✍ 12. With ✭ ✮ and ✮ as in the definition of ✢ q ✝ ✑✡ ✦ ✤ q is✔✟✞✣ equivalent to✔✟✞ the★✼ ✛✣assertion that ✭ ✮ and ✮ for holds, ✔✖✕✣✱☎➇❾❘★q ❤ ✢ ➇➃❘★q ✲✡ ✔✟✞✣✔ ✕✣✱☎➇❴❘ ✄ ◆✢ ✄ . If✔✟✞ this ✼✜✛✣✢❤ ➇❾❘ then for every if ✝ ✝ ⑨✰ , and hence ✝ ✍ ✝ q✕➁ q . Conversely, q ✡ ✮ ✭✯✮ the latter condition holds, then for any there exists such that for ✦ ✤ ➇❾❘❀✍ ❴✈q ✥✤ ✔✟◗ ✞ ➇❾❘✘✄ , and so for ; hence ✝ .

✘✱

✁

✘✱

✴✱

✱

✱

✁ ✴✱

✙

✶ ✁

1.6 Compactness

✴✱

✱

✞✝

✟✝

✟✝

5.

6.

✱

✱

✄

☞ ✒ ☞ ✒ ❵ ❷ t❵ ❷ bounded. ✁❑P✱➂❨✁ ✟ ➂✆✄✆✄✆✄ be a sequence of distinct points of ☎ . If ❷ is compact and ☎ is an infinite subset of ❷ , let ✚ ✚ This sequence has a convergent subsequence whose limit lies in ❷ , and is an accumulation point of ☎ ❷ is not compact. If ❷ is not closed, there ✁■ 6, ✂ 1.4). Conversely, suppose is❘✜a➅ sequence ✚✿✐ ✁❙❘☎(Exercise ➅ ➀ ✁❙❘❬ in ❷ that converges to a point ✁❁P✱➂❨✁ ❷ ➂✆,✄✆and if ❷ is not bounded, there is a sequence ➀ in ❷ with ✟ ✄✆✄ ➅ is an infinite subset of ❷ with no accumulation point in . In either case, the set ➀ ✚ ❷ . (In the first case, the only accumulation point is ; in the second case, there is no accumulation point at all.) ✁✬❘✾➅ ☞❊✁❄❘▲✒ ✌✏✓✽ ✁✬❘✖✕✠➅ ✚●✐ ➀ ❷ ❵ ➀ ❷ ; If not, there☞ ✚ ✒✿ is✄ a sequence in such that . Some subsequence has a limit ✔✟✞ ☞❊✁❙❘✗✕✩✒✿✄✖✰ ✝ ❵ , contrary to assumption. but then ❵ ✪✤✌✏ ✁■❘✲✐ ❘ ✁❄❘✲✐ P ✁●✖❘ ✕✜➅ , pick . Then for all , so some subsequence By Bolzano-Weierstrass: For ❷ ❷ ➀ ✚❤✐ P ✁ ✐ ❘ ✚ ✚❤✐ ❘ ❘ ❷ . But since ❯ ❷ for P ❷ P. converges to a point , is actually in ❷ for all , i.e., ✄ ✁ ✁ ❘ ⑨➆❝❷ ❯ . If the sets covered ❷ , By Heine-Borel: Let be an open ball containing ❷ , and let ❯ P ✄ ✄✌ ❶ ❼ ✁ P ✁ ❘✘✄ ✁ ❷ there would be a❘ finite subcover; that is, ❷ this is false since ❷❧➆ . ❘✘✄ ❶ . But ❼ ✁ ✁ ❘ ✄✌ do not cover ❷ , that is, ❷ . ❷★➆ Thus the sets ✁⑩✐ ✁ ✢●❘✾➅ ✁ ✁❝✍✘✢●❘❬ ✰ ☞ ✁ ➂ ✒☛✄✖✰ in ☎ that converges to , i.e., ; thus . (a) If ☎ , there is a sequence ➀ ☎ ☞ ✁ ➂ ✒❝✄✳✰ ✁✿❘ ✐ ✁ ✢❙❘❢✐ ✁ ☎ ☎ (b) is closed, but . Then there exist , ✁❙❘✑Suppose ✍ ✢❙❘✣ ✰ is compact, ✁❁❘ ☎ such ✁t✐ that ✁ ✁ is by passing to a subsequence we may assume that . ✢❄❘. Since ❏✁ ✁⑩compact, ✐ ✄ ✄✌❼ ✁ ☎ ☎ , contradicting ☎ But then also , so . ✄ ☞❊➇❄➂❨➁❬✒❤➄▼➁❢q ✰✈➅ ✄ ☞❊➇✬➂❨➁✫✒✯➄▼➁ ✤ ☎➅ ✁ ➀ ➀ ,☎ . (c) One example is

✆

✣ ✥

4.

✱

✶

✄✂✆☎

✄✖❥ ☞❊➇❴✒✿✄ ,❵ . ✄❈❥ ☞❊➇❴✒❁✄✖➇ ✟ ,❵ . (b) One example is ❷ ✄♠☞♦✰✣➂✱✏✓✒ ☞❊➇❴✒✿✄ ✏✓✽✩➇ ,❵ . 2. (a) One example is ❷ ✄ ✄ ☞ ✒ ✄ ❷ compact ❵ ❷ compact (b) ❷ bounded

1. (a) One example is ❷

3.

✱

✕ ✩✶

✶

✌ ✙ ✶

✠ ✄ ✑☞☛ ✑

✟

✣

✣

1. (a) The two branches (

☛

✌

1.7 Connectedness ➇

✙ ✰

and

✶

➇

✕ ✰

✟

✯✣

✍✂✆☎

).

✙

(b) One point in the set and the rest of the set. ➇ ✰ ➇ (c) The intersections with the half-spaces and

✕ ✰

.

✡✠

✶

☛

✌

☛

✮✣

Chapter 1. Setting the Stage

6

❡❫➂❜❣

❡ ❣

❣ are points in the unit sphere ❷ , the plane through , , and the ❡ origin ❣ (that is, the linear span of and ) intersects in ✄❖ a circle, ❡ ❣ ❷ ❣⑩ ✍❝❡ and either of the two arcs ❡ between and provides a continuous path in ❷ from to . (If , any great circle through will do.) This argument works in any number of dimensions.

2. If ❡

➇❄➂❨➁❾➂✁ ✐✄✂ one ❵ ✕ is✴ ☞❊➇❫✒ nor q strictly ☞❊➁❬✒ decreasing, ☞❊➁✫✒ ☞✆☎✒ can find ☞❊➇❴✒ points☞❊➁❬✒ ☞❊➁❬✒ such q ☞✆that ☎✒ (i) ➁ neither ✕☎ strictly increasing ❵ ❵ ❵ ❵ ❵ ❵ ❵ , and (ii) either ❵ and , or and ; we ☞❊➇❫✒✭✄ ☞❊➁❬✒ ☞❊➁❬✒✭✄ ✆☞ ✈✒ ❵ ❵ or ❵ ❵ , then ❵ is not one-to-one. Otherwise, assume the former alternative. If ☞ ☞❊➇❴✒ ➂ ☞❊➁❬✒❨✒ ☞ ☞✆☎✒ ➂ ☞❊➁❬✒❨✒ and ❵ are nonempty, and ☞❨one in the other. Assuming the intervals ❵ ❵ ❵ ☞❊➇❄➂❨➁❬is✒❨✒✞contained ✳ ✝ ☞ ☞❊➇❫✒ ➂ ☞❊➁✫✒❨✒ ☞❨☞❊➁❾➂✁☎✒❨✒✞✝ ❵ ☞ is✆☞ ☎✒continuous, ❵ ❵ ❵ the intermediate value theorem implies that and ❵ ➂ ☞❊➁❬✒❨✒ ☞❊➇✬➂❨➁✫✒ ☞❊➁❾➂✁☎✒ , so there are points in and at which ❵ takes the same value, and again ❵ is not ❵ ❵

3. If ➇

✤

one-to-one.

P

P

✤

✁ ☎

✄

✁

☎

✟ 4. Suppose ❷ ❿★❷ ✟ is disconnected, P✂✄ ☞ P ✟ so ✒ ❷ ☞ ❿❦ P ✟ ❷ ✒ ❿ where neither P nor intersectsP ✟ the closure P ✟ of is a disconnection of ❷ unless either ❷ or ❷ P the other one. Then ❿ ❷ P ❷ P❷ is empty, i.e., ❷ ✄ or ❷ ✄ . Likewise, we must have ❷ ✟ ✄ or ❷ ✟ ✄ . ItP cannot be that ❷ and ✟ ❷ ✟ are both contained in (resp. ), for then (resp. ) would P ✟ be✟ empty; ✄✌ ❶ ❼ so ❷ ✄ and ❷ ✄ or vice versa. Either alternative contradicts the assumption that ❷ ❷ P✢✟ ✟ ✄✳✏ ✙✌✏ . P ❷ ❷ is connected when ✟ by Theorem 1.25, but not when ✟ . For example, take ❷ to be the unit sphere (Exercise 2) and ❷ ✟ to be a line through the origin; the intersection consists of two points.

✁

☎

✁

✁ ☎ ✁

✄

✁

☎

☎

✁

✁

☎

☎

✁

☎

✁

✁✥✐

✁

☎ ✁ ✁

✁

☎

☞☎

✁ ☎

✁

, there is✟ a ball at that 5. Suppose ❷ ❿ where and are open and disjoint. ✁✝✐ ✽ If ✄ centered ❼ is contained in and hence is disjoint from ; hence . Likewise , so ❿ is a disconnection of ❷ . ✄ ✁✕✐ , there is a ball centered Conversely, suppose ❷ is open and ❷ ❿ is a disconnection. ✁ If ✁ at ✟ that✄ is❼ contained in ❷ (since ❷ is open) and a ball centered at that does not intersect (since ✁ ). The smaller of these two is a ball centered at that is contained in . Thus every point of is an interior point of , so is open; likewise, is open.

☎

✁ ☎

✁ ☎ ✁

✁ ❿ ☎

✄

✁

✁

✁

☎ ✁ ☎

☎

☎

✁

✁ ☎ ✁

where and are closed and disjoint, it is immediate that❡❋✐ ❿ is a disconnection of if ❷ is✟ closed . Since ❷ is closed, we ❷ . Conversely, ❡★✐ ✄✌❼ and ❿ ❡ is ✐ ✽ a disconnection ❡★✐ of ❷ , suppose ❷ ; since have , we have . Hence , so is closed. Likewise, is closed.

6. If

❷

☎

✄

P

☎ ✁

☎

✁

☞❊✁✬✒❝✄ ✰

✁t✐

✁

P

☞❊✁✬✒❀✄ ✏

☎

✁t✐

❷ P ❿❋❷ ✟ is a disconnection of ❷ , define ❵ ❷ and ❵ ❷ ✟ . Each for for ✟ 7. If ❷ ✟ ; likewise point ofP ❷ has a neighborhood that does not intersect ❷ , so that ❵ is constant on ❷ with ❷ and ❷ ✟ switched. It follows that ❵ is continuous on ❷ . P ✰✣➂✱✏▲➅ P♣✄ ✎ P ☞ ✰✈➅▲✒ ✟ ✄ ❵ ✎ ☞ ➀ ✏▲➅▲✒ . If ✁♠✐ ❵ maps ❷ continuously onto ➀ , let ❷ ❵ ➀ and ❷ Conversely, if P ✟ ✟ ✄⑨❼ P ✟ P ☞❊✁✬✒☛✄✕✰ ✁✕✐ ✽ ✟ since ❵ is continuous, so , and likewise with ❷ and ❷ ❷ , then ❵ P ❷ . Thus ❷ ❷ ❷ ✟ switched, so ❷ ❿★❷ ✟ is a disconnection of ❷ .

✁

✄

✁ ☎ ☎

✁

☞ ✟

✁

✒

☞ ✟

☎ ✟ ✒ is✁ a✄ disconnection of ❷ unless ❼ ✄⑨❼ then ❷✴✄ ☎ ; but ✁ ❷✘✄ ✁ ✟ ☎ , say, , contrary to the

❷ ✟ ❿ is a disconnection of ❷ . Then ❷ ❿ ❷ 8. Suppose ✟ or ❷ is empty. The latter alternatives are impossible: ❷ ✄ If ✟ ❷ since does not intersect the closure of , we would have definition of disconnection.

✁

✁✪✐

☞❊✁✬✒☛✄✖✰

☎

✁

☞❊✁✬✒ ✙ ✰

✁

☞❊✁✬✒ ✕ ✰

☞✎✍✯✁✬✒☞✕⑤✰

. If✰ ✠ we are done. Otherwise, either ✠ or ✠ , in which☞❊✢●case 9. Pick☞✎✍✯✁✬✒✬❷ ✙✖ ✒✯✄✚✠ ✰ ✠ ✠ or respectively; either way, the intermediate value theorem implies that for some ✢ ✐ ❷ .

1.8. Uniform Continuity

☞ ✫➂✱✍✑✏✓✒☛✄✕❂ ☞❊➇❄➂❨➁❬✒❤✐ ☞❊➇❄➂❨➁❬✒❁✄✖✰ ➇❢✄✖➁ ❷ such that ❵ , so there is a point , i.e., . ➁✉✄✖✼✄✔ ✕❙☞♦❃❄✽✩➇❫✒ ✮ ✰ ✕⑤➇❇q⑤✙ , , is arcwise connected almost by definition (it’s an arc!), and ❷ 11. (a) The graph is its closure. (Check that every point in ❷ ✆ has a neighborhood ✰✈➅❢that ② ✍✑does ✏✠➂✱✏✮➉ not intersect ❷ , and that every contains points of the graph neighborhood ➈ ➁✉✄❈✼✄✔ ✕❙☞♦❃❄✽✩➇❴✒ of every point on the vertical line segment ➀ .) So ❷ is connected by Exercise 8. ✂⑤➄ ✰✣➂✱✏✮✓ ➉ ✣ ✂✈☞♦✰✾✒❦✄ ☞ ✙✈➂❜✰✾✒ ✂☎☞✎✏✓✒★✄ ☞♦✰✣➂✱✏✓✒ and ✶ . ❛✗❘ The ➈ ❷ is continuous and satisfies (b) Suppose P ✂ ✐ ✰✣first ➂✱✏✮➉ of is continuous, so by the intermediate value theroem, for each there exists component ❵ ➈ P✓☞❊❛✎❘✠✒❁✄ ✏✓✽✠✙ ✶ ✂☎☞❊❛✎❘▼✒❁✄ ☞✎✏✓✽✠✙ ✶❾➂❜✰✾✒ ➇ ✏✓✽✠✙ ✶ ❛ so that ❵ ❛❨❘ (= the only✏✓✽✠point ). As ❛ ❯ ✌❦✄ ❶ and ✶ hence P ✙ ✶ in ❷ ✏✓✽✠with ✙ ✌ -coordinate goes from to ( ✍✑✏ ,) ❵ ✏ must assume all values between and , and hence ➇ ❵ ✟ must assume all values between ➁ and (again because there is only one ✍✑✏ point✏ in ❷ with a given -coordinate in range from to ). By passing to a subsequence, by this range, and the -coordinates of these points ❛ ❘ ✣ ✁❛ ❛ ✂ . Every neighborhood of contains points at which Bolzano-Weierstrass we may assume✍ that ✟❵ assumes any given value between ✏ and ✏ , so ❵ ✟ cannot be continuous at ❛ , contrary to assumption. 10.

❵

☞✎✏✠➂✹✞✜✒❑✄✳✍✤✙

7

and ❵

1.8 Uniform Continuity 1. 2.

3.

4.

5.

✁❦✍✇✢✂✏✕✚☞ ✽☎✄ ✒ ✓P ✒✁✆ ✂☎☞❊✁✬✒■✍ ✂✈☞❊✢●✒✗✸✕ Given , if then . ✆ ✆ ✆ ✆ P ✆ P ☞✳✱✘✡❲❛❨✒ ✎ ❛ ✕ ❛ ✎ ❛ ☞✳✱✘✡✣✗✻✒ ✍ ✱ ✕ ✗ , so . (a) ✝✟✞ ✝✠✞ ➇❄➂❨➁ ➇✿ ✆ q✚☞❜ ➇⑧✍⑥➁❙❨✡⑤ ➁❙❺✒ ✆ q✚ ➇❧✍⑥➁❙ ✆ ✡ ➁❙ ✆ ➇ ➁ (b) For any , and likewise with and switched; ➇■ ✆ ✍t ➁❫we✆ have q✚ ➇♣✍✇➁❙ ✆ ✡ . hence ✡ ✡ ✡ P ✂☎☞❊✁✬✒☛✍ ✂☎☞❊✢●✒✗ ✕ ✙✕✰ ✱ P ➂ ✙✌✰ ✟ ✟ ✕ whenever ✁✿➂❨✄ ✢⑤✐ ✞ ✔❷ ✕❙☞andP✱➂ ✁◆✒ ✍⑩✢❤ ✕ P ✵ ✵ Given , we so that P ☛●☞❊✁✬✒✦☞ ✍ ☛■can ☞❊✢●✒✗choose ✹✕ ✿ ✁ ❨ ➂ ✢ ✐ ✥ ✁ ✖ ✍ ✂ ✢ ✟ . Then ✵ ♥☞✎✂❤and ❷ and ✵ ✟ . Let✁✿➂❨✵ ✢◗✐ ✵ ✁★✵ ✍✇ ✌ ✡ ☛❴✒✻☞❊✁✬✒❁✍⑤✎☞ ✂✂✡✍☛❫✒✻☞❊✢●✒✗❍q✟ ✚ whenever ✂☎☞❊✁✬✒✿✍ ✂☎☞❊✢●✒✗✩✡✌ ☛■☞❊✁✬✒✿✍✎☛■☞❊✢●✒✗✸✕ ✢❤✏✕ whenever ❷ and ✵ . ✂ ✁●❘☎➅ ✙ ✰ ✙❏✰ Suppose continuous is Cauchy. Given , there exists so that ✂☎☞❊✁✬✒✭☞ ✍ ☎✂ ☞❊✢■is✒✗ uniformly ✕ ✁✥✍❈✢✂and✕ ➀ ✁ ❯ ✍✖ ✁❙❘✈ ✵ ✕ whenever ✵ , and there exists ✵ whenever ✌▼➂ ✶ ✙ ✂☎☞❊✁ ✒❤✍ ✂✈☞❊✁❙❘✠✒✗ ✕ ✌▼➂ ✶ ✢ ✙ such that ✂✈☞❊✁❄❘▼✒✹➅ ❯ ✢ . It follows that ✢ , so ➀ whenever is Cauchy. For the ➇❫❘✦✄ ✏✓✩✽ ✶ ☞❊➇❴✒✿✄✳✏✓✽✩➇ ☞❊➇❴✒☛✄❈✸✻✺✜✼✩☞♦❃❄✽✩➇❴✒ and ❵ or ❵ . counterexample, take ✂✈☞ ✒ ✁■❘✜➅ ✂☎☞❊✁❄❘▼✒✗ ✣ ✥ If ❷ is unbounded, we can find a sequence ➀ . If also ❷ is bounded, ✁■in❘✾➅ ❷ such that converges by passing✁ to✍✉ a ✁❙subsequence we may assume that ➀ ❘✣ ✌ ✶ to some limit (which may not be ✌ in will be✂☎☞❊as small as we please provided and are sufficiently large, but for any we ❷ ). Then✶✚✙ ❯ ✌ ✁✬❘✜✒✗✏✙✚ ✂✈☞❊✁ ❯ ✒✗✱✡✖✏ ✂☎☞❊➇ ✒●✍ ✂✈☞❊✁❙❘✠✒✗☎✤⑨✏ ✂ ❯ can find such that and hence . Thus is not uniformly continuous on ❷ . ✙

✰

✌

✌

Chapter 2

Differential Calculus 2.1 Differentiability in One Variable

✱➃➂✙◗ ✗ ✐

✱ ✕

✙

✗

1. Suppose . If ❵✁ ➇✬P✱➂✆✄✆and ✄✆✱ ✄ ➂❨➇❾❘ be the Otherwise, let ☞ ✣✒☛✄ ☞ ✗✗✒❾✍ ☞❊➇➃❘▼✒ ➉✠✡ ☞❊➇❾❘▲✒❴points ✍ ☞❊➇❾❘ of

❵

✳✱

✂

➈❺❵

❵

➈❺❵

✎

❵

is positive by the mean value theorem.

☞♦✰✾✒✦✄

☞❊➇❴✒❨✽✩➇ ✄

✔✟✞

✳☞ ✱➃✙➂ ✗✻✒ , then ❵ ☞✳✱✣✒ ✕ ❵ ☞ ✗✗✒ by the mean value theorem. ☞✳✱❍on ➂✙✗✗✒ ☞ ✗✗✒■✍ at which ❵ vanishes, in increasing order. Then ❵ P ➉▲✡❲✧✗✧✗✧✻✡ ☞❊➇ P ✒❫✍ ☞✳✱✣✒ ➉ . Each of the differences on the right ➈❺❵ ❵ ✰

➃➇✭✼✄✔ ✕❫☞✎✏✓✽✩➇❴✒✘✄✝✰

✔✟✞

➇❀✙✼ ✔ ✕❙☞✎✏✓✽✩➇❴✒✗●q ➇■ ✍✣ ✰ .

2. We have☞❊➇❫❵ ✒❁✄✕✙▲➇✭✼✄✔ ✝ ✕❙☞✎✏✓✄✽✩✂ ➇❴✒❾❵✍❦✸✻✺✜✼▲☞✎✏✓✽✩➇❫✒ ✝ ✄✂ since ➇ have ❵ ; the first term on the right approaches 0 as term has no limit. ☎

3.

☎

➇✌✄✝ ❶ ✰

For we , but the second

☞♦P ✰✾✒☛✄ ☞♦✰✾✒✜✡ ✟P ✄ ✟P ➇ ✄✖ ❶ ✰ ☞❊➇❴✒❁✄✌✙▲➇❀✼✄✔ ✕❙☞✎✏✓✽✩➇❫✒✣✍✉✸✻✺✜✼✠☞✎✏✓✽✩➇❴✒✾✡ ✟P ☞✎✏✓✽✠✙ ✶✣❃❄✒❑✄ ❵ , and for , . In particular, ✍ ✟ ✕ ✰. , so every interval about 0 contains small subintervals on which ➇ ☎ ☞♦✰✾✒☛✄ ✔✟✞ ✄✂ ☎ ☞❊➇❫✒❨✽✩➇⑥✄✖✰ ☎ ☞❊➇❴✒✗❬q✥➇ ✟ since for all . ✝ ✳☞ ✱⑧✡ ☎ ✒✂✍ ❵ ✳☞ ✱✫✒ ➉❊✽ ☎ ✄ ✁❵ ✝☞ ✆✱✒ for some ✆★✐r✳☞ ✱❍➂ ✱❧✡ ☎ ✒ . As ☎ ✣ ✰ , ✆ ✣ ✰ also, and so ☎ ✙❖✰ For✝☞ ✆✱✰ ✒ ✣✟✞ , ➈❺❵ . ❵ ✠

✠

✠

✠

4. 5.

☎

☛ ✳✱ ☛ ☛ ✳✱

☛ ✳✱

✳✱ ☛ ✎ ☛ ✳✱ ✌ ☞ ✗✗✽✩➇❴✒ ✺✍✓❫☞✎✏❑✡✗✱☎➇❴✒ ✱ ✝ . By l’Hˆopital, the latter quantity tends to

✳✱

✳✱

6. These are obtained ☞ ✍✌❜☞ ✫by ✔✟✞✡✠ ✒ applying ☞ ✫✒ general ✄ ☞ ✡ result ☞ ✣✒ ✁☛formulas ☛ ✠ l’Hˆopital’s rule 2 or 3 times. ☛ ✠ The ☎ ✒❝✍ is that ✠ ☞ ✣✒❨✽ ☎ ✖✄ ✂ is✡ the ✒ operator defined by ✝ ❵ ☛ ✠ ☞ ✫✒✿✄ ❚ ❵ ☞✎✍✑,✏✓where ❵ ❵ ❵ . ✏✎ ❯ ✒ ☞ ✘ ☎ ❯✮❱ ❯✒✑ ❵ ❵ (Explicitly, .) 7. ✝

✺✍✓ ☞✎✏❑✡✖✱✾➇❫✒ ➈ ☞✎✂✤✧ ❫✒ ✄ ☞ ❚

✒

☎

➉❾✄

☛

☛

8.

☛

✠

✂

9.

✗

as

➇✚✣ ✰

.

✒ ✄ ❚ ☞ ❯ ❯ ✡ ❯ ❯ ✒☛✄☞✂ ✧ s✡ ✂❝✧ ❵❯ ❯ ❵✔ ❵ ✕ . The calculation for cross products is similar. ✓✒ P ✒ ✗ ✔✟✞ ✎ ✄✖✗✽✩➇ ☛ ✄ ✝ ✔ ✞✘✗ ✂ ➁ ☛ ✟ ✽ ✄✖✰ by Corollary 2.12. (a) ✝ ✄✂ ✒ ✄✳✏ of (a). (b) This is the case ☞ ❘ ✌ ☞❊➇❴✒❲✄✚✙✉☞✎✏✓✽✩➇❫✒ ✎ P✓✒ ✄✖ ☞ ❘☞☛ P ✌ ☞❊➇❴✒❲✄ ✶ ✄ ✏ ☞❊➇❫✒◗✄ ☞✎✍✤✙✜✽✩➇ ✜ ✒ ✎ P✓✒ ✄✖ (c) ; then ❵ ☞✎✍✑✏✓For ✽✩➇ ✟ ✒✛✙ ☞✎✏✓✽✩➇❴, ✒✮✁❵ ✍ ☞ ✙✜✽✩➇ ✜▲✛✒ ✙✉☞✎✏✓✽✩➇❫✒ ➉ ✩✎ P✓✒ ✖ . Assume ❵ ☞ ✞ ✶ ✍ ✏✓✒❜✡ . The first term in brackets is a polynomial of degree ➈✙ ✄✕✞ ✶ ✡✕✏ ✏✓✽✩➇ ✞ ✶✑✡✥✞ ✏✓✽✩➇ in , and the second term is a polynomial of degree . ☞ ❘ ✎ P ✌❜in ☞ ❘ ✌❜☞♦✰✾✒✦✄ ◆✄❞✏ ✶ ☞♦✰✾✒✘✄✝✰ is (b). Assuming , we have ❵ (d) The case ☞❘ P ☞ ❘ P by inductive ✔✟✞ ✄ s ✜ ☞✎✏✓✽✩hypothesis ➇❫✒ ✎ ✓P ✒ ✄✖ that ❵ ✜ ✄✂ ❵ ✎ ✌ ☞❊➇❴✒❨✽✩➇ . By (c), ❵ ✎ ✌ ☞❊➇❴✒❨✽✩➇✪✢ ✝ where is a polynomial; hence the limit ➇✧✣ ✰ ✞

✠

☛

✠

☎

✂

☎

✟

✂

✂

☎

✂

☎

✂

☎

as

is 0 by (a).

8

☎

☎

2.2. Differentiability in Several Variables 10.

9

➇✥✄ ✏

☎

☞♦✰✣➂✹✙✜✒

is well defined by inspection and ✰✾✒ at ☞✎✏✓✒ ; it is continuous on ✰ since ✙ the two formulas ☞♦agree . By ☞ the✒✑intermediate value theorem, for any continuous ☞♦at✰✾✒ and ☞✎by ✏✓✒ definition of ❵ ✐⑨☞♦✰✣➂✹and ✙✜✒ ❵ ✄ ☎ ✁ ❵ ✔ ❵ ✁ ✁ between and there exists such that , and by the mean value theorem, ☞✝✆✗✒ q⑨✏ ✤✌✏ ✆✘✐⑩☞♦✰✣➂ ✒ ✆✘✐✪☞ ✍ ✏✠➂✬✏✓✒ ☎ ☞ ✒☛✄ ✁ ❵ for some (if ✁ ) or (if ✁ ). ✁ ✁

2.2 Differentiability in Several Variables 1. (a) See the answer in the ✗ back of the✗ text.

☞❊➇✬➂❨➁✫✒✿✄♠☞ ✂ ✪ ☎ ✎ ✖ ➂❄✍✤✙▲➁ ✂ ✪ ☎ ✎ ✖ ✒ ✂ ☞✎✏✠➂✱✍✤✙✜✒☛✄✳☞ ✫➂✚☎✒ ☞ ✜ ✒☎✄✝✆ ✪ ✒☎✄ ✌ ✎☞ ✏✠➂✱✍✤✙✜✒❁✄ ✄✜ ✧✆✦✡ ✪✄ ✙✧ ✲✄ ✟✟✄ ✞ ; ❵ ;❽ . ❵ ☞❊➇✬➂❨➁✫✒ ✄ ☞✎✍✑▼✏ ✏✻➁❦✍r✏☞✫➂❄✏▼✏✻➇✪✍r✏✱✙✜✒❨✽✣☞✡✠✩➇◆✡✚✞▲➁✫✒ ✟ ✂ ☞✎✏✠➂✱✍✤✙✜✒❢✄ ☞ ★✈➂✱✍✑✏✓✒ ☞ ✜ ✒☎✄✝✆ ✪ ✒☎✄ ✌ ☞✎✠✏ ➂✱✍✤✙✜✒❢✄ ✂ ❵ (c) ❵ ; ❵ ; ❽ ✜✄ ✧ ★✘✍ ✪✄ ✧☎✏✤✄ (b)

✂

❵

.

2. (a) See the answer in the back P of the text.

➇❁✡s✞▲➁ ✟ ➁❄✡✉✙ ❍☞❊➇✿✡ ✟ ✒ ✎ P

☞❊➇❄➂❨➁❾➂✁☎✒☛✄♠☞❊➇✿✡ ✟ ✒ ✎

(b)☞✎✏✠✌✔➂✱❵ ✏✠➂❜✰✾✒☞☛✕☎ ✰ ✄ ✏❑✡✥✞✫☞♦✰☎✄ ✙✜✒✂✄✖✌ ✰☎✄✌✠

❵

3.

5. 6.

7.

8.

✌ ;✌ ❵

☞✎✏✠➂✱✏✠➂❜✰✾✒✂✄ ✌

➇❁✡s✞ ➁

✌ ;❵

☞✎✏ ✄ ✠✏ ➂✱✏ ✄ ✈✙ ➂✱✍✤✰☎✄ ✏✓✒✠✍

.

✜ ✟ ✞▲➇ ✟ ➁ ✓P ✒ ✟ ✑ ➇ ✟➁ ✜✒✟ ✄ ✄ ✙▲➇❍❀✡✏➁ ✖ ➇ ✡ ➁ ✡ ➇ ✡ ✑ ✏ ✄✝✆ ✆❺P ✄ ✜✰ ☞ ✪ ✌ ✟ ✌ ➁✑✡✥❂✠✰ ✌ ✏ ✌ ✙✫☞✆✦✡✖✏✓✒ ✌ ✆☞ ✭✡❈✏✓✒ ✟ ✌ , so ✌✎✍ ✌✎✍ ✌ ✰✲✄✡✜✰❬☞✗✄ ✰✜✞✜✒✬✡✒✏ ✟ ✄ ☞✎✍ ✄ ✯✰ ★✜✒❄✡ ❂✠✰ ✄✞✝ ❧✄ ✑ ✍ ✏ ✄ ✙❀✡❲✞✩➉❊✽✠❂✠✰ ✄ ✄ ✰✜✞▼❅ (a) . ✌ ✌ ➈

✒

(b) The coefficient of ✌

4.

✌

is largest.

.

➇ ☎ ✥ ✡ ✙▲➁ ✗ ✡ ✄❈➇ ✂ ✟ ✓ ✍⑩✙▲➁ ✎ P ✂ ✄ ✓ ✡✕☞ ✙▲➇ ✂ ✟ ✓ ✡✥❂▲➁ ✎ P ✂ ✄ ✓ ✒☛✄✕✞▲➇ ✂ ✟ ✓ ✡❲✞▲➁✢✎ P ✂ ✄ ✓ ✄✕✞ ❽ ✁ ❽ ✁ ❽✔✓✕✁ ✁ . ✟ ✟ ✟ ✟ ✳ ✄ ✍ ✯ ➁ ✣ ✽ ❊ ☞ ❍ ➇ ➁ ✇ ✍ ✑ ➁ ❲ ✡ ▲ ✙ ❴ ➇ ✒ ✗ ✕ ✄ ▲ ✙ ➇ ✣ ✽ ❊ ☞ ➃ ➇ ➁ ✇ ✍ ✘ ➁ ✡ ▲ ✙ ❴ ➇ ✒ ❽☎✁ and ❽ ✁ ; the result follows. P P ❩❜☞❜ ✁❑ ✎ ✒❁✄ ✍✯➇❍❩❨ ✁❑ ✎✢✜ ✄❖ ✁❑ ✎ ➇ ❯ ✍ ❚ ☛❩ ❱ P ➇❍❩❊➇ ❯ ✁❑ ✎✢✜ ➇❍❩ , we have ✌ ❵ ❯ ✌ ✌ and hence Since ❽ P ✖ ➇ ❯ ❯ ✄ ✖ ➇ ❯ ✁❑ ✎ ➇ ❯ ✍ ✖ ✖ ➇❍❩❊➇ ❯ ✟ ✁❑ ✎✢✜ ➇❍❩ ✌✔❵ ✌ ✌ ❩ ❯ ❯ ❯ ✄ ✖ ➇ ❯ ✁❑ ✎ P ➇ ❯ ✍ ✖ ➇❍❩✹ ✁❑ ✎ P ➇➃❩❫✄✖☎✰ ✄ ✌ ✌ ❩ ❯ ☞❊➇❄➂❨➁❬✒✗✣q ✟P ➇■ ☞❊➇❄➂❨➁❬✒✔✣ ✰ ☞❊➇✬➂❨➁✫✒✭✣ ☞♦✰✣➂❜✰✾✒ (a) ❵ , so ❵ as . ✄ ☞♦✸✻✺✜✼ ➂❜✄✼ ✔ ✕ ✒ ☞♦✰✣➂❜✰✾✒☛✄ ✔✟✞✛✚ ✂ ☞❊❛✣✸✻✺✜✼ ➂✣❛✣✼✄✔ ✕ ✒❨✽✩❛❁✄✖✸✻✺✜✼ ✟ ✼✄✔ ✕ (b) With ✗ ✵ ✵ , ✙❽ ✘✫❵ ✝ ❵ ✵ ✵ ✵ ✵. ✄♠✰ ✄ ✟P ❃ P ☞♦✰✣➂❜✰✾✒✦✄ ✟ ☞♦✰✣➂❜✰✾✒✭✄ ✰ ❽ ✁ (c) Taking ✵ . If ❵ were differentiable it would ☞♦✰✣or ➂❜✰✾✒☛✵ ✄❈✰ we see that ❽ ✁ ❽ ✘✫❵ for all ✗ ; but this is false. follow that ✜ ✄✕✙ Assume ✟ for simplicity;P the✈q general case follows by an elaboration of the argument as in the proof ✟ ❵ ✈q on ❷ . Given ❡❦✐ ❷ , let ✁✓✙ ✰ be small enough so ❽ ❵ and ❽ of Theorem 2.19. Suppose ☞✂✁▲➂❜❡➃✒ ✢❤✏✕✖✁ , by the mean value theorem we have that ✄t❷ . If ☞♦❡✲✣ ✡ ✢●✒■✍ ☞♦❡➃✒✗✈q✚ ✳☞ ✱ P ✡ ☎ P ✢➂ ✱ ✟ ✡ ☎ ✟ ✒■✍ ✳☞ ✱ P ✠➂ ✱ ✟ ✡ ☎ ✟ ✒✗✱✡⑨ ✳☞ ✱ P ✢➂ ✱ ✟ ✡ ☎ ✟ ✒■✍ ✳☞ ✱ P ✲➂ ✱ ✟ ✒✗ ❵ ❵ ❵ ❵ ❵ ❵ q s☞❜ ☎ P▲✗✡✌ ☎ ✟ ❺✒ ➂ ❡ which implies the continuity of ❵ at . ✄

✄

✄

Chapter 2. Differential Calculus

10

2.3 The Chain Rule 1. See the answers in the back of the text.

☎ ✄✂ ☎ ☎ ☎ ☎

2. (a) and (c) See the answer in the back of the text. ✄ ✎✢✜ ✗ P❄✡ ✙▲➇ ✟ ✽✣☞❊➇ ✟ ✡❈✏✓✒ ✗ ✄✳✍❀✞ ❵ ❵ (b) ❽ ✍ ,❽ ✍ ✙ ✍⑩✞ ✗ ✄ ☞ ❅✑✍⑩❅✜✒ ☞ ✞▲➇❧✡✥✙▲➁❬✒❁✄✖✰ . 3. (a) ❽ ✁ ❽ ✁ ❵ ➇ ✡⑩➁ ✗ ✍ ✄❈➇ ➁✭✡ ☞❊➁✈➇ ✎ P ✒❙✍❇➁✈➇ ✎ P ❽ ✁ ✁ ➈ ❵ ❵✔ (b) ❽ ✁ ➇ ✡◗➁ ✗ ✄❈➇ P✬✡◗➁ ✟ ✄ ❽ ✁ ❵ ❵ ❽ ✓✕✁ . (c) ❽ ✁ 4. 5. 6.

7.

✂☎

✎✢✜

✗

❵

P✬✡✥✙▲➁ ✜

❵

✽✆ ➁✪ ✌ ✡ ✜

.

➇ ✡ ☞❊➁☎➇ ✎ P ✒ ➉❬✍❋➇❍➁ ✍❋➇ ☞❊➁✈➇ ✎ P ❁✒ ✄t➇➃➁ ☞❊➁✈➇ ✎ P ✒ ➉✾✡⑩➁ ✲ ➈ ❵✔ ❵

.

✽ ➇ ❯✒✟ ✳ ✄ ☞ ❚ ➇ ❯ ✟ ✒ ☞✂✁✾✒ ✟ ✽ ✁ ✟ ✄ ☞✂✁✜✒ ✟ ❽ ❵✔ ❵✔ . ☞ ✱➃➂✙✗✓➂ ☞✳✱➃➂✙✗✗✒❨✒ ✳ s✍ ☞✳✱➃➂✙✗✗✒✘✄ ☞❊➇❇✍✖✱✣✒✿✡ ☞❊➁ ✍✣✗✻✒ ❵ ❵ ✟ Both formulas for the tangent plane at amount to ✄ ☎ ☞✳✱➃➂✙✗✗✒ ✄ ✗ ✳☞ ✱❍✙➂ ✗✗✒ where ✟ ❽ ❵ and ❽ ❵ . ✁ ☞❊➇❄➂❨➁❾➂✁☎✒☛✄✖➇❍➁ ✟ ✍ ✝ ✺✍✓➃☞✆❴✍❧✏✓✒ , we have ❽ ☎ ✁ ✄❈➁ ✟ , ❽ ✗ ✁ ✄❈➇ ✟ , P These are✄✚ all✙▲similar; we just do (d). If ➇❍➁ ✲✍t☞✆✲✍t✏✓✒ ✎ ✁ ✂✂✁ ☞✎✍✤✙✈➂✱✍✑✏✠➂✹✙✜✒✯✄✝☞✎✍ ✫➂✱✍✤★✈➂ ✠▼✒ , so and ❽ ✓ ✡✥ ✍ ❍☞❊➇s ✙✜✒■✍ ★✫☞❊➁ ✡✖✏✓✒❙✡ ✠✣✆☞ ✍⑩ ✙✜✒❁✄✕✰ ✠ ⑧✄✡▼➇ ✖ ✡ ★▲➁✑✡✥✞✠✰ . Hence the tangent plane is given by , or . ✍✂☞❊➇❴✒☛✄ ✁ ☞❊➇❄➂❨➇❄✆ ➂ ✄✆✄✆▲✄ ➂❨➇❴✒ ✍ ☞❊➇❴✒☛✄ P ✁ ✡ ✟ ✁ ✡⑩✧✗✧✗✧❨✡ ☛ ✁ ❽ ❽ ❽ , these derivatives being evaluated We☞❊➇✬have , so ➂❨➇❄✆➂ ✄✆✄✆✩✄ ➂❨➇❴✒ ❽

✁

✽ ➇ ❯ ✄✖➇ ❯ ☞✂✁✾✒❨✽ ❽ ❵✔

✁

, so

❚ ☞

❽

✁

.

at

2.4 The Mean Value Theorem 1. (a) The conclusion that point line ✄ ☞♦❣✇✍ ❡➃✒❨is✽✫ ❣⑩ ✍ ❡✬the directional derivative ❽ ✘✫❵ vanishes at some ✍✂☞❊❛❜✒❝✄ ☞❊❛✎❣❦on✡✌the ☞✎✏✭✍⑩ ❛❜✒✔❡➃segment, ✒ . (Apply Rolle’s theorem to the function on the where ✗ ✰✣➂✱✏✮➉ ❵ interval ➈ .) ❡★✐ ☞♦❡➃✒■✄✕③ ✄ ✆ ✆ ✂ ❷ ❵ ❵ ❾ ❽ ❷ ❵ such that . (Suppose on . If (b) The conclusion is that there is a point ③ ✂ on ❷ then ❵ on ❷ . Otherwise,❡★either the③ minimum of ❵ on ❷ [which exist since ✐ the maximum☞♦❡❾or✒✿✄✕ ❷ is compact] is achieved at a point ❷ , and then ✂ ❵ .)

☎✄

✆✄

☞♦❣■✒✿✍

☞♦❡➃✒✿✄

P ☞♦④☎✒✻☞ ✗✓P❑✍ ❬✱ P ✒❁✄✖✰

❵ ❽ ❵ 2. (a) If ❷ is convex, we can apply Theorem 2.39 to get ❵ . ✄✕✙ ☞✎✍✑✏✠➂✱✏✓✒✂②✪☞✎✍✑✏✠➂✱✏✓✒ ✰✣➂✱✏✓✒ ➁ ❷ ➈ : let be the square with the segment on the (b) An example with ✟ ✟ ☞❊➇❄➂❨➁❬✒❁✄✌✰ ➁⑥q⑤✰ ☞❊➇✬➂❨➁✫✒☛✄ ✍✯➁ ➁✍✙⑤✰ ➇ ✕t✰ ☞❊➁✫✒❁✄✕-axis ➁✟ on by if , if and , and if removed. Define ❵ ❷ ❵ ❵ ❵ ➁✧✙✥✰ ➇ ✙ ✰ ✰ ☞✎✍✯➇✬➂❨➁✫✒✭✄ ❶ ☞❊➇✬➂❨➁✫✒ ➁✧✙ ✰ ❵ and . Then ❽ ❵ on ❷ , but ❵ for .

☎ ✝✄

2.5 Functional Relations and Implicit Functions: A First Look

✂

1. (a) See the answer in the back of the text. ➇ ▼➇❧✡✖☞ ✙ ✤✡ ✎ ✓ (b) Differentiation ➁ ✗ in⑧✄✳gives ✍❀❅▲➁✫✽✣☞ ✙ ✦✡ ✩✎ ✓ ✒ . tiation in gives ❽

✂

✒ ☎ ❧✄✖✰ ❽

➁ ✟ ✡✧▼➁❦✄r➇ ✟ ✍❲✙▲➇

2. (a) ✛ ➇Elimination ✟ ✍ ✙▲➇ ✡ of gives ➁✫✽ ➇⑥✄⑨❆ ☞❊➇✬✍✑✏✓✒❨✽ ,✛ so➇ ✟ . Therefore ➇ (b) in gives ✙✤✡ Differentiating ▼➁ ➁ ✄ ☞❊➇♣the ✍ original ✏✓✒❨✽✣☞❊➁✲✡ equations ✙✜✒ ✄✕✙✤✡✌ ❍☞❊➇ , so and

✌ ✌

, or ❽

☎ ✲✄✳✍

▼➇❴✽✣☞ ✙ ❀✡

✂ ✎

✓

✒

. Likewise, differen-

➁❦✄ ✍✤✙✦❆ ✛ ➇➃✟❤✍⑩✙▲➇s✡ ❢✄♠✙▲➇⑥✍ ★✦❆ ✍⑩✙▲➇s✡ ☎✽ ➇⑥✄✕✙✜and ❆ ❍☞❊hence ➇✬✍✑✏✓✒❨✽ ✛ ➇ ✟ ✍✪✙▲➇ ✡ , and ✌ ✌ ✄✕✙✿✡ ▼➁ ✄✕✙▲➇ ✍★✙▲➁✈➁ ✙▲➇ ✍★✙▲➁✈➁ ✄ . Thus ✍ ✏✓✒❨✽✣☞❊➁ ✡✥✙✜✒ and .

.

2.6. Higher-Order Partial Derivatives

11

❂▲➁ ✪ ➁ ✡r☞❊➁

3. Differentiating the equations ➁ gives which are linear equations in and 4.

✗

✓

❛ ✟ ✡❈✙ ▼❛✘✄❖✰ ✡

and

✙▲➁✈➁ ✡

✄⑦✙▲❛ ✜

,

to be solved simultaneously.

☞❊➇✬➂❨➁✫✒ ✄❈➇ ✟ ✡♣✞▲➁ ✟ ✄✞✝ ☎ ✄✖✙▲➇ ☞❊➇✬➂✁☎✒ ✄❈➇ ✟ ✡✉✞✫☞❊➇ ☎✒ ✟ ✄✞✝ ☎ ✄✕✙▲➇☛✡♣❅▲➇ ✟ If are IVs, . If are IVs, . ✟ ✄✌❃ ✁ ☞ ✽ ✒✗ ☞ ✽ ✒✗ ✁ ✁ For✄✕✙✠❽ ❃ ☎ ✁❬☞✂❽ ➃ ,✽ is ✄✖ implicitly ✁ ✡ ✒ , just use ✁✾☎ ✽ ✄ ✍☞✾✁ . ✽✣For ☞ ✙ ✁➃✡ ❽ ☎ ✒ ❽ ❃ ✁ ✟ ✡♣✙✠❃ a✁❬function ☞ ✁✾✽ ✒ of ✄✖and ❃ ✁ ✟ ❷ ✍❧;✙✠the ❃ ✁ ✟ equation ✽✣☞ ✙ ✁➃✡ yields , so ❷ ✒ ❽ ❽ ❽ ☎ ❽ ❽ ❽ ☞ ✽ ✒✗ ✁ . For ❽ ☎ ✽ ❽❾❷ ✄✕❃ , ✁ ☎ ✟ ☞ and✽ are in ✒ implicitly ✏✭✄✕✙✠❃ functions ✁❬☞ ✽ ✒of ❷ and ; differentiating ✽ ✄ ✁✾✽✠✙ the given ☞ ✽ equations ✒✗ . For ✏ ❽❴✄✳ ❷ gives ❽ ☎ ❽❾❷ ❽ ❽❴❷ and✁ ❽ ❽❴❷ , whence ❽ ☎ ❽❴❷ ❷ ❃ ❽ ✁ ☎ ✟ ☞ , ❷ ✽ and✒ ☎ and ; differentiating the given equations in ☎ gives ❽ ❽ ☎ are implicitly functions of ✽ ✄✕✙✠❃ ✁❬☞ ✽ ✒ ✽ ✄✕✙✜✽ ✁ and ❽❴❷ ❽ ☎ ❽ ❽ ☎ , whence ❽❾❷ ❽ ☎ . ➇❫✽ ➁ ➇ ➁ ✍ ✽ ☎ ❽ ❽ is the derivative when is considered as a function of and ; by (2.44) it equals ❽ ➁❬✽ ✄✳✍ ✽ ☎✽ ➇❦✄✳✍ ☎ ✽ ✍✑✏ ❽ . Likewise ❽ ❽ ❽ ❽ and ❽ ❽ ❽ ❽ . The product of these quantities is . ✁

✁

✁

☎

☎

5.

✡⑤➁ ✒ ✂

☎

☎

☎

☎

✁

☎

☎

☎

☎

☎

☎

☎

☎

☎

☎

☎

☎

✗

✁

6.

✁

✗

✁

7. Taking say

☎

and

✂

✁

✁

✁

✓

✂

✓

as independent variables means that

✄

✙

and

are determined as functions of

☎

and ,

✳✄✆☎✭☞ ➂ ✂ ✒ ➂ (*) ☎ ✍ ✂ ✡ P ✍✥✍ ✂ ✟ ☎t✡ ✄❉✰ and the equation ❽✞✝✟✄ then becomes ❽ ❽✡✠ ❽ ✄ . Now ☞ P ✍■✒✻take ☞ ✒ and ✂ as the independent variables. ☞ ❽ ✌ ☛ ✄ ❽ ✍ ❽ ☛ ☎ Differentiating (*) with respect to gives and ✏◆✄ ☞ P✎☎✯✒✻☞ ✒ P★✍❖✄ ✽ P✎☎ ✄ ✏✓✽ ✂ ❽ ✄ ☞ ❽✍☛ P ☎❤☎ ✒✻☞ , so ❽ ✒✂✡ ☎ ❽✏☛✌✄ ✏❽ ☛ ☎ ☎ ✄ and✍ ❽☞ P ☎❤✒✻☞ ✍❽ ✒★ ☛ ☎ . Differentiating (*) with respect to ✰t ✄ ✍ ✽ gives ❽ ✠ ✰☎ ❽ ✟ , so✡ ✂ ❽ ✟ ❽ ❽✠ ☎ ❽ ✠ ☎ ❽ ☛ ☎ . Substituting these into P ✍❇✍ ✂ ✟ ☎✪❽ ✡ ✚✄✖ ❽ ❽ ✡❽ ✠ ☎ ✡ ✏❽ ☛ ☎ ✄✖✰ . gives ❽✏☛✌✄

✄

✙

✙

✄ ✍✂☞ ➂ ✂ ✒ ➂ ☎ ✄❉✰

✙

✙

✙

✙

✙

✙

2.6 Higher-Order Partial Derivatives 1. These are routine exercises in elementary calculus. Half of the calculations were performed in Exercise 1, 2.2.

✂

✄ ♦☞ ✸✻✺✜✼ ✒ ☎ ✡ ☞♦✼✙✔ ✕ ✒ ✒✑ ✄ ✍✲☞♦✼✄✔ ✕ ✵ ✒ ❵ ☎ ✡ ☞♦✸✻✺✜✼ ✵ ✒✻☞ ❽➃❵ ☎ ✽ ❽➃✵ ✒✯✡ and hence ✵ ❵ ✵ ❵ ▲✽ ✒ ✄ ✍ ☞✂✁✿✼✄✔ ✕ ✒ ☎ ☎ ✡ ☞✂✁✿✸✻✺✜✼ ✒ ☎ ▲✽ ✄ ☎ ✽ ✵ ❵ ✵ ❵ and ❽❾❵ ❽➃✵ ✍ ☞✂✿✁ ✄✼ ✵ ✔ ✕ ❵ ✒ ☎ ❑✡✌☞✂✵ ■✁ ✸✻✺✜❽❾✼ ❵ ✒ ❽➃✵ ; furthermore, ❽➃❵ ❽❍✵ ✵ ❵ ✵ ❵ . ✟ ✄⑨✙ ☎ P●✡✌☞♦✼✄✔ ✕❝✞▲➁❬✒ ☎ ✟ ✡✧▼➇ ✜ ☎ ✜ ✡✖✏✱✙▲➇ ✟ ✜ ✄✚✙ P●✡✕☞♦✼✄✔ ✕✤✞▲➁❬✒ ✟ ✡ ▼➇ ✜ ✜ (a) ❽ ☎ ❵ ❵ ❵ . Hence ❽ ☎ ❽ ❵ ❽ ❵ ❽ ❵ ❵ . The ☎ ❵ ❯ are just like ❽ ☎ but with an extra subscript ✌ on the ❵ ’s: ❽ ☎ ❵ P❑✄✕✙ ❵ P❨P❴✡⑤☞♦✼✄✔ ✕❝✞▲➁❬✒ ❵ P ✟ ✡ ❽ derivatives ▼➇ ✜ P ✜ ✄ ✙ P☛✡✳☞♦✼✄✔ ✕❀✞▲➁✫✒ ✟ ✡ ☎ ❽ ❵ , etc. Now it’s just a matter of collecting terms. Similarly, ❽ ❽ ❵ ❽ ❵ P✟ ✞✫☞♦✸✻✺✜✼❾✞▲➁❬✒ ✟ ✡✚☞ ✞■✸✻✺✜✼❾✞▲➁❬✒ ✟ ✞▲➇✭✸✻✺✜✼➃✞▲➁ multiplying ❵ in the answer in the book should be ❽ ❵ , etc. (The ❅▲➇❀✸✻✺✜✼❴✞▲➁ ❵ .) P ✄ ✍✤✞ ✂ ☎ ✎✢✜ P ✡ ✙▲➁ ✜▼☞❊➁ ✪✫✡ ☎✒ ✎ ✓P ✒ ✟ ✜ ☎ ❽ P ✄✳✍❀✞ ✂ ☎ ✎✢✜ ☞ ✂ ☎ ✎✢✜ ❵ P❨P ✡♣✙▲➇■☞❊➇ ✟ ✡❦✏✓✒ ✎ ❵ P ✟ ✒✾✍ ✓✒ P . Hence (b) ❽ ❵ ❵ ❽ ✟ ✄ ✍✤✞ ✂ ☎ ✎✢✜ ✞ ✂ ☎ ✎✢✜ P✦✡♠✙▲➁ ✜ ☞❊➁ ✪ ✡ ☎✒ ✎ ✟ ☞ ✂ ☎ ✎✢✜ P ✜ ✡❖✙▲➇■☞❊➇ ✟ ✡ ✏✓✒ ✎ ✟ ✜ ✒ P✦✡ ❵ ❵ ❽ ❵ . Similarly, ❽ ✓P ✒ ✓P ✒ ❵ ✟ ✟ ✟ ☎ P ☎ ❨ P ✴ ✂ ✎✢✜ ✡♠✙▲➁✏✠✜ ☞❊➁ ✪⑧✡ ☎✒ ✎ ✡ ✙▲➁ ✜✜☞❊➁ ✪⑧✡ ☎✒ ✎ ➉ ✜ ✴ ✂ ✎✔✓ P ✍ ✜ ✓P ✒ P ✒ , which works out to be ❵ ❽ ❵ ❽ ➈ ❵ ✏✱✙▲➁ ✜ ✂ ☎ ✎✢✜ ☞❊➁ ✪ ✡ ☎✒ ✎ ✟ P ✜ ✡ ▼➁ ✓ ☞❊➁ ✪ ✡ ☎✒ ✎ ✜ ✜ ✡ ✴ ✂ ☎ ✎✢✜ P✬✡✕☞ ✙▲➁ ✓ ✡✥✙ ▼➁ ✟ ✒✻☞❊➁ ✪ ✡ ☎✒ ✎✢✜ ✟ ✜ ❵ ❵ ❵ ❵ ❵ . ✄ ❊ ☞ ➇ ✡ ❊ ☞ ❬ ➁ ❨ ✒ ✒ ♣ ✄ ❊ ☞ ❢ ➇ ✡ ❊ ☞ ✫ ➁ ❨ ✒ ✒ ❊ ☞ ✫ ➁ ✒ ✄ ❊ ☞ ♣ ➇ ✡ ❊ ☞ ❬ ➁ ❨ ✒ ✒ ♣✄ ☎ ☎ ☎☎ ✠ ✠ ✠ ✠ We☞❊➇❧ have ; hence and ; also ✡ ☞❊➁✫✒❨✒ ☞❊➁✫✒ . The result follows. ✠ ✠ ❚ ❯ ❘ ➇ ❯ ➇➃❘ ❯ ❘ ☞❊✁✬✒ ✱❴✳ ☞ ✱⑧✍⑤✏✓✒ ☞❊✁✬✒ ☞ ✟ ✽ ❛ ✟ ✒ ☞❊❛✔✁✬✒✗ ❱ P ❽ ❽ ❵ and ❵ are both equal to ✌ ✌ ❵ .

2. From ☞♦✸✻✺✜✼ ✒ Example ✲✡ ☞♦✼✄✔ ✕ 4, ✒✻☞

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3.

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✆

Chapter 2. Differential Calculus

12

✄ ✙✄✂ ✡✕✙▲❛ ✗ ✄ ✙ ✡ ✆✂ ✟ ✡ ▼❛ ✟ ✗ ✗ ✚✘✄❉✍✤✙▲❛ ✡✌✄✙ ✂ ✗ ✚ ✚✘✄ ✍✤✙ ✡ ❵✡ ✆✂ ✟ ✗ ✗ ❵ , so ✁✁☎ ❵ ❵ ❵ . Also ✁ ❵ ❵ , so ✁ ❵ ❵ ❵ . The result follows. ✟ ✚ ✚●✄ ✆ ☞❊➇♣✍ ✆✮❛❜✒❙✡ ☞❊➇❧✡ ✆ ❛❜✒ ➉❴✄ ✆ ✟ ✁ ➈❺❵✔ ✁ . ✁✾✽ ➇⑥✄❈➇❫✽ ✁ ✄✳✍✯➇ ✁ ✎✢✜ ✝☞ ✆✮❛✿✍ ✁✾✒●✍✇➇✢✁ ✎ ✟ ✝☞ ✆✮❛✿✍ ✁✾✒ We have ❽ , so , and hence ❽ ✟ ✟ ✟ ✟ ✄✳✍ ✝☞ ✆✮❛✿✁ ✍ ✁✾✒ ✡ ✞▲➇ ✝☞ ✁ ✆ ✄ ❛✿✍ ✁✾✒ ✡ ➇ ✝☞ ✁ ✆ ❛✿✍ ✁✾✒ ✍ ☞✝✆✮✁▼■❛ ✟ ✍ ✁✾✒ ✡ ✙▲➇ ✁☞✝✆✮❛✿✍ ✁✾✒ ✡ ➇ ✝☞ ✁ ✆✮❛✿✍ ✾✁ ✒ ✄ ✜ ✪ ✪ ✜ ✗ ✗ ➇ ➁ ➇ ✟ ✡◗➁ ✟ ✡ ✟ ✄ ✁ ✟

✁ 6. ✁ ▼❛ ✟

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7. 8.

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and

✓ ✓

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are the same, with

✍ ✞ ☞✝✆✮✁ ❛■✍ ✁✜✒ ✡ ✞ ☞✝✆ ✁ ❛❁✍ ✁✜✒ ✡ ✠

✠

✜

replaced by

and . Since

, adding these gives

☞✝✆✮❛✿✍ ✁✾✒ ✍ ✞ ✝☞ ✆✮❛✿✍ ✁✾✒ ✡ ✙ ✝☞ ✆ ❛✿✍ ✁✾✒ ✡ ✁ ✟ ✁ ✟ ✁ ✟ ✠

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☞✝✆✮❛✿✍ ✁✜✒ ✄ ✁ ✠

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✆ ✎

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✚ ✚✜✄ ✁

P ☞✂✁✾✒❍✍ ✄ ➇ ❯ ✁ ✎ P ✂☞ ✁✾✒ ✄ ✁ ✎ P ✂☞ ✁✜✒❍✍ ➇ ❯ ✟ ✁ ✎✢✜ ☞✂✁✾✒☎✡❦➇ ❯ ✟ ✁ ✎ ✟ ☞✂✁✾✒ ✴ ✁✸✎ ❯ P❈ ❵ , so ❯✔❯ ❵ ❵ ❵ . Adding these up gives ❵ ✁ ✎ ☞✂✾✁ ✒❫✡ ☞ ✾✁ ✒ ✂ ❚ ➇ ❯✟ ✄ ✁ ✟ since . ❵ ❵ ☞ ✒ ☞ ❘ ✌ ✄ ❚ ❯✮❘ ❱ ✎ ❯❘ ✑ ☞ ❯ ✌ ☞ ❘ ✎ ❯ ✌ 10. In one variable, the assertion is that ❵ ❘ ✡ ❵ ✎ ❘ ✄ , which ❘☞☛ P is proved just like the ✎ ✎ ✶ P ❯✎ ✑ ❯ ✑ .) The ✄✟ -variable binomial theorem. (Induction on , using the fact that ❯ ✑ ✞★✡✡✠☞result ☛ follows by applying the one-variable result in each variable separately; the facts that ✝ ✄ ✌ ✞ ✡ ✍ ✠ ✌ ✄ P ✗ ✧ ✗ ✧ ✧ ❯ ❯ ❯ ☛ ✁

✁

9.

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and ✝✏✎

for all an induction on .)

✝ ✎

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✎ make everything turn out right. (This could be phrased as

✟

11. This follows by applying the one-dimensional binomial theorem in each variable as in the preceding problem.

2.7 Taylor’s Theorem

☞❊➇❫✒✿✄❈➇ ✟ ➇✤✍★☞❊➇✤✍ P ➇ ✜ ✡❋✧✗✧✗✧✮✒ ➉❫✄ P ➇ ✄ ✡❋✧✗✧✗✧ ☞❊➇❴✒❁✄ ☞✎✏❬✡♣➇✂✡❋✧✗✧✗✧✻✒✾✍⑥✏✮➉ ☞✎✏❴✍s✙▲➇ ✟ ✡❋✧✗✧✗✧✮✒☎✍⑥✏✮➉ ✟ ✄ ✄ ➈ ➈ ➈ , and ▼➇✢✪✂✡✖✧✗✧✗✧✮✒❁✄ ▼➇ ✡t✧✗✧✗✧ . ☞❊➇❫✒❨✽ ☞❊➇❴✒❑✄ ☞ P ➇ ✄ ✡❈✧✗✧✗✧✮✒❨✽✣☞ ▼➇ ✄ ✡t✧✗✧✗✧✻✒✿✄ ☞ P ✡❈✧✗✧✗✧✮✒❨✽✣☞ ✦✡✖✧✗✧✗✧✮✰✒ ✣ ✟ P ✪ . (b) ❵ P☎ ✟ ✡ P☎ ✜ P ☞ ✜ ✌❜☞❊➇❫✒❑✄⑨✙▲➇ ✎✢✜ ☞ ✪ ✌✹☞❊➇❴✒✗☎✄ ☞❊➇❫✒❑✄✕➇ ✎ ☞❊➇❴✒☛✄❖✍✯✛➇ P✎ ✟ ✙☛P✟✆ ☞ ☎ ✒✂✄ ☎ ✍ ✜ ✟ ✜ , , and , so ; also (a) ❵ ❵ ❵ ❵ ✠✍⑩❅▲➇ ✎✠✪ ✈q⑤✴▼❅ ➇ ✍⑤✏✾✣q ✟ ⑨✄✕✴▼❅✜✽✁ ✄ ✎ for , so . P P P☎ ✍ P☎ ✟ ✡ P ☎ ✜ ✓P ✒ ☞ ✜ ✌✹☞❊➇❫✒✂✄ ✜ ✛ ☞❊➇❴✒❤✄ ✟ ➇ ✎ ✟ ☞❊➇❴✒❤✄⑦✍ ✛➇ ✎✢✜ ✒ ✟ ➇ ✎ ✄✄✒ ✟ ✙☛P✟✆ ☞ ☎ ✒❤✄⑦✏❤✡ P ; ✜ ✞ ✟ ✞ ✪ ,P ❵ ✄ , and , so (b) ❵ ☞ ❵ P ✄ P P ✄ ✒ ✒ ✒ ✒ ✟ ✟ ✟ ✟ ➇ ✣q ✙ ❢ ➇ ✍ ✾ ✏ ✈q ⑨ ✄ ✙ ✽ ✁ ✕ ✄ ✭ ❂ ✩ ✧ ✙ ✪ ✌ ☞❊➇❫✒✗✾✄♠▲✍ ✎ ✎ ✏ ✏ ✏ ✏ P P P ✟ , so ✎ also ❵ for . ☞ ✟ P ✆ ✜ ☞ ☎ ✒✂✄ P ✍ P P ☎ ✡ ☞❊➇❫✒☛✄❖✍ ☞❊➇s✡ ✞✜✒ ✎ ☞❊➇❴✒✂✄⑨✙✫☞❊➇✉✡✥✞✜✒ ✎✢✜ ✜ ✌✹☞❊➇❫✒❑✄♠✍✤❅✫☞❊➇✉✡ ✞✜✒ ✎✠✪ ✙ ✟ ,❵ (c) P ☎ ❵✟ ✍ P ☎ ✜ P , so ✄ , and ❵ ✄ ✄✪ ☞ ✟ ✄ ; also ❵ ✪ ✌ ☞❊➇❫✒✗✜✄♠ ✙ ❍☞❊➇s✡✥✞✜✒ ✎ ✣q⑤✙ ❍☞ ✞ ✄ ❂✜✒ ✎ for ➇ ✍ ✏✾✫q ✟ , so ⑨✄✳☞ ✞ ✄ ❂✜✒ ✎ . ✪ P ➇ ✜✜☎✄✝ ✑ ✆ ☞❊➇❴✒✗❍q✳ ➇✿ ✄ ✽✠❂ ✄ ✔✼ ✄✔ ✕✯➇ ✍✪➇❢✍ ✔✼ ✄✔ ✕ ☞ ✌ ☞❊➇❴✒✗✈✄✝✔✼✙✔ ✕❤➇■❍q✚✏ ✪ , and By ☞ ✟P ❃❄Lagrange’s ✒ ✄ ✽✠❂ ☛⑦☎✰ ✄ ✔✰ formula, ✠✠✎ ✴✠ ✑ ✆ ✟ ✎ P▲☞❊➇❫✒✗❫✄ ✑ ✆ ✟ ✎ since ☞❊➇❴✒✗●q❞ ➇■ ✟ ✮ ☛ P ✽✣☞ ✙ ✦❞✡r✏✓✒ P ✎ ✰☎✄ ✰✫✏ ✮ ✮ ✎ . In general, we have . This is ➇■✫q ✟ ❃ ✦ ✤⑤✞ less than for provided ; so the 5th order polynomial suffices.

❵ ✧✗✧✗✧✮✒✻☞ 1. (a) ☞❊➇ ✡✖

✓

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2.7. Taylor’s Theorem

☞❊❛❜✒✿✄✄✂ ✎ ✣❘ ☞❊❛❜✒ ❛ ✆❘ ☛ P ✽✣☞ ✶✯✡ ✏✓✒ ❛❤✐ ✰✣➂✱✏✮➉ ❘☎☞❊➇ ✟ ✒ ➇✿✣q ➈ ✌ the remainder is bounded by for , so ➇ ✟ ➇❫✽✣☞ ✶ ⑨ ✡ ✏✓✒ ✄ ✏✓✽✣☞ ✶ ✡✚✓✏ ✒ ☞ ✙ ✶✉✡⑤✞✜✒ ✰☎✄ ✰▼✰▼✰✜❂ ✶ ✤r❅ , so to three decimal P ✌ ✂ ✎ ☎✄✖ ➇★✄ ❚ ☞✎✍✑✏✓✒ ❘ P ➇ ✟ ❘ ❴➇ ✽✩✶ ✄. This ✏✓✒ ❘ ✽✩than ✶ ☞ ✙ ✶✲✡✖✏✓✒❁✄✕for✰☎✄ ✁ ❚ ☞✎is✍✑less ✌ ✌ places . P P P ➇❀✼✙✔ ✕❙☞❊➇❧✡◗➁❬✒❁✄❈➇ ☞❊➇❧✡◗➁❬✒■✍ ☞❊➇❧✡❲➁✫✒✫✜☛✡❈✧✗✧✗✧❜➉❴✄❈➇ ✟ ✡◗➇➃➁⑧✍ ➇✢✪❝✍ ✟ ➇ ✜ ➁❧✍ ✟P ➇ ✟ ➁ ✟ ✍ P ➇➃➁ ✜☛✡❈✧✗✧✗✧ . ➈ (a) ✗ ☎ ✂ ✸✻✺✜✼✓☞❊➇ ✟ ✡⑧➁ ✟ ✒❁✄ ✏✾✡⑧➇➃➁❫✡ ✟ P ☞❊➇❍➁❬✒ ✟ ✡ ✧✗✧✗✧✎➉ ✏✫✍ ✟P ☞❊➇ ✟ ✡⑧➁ ✟ ✒ ✟ ✡ ✧✗✧✗✧❨➉❾✄ ✏✾✡⑧➇❍➁■✍ ✟ P ☞❊➇✢✪☎✡⑧➇ ✟ ➁ ✟ ✡⑧➁ ✪✓✒✗✡ ✧✗✧✗✧ . (b) ➈ ➈ ✚

4. For P ❵ ✟ ❘✆☛

✆

✑

✄

✎

✌✠

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13

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✂☎✎ ✟✗ ✁ ✏❑✡◗➇❾✟✯✍◆➁ ✄ ✎✏❑✡✕☞❊➇

✄✂

✍⑩✙▲➁❬✒❙✡ ✙ ✏ ☞❊➇❢✍⑩✙▲➁✫✒ ✟ ✡ ❅ ✏ ☞❊➇ ✍⑩✙▲➁❬✒ ✜ ✡ ✙ ✏ ☞❊➇ ✍⑩✙▲➁❬✒ ✪ ✡❈✧✗✧✗✧ ② ② ❪✏❑✡✌☞❊➁⑧✍✇➇ ✟ ✒❙✡✕☞❊➁❧✍✇➇ ✟ ✒ ✟ ✡✕☞❊➁❧✍✇➇ ✟ ✒ ✜ ✡✌☞❊➁❧✍◆➇ ✟ ✒ ✪ ✡❈✧✗✧✗✧ ✄ ✏❑✡◗➇ ✍⑩✙▲➁✑✡ ✙ ✏ ➇ ✟ ✍⑩✙▲➇➃➁ ✡✥✙▲➁ ✟ ✡ ❅ ✏ ➇ ✜ ✍✇➇ ✟ ➁✑✡✥✙▲➇➃➁ ✟ ✍ ✞ ➁ ✜ ✡ ✙ ✏ ➇ ✪ ✍ ✞ ✏ ➇ ✜ ➁✑✡◗➇ ✟ ➁ ✟ ✍ ✞ ➇❍➁ ✜ ✡ ✞✙ ➁ ✪ ✡✖✧✗✧✗✧ ❢② ❪✏☛✡◗➁❧✍✇➇ ✟ ✡◗➁ ✟ ✍⑩✙▲➇ ✟ ➁✘✡❲➁ ✜ ✡◗➇ ✪ ✍⑩✞▲➇ ✟ ➁ ✟ ✡◗➁ ✪ ✡❈✗✧ ✧✗✧ ✄⑧✏✂✡◗➇♣✍✇➁❧✍ ✙ ✏ ➇ ✟ ✍✇➇❍➁✑✡◗➁ ✟ ✍ ❂❅ ➇ ✜ ✍ ✙ ✏ ➇ ✟ ➁✘✡◗➇➃➁ ✟ ✍ ✞ ✏ ➁ ✜ ✡ ✙ ✏✱✞ ➇ ✪ ✍ ❅ ✏ ➇ ✜ ➁ ✍ ✙ ✏ ➇ ✟ ➁ ✟ ✍ ✞ ✏ ➇❍➁ ✜ ✡ ✞ ✏ ➁ ✪ ✡✖✧✗✧✗✧ ✄

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➇❢✄✕✞❤✡ ☎ ➁s✄✳✏■✡ ✶ P ☞❊➇❄➂❨➁❬✒✿✄✕✞❤✡ ☎ ✍❇✸✻✺✜✼❍❃ ✶✦✡ P ☞ ✞❤✡ ☎ ✒ ✺✍✓❴☞✎✏✿✡ P ✶❬✒✿✄✕✞✂✡ ☎ ✍◗☞✎✏✂✍ P , , we have ❵ ✟ ✡❈✧✗✧✗✧✮✒❄✡✕☞ ✞✤✡ ☎ ✒✻☞ ✶ ✍ ✟ ✶ ✡ ✶ ✜ ✡❈✧✗✧✗✧✻✒❁✄✕✙❝✡ ☎ ✡✥✞ ✶ ✡ ✟ ☞♦❃ ✟ ✍ ✞✜✝ ✒ ✶ ✟ ✍ ✟ ☎ ✶ ✟ ✡✘✶ ✜ ✡t✧✗✧✗✧ ✟ ✜ . ➇ ✄ ✏✲✡ ☎ ➁ ✄ ✙✉✡ ✶ ✚✄ ✏ ✡ ✠ ☞❊➇✬➂❨➁➃➂✁☎✒⑩✄ ☞✎✏✲✡ ☎ ✒ ✟ ☞ ✙✉✡ ✶❬✒✤✡ ☞✎✏✲✡ ✠ ✒✇✄ 7. With , we have ❵ ✞❀✡✧ ✶✲✘ ✡ ✶⑧✧ ✡ ✠❍✡⑤,✙ ☎ ✟ ✡⑤✙ ☎ ✶⑧✡ ,☎ ✟ ✶ , with no remainder. (The remainder is also known to vanish since all 4th order derivatives of ❵ vanish.) ☞ ✶ ✍✥✏✓✒ 8. A -fold application of l’Hˆopital gives ☞✳✱✘✡ ☎ ✒■✍ ✙ ❘✣☞ ☎ ✒ ✄ ✔✟✞ ☞ ❘ ✎ P ✌❜✳☞ ✱✑✡ ☎ ✒■✍ ☞ ❘ ✎ P ✌✹☞✳✱✫✒●✍ ☞ ❘ ✌✹☞✳✱✣✒ ☎ ✔✞ ❵ ❵ ❵ ☎ ❵ ❘ ✠✝ ✠✝ ☎ ✂ ✂ ☞ ❘ ✎ P ✌ ☞✳✱✑✡ ☎ ✒■✍ ☞ ❘ ✎ P ✌ ☞✳✱✫✒ ✄ ✠ ✔✟✞ ❵ ✍ ☞ ❘ ✌ ☞✳✱✣✒❁✄✕✰ ❵ ☎ ✝✂ ❵ ☞ ❘ ✌❜☞✳✱✫✒ by definition of ❵ . ☞✳✱✲✡ ☎ ✒❀✄ P ☞✳✱✫✒●✡ ☞ ❘ ✌ ☞✳✱✫✒ ☎ ❘ ✽✩✶ ✡ ❘✈☞ ☎ ✒ ☎ 9. We have ❵ , and by ☞ ❘ ✌✹Corollary ❘✈☞ ☎ ✒✗❑q ❵ ✟ ☞ ❘ ✌ ☞✳✱✣❵ ✒ ☎ ❘ ❺✽✩✶ ✶ ☞✳✱✣✒ ✙ ✰ 2.60, for ☞✳✱❧sufficiently ✡ ☎ ✒❤✍ ☞✳small ✱✫✒ ✤ ❵ ☎ Thus for ☞ ❘ even, if ❵ we have ❵ ❵ we P ☞ have ❘ ✌ ☞✳✱✫✒ ☎ ❘ ✽✩✶ ✙⑤✰ ☞✳✱✘✡ ☎ ✒■✍ ☞✳✱✫✒✝✕⑤✰ ✌ ☞✳✱✫✒☞✕ ✰ ☎ ✟❵ ✶ ❵ ☎ ❘ ☎ for for small , and likewise☞✳✱✘if✡ ❵ ✒✿✍ ☞✳✱✣✒ we have ❵ ✄✖✰small . ☎ ❵ changes sign along with at . For odd, the same reasoning shows that ❵ P ❘ P ❘☎☞ ■✒✗✫q✘✶ ❚ ❱ ❘ ☞❜ ❺✽ ✒ ☞✎✏✤✍◆❛❨✒ ✎ ✦❛ ✂ ❛ 10. By (2.70) we have ❤ ■q ❤ ❘ ✬✄ ✶ ✌ . Since each ❘✈☞ ■✒✗component ■q ✂ ✆❘ ☛ of P ☞✎✏❝✍◆we is less than for . Hence ❘ have P value, P❛ ❛ ✄ ✶ ❚ in absolute ❜ ❛ ✒ ✎ ✎ ❘ ❱ ✌ . where P ☞♦❃ ✶❬✒ ✟ 6. Setting

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Chapter 2. Differential Calculus

14

2.8 Critical Points ✞

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1. We employ the notation , , as in Theorem 2.82. ✄ ✙▲➇ ✗⑥✄❉✏✱✙▲➁❫☞❊➁s✡✌✙✜✒✻☞❊➁ ✍✌✏✓✒ ☞♦✰✣➂❜✰✾✒ ☞♦✰✣➂✱✍❀✙✜✒ ☞♦✰✣➂✱✏✓✒ ❵ (a)✄ ❵ ✙ ✚✄ and , so the critical points are , , and . Also ✰ ✖✄ ✞▼❅▲➁ ✟ ✡✌✙ ▼➁⑥✍t✙ , , , so these points are respectively a saddle, a minimum, and a minimum. ✗ ✄✕✞✫☞❊➁ ✟ ✍♣✙✜✒ ✄ ▼➇●☞❊➇ ✟ ✍❦✏✓✒ ☞♦✰✣➂ ❆ ✛ ✙✜✒ ☞ ❆❧✏✠➂ ✛ ✙✜✒ ☞ ❆❧✏✠➂✱✍ ✛ ✙✾✒ and , so the critical points are , , and . (b) ❵ ❵ ✟ ✄✝✏✱✙▲➇ ✍ ✄r✰ ◆✄✳❅▲➁ ☞♦✰✣➂ ✛ ✙☎✒ ☞ ❆ ✏✠➂✱✍ ✛ ✙☎✒ ☞♦✰✣➂✱✍ ✛ ✙☎✒ Also☞ ❆❧✏✠➂✮✛ ✙✾✒ , , , so and are saddles, is a maximum, are minima. and ✗ ✄❖✙▲➁❴☞✎✏✭✍ ➇❴✒ ✗ ✄✳✰ ✄ ✞▲➇ ✟ ✍❲✙▲➇★✍⑩➁ ✟ ➁★✄ ✰ ➇◆✄ ✏ (c) ✄r . If ❵ ✄r✰ ✄ then➁❢ either or . In the first☞♦✰✣case, ❵ ✰ ✄ ✟ ❵ ➇❋✄✚✰ ➇❋✄ and ✄r❆ ✏ ➂❜✰✾✒ ❵ ☞ ✟ ➂❜✰✾✒ ❵ or ; in the second case, . So the critical points are , ✄r❅▲➇⑥✍ ✙ ☞✎✏✠➂ ❆❧✏✓✒ ✄⑦✍✤✙▲➁ ◆✄r✙✫☞✎✏❀✍ ➇❴✒ ☞✎✏✠➂ ❆ ✏✓✒ ☞♦✰✣➂❜✰✾✒ . Also, , , and , so and are saddles ☞ ✟ , ➂❜and ✰✾✒ and is a minimum. ✄✚➇❍➁ ✟ ☞ ⑧✍◗✞▲➇⑥✍◗✙▲➁✫✒ ✗✲✄✚➇ ✟ ➁❴☞ ❧✍ ✙▲➇⑥✍❲✞▲➁✫✒ ➇◆✄r✰ ➁❢✄✚✰ ✄ ✗✲✄r✰ ❵ ❵ ❵ and . If either or then ; (d) ❵ ✄ ✗✦✄✖✰ ✞▲➇ ✡✪✙▲➁✉✄✕✙▲➇ ✡✪✞▲➁✉✄ ➇❢✄✖➁ ✄ only when otherwise, ❵ ☞ ➂ ✒ ❵ ➇ ➁ ✄✕,✰ that is, ➇❢✄✕✰ . Thus ➁s✄✖✰ the critical ➇✑✡✇➁ points ✄✌✙ ❵ are and all✰ points➇✤on✡⑥the and axes. Note that on the lines , , and ; ➁ ⑤✙ ➇❝✡★➁ ⑤✙ ☞ ➃➂❜✰✾✒ ☞♦✰✣➂ ✫✒ ✰ when and ❵ when . Thus the points and are local elsewhere, ❵ ⑨✙ ⑨✙ ⑥✄ ✙ , and saddle points . (nonstrict) ☞ ➂ minima ✒ t✰ when☞ ➂ ✒ , local (nonstrict) maxima when ✄⑨✰ when Also, ❵ , and is inside the triangle bounded by the lines on which ❵ , so it must be a maximum. (One could also check✗ this by Theorem 2.82.) ✗ ✗ ✄♠✙▲➁❫☞✎✏✦✍✥✙▲➇ ✟ ✍⑩➁ ✟ ✒ ✄✝✙▲➇●☞ ✙⑧✍❲✙▲➇ ✟ ✍◗➁ ✟ ✒ ✖ ✖ ✖ ✖ ✄♠✰ ➇✪✄❖✰ and (e) ✙▲❵ ➇ ✟ ✡✌➁ ✟ ✄❏✙ ❵ ❵ ❆❧✏ ✗✪✄ ✙▲➁❫☞✎✏⑧✍t➁ ✟ ✒ ✄ ✰❢✄ ➁t✄ ✰ . Thus ➁❈✄❏ ❵ or . In the first or . In the second ✗ ✄ ✍✤✙▲➁✪✄ ✰s✄ ➁✇✄ case, ✰ ✙▲➇ ✟ ✄ ✙ ➇❲✄ ❆❧✏ ☞♦✰✣➂❜✰✾✒ and hence , or . Thus the critical points are , case, ❵ ☞♦✰✣➂ ❆❧✏✓✒ ☞ ❆ ✏✠➂❜✰✾✒ ☞♦✰✣➂❜✰✾✒ , and . is obviously the global minimum. ☞ ❆❧✏✠➂❜✰✾✒ ☞♦✰✣➂ ❆❧✏✓A ✒ straightforward but tedious application are maxima and are saddles. (See also the solution to of Theorem 2.82 shows that Exercise 1h below.) ✄✳✍ ✾➇ ✟ ✡◆➁ ✗✘✄✳✍ ✗ ➁ ✟ ✡◆➇ ✄✖✰ ➁ ✄ ☎➇ ✟ ✗✘✄✖✰ P ✓ ✒ P ✓ ✒ ❵ ❵ ❵ and . If then substituting gives (f) ➇❇✄✝☞ ✟ ✽ ✗✗✒ ➁ ✄⑦☞ ✗ ✟ ✽ ✣✒ ✄r✙ ✗✗✽ ;⑩ ✄✝✏ ❇this ✄r✙ into ✣✽ ✗ ❵ and . At this critical point, , , and , so the point ✗✗✽ ✰ ✗✗✽ ✰ and a maximum if . is a minimum if ✄ ✞✫☞❊➇ ✟ ✍✖✏✓✒ ✗ ✄ ✍✤✞✫☞❊➁ ✟ ✍✥✞✜✒ ✄ ✙ ➇◗✄❞❆❧✏ ❵ ❵ ❵ (g) , , and , so the critical points are those where , ➁❖✄ ❆✑✛ ✞ r✄ ✰ ❅▲➇ ✍✤❅▲➁ ✙ . The entries , , . Thus ☞✎✏✠➂✱✍❀✛ ✞✫➂❜✰✾✒ , and ☞✎✍✑✏✠➂✻Hessian ✛ ✞✈➂❜✰✾✒ matrix is diagonal❆ with ☞✎✏✠➂✻✛ diagonal ✞✾✒ is a minimum, is a maximum, and are saddles. ✗ ✟ ✒ ✄✖ ✖ ✖ ✗ ✄⑨✙▲➁❫☞ ✙✑✍◗✞▲➇ ✟ ✍ ✙▲➁ ✟ ✍ ✄ ✄✚✙▲➇●☞ ✞✲✍⑩✞▲➇ ✟ ✍ ✙▲➁ ✟ ✍ ✟ ✒ (h) With✄ ✙ ❍☞✎✏✲✍t✞▲➇ ✟ ✍⑤✙▲we have , , ❵ ❵ ✟ ✟ ✟ ✟ ✟ ➁ ✍ ✒ ✄ ✰ ➇❈✄ ✰ ✞▲➇ ✡✖➁ ✡ ✄ ✞ and ❵ ✟ ✡ ✟ ✄❏✙ or ✥✄ ✰ ✗ ✄ ✄ ✰ . If ❵ ➁❈✄❉✰ , then✙▲➁ either ✙▲➁ ✟ ✡ . ✟ In✄ the ✏ give or , and or ; first case the equations ❵ ❵ ➁⑤✄ ◗✄ ✰ ➁ ✄ ✰ ◗✄ ❆❧✏ ➁ ✄❉❆❧✏ ◗✄ ✰ the solutions are ; , ; , . In the second case, the equations ✗✦✄ ✄✖✰ ➁ ✄ ❧✄✖✰ ➇❢✄⑨❆❧✏ ☞♦✰✣➂❜✰✣➂❜✰✾✒ ☞ ❆❧✏✠➂❜✰✣➂❜✰✾✒ ☞♦✰✣➂ ❆❧✏✠➂❜✰✾✒ and hence . So the critical points are , , , ❵ ☞♦✰✣❵ ➂❜✰✣➂ ❆❧✏✓give ✒ and . One can analyze them without the tedium of computing all the second derivatives as ☞❊✁✬✒ ⑦✰ ✁⑦✄❞ ❶ ③ ☞♦✰✣➂❜✰✣➂❜✰✾✒ ☞❊✁✬✒ ✰ ❵ for , is obviously the global minimum. Since ❵ as follows. Since ✁ , ❵ must☞ ❆❧ have a global maximum; by examining the values at the critical points one sees that the ✏✠➂❜✰✣➂❜✰✾✒ ☞❊✁✬✒ ✁ ☞♦✰✣➂ ❆❧✏✠➂❜✰✾✒ ❛ ❛❁✄✳✏ ✗ ❵ maximum is at . Now consider for near . Since has a maximum at ✟ ✟ ✖ ☞♦✰✣➂❨➁➃➂❜✰✾✒❑✄✕✙▲➁ ➁ ✄⑨❆❧✏ ☞❊➇✬➂ ❆❧✏✠➂❜✰✾✒☛✄♠☞ ✞▲➇ ✡❇✙✜✒ ✄✖ P ✄ , P ☞ ✙❤ at✡✪➇ ✟ ✡✥✧✗.✧✗On ❵ P ☞ ✞▲➇ ✟ ✡ ✙✜✒✻☞✎✏❤✍❋➇ ✟ has ✡ ✧✗a✧✗✧✮maximum ✒ ➉❴✄ ✧✻✒ the other hand, ❵ ➇⑥✄❈✰ ☞♦✰✣➂ ❆❧✏✠➂❜✰✾✒ ➈ has a local minimum at . Thus is a ☞♦✰✣➂❜✰✣➂ ❆❧✏✓✒ saddle, and likewise so is . ✝

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✙

☎

✄

✂✩✎ ☎

✟✝

✜

✎

✴✱ ✎

☎

✱✚✕

✱

✎

☎

✞

✠

☛

✝

✩✱

✝

✓

✂✎☎

✎ ✎

☎

✓

✄

✓

✄

☎

✄

✄

✄

✓

✓

✣✦✥

✂✎

✓✙

✂✩✎

✄✂ ✎

✂✎

✚

✣

✂✩✎ ☎

✎

2.9. Extreme Value Problems

✄ ✰

15

➇★✡❈➁s✡✄⑩✄

➇✖✄ ✰ ➁◗✄ ✰ ⑩✄ ✰

(i) Note that ❵ precisely on the planes ③ ✁ ✄✂, , and☞ P ✒ ✄ ③ that include ✆☎, ✂ ✟ on the set where the P✂✄ faces ✄✖of✰ the tetrahedron with vertices at , , , and . Since ❵ ❵ ❵ ❵✟ , all points on the lines where these planes intersect are degenerate critical points, and none of them ✄✕➁ ❍☞are ⑧✍⑩maxima ✙▲➇ ✍⑩➁⑧or✍ minima ☎✒ ✗ ✄✕since ➇ ❍☞ ❵ ✍⑩changes ➇ ✍⑩✙▲➁ sign ✍ ☎✒ whenever✄✌one ➇❍➁❫☞ crosses ✍✪➇ ✍✪one ➁❧✍ of✙ ☎✒these planes. Since ✓ ➇⑧✡⑩➁✦✡❲✙ ⑧✄✡ ,❵ ❵ ✙▲➇❧✡⑩➁✦✡ ✲✄✖➇⑧✡◗✙▲➁✘, and ✡ ⑧✄❵ ✖ ☞✎✏✠,➂✱one ✏✠➂✱✏✓✒ sees that the only other critical point is where , i.e., . This point ✙✚✰ ✄♠✰ there, and ❵ on the faces of the tetrahedron; hence this point is a is inside the tetrahedron, ❵ local maximum. ✄ ✙✩✱ ❋ ✗ ✄ ✗ ✗❋ ✗ ✄ ✙ ✆ ✯✱✕✆ ✤ ✗ ✟ ✱➃➂ ✆ ✤ ✰ ❵ ✗ ✟ , and 2. We have ❵ , so the origin✯✱ is✆ a✕ minimum and ☞❊➇❄➂❨➁❬✒❋✄ , ✯✱✕✆ , ✤✛ ✱➃➂ t ✆ ❵ q❏✰ ✗ ✟ ✯✱if ◗ ✆ ✄ ✗ ✟ and , and a saddle if . (If , then ❵ a❆ maximum ☞❜ ✱❫ P✓✒ ✟ ➇⑧❆✕if✜✆ P✓✒ ✟ ➁✫✒ ✟ [any combination of the two signs can occur], so the origin is still an extremum.) ☎

☎

☎

☎

P

3. The origin is a global minimum for ❵ and a saddle point for ❵ ✟ . For ❵ ✜ it is a “shoulder point” as in the right-hand graph of Figure 2.5 (but upside down). 4. (a) The 2nd-order Taylor polynomial of

❵

at the origin is

➁ ✟

, so the origin is a degenerate critical point.

☞✳✱☎❛ ➂✙✗ ❛❜✒✿✄✏✗ ✟ ❛ ✟ ☞✳✱✾❛✮➂❜✰✾✒❁✄✕✩✙ ✱✏✪✗❛ ✪ (b) For we have ❵ + higher order, and ❵ ; these ➁ ✄✌all➇ ✟ have local ➁ ✄r✙▲minima ➇ ✟ at the origin. However, ❵ is negative in the region between the two parabolas and and ❧✄✖ ❶ ✰ ✗

positive in the regions inside both or outside both. The origin is on the boundary of all these regions, so ❵ has neither a maximum or a minimum there. 5. The second directional derivative of ❵ in the direction ✗ at ✌

✟ ✔❛ ✟ ❵ ♦☞ ❡ ✡◗❛ ✗ ✒ ✚ ❱ ✄

✌

✡

✡

❛

✖ ✁

✡

✌

❡

is

✝ ☞♦❡➃✒ ✧ ❯ ❽ ❯ ❵ ☞♦❡✲✡◗❛ ✗ ✒ ✚ ✄ ✖ ✆ ❘ ✁ ❯ ✁ ❘ ❽ ❯ ❽ ❘ ❵ ☞♦❡➃✒✿✄✞❲ ✗ ❱ ❯ ✡

✡

✡

2.9 Extreme Value Problems

2.

✗ ✄✕✙▲➁ ✄ ▼➇ ✡ ✙ ☞✎✍ ✟P ➂❜✰✾✒ ☞✎✍ ✟P ➂❜✰✾✒❁✄✳✍ ✟P only . On the unit circle, ❵ ☞♦✸✻✺✜✼ ➂❜✼✙✔ ✕ and ✒❑✄✚❵ ✙■✸✻✺✜✼ ✟ ,✡✥so✄✼ ✔ the ✕ ✟ ✡ ✙■critical ✸✻✺✜✼ ✄⑦point ☞✎✏✂✡ is✸✻✺✜✼ ✒ ✟ , and ❵ P ❵✰ ✵ ✄✖✰ ✵ ✵ ✵ ✵ ✵ , whose maximum and minimum are and ✍ ✟ ✄✕❃ (at ✵ and ✵ ). So the maximum is and the minimum is . ✄ ❅▲➇ ✗❈✄ ✍ ▼➁❦✡✳✙ ☞♦✰✣➂ ✟P ✒ ☞♦✰✣➂ ✟P ✒✇✄ ✟P ❵ and ❵ , so the only critical point is , and ❵ ☞♦✸✻✺✜✼ ➂❜✼✄✔ ✕ ✒✯✄❖✞✑✍❲❂■✼✄✔ ✕ ✟ ✡t✙■✄✼ ✔ ✕ ☞ P ✽ ✒ ☞♦✸✻✺✜✼ ➂❜✼✄✔ ✕ ✒✯✄ ✍✑✏✗✰✿✸✻✺✜✼ ✼✙✔ ✕ . On ✡t✙■the ✸✻✺✜✼ unit ✄ circle, ✵ ✵ ✼✄✔ ✕ ✄✚❆❧✏ ✵ ✄✕ ✵ ✰ ✍ ✵ ✙■✸✻✺✜✼ ☞✎❵ ✏❝✍⑩❂■✵ ✼✄✔ ✕ ✒☛✵ ✄✕P ✰ ✸✻✺✜✼ ✵ ✄✕✰ ✼✙✔ ✵ ✕ , so✄ ✄ ✾✵ ❵ ✵ ✵ ✄ ✄ when ✵ ✵ or ✍ ✵ . In the first case,P , so ❵ or . In ✄ . So the minimum is and the maximum is . the second case, ❵ ✟ ✄ ✞▲➇ ✍✕✏ ✗♣✄ ✙▲➁♣✍ ✙ ☞ ❆❧✏✓✽☎✛ ✞✈➂❄✏✓✒ and ❵ , so the critical points are at➁❋✄❖✰ ➁❋✄⑦✙✲,✍❲ which ❵ ✙▲➇ are➁❇both ✄✝✙✭outside ✡❈✙▲➇ the triangle.☞❊➇✬The sides of the triangle are segments in the lines , , and ➂❜✰✾✒✤✄ ➇ ✜❀✍⑩➇ ➇✪✄❖❆ ✏✓✽✾✛ ✞ ☞ ❆❧✏✓✛ ✞✫➂❜✰✾✒✤✠ ✄ ✟✑✙✜✽✠✞✜✛ ✞ . We whose , and✰✉❵ q✥➇❋q✌✏ have ☞❊➇❄➂✹✙❄❵ ✍⑧✙▲➇❫✒❁✄❈➇ ✜✣✡ ▼➇ ✟ , ✍❧ ❂▲➇ critical points are at ➇❢✄ ☞❨✛ ✞✣✏❾✍ . Also, ☎✒❨✽✠✞ , whose only critical point in the interval is at , ❵ ✄⑦☞ ✞✠✯✰ ★ ✍◗❅▼✙✜✛ ✞✣✏▲✒❨✽✠✎✙ ✠ ☞❊➇❄➂✹✙✭✡✥✙▲➇❴✒❤✄✌➇ ✜ ✌ ✡ ▼➇ ✟ ✡✥✞▲➇ ❵ where ❵ ✍ ✏✉q✚➇❲qr✰ . Next, , whose only critical point in the ➇⑩✄ ☞ ✛ ✠✲✍ ☎✒❨✽✠✞ ✄ ☞ ✙✠✰❧✍✖☞✏ ☎✛ ✠▼✒❨✽✠✎✙ ✠ ✄ ✰ is at , where ❵ interval ☞ ✞✠✯✰ ★✂✍★❅▼✙❬✛ . Finally, ✞✣✏✩✒❨✽✠✙✎✠ ☛ ❵ ✍✑✏ ✄ ✞✎at✠✩★ all three and the vertices. Comparing ✙✜✽✠✞ ✛ ✞ all ☛✕☎ ✰ these ✄ ✞ ★ ✾✴ values, we see that the minimum is ☎

☎

✌

✓

3.

✄

✡

✌

✡

1.

✗

✓

☎

maximum is

✌

.

Chapter 2. Differential Calculus

16 4.

☎

✄✌❅▲➇❢✍ ★▲➁ ✡✥✙ ✗✑✄ ✍✤★▲➇❢✍ ★▲➁ ✡✖✏✱❅ ➇★✄✌➁s✄❖✏ ❵ and ; setting these simultaneously equal to 0 gives , ✟ ☞✎✏✠➂✱✏✓✒⑥✄❏✴ ☞♦✰✣➂❨➁❬✒❢✄ ✍ ▼➁ P ✡❖✏✱❅▲➁ (i) ❵ ; critical point and➁◗❵ ✄ ✙ ☞♦.✰✣➂✹Next ✙✜✒⑧✄ one ✏✱❅ analyzes ☞❊➇✬➂❜❵ ✰✾✒❧on✄ the ✞▲➇ ✟ four ✡✕✙▲sides: ➇ ➇t✄ ✍ ✜ ❵ at , and ❵ ☞ ✫➂❨➁❬✒s✄ ❂▼❅s✍✚ . (ii) ; critical point at , not in the range of ✏✱❅▲➁★✍✧▼➁ ✟ ➁ ✄❏✍✤✙ ❨ P P ❨ P P ❵ ; critical point at , not in the range of interest. (iv) interest. (iii) ☞❊➇❄➂✹✞✜✒❀✄ ✞▲➇ ✟ ✍◗✙▼✙▲➇♣✡⑨✏✱✙ ➇✇✄ ☞ ➂✹✞✜✒❝✄ ✍ ✜ . Finally one ❵ ☞♦✰✣➂❜✰✾✒✯✄✚✰ ☞♦✰✣➂✹✞✜✒❤✄✝✏✱✙ ; critical ☞ ✫➂❜✰✾✒❤✄rpoint ❂▼❅ at ☞ ✫➂✹✜ ✞✜✒✯; ❵ ✄✝✍✤✜ ✙ ★ ✍ checks the corners: ❵ ,❵ ,❵ , and ❵ . The minimum is ✜ and the maximum ❵

✄

✞

✄

✞

is 56.

✗

☞❊➇✬➂❨➁✫✒✗ ✝ ✤ ♥☞❊➇❄➂❨➁❬✒✗ ✟ ❥ ✟ ✄⑦✍✤✙ ✗✠☞ ✍ , so ❵ has a minimum on by Theorem 2.83a. We have ❵ ☎ ✟ ✗ ✄ ✍✤✙ ✆▼☞ ✍ ✗ ➇✪✍ ✆✮➁❬✒✂✡r✙▲➁ and . Setting these equal to 0 simultaneously gives ❵ ✟ ✗✗✽✣☞✎✏❙✡ ✗ ✟ ✡ ✆ ✟ ✒ and ➁s✄ ✟ ✆✗✽✣☞✎✏❙✡ ✗ ✟ ✡ ✆ ✟ ✒ . Substituting these values into ❵ ☞❊➇✬➂❨➁✫✒ and simplifying ✄ ✟ ✽✣☞✎✏❑✞ ✡ ✗✟ ✡ ✆✟✒

5. Clearly ➇⑩✍ ✆ ➁❬✒❤❵ ✡✳✙▲➇

➇⑥✄

✟

gives ❵

✟

.

☞❊➇✬➂❨➁✫✒☞✙ ✰

➇⑥✄❈➁s✄✖✰ ☞♦✰✣➂❜✰✾✒☛✄✖✰ ☞♦✰✣➂ ❆ , so ✏✓✒❁❵✄✕✙ ✂ ✎ P

6. Clearly ❵ except when maximum by Theorem 2.83b, namely ❵ critical points.) 7.

is the absolute minimum. ❵ has an absolute . (See Exercise 1e in ✂ 2.8 for the analysis of the

☞ ❆ ✏✠➂❜✰✾✒ ☞♦✰✣➂ ❆❧✏✓✒ ☞ ❆ ✏✠➂❜✰✾✒❁✄ ✄ ✂✎ P P As in Exercise 1e✍✤in✙ ✂ ✂ ✎ 2.8, one finds that the critical points are and , and ❵ ☞♦✰✣➂ ❆❧✏✓✒❑✄❖ and ❵ . Since ❵ assumes both positive and negative values and vanishes at infinity, by Theorem 2.83b it has both an absolute ✂ ✎ P maximum and an ✍❀absolute ✙ ✂ ✎ P minimum, which must occur at critical points; hence the maximum is

☞❊➇❄➂❨➁❬✒

and the minimum is

.

➇❢✄ ✏✓✽✠✞

➁s✄✳✏✓✽✁

8. If is in the first but outside and ➇❍➁❦quadrant ✄ ☞❊➇❄➂❨➁❬✒ ✙ the “triangle” bounded by the lines P and the hyperbola , then ❵ ➇❢✄ ☞ ✴✜✽✁☎✒ P ;✜ hence ❵ has a minimum but no maximum by Theorem ➁ ✄ ☞✎✏✱❅✜✽✠✞✜✒ ✜ P , (see Exercise 1f in ✂ 2.8), and the value 2.83a. The only ☞✎✏✱✙✜✒ critical ✜ point is at of ❵ there is ; this has to be the minimum. ✙▲➇❦✄⑨✙✁❬➇ ▼➁♣✄✚✙✁✫➁ ❅ s✄⑨✁✙ 9. Lagrange’s method , and ✟ ✄⑦✏ here: one has to solve ➇ ✟ ✡✥works ➁ ✟ ✡ easily ❋✄, ⑦✏ ➇◆✄r ✰ ❦✄✝✏ subject to . If , the second the constraint ➁ . ✄ The✇✄first✰ equation implies ➇⑤✄❞❆ that ✏ ➇ ✄ or✰ and equations imply✙ that ♣✄✳✰ ◆✄✳third ✙ ➁⑥ ✄ ✰ ❇✄ ➁⑥and ✄❖❆❧hence ✏ ➁★✄r✰ . If ❢✄♠❆❧✏ , the second equation forces or ❆ ☞✎✏✠➂❜.✰✣➂❜If✰✾✒ ❆ ☞♦✰✣,➂✱✏✠then , so , then . So the constrained ➂❜✰✾✒ ❆ ☞♦✰✣➂❜✰✣➂✱✏✓✒ ; if ✏ critical , . Clearly the first pair gives the minimum of and the points are ✞ , and last pair gives the maximum of . ✄

✄

✓✒

✓✒

✓✒

✄

✄

☞✳✱❍➂✙✗✗✒❁✄ ❚ ❊☞ ➁ ❯ ✍ ✾✱ ➇ ❯ ✍ ✗✗✒ ✟ ✄ ✍✤✙ ❚ ➇ ❯ ☞❊➁ ❯ ✍ ✱✾➇ ❯ ✍ ✗✗✒❁✄✕✙ ✱ ❚ ➇ ❯ ✟ ✡ ✗ ✶ ➇✘✍ ❚ ➇ ❯ ➁ ❯ ➉ ✳ ➈ ✍ ✱☎➇ ❯ ✍✑✗✗✒❁✄✕✙ , ✶ we✱ ➇✑have ✧ ✡ ✗❄❵ ✍ ➁☎➉ ✱ ✗ ➈ . Solving this pair of linear equations for and yields

10. With ❵ ✄✳✍❀✙ ❚ ☞❊➁ ❯ and ❵ the asserted result.

✠

✞

✗

✗

✏✘✄✝✍✤✙✁✠✱☎➇✞✎ ✟ ✄✝✍✤✙✁ ✹➁ ✎ ✟ ✄✝✍✤✁✙ ✆ ✏✎ ✟ ☞✳✱✫✽✩➇❴✒✬✡✌☞ ✗✽✩➁✫✒✬✡ to✇✄ 11. ☞✝By Lagrange’s method, we solve ✆✱✽ ☎✒❧✄ ✏ ✄ ✍✤✙✁ ➇ ✄ ✛ ✱ ➁ ✄ subject ✛ ✛ ✂ ✆ , and the . With ✂ , the first set of equations gives ✂ , ✂ , ✛ ✄ ✛ ✱❀✡ ✛ ●✡ ✛ ✆ , and hence ➇✲✡✇➁❀✡ ✲✄ ☞ ✛ ✱❀✡ ✛ ❄✡ ✛ ✆✓✒ ✟ . (This constraint equation then➇❧gives ✡◗➁✑✡ ✹✣✂ ✥ ☞❊➇❄➂❨➁❾➂✁☎✒✭✣ ✥ as on the constraint surface.) is a minimum because

✗

✗

➇♣✡⑤➁⑧✡

➇➃➁ ★✄

☎

➇❄➂❨➁❾➂✁ ✙✳✰

✗

12. We wish to minimize ). By Lagrange’s ✏✑✄✄✫➁ s✄✄❬➇ s✄subject ☎❬➇❍➁ to the constraint ➇❍➁ (and s✄ P ✜ to the constraint method, ➇✥✄ ➁✪✄ we❇✄solve P ✜ ➇ ✡t➁❧✡ ◆✄ ✞ subject ☞❊➇✬➂❨➁➃➂✁;✈✒the✄ solution ☞ ➂ ✆▲➂ ✆ is✎ P obviously ✒ , so . There is no maximum; satisfies ➇❍➁ ⑧✄ ✆ no matter how large is.

☎

☎

✓✒

☎

✓✒

☎

☎

2.10. Vector-Valued Functions and Their Derivatives

17

✂✈☞ ▲ ✂ ✒✭✄ ☞✎✏✭✍ ✂✾➂ ✂✜➂❜✰✾✒ ●☞❊❛❜✒✦✄ ☞❊❛✮➂❨❛✮➂❨❛❨✒ 13. Parametrizing the first line by and the second one by , we ✟ ✟ ✟ ✟ ✟ ✍✂☞ ✾ ✂ ➂❨❛❨✒❁✄❖ ✂☎☞ ✂✠✒❄✍ ■☞❊❛❨✒✗ ✄♠☞✎✏❑✍ ✂✯✍❋❛❨✒ ✡❈☞ ✂✯✍❋❛❨✒ ✡⑩❛ ✄✕✙✄✂ ✡ ✞▲❛ ✟ ✍✇✙✄✂✯✍◆✙▲❛❴✡⑤ ✏ wish to P P . We have minimize ✍ ✄ ✆✂❀✍⑩✙ ✍ ■✄✕❅▲❛●✍ ✙ ✂✭✄ ✟ , ❛❁✄ . The point on the first line closest to and P . so the critical point is P P ✂☎☞ ✒❁✄ ☞ ➂ ➂❜✰✾✒ ✟ ✟ (and the minimum distance is ✆ ✏✓✽✠❅ ). the second one is ✟ ☛

✜

☛

✚

✄⑨➇❍➁ ✄❞➇❍➁❫☞

➇ ✟ ✡✥✙▲➇➃➁ ✄

✄✖➁ ✟

✟

➇❍➁ ✡ ✙▲➇ ✘✡⑤✙▲➁ ✉✄

subject ✟ . Solving the constraint ✍✥➇➃➁✫✒❨✽✣to☞ ✙▲the ➇❦✡✕constraint ✙▲➁❬✒ ✄ ✙▲➁ ✟ ☞ ✍ ✟ . After a little calculation we find ☎ ✟ ✗ ✄ ✙▲➇ ✟ ☞ ✍t➁ ✟ ✍✕✙▲➇❍➁❬✒❨✽✣☞ ✙▲➇◆✡✚P ✙▲➁✫✒ ✟ ✡✥✙▲➇➃and ➁ ☎ ➇⑥✄❈➁ ✟ ✄ ✆ ✽✠✞ ❧✄ ✟ ✆ .✽✠✞ Setting these ✄ ✟P ☞ ✆ equal✽✠✞✾✒ to 0 gives ✟ ✟ ✟ , so that ; hence and ☎ .

14. We wish to minimize ☎ equation ➇ ✟ ✍❈✙▲➇➃➁✫for ✒❨✽✣☞ ✙▲➇✇gives ✡r✙▲➁✫☎ ✒ ✟

☎

✜

➇ ✟ ✡✇➁ ✟ ✡

✟

➇✑✡ ⑧✄

✞▲➇❧✍★➁s✄✕❅ ✄ ✙ ❦✍✚❅▲. ➁ By Lagrange’s

15. We wish to minimize and ✙▲➇ ✄ ❋✡♠✞ subject ✙▲➁❖✄ to✍ the constraints ✙ r✄ ✙▲➇ ✂ , ✂ , and to obtain method, we solve ➇⑥✄ ❧✄✕✙ ➁s✄✖✰ simultaneously with the constraint equations gives , .

; solving this

✙ ☞❊➇ ✍✑➁ ✒☛✄✕✙✁❬➇ ✍❀✙ ☞❊➇ ✍✘➁ ✒❁✄✌✙✁✫➁ ✍❀✙▲➁❴☞❊➇ ✍✘➁ ✒✿✄ , ➇ ✍❲➁ ✄⑦ , 16. ✙ (a) Lagrange’s ✙▲➇■☞❊➇ ✍ ➁method ✒✦✄ ✙ gives the equations ❶ ✰ ✂ , ✂ . Eliminating ✽✩and ✂ (and assuming , which is OK since the ➇❲✄ ✍ ✽✩➁ ➇❫✽ ✄ ✍✯➁❬✽ or (whichever avoids maximum is clearly positive) we obtain ☞❊➇❄➂❨➁❾➂ ➂ ✒ ☞ ➂ ✒ ☞✎division ✍✯➁❾➂❨➇❴✒ by to zero); either way, the critical points are those ✟ ✗. ✟ By ❆ ✗✻✽ such☞❊➇ that✍✇➁ ✒ ✟ ✄ is ☞ proportional ✗✗✽ ✫✒ ✟ ☞❊➇ ✟ ✡◗➁ ✟ ✒ ✟ ✄ the constraints, the constant of proportionality is , and . ✁

✁

✁

✁

✁

✁

✁

✁

✁

✱

(b) Using the parametrization, ✟✗✟ . maximum is obviously

✂P

✱

✁

✁

☞❊➇ ✍❧➁ ✒ ✟ ✄✴✱ ✟ ✗ ✟ ♦☞ ✸✻✺✜✼ ✄✼ ✔ ✕ ✵ ✁

✍

✍ ✼✄✔ ✕ ✻✸ ✺✜✼ ✍●✒ ✟ ✄ ✵ ✁

➁✣P❍✼ ✸ P

✱

✱✟

✴✱

✗

✟ ✼✄✔ ✕ ✟ ☞ ✍ ■✍ ✒ ✵ , whose

✍✯➁ ✟ ✼ ✁ ✸ ✟ ✵ (remember that ✟ ✒✈✼ ✁ ✸ ✵ ✟ . The constraint ✍✖➇❄P✩ ✸ ✟ ✂❜✶ . ✕ Thus, ✄ ❬➁Lagrange’s ✟ ✼✁✸✟ ✵✟, ✵ ✵✟

✁ ✵ , and the distance from to ✟ is 17. ➁ The distance from to is ✰ ✙❤P ✙ ✟ ✟ ☞❊➁✣P ✽ P✮✒✈✼ ✁ ✸ P❝✍⑨☞❊➁ ✟ ✽ ). Thus ➇ the total travel time from to➁➃P✄✂❜✶ ✕ is P✘✍❈➁ ✟ ✂❜✶ ✕ ✵ ✟ ✄ ➇ ✟ is that the -distances have☞❊➁❍to P✹✽ add P✮✒✈✼ ✁up✸ right: P☎✂❜✶ ✕ P✂✄ ❬➁✣✵ P❬✼ ✁ ✸ ✟ P ✵ ☞❊➁ ✟ ✽ ✟ ✒✈✼ ✁ method gives the equations ✵ ✵ ✵ and ✼✙✔ ✕ P ✽ P✂✄ ♣✄✕✼✄✔ ✕ ✟ ✽ ✟ ✵ ✵ whence .

✕

18. ✙ Let✄

✙

✙

❯

✜

✜

✌

➇ ❩

✌

✙

☛

be the product of the ’s with the th➇ term omitted. Lagrange’s method gives the equations ➇■P❙✧✗✧✗✧❜➇ ❯ ❯ ’s are all equal. Thus the maximum P✓value ✒ for➇❄P✂ all✄ ✧✗, ✧✗from which it follows that the of ✧✣✄✖➇ ✄ ✆✱✽ ☞✝✆✱✽ ✒ ☞❊➇✬P❙✧✗✧✗✧✹➇ ✒ ✆✱✽ occurs➇❄at is at most ➇ ✟ ✟ , and that value P❄✡⑤✧✗✧✗✧✩✡✪➇ ✄ ✆ ☞❊➇✬P❙✧✗✧✗✧❜➇ ✒ P✓✒ isq⑨☞❊➇❄P❙✟ ✡t.✧✗✧✗In✧✓✡ other ➇ ✒❨✽ words, ✟ , with equality only when the ❯ ’s when , that is, are all equal.

☛

P✗➂✆✄✆✄✆✓✄ ➂

☛

☛ ☛

☛

☛ ☛

☛

☛

●P✗➂✆✄✆✄✆✓✄ ➂

☛

✁t✄ ❚ ❛

❯ ❯ for ✟ , with eigenvalues . If 19. Let ✆ ✁❑ ✟ ✄ ✆ be ✁◆✧✠✁t✄ ❚ eigenbasis ❛❯ ❯ ✟ ❚ ❛ ❯ ✟ an orthonormal ❚ ❯ ❛ ❯ ✟ ✆ , we ; thus we wish to maximize and minimize subject to have ✄✳✏ ✟ ❚ ❛ ❯ ✟ and the constraint . By an easy extension of the argument in Exercise 9, Lagrange’s method shows messier that the critical points are the unit eigenvectors of ✟ . (Things are a little ✁ ✁❋✧✩✁◗ ✄ ❯ if the eigenvalues are ❯ not all distinct.) If is a unit eigenvector with eigenvalue , then ✟ , so the maximum and minimum are the largest and smallest of the ❯ ’s. 2.10 Vector-Valued Functions and Their Derivatives 1. ✝

✈☞❊➇❄➂❨➁❾➂✁☎✒☛✄✟✞ ➁✞▲➁ ✟✟ ➇ ❅▲➇➃✟ ➁❧✍ ✍ ▲★ ➁ ✂

✙▲➇➃➁ ✍✯➁✡✠

.

18

2.

✈☞❊➇❄➂❨➁❬✒❁✄ ✂ ✝

✁✂ ✙▲✞▲✏ ➇➁

✄☎

✙ ▲✞ ➇ ✍✤❅▲➁ ✙

Chapter 2. Differential Calculus

.

✆✂ ✞✝ ✂

3. (a) ✝

✠✟

(b)

4.

✝

✂

✄☎

✂

☛

✞✝ ✞

✠

✝

✍❀❂ ✏ ✰

✆✂ ✰✙ ✍✤✙✠✰ ✄☎

☎☞♦✰✣➂❜✰✾✒☛✄ ✂ ✝

✂ ✍✑✞✙ ✏✱❂

✗

✡✟

✗

✂

✂

✞

✠

✞

✞

✠

for all

✠

and , so ✟

✝

❅ ✏✗✰

✍✤✙✣✏ ✍ ✾✞

✄ ❚ ❘ ❘ ❘ ❵ ✠

, so ❽

❯ ✄ ❚ ❘ ☞ ❽ ❯ ❵ ❘✠✒ ✠ ❘❝✡ ❚ ❘ ☞ ❽ ❯ ✠ ✠❘ ✒ ❵ ❘

✠

❡

; this is the th component of

8. By (2.86),

✌✌✂ ❽❽ ✏ ➇➁ ✚

❽

✚

❽ ✍

✟✍

✄ ✎

✑

P ❽ ❵

❽✟❵

☞ ▲✂ ✒

✌

☎

✎

.

for all .

6. This is immediate from the definitions. ☎

.

✞

✟

7.

✄☎

.

✙✫☞❊➁ ✍ ✏✓✒ ✍★✸✻✺✜✼ ✄ ✟ ✞ ✍ ✏✗✰ ✟ ✎ ✄ ✓ . ✎ ✓ ✑ (a) ✙ ✤ ✍ ✙ ✍ ✏ ✄ ★ ☞ ✂▲✒✻☞♦✰✣➂❜✰✣➂❜✰✾✒✂✄ ✞ ✏ ✙ ✞ ✍✑✗✏ ✰ ✏★ (b) ✂☎☞♦❡⑧✡✣● ✢ ✒●✍ ✂✈☞♦❡➃✒❁✄ ✢ ❡ ✢ ✄ ✂☎♦☞ ❡➃✒ ✝

✰

✙

☎☞❊➇❄➂❨➁❾➂✁☎✒❑✄ ✂

☛

5.

✄☎

✍✤❂ ✟ ☎✂ ☞ ➂ ✒❁✄ ✙ ✟ ✁ ✙ ✍❀✙ ✽✣☞✎✏✂✡ ✟ ✒ , ✰ ✍✤❂ ✏ ✙ ✄ ☞✎✂ ❫✒✻☞✎✏✠➂✹✙✜✒✂✄ ✰ ✏ ✞ ✙ ✰ ✁

❽✜❵

❽✪❵

✚

✑

✰

✄✎☎ ✍✍

➇ ❽ ➁ ❽✰ ✏

✝

☞☛

✞☛

✡✌☞ ❫✒ ☛

☛

✝

✂

.

➂

which gives the desired result. ✍

(b) ✁❑ Obviously ✟ ✄ , with equality if ✟ . ✁

✁

✍

❷ is compact, and is continuous on it, so achieves a maximum ●③ ❍✄ ✰ ✄ ③✬ ✁r✄ ❶ ③ ✁♣✄ ✁❑ ✄ ✁✬✽ ✁ ✳ ✁❑❍✄ ✁❬ , let and ✗ . Then✁❑✟✬q ✁ ✄✝❡ . If ✁❑ ✟

9. (a) The ❡★unit ✐ sphere point ❷ .

✑✏ ✏ ✏ ✁⑩❀✏ q✐⑥✔❥ ✛ ☛ s✶✓✒ ❯ ❚

10. (a) If so ✟

✦

✁

. Thus

✟

(b) If ✟ ❯ ✁ and ✟ for any ; in particular, if ✤ ✛ ✦ ✟ .

✏ ✏

✁

✓✒

is the smallest constant such that

✓✒

✁

✟

✓✒

on ❷ at some ✗

❫q ✁ ✁

✁

✄

for all , i.e.,

✁❑❬q ✛ ✦ ✞s✶ ❯ ♥☞ ✁✬✒ ❯ ▼✄ ✛ ✦ ✞s✶ ❯ ❚ ❘ ❯ ❘✱➇❾❘✣✣q ✛ ✦ ✞s✶ ❯ ❚ ❘ ❯ ❘✈✗ ✁❑ ✟ ✟ ✟ , ❘ ✟ ❯ ❘✈ . ❘ ✄r✰ ✶ ✙ ✏ ✁ ✄ ☞❊➇❄P✗➂❨➇❙P✱➂✆✄✆✄✆✄✩➂❨➇❙P✮✒ ✛ ✦ ❯⑧ bound for ✁❑✤✄ ✟ in✏ (a) is ✁❑❀✄. Also, ✁ for✄ ☞✎✏✠➂❜✰✣➂✆, ✄✆the ✄✆✩ ✄ ➂❜✰✾✒ ♥☞✎✏✠➂✱✟ ✏✠➂✆✄✆✄✆✄▲➂✱✏✓✒✗❀✄ ✛ ✦

, we have ✞

P❀✄ ✏

✁

✏ ✏

, then

and

✟

, so

Chapter 3

The Implicit Function Theorem and its Applications 3.1 The Implicit Function Theorem

✁ ☞❊➇❄➂❨➁❾➂✁☎✒❑✄❈➇ ✟ ✍ ▼➇❁✡♣✙▲➁ ✟ ✍ ➁ we have ✁ ✄✕✙▲➇✯✍ , ✁ ✦✄ ▼➁❁✍ , ✁ ✄ ✍✯➁ , so ✂✁ ☞ ✙✈➂✱✍✑✏✠➂✹✞✜✒☛✄ ✁ ✄ ✏ can be solved locally for ➁ or but not ➇ . Explicitly, t✄ . Hence the equation

1. With ☞♦✰✣➂✱✍ ☎➂✱✏✓✒

✗

✂

☎

☞❊➇ ✟ ✍ ▼➇✘✡⑩✙▲➁ ✟ ✍ ✏✓✒❨✽✩➁ ✠

square root vanishes at and none for others.

➁s✄ ✆☞ ❀✍ ✆ ✜✟☛✡✘★✫☞✎✏❝✍✇➇➃✟☛✡✌▼➇❴✒❜✒❨✽✁ ☞✎✍✑✏✠➂✹✞✜✒

☞❊and ➁➃✁➂ ☎✒❑✄

✓

➇❢✄✕✙✂❆ ✆ ❂✑✍⑩✙▲➁✈✟☛✡ ➁

; but , ➁ and the ➇ , so there are two values of for some nearby values of and

✡ ✙▲➇➃➁✯✡◆✞▲➁ ✟ ✁ ☞❊➇✬➂❨➁✫✒☛✄❈➇ ✟ ◆ ✁

✁ ✄✕✙▲➇✭✡◆✙▲➁

✁ ✘✄✕✙▲➇✭✡◆❅▲➁

✁

✹✙

✁

✁

✁

✁

and☞❊➇❄➂❨➁❬✒✂✄ 2. With ☞❊➇❄➂❨➁❬✒✲✄❖ ❶ ☞♦✰✣➂❜✰✾✒ we have t✰ , so ✭at✄✚least ✰ one of ✄ and✄✚✰ is nonzero when , which is the case when . If ✄ ♠✰ ✄ then and the set where is a single point, and if then the set where is empty. (Clearly ☞❊➇❄➂❨➁❬✒☛✄♠☞❊➇❧✡◗➁❬✒ ✟ ✡✥✙▲➁ ✟ ✰ ☞❊➇❄➂❨➁❬✒✦✄ ❶ ☞♦✰✣➂❜✰✾✒ when .)

✁

✚✕

✙

✆

✁

☎

✁

✗

➇

✁

✆

✆

✄✝☞❊➇ ✟ ✡◗➁ ✟ ✡ ✙ ✟ ✒ ✓P ✒ ✟ ✁ ☞❊➇✬➂❨➁➃➂✁☎✒❤✄ and ✁ ☞♦✰✣➂✱✏✠➂❜✰✾✒❤✄❈✰

3. With✣☞♦✰✣➂✱✏✠➂❜✰✾✒❑✄✳✏ and so

✗

✆

✁ ✄✚✙ ☎✽ ✡❲✼✄✔ ✕

, we have ➁ and . Hence the equation can be solved for but not .

✓

➁

✎

✗

✁

☞♦✰✣➂✱✍ ✓P ✒ ✜✩✒

☞✎✏✠➂❜✰✾✒

✜

☎

✗

✗

✆

✁ ✄✕➁❬✽

✍ ✸✻✺✜✼

✁

☎

✓

,

✁

✜

P ✾at✒ P✓✒ and ; the tangent line is vertical at the former 4. The and intercepts ➁❦✄ ☞❊➇❫of✒✤the ✄❞☞❊graph ➇★✍ are ➁❋✄✳ ➇ P✓✒ P ✒ the graph is asymptotic to the curve ❵ point. Writing , we see that ➇ ✡ ➁⑥✄⑦✍ ➇ ✍ ❥ ❥ P ✈✒✿asymptotic as as ; hence ❵ maps ontoP ➄▼❥ . Also, ☞ ✽ ➇❴✒✻☞❊➇✘✍ and ✄✳✏❴✡ P to⑨the✏ curve ❏❥ , so ❵ is one-to-one (in fact, strictly increasing). Hence ❵ exists.

✣

✥

✂

✂

✌

✎

☎

✂

✎ ✙

☎

✂

✎ ✜

☞

☎

✂

✌

✣

✥

✎ ✴✣

☎

✌

☞❊➇✬➂❨➁✫✒☛✄ ✁ ☞ ✁ ☞❊➇✬➂❨➁✫✒ ➂❨➁❬✒ ✘✄ ✁ P✩☞ ✁ ❊☞ ➇❄➂❨➁❬✒ ➂❨➁✫✒ ✁ ✟ ☞❊➇❄➂❨➁❬✒✣✡ ✁ ✟ ☞ ✁ ☞❊➇❄➂❨➁❬✒ ➂❨➁✫✒ ✈☞♦✰✣➂❜✰✾✒❑✄ , we have , so ✜ P ♦ ☞ ✣ ✰ ❜ ➂ ✾ ✰ ❄ ✒ ✖ ✡ ✮ ✏ ❑ ➉ ✕ ✄ ❶ ✰ ♦ ☞ ✣ ✰ ❜ ➂ ✾ ✰ ✑ ✒ ❈ ✄ ❶ ✰ P ✩ ♦ ☞ ✣ ✰ ❜ ➂ ✾ ✰ ✑ ✒ ✳ ✄ ❶ ✑ ✍ ✏ ✁✟ ✁ ✁ ➈✁ when ✟ and . ☞ ➂ ✒❑✄✂✁ ☞❊➇❄➂❨➁❾✁➂ ☎✒❁✄♠☞❊➇❍➁❬✡✘✙▲➁ ❍✍✤✞▲➇ ✫➂➃➇❍➁ ☎✡✭➇●✍✯➁✫✒ ✁t✄ ➁❧➁ ❀✡✍⑩✖✞ ✏ ➇❧➇ ✡ ✍ ✙ ✏ ✙▲➁ ➇❍✍⑩➁ ✞▲➇ , we have With ☞✎✏✠➂✱✏✠➂✱✏✓✒ ✁ ✄ ✍✤✙ ✙ ✰✞ ✍✑✏ ✏ ; ❽ ☞ ➂ ✒❨✽ ❽ ☞❊➇❄➂❨➁❬✒✥✄ ✍✤❅ , ❽ ☞ ➂ ✒❨✽ ❽ ☞❊➇❄➂✁✈✒ ✄ ✰ At , then, we have ☞ ➂ ✒❨✽ ☞❊➁➃✁➂ ✈✒✂✄✌✞ ➇ ➁ ➁ ❽ ❽ . So the equations can be solved for and or for and .

5. With ☞♦✰✣➂❜✰✾✒

✗

✗

✞

6.

✁

. ✠

✝

✞

✠

✝

✁

✁

19

✁

,

Chapter 3. The Implicit Function Theorem and its Applications

20

✂✁

☞✆✫➂

7. With

✒☛✄ ✍

☞♦✰✣➂✱✏✠➂✱✏✠➂✱✍✑✏✓✒

☞❊➇✬➂❨➁➃➂ ➂ ✒❑✄

☞ ✜❾✡★➇

✍♣➁➃➂

✍ ✚✄ ✞ ✁ ✍ , then, we have ✝ ✁

✜❴✡⑥➁

✁t✄✟✞ ✍ , we have ✝

✍✉➇❫✒

✁

✁

✏

✍ ✏ ✏ ✏

✞

✰ ✏

✏

✍✑✏

✞ ➁

✟

➇ ✞ ✟

✁

✠ . The determinants of all the ✙✉②◆✙ ✞

✁

✠ . At

submatrices

of this matrix are nonzero, so the equations can be solved for any pair of variables. ☞ ✜➂❨❛❜✒ ➇ ✙▲➁ ➇❍➁ ✟ ✡◗➇ ✡◗➁ ✟ ✄ ✄ ❽ ✟ ✟ , we have ☞ ➂ ✒ ✟ ✍❀✙ ✟ ✡ ✙▲➇ , which ✜✮➁ ✦✡✥✙▲➇ ✍ ✞ ✟ ✑✍⑩✙ 8. With ❛ ❽ ✍✤✙ when all variables are set equal to 1. Hence the equations can be solved for and . equals ✙▲➇ ✙▲➁ ✙ ☞ ➂ ➂ ✒ ✄✖➇ ✟ ✡❢➁ ✟ ✡ ✟ ✄❈➇❍➁❁✡ ❛ ✄❈➇ ●✡ ❛ ➁❁✡ ✄ ➁ ➇ ❛ ❽ 9. With , , and , we have ☞❊➇❄➂❨➁❾➂✁☎✒ , ❛ ➇ ❽ ★ ☞❊➇❄➂❨➁❾✁➂ ✫➂❨❛❜✒☛✄ ☞✎✍✑✏✠➂✱✍❀✙✈➂✱✏✠➂❜✰✾✒ ➇❄➂❨➁❾✁➂ . So the equations can be solved for . which equals at

✞ ✠ ✞

✠

✂

✁

✁

✁

✁✁✂ ✞

✂

✁

✁

✠

✁

✁

✁

✚

✂

✁

✁

✍

✍

3.2 Curves in the Plane

✂✁✂

✄☎

✂

✁

✄❉③ ☞♦✰✣➂❜✰✾✒❲✐ ✽ ➁⑤✄ ☞❊➇❫✒ 1. (a) is a smooth curve (an ellipse); only at ; near any point except ❷ ❷ ❵ ☞ ❆✑✛ ✞✣➂❜✰✾✒ ➇❢✄ ☞❊➁✫✒ ☞♦✰✣➂ ❆❧✏✓✒ ; ❵ near any point except ✄r③ ☞♦✰✣➂❜✰✾✒ ✐ ✽ ➁ ✄ ☞❊➇❫✒ ❷ ; ❵ (b) ❷ is the union only at near any ☞ ❆✲✛ ✞✈of➂❜✰✾two ✒ ➇❢smooth ✄ ☞❊➁❬✒ curves (a hyperbola); ; near any point. point except ❵ ➁s✄ ☞❊➇❫✒ ☞✹✛ ✞✣➂❜✰✾✒ (c) ❷ is a smooth curve (one branch of a hyperbola); never vanishes; ❵ except near ; ➇⑥✄ ☞❊➁✫✒ globally. ❵ ➇✪✄ ✰ ➁❋✄ ✰ ➇s✡⑤➁❦✄ ✏ ✄♠③ (d) ❷ is the union of the three lines , and ; ➁s✄✖✰ at➇❀the ➁ ✄ ☞❊➇❴✒ , ✡❦➁✉three ✄✳✏ points ➇⑥✄ where ☞❊➁❬✒ ❵ ❵ these lines intersect; elsewhere, near all points of the lines and and ➇⑥✄✖✰ ➇❧✡❲➁ ✄✳✏ near all points of the lines and . ☞♦✰✣➂❜✰✾✒✹➅ ➁⑥✄r➇ ✟ ✡⑨✏ ✄ ③ ☞♦✰✣➂❜✰✾✒ (e) the ; at ☞♦✰✣➂✱✏✓but ➁✉✄ ❷ is☞❊➇❴the ✒ union of ➀ ➇❢✄ and☞❊➁❬ ✒ parabola ✒ not on the parabola; on the parabola; near any point of the parabola except . ❵ ❵ ➁ ✄ ➇ ✟ ☞♦✰✣➂❜✰✾✒❧✄ ③ ➁ ✄ ☞❊➇❴✒ (f) ; ☞♦✰✣➂❜✰✾✒ but ❷ is still a smooth curve there; globally; ❵ ➇⑥✄ ❷ is☞❊➁✫the ✒ parabola ❵ near any point except P . ☞♦✰✣➂●☞ ✙ ✡ ✒✔❃❄✒❤➄ ✐ ✭➅ ✄✕③ ✟ ; at each point of ❷ . (g) ❷ is the discrete set ➀ ✟ ✟ ☞❊➇ ✬✡❢➁ ✜✒❁✄ ●☞❊➇ ✩✎ P ➂❨➁ ✎ P ✒✭✄♠ ❶ ☞♦✰✣➂❜✰✾✒ ☞❊➇❄➂❨➁❬✒❁✄ ☞♦✰✣➂❜✰✾✒✲✐ ✽ 2. (a) except at ❷ , so ❷ is locally a smooth curve. ❷ is also connected; see part (b). ❲✄ ✏ ❲✄ ✙ ✺✙❞✏ P ✓ ✒ is a straight line. If (b) If , , is the unit circle. If ❷ ❷ ➁❇✄ ☞✎✏✦✍ ➇ ✒ ➁◆✄ ✍✯➇ ➇ ✣ ❆ ✥ is odd, ❷ is the graph of , a curve that is asymptotic to the line ☞♦✰✣➂✱✏✓✒ as☞✎✏✠➂❜✰✾✒ ✙tbut ✙ has a “bump” in the and . If even, is❆❧ a ✏ simple middle where it goes through the first quadrant between ➇❢is✄⑨ ❆❧✏ ❷ ➁s✄✌ closed curve intermediate between the unit circle and the square with vertices ✓P at , . ➁s✄⑨❆ ☞✎✏✯✍✪➇ ✾✒ ✒ (c) If✍✑✏✠➂✱is✏✮➉ even, the top and bottom halves of ❷ are graphs of ➇ ➁ , a continuous function P ✓ ✒ except at the endpoints; likewise with on ➈ that is differentiable and switched.➇⑩If✄ ✏ is odd, ❷ ➁❋✄ ☞✎✏✭✍ ➇ ✜✒ is the➇ graph➁ of , a continuous function that is differentiable except at ; likewise with and switched. ➇❢✄♠☞❊➁⑧✍ ✏✓✒ ✟ ✍ ✏ . 3. (a) ❷ is the parabola ➁s✄❈➇❧✡ ✙ ➇ ✌✍✑✏ ☞✎✍✑✏✠➂✱✏✓✒ ✂✈☞♦✰✾✒ ✂ ☞♦✰✾✒❑✄✖✰ (b) ❷ is the half-line , ; the endpoint is , and . ✂

✂

✂

✁

✁

✁

✂

✂✁ ✂

✂

✁

✝✆ ✞✆ ✠✟ ✝✆ ✡✆ ✆ ✟ ✆ ✆✆

☎✄

✂✁ ✂

✂

✟

✟

☛✆ ✆

✆

✆

✤

✟

✆

✟ ✝✆ ✆

✆

✟

✆

✆

3.3. Surfaces and Curves in Space

21

➁s✄❈➇❧✡ ✙ ✂ ☞♦✰✾✒❁✄✖✰ ✂✈☞♦✰✾✒☛✄ ☞✎✍✑✏✠➂✱✏✓✒ ; ✒ but ❷ is still smooth at . ✒ ➇ ✟ ✡✇➁ ✟ ✄✳✏ ❆ ☞✎✏✠➂❜✰✾✒ (d) ❷ is the✂☎astroid , aP simple closed curve with cusps at ☞❊❛❜✒ ✂ ☞❊❛❜✒❁✄✕③ ❛❁✄ ✟ ❃ ✐ ❷

(c)

is the line

✜

✜

where

the points

✟ ,✟

(

and

❆ ☞♦✰✣➂✱✏✓✒

, which are

☞ ✙✈➂❜✰✾✒

☞♦✰✣➂▲✛ ✞✾✒

✄

).

(e)☞✎❷ ✍ is ✏✠➂❜a✰✾✒ limac☞♦¸✰✣on; ➂❜✰✾✒ a reasonable ☞✎✍✑✏✠➂❜✰✾✒ sketch ☞♦✰✣➂✱✍✭✛can✞✾✒ be obtained ☞ ✙✈➂❜✰✾✒ by drawing a smooth ❛☛✄ curve ❃❄✽✠✞ from ✄✖✰✣➂✆✄✆✄✆to✓✄ ➂✹❅ to to to to (corresponding✂ to for ). It is ☞✎✍ ✏✠➂❜to✰✾✒ ☞❊❛❜✒ a smooth curve except at , where it has a self-intersection; never vanishes.

✟

✟

✍

✦✄✖✰

are both 0. ❛ ✰✣➂ ✟P ❃❾➉ ✟P ❃■➂❜❃❾➉ ❃■➂ ✟ ❃❾➉ ❃■➂✹✙✠❃❾➉ ✂☎☞❊❛❜✒ ✟ ➈ ➈ ➈ ➈ (b) Since , as☞✎✏✠➂❜traverses the intervals , , , and ✰✾✒ ☞♦✰✣➂✱✏✓✒ ☞✎✍✑✏✠➂❜✰✾✒ ☞♦✰✣➂✱✍✑✏✓✒ ☞✎✏✠➂❜P ✰✾✒ ☞ ✽ ❛❜✒✚✍✂☞♦✸✻✺✜✼❾, ❛❨✒✿✄ traces line segments to ✄ .❃ Since ✐ ❆✘✙■P ✸✻✺✜✼➃out ❛✈✼✄✔✖the ✕❑❛ ☞ ✽ ❛❨✒✚✍✂☞♦✼✄from ✔✖✕❝❛❜✒ ✄ ❆✘✙■to✼✙✔ ✕❤❛✈✸✻✺✜✼❍to❛ ✂ ☞❊❛❜✒❢✄ to ✰ ❛❢ ✟ and , when for ✟ ; the points ✟ ✂☎☞ ✟ ✟ ❃❄✒ are the corners of the square. ➁ ✟ ☞✎✏✲✍t➇❴✒♣✄ ➇ ✟ ☞✎✏✭✡✌➇❴✒ ➇⑨✄ ☞❊❛ ✟ ✍r✏✓✒❨✽✣☞❊❛ ✟ ✡♠✏✓✒ ➁t✄ on ❷ by substituting in and (a) One sees that ☞❊➇❄➂❨➁❬✒ ☞❊➇✬➂❨➁✫✒❁✄✏✂☎☞❊❛❜✒ ❛✻☞❊❛ ✟ ✍❋✏✓✒❨✽✣☞❊❛ ✟ ✡✇✏✓✒ this ✟ ✒❨✽✣☞❊➁ ✟ ✡❋ ❛☛✄t➁✫✽✩➇ ❛✿✄⑨❆❧✏ and ☞❊➇✬➂❨simplifying. ➁✫✒❁✄♠☞♦✰✣➂❜✰✾✒ Conversely, ➇❢✄ if ☞❊➁ ✟ ✍❢➇satisfies ➇ ✟ equation, ✒✿✄ ☞❨☞❊➁✫✽✩➇❫then ✒ ✟ ✍✇✏✓✒❨✽✣☞❨☞❊➁✫✽✩➇❫✒ ✟ ✡◗where ✏✓✒✿✄ ✟☞❊❛ ✍⑤✏✓✒❨✽✣(☞❊❛ ✟ ✡✕✏✓✒ if ➁ ✄♠☞❊➁✫✽✩➇❫✒ ➇⑥),✄❈because ❛✔➇ and . ❛✱ ☞❊➇❄➂❨➁❬✒ ☞✎✏✠➂❨❛❜✒ ➇⑥✄✳✏ ❛ ✍ ✍✑✏ ✏ (b) For very ☞✎large, to to to ✂☎☞❊❛❜✒ ✏✠➂✱✍ ✒ ☞♦✰✣➂❜✰✾✒ , so ❷ is asymptotic ☞✎to✍✑✏✠➂❜✰✾✒ . As goes from ☞♦✰✣➂❜✰✾✒ ☞♦✰✣➂❜✰✾✒ , goes from to , makes a loop through and back to , and goes from ☞✎✏✠➂ ⑤✒ to . ☞♦✰✣➂❜✰✾✒❤✄ ✂✈☞ ❆❧✏✓✒ ✂☎☞❊❛❜✒ ❛ ✍✑✏ ✏ (c) We have . The for near or are both smooth; they are ❆❧✏ curves described by at the origin. One can see this from the tangent➇ to the lines with✏✂slope ❆◗➇ ✳✏ ➁ ✟ ☞✎✏❝✍✇➇❴✒☛✄❈➇ ✟ ☞✎✏❑✡◗➇❫✒ nonparametric ➁ ✟ t➇ ✟ representation ➁ ✌❆✭➇

4. (a) The left-hand and right-hand derivatives of

✼✄✔ ✕ ✟ ❛☛✡✕✸✻✺✜✼ ✟ ❛ ✄ ✏

✂

at

✜

✜

✌

✌

✄

✌

5.

✌

☛

✥

✥

✥

✥

☛

is small, then

too: if

✄✖✰ P ✄✖✰ ✂ ✄✕✰ if and only if either or ✟ ✄ ✰ ✟ ☛P ✄✌③ when P✂✄ ✟ ✖ .

✁

✁

6. (a) This is obvious: (b)

✂

✄

✁

✁

P

✂

✁

✜

☛

, so the equation

✟ ✡ ✜

✁

✂

✁

✁

gives

.

✁

.

✁

3.3 Surfaces and Curves in Space

➇ ✡◗➁ ✄ ✄ ✂ ✂ ;❽ and ❽ everywhere independent. ❊☞ ➇❫✽ ✣✒ ✟ ✕ ✡ ☞❊➁❬✽ ✗✻✒ ✟ ✄ ✟ ✂✦✄✕③ (b) ❷ is the cone ;❽ at the origin (the vertex of the cone). ➇ ✟ ✡◗➁ ✟ ✍ ✟ ✄✳✏ ✂ ✂ (c) ❷ is the one-sheeted hyperboloid ;❽ and ❽ everywhere independent. ✲✄✖➇ ✟ ✡◗➁ ✟ ✂✭✄✌③ ✄✖✰ (d) ❷ is the paraboloid ;❽ when , but the surface is nonsingular. P ✂☎☞ ➂ ✒❁✄ ☞ ➂ ✍◆✞ ➂ ✟ ☞ ✟ ✡ ✟ ✒❨✒ ☞✎✏✠➂✱✍✤✙✈➂✱✏✓✒✂✄☞✈✂ ☞✎✏✠➂✱✏✓✒ ✂✈☞✎✏✠➂✱✏✓✒☛✄ ☞✎✏✠➂✱✏✠➂✱✏✓✒ we have , and (a)✂☎With ☞✎✏✠➂✱✏✓✒❝✄ ☞✎✍✑✏✠➂✱✍✤✞✈➂✱✏✓✒ ☞ ✫➂✱✍✤✙✈➂✱✍✤,✙✜❽ ✒ The ✏✓cross , so the tangent plane is ❍☞❽ ❊➇ ✍⑤✏✓✒■✍⑩✙✫☞❊➁ ✡✥✙✜✒■✍⑩✙✫. ✆☞ ✲✍ ✒❁✄✖✰ product ✙▲➇♣✍✇of➁❧the ✍ ⑧✄latter ✌✞ vectors is or . ✂☎☞ ➂ ✒✯✄⑦☞❨☞ ✡ ✒ P ➂❄✍ ✍ ➂ ✓✒ ☞✎✏✠➂✱✍✤✙✈➂✱✏✓✒✤✄ ✂✈☞✎✏✠➂❜✰✾✒ ✂✈☞✎✏✠➂❜✰✾✒❝✄⑦☞✎✍✑✏✠➂✱✍✑✏✠➂✹✞✜✒ , we have , (b) With ✂☎☞✎✏✠➂❜✰✾✒❀✄ ☞✎✍✑✏✠➂✱✍✑✏✠➂❜✰✾✒ ☞ ✞✈➂✱✍✤,✞✈❽ ➂❜✰✾✒ and✞✫☞❊❽ ➇❢✍ ✏✓✒■✍ ✞✫☞❊➁✘✡✥✙✜✒☛✄✖✰ . The , so the tangent plane ➇♣✍✪cross ➁ ✄✕✞ product of the latter vectors is

1. (a)

❷

is the plane

✝

✁

✱

✁

✝

✁

✁

✁

✝

✂

✎

✁

2.

☛

, or

✝

✁

✁

✁

✁

✎

✂

✜

✁

✝

✁

✁

✁

✁

✁

is

or

3. (a) Using polar coordinates in the (b) Using polar coordinates in the

➇❍➁ ➁

.

☎☞ ➂ ✒☛✄ ☞ ✸✻✺✜✼ ➂ ✼✄✔✖✕ ➂ ☞ ✒❨✒ . ❵ ✂☎☞ ➂ ☛ ✒ ✄ ☞ ➂ ☞ ✒✈✻✸ ✺✜✼ ➂ ☞ ✈✒ ✼✙✔ ✕ ✒ -plane, ❵ ✵ ❵ ✵ . -plane,

✂

✁

✁

✁

✁

✁

✁

Chapter 3. The Implicit Function Theorem and its Applications

22

☎☞❊➇❄➂❨➁❬✒⑧✄❉☞❊➇✬➂❨➁➃➂✱✍ ✆ ✏❑✡❲✙▲➇❾✟☛✡◗➁✣✟✓✒ ✂

(c) two of the possibilities.

✈☞ ➂ ✒❑✄ ☞ ✞■✸✻✺✜✼ ➂ ➂❾✞■✼✄✔ ✕ ✒ ✂

(d)

✁

✁

✁

and

.

■☞ ➂ ✒ ✄❉☞♦✼✄✔✖✕✁

✸✻✺✜✼▲☞ ✽▼✛ ✙✜✒ ➂❍✼✄✔ ✕✂

☛

✁

✁

are

☞ ✙✈➂✱✍✑✏✠➂✱✍✑✏✓✒ P ✞✫☞✎✏✠➂✱✏✠➂✱✏✓✒ ☞✎✏✠➂ is ➂ ✒❫✡◗❛✗☞✎✏✠, ➂✱and ✏✠➂✱✏✓✒ one point in the ✜ ✜ works. ☞✎✏✠➂✹✙✈➂❜✰✾✒ ☞♦✰✣➂✱✏✠➂✱✍✤✞✜✒ ☞✎✍✤❅✈➂✹✞✈➂✱✏✓✒ (b) The cross product ❛✻☞✎✍✤❅✈one ➂✹✞✈➂✱point ✏✓✒ in the ☞ ✞✈➂❜✰✣➂✱✍ ✟ of✒ the normal ➁ ✄✖✰ vectors ➇ and ✂☎☞❊❛❜✒❁✄ is☞ ✞✈➂❜✰✣➂✱✍ ✟ ✒❫✡❲, and ✜ (set ✜ and solve for and ), so works. intersection is P ⑧✄✳✏❄✍♣➇ ✙▲➇ ✟ ✍ ✙▲➇✤✡★➁ ✟ ✄✖✰ ☞❊➇✘✍ ✟ ✒ ✟ ✡ ✟P ➁ ✟ ✄ P P P (a) Substituting in the equation of the sphere gives or ✏✯✍✇➇❢✄ P ✍ P ✸✻✺✜✼✫✪ ❛ . ➇❢✄ ✟ ✡ ✟ ✸✻✺✜✼❬❛ ➁ ✄ ☎ P ✼✄✔ ✕✂❛ ❧✄✳ ✟ ✟ , ; then . Parametrize this ellipse in the usual way: ✟ ☞ ✟P ➂✱✍ ☎ P ➂ ✟P ✒ ❛ ✄❉✍ ✟P ❃ ☞❊➇✬➂❨➁➃➂✁☎✒ ☞❊❛❜✒ ✄ ☞✎✍ ✟P ✼✄✔ ✕❤❛✮➂ ☎ P ✸✻✺✜✼❬❛✮➂ ✟P ✼✄✔ ✕✂❛❜✒ (b) corresponds to ;P we have , which ✟ ✟ P ➂ P ✒❙✡◗❛✗☞ P ➂❜✰✣➂✱✍ P ✒ ☞ ✟P ➂❜✰✣➂✱✍ ✟P ✒ ❛✿✄✳✍ ✟P ❃ ✂☎☞❊❛❜✒❁✄ ☞ ➂✱✍ ☎ ✟ ✟ ✟ ✟ at ; so works. equals ✟ ➁ Perhaps the best ✟ this is to perform a rotation around the -axis to make the plane horizontal. ✤✄ ✛way ✏☛✡✖to✱ nail ✞

6.

✁

☞✎✏✠➂✱✍✤✙✈➂✱✏✓✒

4. (a) The cross product normal and ✂☎☞❊❛❜✒❁✄ ☞✎✏✠➂ P ➂ of✒ the ➇⑥ ✄✳✏ vectors ➁ ✜ ✜ (set intersection is and solve for and ), so

5.

✼✄✔ ✕ ➂❄✍★✸✻✺✜✼✄ ✒ ✁

✞

✆

and

Namely, let

✄ ✏

✞

✁

✏

✱

✍ ✱ ✞

✠

✆

✄ ✏ ➇

✏

✞

✠

➇ ✡✗✱ ❧ ✍✤✱✾➇ ✡ ✞

✠

✆

✄

✠

(The matrix is orthogonal since its✄✳ columns are orthonormal, ✟ ✄ transformation ✟ ➁ ✟ shapes.) ✏✓✽ ☞✳✱ ✡ so ✒ this ☞ ✍ ✱ ✒ ✟ ✡ ✟ preserves ➁✟ ✡❈☞✎✏✂✍ In these coordinates the plane is and the cone is or ✟ ➁ ✟ ✡◆☞✎✏❴✓ ✱ ✟ ✒ ✟ ✍ ✯✱ ✄ ☞✎✏❴✍✓✱ ✟ ✒ ✟ ✍ ✱ ✟ ✒ ✟ ✍ ✯✱ ✽ ✤✄♠☞✎✏❾✓ ✍ ✱ ✟ ✒❨✽ ✟ . Thus the intersection is the curve ✄♠✏✓✽ ☞❊➁➃➂ ✒ ✱✉✄✌✰ ✰ ✕✳ ✱❫✸✕r✏ in the plane ✱❴✭✄ with . ✱❫This , an ellipse if , ✏ coordinates ✹✙ is✏ clearly a circle if , and a hyperbola if . Parametrizations may be obtained in the form a➁❋parabola if ✄ ✼✄✔ ✕❤❛ ✄ ✸✻✺✜✼❍❛✬✡ ➁❋✄ ✼✄✔ ✕✁✂❛ ✄ ✲✸✻✺✜✼✄❝❛●✡ P , in the elliptic case and , in the hyperbolic ➁s✄❈❛ ✄✌❆ ☎ ❛ ✟ ➂ ☛➂ ), and , case (for suitable ✟ in the parabolic case. ✆

✆

✁

✆

✁

✆

✁

✆

✁

✁

✆

✁

✆

✞

✝

✝ ✞ ✠

✁

✠

✞

✁

✝

✍✠

✁

✁

P the proof is as outlined 7. The statement is as follows, and in the ✄ text✁⑩ on➄ p.☞❊✁✬130. ✄ ❥ ✜ ✒✂✄ (a)☞❊✁✬Let ✒✂✄✌✰✈➅ and ❡❋✐ be on an open set in , and let . If ✟ ❷ real-valued functions of class ❷ ➀ ☞♦❡❾✒ ☞♦❡➃✒ ❡ ❥ ✜ and and ✄ P are linearly independent, there❥ is a neighborhood ❥ ✟ ☞❊➁❾➂✁☎✒✉✄ ✆ ☛●of☞❊➇❫✒ in such that ❷ ✆ P is the graph of a function✂♣from some interval in into ( , or similarly with the✂ ➄●☞✳✱➃✙➂ ✗✻✒ ✣ ❥✝✜ ✄ ✂ ☞❊❛✁✓✒❧✄♠ ❶ ③ P be of class . If , there is an open interval variables permuted). (b) Let ❛ ✂☎☞❊❛❜✒✂➄✜❛❤✐ ✂✫➅ ✄ containing such that the set ➀ is the graph of a function as in part (a). ✁

✁

✂

✁

✂

3.4 Transformations and Coordinate Systems

➇

➁

1. (a) constant: circles centered at the origin. back of text for the inverse and Jacobian.)

P (b) ✂ ✎

✙ ➇ the lines ☞ ➂ ❁✒ ✄ ☞ ✛ . ➂ ✛ constant: ✒ . ✂ ✰ ✂

✄

✂

✂

✁

✝

✄

✁

✁

constant: half-lines starting at the origin. (See answer in ✆

✆

(

✁

✁

✂

(c) parabola.

. The range of

✁

✝

2. (a)

✄

is the parabola

➇❢✄ ✍ P ☞ ✍⑩✙ ✒ ➁ ✄✳✍ P ☞ ✙ ✍ ✜ ✜ , ✁

✁

✒

(i.e.,

✁

✙ ✰

➁

✂

).

✄ ✱✽ ✟ ✆

constant: the curves

✄ P ✟ ➇❇✄ ✪ , and each line

✯✎ P ✄ ✍ P ✜ ✂

).

✆

or

✁

➁⑥✄

✆

(

✙ ✰

).

✆

maps onto this

3.4. Transformations and Coordinate Systems

➇

23

➁

✄ ✍

✄

(b) , , and ❂ ✍ Substituting ✄✕✞ the formulas for and into the equations for the lines gives bounded by these lines. ☞♦✰✣➂❜✰✾✒✉✄ ;✂☎the ☞♦✰✣➂❜image ✰✾✒ ☞✎✍✑is✏✠➂✹the ✙✜✒ ✄ triangle ✂☎☞ ➂ ✪ ✒ ☞ ✙✈➂✱✏✓✒ ✄ ✂✈☞♦✰✣➂✱✍✑✏✓✒ ✜ ✜ (c) , , and ; the region is the triangle with these vertices. ✁

✁

✁

✄

✼✄✔✖✕❑➇◆✄ ✱ ✸✻✺✜✼❬➇❋✄ ✗ ☞ ✽ ✱✫✒ ✟ ✍❈☞ ✽ ✗✗✒ ✟ ✄✝✏ ✸✻✺✜✼ ✯➁❢✄ ✼✙✔ ✕✁✂➁ ✄ , we have . If and , we have ✟ ✡✕☞ ✽ ●✒ and ✟ ✄✳✏ . ☞ ➂ ✒ ✄ ✸✻✺✜✼❍➇❀✸✻✺✜✼✄✤➁ ✼✙✔ ✕❤➇❀✼✄✔✖✕✁✂➁ ✄❏✸✻✺✜✼ ✟ ➇❀✸✻✺✜✄✼ ✟ ➁❢✡✌✼✄✔✖✕ ✟ ➇✭✼✄✔ ✕✁ ✟ ➁✕✄❏✸✻✺✜✼ ✟ ➇●☞✎✏✘✡ ❽ ✍★✼✄✔ ✕❤➇✭✼✄✔ ✕✁✂➁ ✸✻✺✜✼❍➇❀✸✻✺✜✼✄✤➁ (b) ☞❊➇✬➂❨➁✫✒ ✼✄✔ ✕✁ ❽ ✟ ➁❬✒☛✡✌✼✄✔ ✕ ✟ ➇❀✼✄✔✖✕✁ P ✟ ➁✥✄ ✸✻✺✜✼ ✟ ➇⑥✡✌✼✄✔ ✕✁ ✟ ➁ ✸✻✺✜✼❍➇✕✄ ✼✄✔ ✕✁✂➁⑤✄ ✰ , which vanishes when , i.e., when ➁✉✄❈✰ ➇❢✄ ☞ ✡ ✟ ✒✔❃ ✄✕❆❧✏ ✄✖✰ and ; for these values, ☞ ✽ ✱✫✒ ✟ ✍✖☞ ✽ ✗✗✒ ✟ ✄ ✏ and ☞ ❆ .✆▲➂❜✰✾✒ ✆✟ ✄ ✱ ✟ ✡ ✗✟ ✱➃➂✙✗ (c) The foci ✆⑧ of✄ the✏ hyperbola are at where ; ☞ with ✱❫ ✤✝ ✗▼ ☞ ✽ ✒ ✟ ✡⑨☞ ✽ ●✒ ✟ ✄ ✏ ❆ ●➂❜✰✾✒ as in , the foci of ✏ the ellipse are at where (a) ✟ ✄ we have ✟ ✍ ✟ . When ➂ ⑥✄✳ ; with as in (a) we have . ➇❧✍⑥➁s✄✖✰ ➇❧✍⑥➁✉✄✳✏ ➇❍➁s✄✳✏ ➇❍➁s✄

3. (a) ☞ ✽ If ✒ ✁

✞

✁

✝

✁✂ ✞

✞

✝

✁

✠

✟

✁

✁

✞

✁

✠

✞

✞

✝

✠

✝

✠

✝

and 4. (a) The lines again in the third quadrant.

✏ ✍ ✏ ✁ ✟ ✞ ➁ ➇ ✠ , (b) ✝ ➇ ✍✇➁ ✭✄ ✂

meet the hyperbolas

✄❈➇❧✡❲➁

and

.

in the first quadrant and

➇➃➁

= constant are tangent to the hyperbolas = constantP along the line (c) The lines ➇★✄❈➁❤✡ ➁✉✄ ✟ ☞✎✍ ✍ ✛ ✟ ✡ ✄ ☞❊➁❝✡ ✒ ➁ ➁ ✟ ✡ ➁✭✍ ✄✖✰ (d) We have , so or ; hence ➁ ✄ ✍✤✞ ☞ ➂ ✒❑✄ ☞ ❂✈➂✱✍✤❅✜✒ ➇★✄t➁✫✡ ✄ ✟P ☞ when , and then minus sign is necessary to make ✁

✁

✁

✞

✁

★✄ ✟P ✍✑✏✤✏✤✍ ✍ With , we have ✏ ✍ ✏ ✍ ✙ ✍✑✏ ❀ ✂☎☞ ✙✈➂✱✍✤✞✜✒ ☛■☞ ❂✈➂✱✍✤❅✜✒❤✄ ✍✤✞ ✙ ✍ ✞ ✍✑✏ ❀ (e) ✟✿✡✌

✄✳✏✓✽ ✛

✍

✞

✝

✁

✝

✠ ✞

✍✤✙ ✍ ✙ ✤

✁

✍

✁

✍

✠.

✍

✏ ✰ ✞ ✠ ✰ ✏✠ . ✞ ✍❀✙▲➁✫➁ ✽✩➇ ✜ ✏✓✽✩➇ ➇ ✟ ✠ ✍

✄

✝

✁

✁

✁

.

✁

✁✂✝

☞ ➂ ✒ ✄ ✂☎☞❊➇✬➂❨➁✫✒ ✄ ☞❊➁❬✽✩➇ ✟ ➂❨➇❍➁❬✒ ✂✇✄ ✂✇✄ ✍✤✞▲➁✫✽✩➇ ✟ . Then and . Also ➁✉✄ ➇ ✟ ➇❢✄ ✽✩➁✉✄ ✽ ➇ ✟ ➇❢✄ ☞ ✽ ✒ P✓✒ ✜ ➁s✄ ➇ ✟ ✄ ☞ ✟ ✒ P✓✒ ✜ , so , whence and . ✁❦✄ ✆ ✆ ✰ ✍✖✄ ✆ ❇✄ ♦ ☞ ✻ ✸ ✺ ✏✆✱✒ ✆ ➇➃✟❁✡◗➁✣✟ : the sphere✆❇ of✄❉ radius :❲✄ the (the positive (a) ✰ ✆❇about ✄❉❃ . ❉(half) ✰ ✆❇cone ✄ ✟P ❃ ✄ ✆ , the plane ). ✵ : the vertical half-plane or negative -axis if ✄ or✆ ➇➃if➁ corresponding to the ray ✵ (polar coordinates) in the -plane. ✼✙✔ ✕ ✍✲✸✻✺✜✼ ✁✿✸✻✺✜✼✯✍✲✸✻✺✜✼ ✍☞✁✿✼✄✔ ✕ ✍✲✼✄✔ ✕ ✂ ✄ ✼ ✔ ✕ ✍✲✄✼ ✔ ✕ ✵ ✁✿✸✻✺✜✯✼ ✍✲✄✼ ✔ ✕ ✵ ✁✿✄✼ ✔ ✕ ✍✲✸✻✺✜✼ ✵ ✙ (b) ✵ , so one easily computes that ✸✻✺✜✼ ✍ ✵ ✍ ✁■✙✼ ✔ ✕ ✍ ✵ ✰ ✂✭✄ ✁ ✟ ✄ ✼ ✔ ✕ ✍✂☞♦✸✻✺✜✼ ✟ ✍ ✡✥✄✼ ✔ ✕ ✟ ✍■✒✻☞♦✸✻✺✜✼ ✟ ✡❲✙✼ ✔ ✕ ✟ ✒■✄ ✁ ✟ ✄✼ ✔ ✕ ✍ ✵ ☞❊➇ ✠➂❨➁ ✓✒✵ ✭✄ ❶ ☞♦✰✣➂❜✰✾✒ . ✁ ✄✖ ❶ ✰ ✍ ✄✖❶ ✰ ❃

5. Let

✁

✁

6.

✝

☛

✁

➁s✄✳✍✯➇ . ✒ ✍ ✛ , where ✟❁✡ the✒

✌

✁✂✝

✁

✝

✁

✁

✁

✂

✄☎

✂

and

(c)

or ; in other words,

.

✄✂ ✄ ✞ ✆✝ ☎ ✝✆✝ ✠ and hence ✁ ✂ ✝✞✂ ✄✂ ✁ ✂ ✝✆✝ . If the ✝ ☞♦❡❫➂❜❣✿✒ ☞♦❡❫➂❜❣✿✒✯✄r③ ☛❴❘ there. ❘ P P P latter determinant is nonzero at , where , then ✂ is locally invertible Because

7. With

✂

☞❊✁✿➂❨✢●✒❤✄♠☞❊✁✿➂ ✲☞❊✁✿➂❨✢●✒❨✒

✰

✂

✁

✁

, we have

✁

✁

✎ ✎ ☞ ➂✠✟●✒✦✄ ☞ ➂✂☛■☞ ➂✠✟●✒❨✒ ☛ ➄❾❥✔☛ ✣ ❥ ✄ ✁ is . For of the❡ form✂✈☞❊of ✁✬✒☛✂ ✄ , ✂☛●☞❊✁✿➂✹③❾has ✒ the form☞❊✁✿✂ ➂✄✂☎☞❊✁✬✒❨✒❤✄♠☞❊✁✿➂✹③❾✒ ☞❊where ✁✿➂✄✂☎☞❊✁✬✒❨✒✂✄✚③ ✂ near , let . Then ✂ , that is, ✎ P , so solves the implicit ✁

✗

✗

✗

✁

function problem. Uniqueness follows from the uniqueness of

✂

.

Chapter 3. The Implicit Function Theorem and its Applications

24

✄☎

3.5 Functional Dependence

✏

✂ ✙▲✏ ➇

✑✄ ✂

1. (a) ✝

➇❧✡✥✙▲➁ third is ✙▲➇ ✂✭✄ ✏ (b) ✰

✂

✏ ✍ ✏ ✑ ✙▲➁ ✍✪✙

✍✑✏ ✏ ✍✤✙▲➁✘✡ ✙ ➇❧✡ ✙

✄☎

times the first plus

✙▲➁ ✏ ✏

✙

has rank 2 everywhere (the first two rows are independent; the

✄ P ☞ ✡ ✒✟ ✡ P ☞ ✍ ✪ ❵ ✪ ❵ ➁✘✡ ✲✄✕✙▲➇

☎

times the second).

✒✟

✠

✠

.

✏ . ✍ ✏ is nonsingular except on the plane ✑ P ➁✢✎ P✓✒ ✟ ✼✄✔✖✕❑➇ ✰ ➁ P✓✒ ✟ ✸✻✺✜✼✫➇ ✂ ✄ ➁ ✙✚✰ ✍✤✙▲➁✂✸✻✺✜✼❍➇❀✼✄✔✖✕❑➇ ✟ ✸✻✺✜✼ ✟ ➇ ✍ ✏ ✰ (c) is defined for and has rank 2 there (the second ✰ ✰ ✏ ✍❀✙▲➁ P✓✒ ✟ ✄✼ ✔ ✕❤➇ ✄✳✍ ✟ row is times the first). ❵ ✏. ➁ ➇ ✂✘✄ ✙✫☞❊➇❍➁✑✡ ☎✒ ➁ ✙✫☞❊➇➃➁ ✡ ☎✒ ➇ ✙✫☞❊➇❍➁ ✡ ✈✒ (d) has rank 1 everywhere (the second and third rows ✍✯➁ ✍✯➇ ✍✑✏ ✄✕✙✘✍ ✙✫☞❊➇➃➁✑✡ ☎✒ ✍✑✏ ✄ ✟ ❵ ❵ . are and P times the first).P and ➇ ✎ ✍✯➁ ✎ ✏ ✂✭✄ ➇ ✎ P ✍✯➁ ✎ P ✍ ✏ ➇✬➂❨➁✍✙⑤✰ ✂ where is defined (the (e) ➁ ✎ P ✍⑩✙▲➁☎➇ ✎ ✟ ✙▲➇ ✎ P ✍✇➇❍➁ ✎ ✟ ✰ has rank 2 on the set ✄ ✂ ☛ ✒✟ ✡ ✙ ✂✎ ☛ ✒✟ ✄ ☞❊➇ ✟ ✍ ✙▲➁ ✟ ✒❨✽✠✙▲➇❍➁ first two rows are independent; the third is times their sum). . ✏ ✍✑✏ ✏ ✂✭✄ ✙▲➇ ✍✤✙▲➁ ✰ ➇❢✄✖✰ (f) . ✏ ✰ ✏ is nonsingular except on the plane P ✂◆✄ ☞ ➂ ✒ ❥ ✟ ❥ ✟ ✁ ❵ ✄ (a) T HEOREM . Let be a map from a connected open set into . Suppose ✂✈☞❊✁✬✒ ✏ ✁✚✐ ✁ ✁ ✐ ✁ has rank at every . ✂✈Then every has a neighborhood such that that ☞ ✒ ❥ ✟ ✄ ☞❊➇✬➂❨➁✫❵ ✒ and are functionally dependent on and is a smooth curve in . P ROOF : Let and ❵ ✄ ☞❊➇❄➂❨➁❬✒ ✂☎☞❊✁ ✒ has rank 1, it has at least one✄ nonzero ☞❊✁ ✒ . Since ☞❊➇❄➂❨➁❬✒ entry, which we may✁✥take ✄❖✁ to be .✄ By☞❊the implicit function theorem, the equation can be solved near and ❽ ☎ ✄❵ ❵ ✁ ✒ ➇ ➁ ✄ ☞❊➇❄➂❨➁❬✒ ❵ to yield as a function of and a function ➁ of and ➁ ✄ . ☞❊Then ➇✬➂❨➁✫✒ ✄ ☞❊➇❄becomes ➂❨➁❬✒ too. Implicit differentiation of the equations ❵ and with respect ✰❦✄ ☞ ☎ ✒✻☞ ✩➇❴✒■✡r☞ ✒ ✄ ☞ ☎ ✒✻☞ ▲to ➇❴✒■✡r(taking ☞ ✒ P ❽ ❵ ❽ ❽ ❵ ❽ ❽ ❽ ❽ . as the other independent variable) gives and ✱➇ ✄❖☞ ☎ ✒ ✎ ✂✑✄✌✰ and substituting , so Solving the first of these for ❽ ✄ ✍✂☞ ✒ in the second ☞❊➇✬➂❨➁✫✒✤gives ✄ ✍❑☞ ❽ ☞❊➇✬➂❨➁✫✒❨✒ ❽ ❵ ✂ is a (smooth) function of alone, say . Hence , and the range of near ❵ ✁ ✄ ✍✂☞ ✒ is the smooth curve . P ✂✦✄ ☞ ➂ ➂ ✒ ❥ ✟ ❥ ✜ ✁ ❵ ✄ (b)✂✈☞❊T✁✬HEOREM . Let be a map from a connected open set into . Suppose that ✒ ✏ ✁⑩✐ ✁ ✁ ✁ has rank at every . Then such that ❵ , , and are ✂✈☞ ✒ every in has a❥☞neighborhood ✜ is a smooth curve in . P ROOF : The proof is similar to part (a), functionally dependent on ✄ and ☞❊➇❄➂❨➁❬✒ ✄ ☞❊➇✬➂❨➁✫✒ ✄ ☞❊➇❄➂❨➁❬✒ ☞❊✁ ✒⑧✄r ❶ ✰ so we just sketch it. Let , . We➁ may assume ❽ ☎ ❵ . Then ✄ ❵ ☞❊➇✬➂❨,➁✫✒ ➇ P P for as a function of and , making and into functions we can solve the equation ❵ ➁ ✄ ☞ ☎ ✒ ✎ ☞ ➂ ✒❨✽ ☞❊➇❄➂❨➁❬✒ ➉ ✄♠☞ ☎ ✒ ✎ ☞ ➂ ✒❨✽ ☞❊➇❄➂❨➁❬✒ ➉ ❽ ❽ ❵ ❺ ➈ ❽ ❵ ❽ ❽ ❽ of and . One calculates that and , ✂ ✄ ✍❑☞ ✒ ✄ ☎✭☞ ✒ ❽ ❵ ➈❺❽ ❵ and (which describes a smooth which both vanish ✏✄ ✍❑☞ ❵ since ✒ ✆✄ has☎✭☞ rank ✒ 1. Thus curve), and and ❵ . ✝

✂

✝

✄☎

✂

✝

✄☎

✠

✄☎

☎

✂

✝

✄☎

✆✂

✝

✠

✁

✄✂

✄✂

✁

☞

☞

✌

✌

☎

✄

2.

✠

✝

✆

✠

✆

✆

✠

✁

✝

✁

✁

✁

✁

✠

✁

✁

✠

✗

✗

✗

✁

✗

✗

✠

✗

✠

✗

✂

✁

✝

✁

✁

✠

✁

✄

☎

✠

☎

✝

✆

✆

✠

✆

☎

✁

✠

✍

✁

✁

✍

✗

✗

☎

✁

✠

✝

✁

☎

✠

✍

✍

✁

Chapter 4

Integral Calculus 4.1 Integration on the Line

✙✗

✗

✏✗

1. The sup and inf of ❵ on any nontrivial interval are 1 and 0, respectively. It follows that for any partition ✙ ➃➂ ❜➉ ✄ ❑✍ ✄✖✰ ☞ ✒✿✄ ✂✍ ☞ ✒✿✄✖✰ of any interval ➈ we have ❷ ●❵ and ■❵ , so ✞ ❵ and ✞ ❵ .

✱

✱

✂

✂

✱

✂

♠✰ ✼✜✛✣✢✁✜☞✝✆ ✒✘✄ ✆❴✜✼ ✛✣✢✂✜☞ ✒ ✔✖✕✣✱ ☞✝✆ ❵ ✒✘✄ ✆ ✔ ✕✣✱ ☞ ❵ ✒ ; it follows that , for any interval we have ❵ ❵ and ☞✝✆ ✒❁✄ ✆ ☞ ✒ ☞✝✆ ❁✒ ✄ ✆ ☞ ✒ ✙ ✱➃➂✙✗✹➉ and hence that ✝ ✞ ✆ ❵ ✄ ✆ ✝ ✞ ❵ . If partition of ➈ ❷ ✰❵ ❷ ❵ and ❵ ✜✼ ✛✣✢ ☞✝✆ ❵ ✒❁✄ for✆ ✔✖any ✆ ✕ ✣✱ ☞ ✒ ✕ ❵ ❵ , etc.) but the final result is the same. , the orders are reversed ( ✙ ✰ , let ✙ be a partition of ➈ ✱❍➂✙✗✹➉ such that ❷ ✿❵ ✍ ✿❵ ✕ . Let ✙ We use Lemma 4.5. Given ✆ ✙ be the partition obtained by adding the subdivision points and to (if they are ✙ not already there). ✍ ✕ ✜ ✆▲➂ ➉ ➈ By Lemma 4.3, ❷ ☎✄ ❵ be the ✍ partition obtained from by omitting✆✩➂ the ✱➃☎➂✄ ❵✆ ➉ or ➈ . ✙➂ ✗✹Let ➉ ✕ of. By ➉ ; then ✝ ❷ ✆■❵ ✆■❵ Lemma 4.5, ❵ is integrable on ➈ . subintervals belonging to ➈ q q q ✙ ✱➃✙➂ ✗✹➉ q then ■❵ and ❷ ✿❵ If ❵ ❷ for any partition of ➈ ; it follows that ✝✠✞ ❵ ✝✟✞ . ✱❍➂✙✹✗ ➉ ✦ ✞ ✠ and ✞ be✄ the sup and inf of ❵ on , and the Given an interval ✄❉➈ ✤ , let ✠s✄ ✦ ✞ ✄ ✍ sup ✦ ✰ q ✰ let and be ❵ ❵ ❵ and inf of on . If on then and , and if on then ✠s✄ ✍ ✞✕✍ ✠s✄ ✍ ✦ ✞ ✄ ✞s✓✶ ✒❄☞ ➂✩ ✦ ❺✒ and ✠ ✤✌✰ ; in✍ either case, ✡r ✦ ❍✄ ✞❲✌ ✠✯q ✞ ✕ ✍ ✦ . If ❵ changes sign on , then ✙ ✱➃✙➂ ✗✹➉ have and ✍ , so ✈q . It follows that for any partition of ➈ ❆ weq⑨ ✍ ❷ ❆❵ ❷ ✿❵ ■❵ , and✈q it follows from Lemma 4.5 that ❵ is integrable. Moreover ❵ ❵ , q ❵ so ✝✠✞ ❵ ✝✟✞ ❵ , hence ✝✟✞ ❵ ✝✠✞ ❵ . P ✠❝✄ ✔ ✞❲➇ ♠ ✙ ✰ ✠ ➇ ☛ ☛✙ . Given ✌ ✄ ✏✠✆➂ ,✄✆the ✟ interval of length centered at contains for➇ sufficiently Let ✝ ✄✆✩ ✄ ➂ ✽✠✙ ❯ ❯ large, say . For , let be the interval of length centered at . Then every ✞ ➇❯ ❯ , and the sum of the lengths of the ❯ ’s is . is contained in ✑ ✄ ☞❊➇ ✱✒ ✙✕✰ ➇ ✆▲➂ ➉ ✱❍➂✙✗✹➉ containing ➇ If ✟ and ❵ is continuous at , there is an interval ❵ ➈ ✝ ✄ ➈ P P ✟ ✟ ☞❊➇❴✒ ✤ ✟ ➇◆✐ ✆▲➂ ➉ ✄ ✆ ✡ ✡ ✟ ✤ ✤ ☞ ✍ ✆✱such ✒ ✙ ✰ that ✟ ❵ ✟ for ➈ . By Theorem 4.9c, ✝ ✞ ❵ ✝ ❵ ✝ ✆ ❵ ✝ ✞ ❵ ✝ ✆ ❵ ✟ . ✙⑦✄ ➇ ✆➂ ✄✆✄✆✓✄ ➂❨✡➇ ✠✫➅ ✙ ✄ ➇ ✽ ✆✩✆➂ ✄✆✄✆✩✄ ➂❨☎➇ ✠✫✽ ✆✩➅ ✙☞☛✮✣ ✙ ➀ ➀ is a one-to-one (a) If ❍✙➂ ,✗✹➉ let ✱ ✫✽ ✆▲✙➂ ✗✻✽ ✆ ➉ ; then ✱ ☞❊➇❫✒ ➇ ❯ ✎ correspondence P✗➂❨➇ ❯ ➉ of . The sup and inf of ❵ on ➈ ✆ are equal between partitions of ➈ ☞✝✆✮➇❴✒ and ➇ partitions ➈ ✹ P ✽ ✆▲➂❨➇ ✽ ✆ ➉ ❯ , and the length of the former interval is times the length on ➈ ❯ ✄ ✎ ✆ to the sup and inf of ❵ ✝☞ ✆●✧ ✒ ✄ ✆ ✝☞ ✆✬✧ ✒ of the latter. It follows that ❷ ■❵ ❷ ✄❪❵ and ■❵ ✄♦❵ ; the result follows. ✠

☛

2. If

✆✥✤

✠

☛

✂

✂

✂

✠

☛

☛

☛

✠

☛

✂

3.

☛

☛

✌

✂

☛

✌

☛

✂

✌

✂

4.

✌

✂

✠

✠

☛

☛

✠

☛

☛

✠

✂

5.

✠

✠

✂

✁

✂

✂

✂

✁

✂

✁

✁

✁

✁

✂

☛

✁

✂

☛

✠

☛

☛

✠

✠

✠

✂

6.

✟

✂

✟

✆

✆

✆

✂

7.

✂

✌

✠

✠

✌

8.

✌

✂

☛

✂

☛

☛

25

☛

Chapter 4. Integral Calculus

26

✙ ✄ ✍✯➇✡❍✠ ➂✆✄✆✄✆✱✄ ➂✱✍✯➇ ➅ ✙ ✄ ➇ ✡ ✆▲➂✆✄✆✄✆✄✩➂❨➇☎✠✦✡ ✆✩➅ ➀ ➀ (b) and (c) are similar; one considers or . ❍ ✱ ✙ ➂ ✹ ✗ ➉ ✄ ☞ ❍ ✱ ✙ ➂ ✹ ✗ ❊ ➉ ❾ ✒ ② ☞ ❍ ✱ ✙ ➂ ✹ ✗ ❊ ➉ ✒ ☎ ☎ , and✙ ✢ ✰ 9. and are uniformly continuous on ➈ ➈ ➈ ✙ ✰ is a compact☞ set➂ in ✒✗✍ the☞ plane, ➂ ✒✗✏✕so ❵ ✽✠is✙✫☞ uniformly ❵ ❵ ✕ continuous on it. Thus, given , we can choose such that ✗❤✍ ✱✫✒ ☞ ➂ ✒ ☞ ➂ ✒ ✍ ✏✕ ✍ ✕ whenever and are points in ✢ with and , and we ✙⑤✰ ☞❊➇❫✒✿✍ ☞❊➇ ✒✗✠✕ ➇❄➂❨➇ ✐ ✱➃can ➂✙✗✹➉ ☎ ☞❊➇❫✒■✍ ☎ ☞❊➇ ✒✗✸✕ whenever choose ➈✙ ✰ ➇⑩✵✍✖➇ ✹✕ small enough so that ✍✂☞❊➇❴✒◆✄ ☞ ☞❊➇❴✒ ➂ ☎ ☞❊➇❴✒❨✒ and and ✵ . Next, ❵ . P By Proposition 4.16 we can choose q let ✍✂☞❊➇❴✒ ➇❇✍ ✍ ✕ ✙ ✄ ❞ ➇❯➅✠ ✱➃➂✙✵ ✗✹➉ ✟ small enough so that ✵ ✵ ✍ and for➇ any partition ➀ of ➈ with ✞s✶✓✒❄☞❊➇ ❯ ✍✇➇ ❯ P ✒ ✕ ❯ ✎ P✗➂❨➇ ❯ ➉ is achieved at some point ❯ , so ✎ ✵ ➈ . Since is continuous, its infimum on ✍◆✄ ❚ ✍❑☞ ❯ ✒✻☞❊➇ ❯ ✍✇➇ ❯ P ✒ ✎ . Then for any ➇ ❯ ➂❨➇ ❯ ✐ ➈ ➇ ❯ ✎ P✗➂❨➇ ❯ ➉ we have ✠

✠

✁

✁

✁

✁

✠

✠

✂

✠

✠

✂

✝

✌

✞

✡

✞

✡

✡

☞ ☞❊➇❴✒ ➂ ☎ ☞❊➇❴✒❨✒ ➇★✍ ❵

q ✂ ✂✍ ☞❊➇❴✒ ➇⑥✍ P ✠

✡

✁

☛

✁

✂

☛

✁

✁

✠

✡

✞

❵ ✖

✌

☞ ❊☞ ➇ ❯ ✒ ➂ ☎ ❊☞ ➇ ❯ ✒❨✒✻☞❊➇ ❯ ✍◆➇ ❯ ✎ P ✒

q ✙ ✡ ✡

✡

✡

✍❑☞

✁ ❯ ✒✻☞❊➇ ❯ ✍✇➇ ❯ ✎ P✹✒ ✡ ✙✫☞ ✗❤✍ ✫✱ ✒ ☞❊➇ ❯ ✍✇➇ ❯ ✎ P✹✒❁✄ ✄ ✖

✍✂☞ ✖

✡

✌

✡

✡

✠

✁ ❯ ✒●✍ ❵ ☞ ❊☞ ➇ ❯ ✒ ➂ ☎ ❊☞ ➇ ❯ ✒❨✒✗♥☞❊➇ ❯ ✍✪➇ ❯ ✎ P✹✒ ✡

✠

✡

✡

✡

✡

✖

4.2 Integration in Higher Dimensions

✑ ❚ ☞✑ ✒ ✕ ✁ ✑✆☎P ✮ where ✟ ✮ , the same is true of . ✑ ✌❧✄✳✏✠➂✆✄✆✄✆✓✄ ➂ ✶ ✙ ✰ , choose a finite collection of rectangles ❯ ✮ ( ❘ ✄ ❯ ✄ ✑ ✮☎ ❱ P ✑ ❯ ✮ and ❚ ✮ ✟ ☞ ✑ ❯ ✮ ✒ ✕ ✽✩✶ for each ✌ . Then ✑ P ✄ ❯ ✄ ✑ ❯ ✄

1. (a) If ✄ (b) Given ✕ .

✠

✆

✑

and ✮

✄✳✏✠➂✆✄✆✄✆✄ ❯ ) such that ❯ ✮ and ❚ ❯ ✮ ✟ ☞ ✑ ❯ ✮ ✒ ✕ ✦

✁

✆

✙ ✄ ➇ ➅ ✱❍➂✙✗✹➉ ✦ ➇ P✻➂❨➇ ❯ ➉ ➀ ❯ ✠ of ➈ , let ❯ ✄ and➇ P✗❯ ➂❨➇ be➉❁the inf and sup of ❵ on ➈ ❯ ✎ . Then 2. (a) Given a partition ② ✦ ❯➂ ❯➉ P ❯ ❯ ❯ ❯ ✎ ➈ ➈ the graph of ❵ is contained where , and the sum of the areas of ✍ in ✑ ■❵ . This can be made arbitrarily small (Lemma 4.5), so the graph has zero these rectangles is ❷ ■❵ content. graph of ❵ and three line (b) The boundary of ❷ is the union✄ of✞sthe ✓✶ ✒ ✙rsegments, ✄ ➇ ❯ ➅ all✱❍➂✙✗✹of➉ which ✜✚✄ have ➇ ❯ zero ➁✜❘✾➅ ❵ ➀ ➈ ➀ content, so ❷ is measurable. Let . Given a partition of , let ✱❍➂✙✗✹➉●② ✰✣➂ ➉ ➇ ✦ be a➁✜partition of ➈ with the same ❯ ’s, such that each ❯ and ❯ (as in part (a)) is among ➈ ❘ the ’s. Then the upper and lower approximations to the area of ❷ corresponding to this partition are ❵ , it follows that ❵ is the area P of ❷ . just ❷ ■❵ and ■❵ ; as these can be made arbitrarily close to ✙ ✰ ✙ ☞ ✒❝✍ ✕ ✟ 3. Given , choose a partition of a rectangle ✙ containing ❷ such that ✟ ❷ . Let ✲P✱✆➂ ✄✆✄✆✄✓➂ be the subrectangles of the partition that are contained in ❷ (so is the sum ✦ ✛ ✏✯✍ of ✮ and side lengths with the same center as their areas). For each ✕ , let✽✠✙ ✮ ☞ be✒ the rectangle ✵ ☞ ✒✧✙ ☞✎✏✲✍ ✒ ☞ ✒ ✜ ✟ ❷ ; then ✟ ✵ ✟ ✮ , and ✮ ✄ ❷■❸❺❹❜❻ . Let be the times as big, where ✵ ➇ ➁ ✮ (resp. ) coordinates of the sides of the ✮ ’s into partition obtained by adding all the ➇❯ ➁✜❘ ✙ ✜ the collection ✤ of ’s (resp. ’s) in . Then the ✮ ’s are among theP subrectangles of the partition , so ✆ ✧✙⑨☞✎✏✫✍✉☞ ✽✠✙ ☞ ✒❨✒❨✒✻☞ ☞ ✒▲✍ ✟ ✒✝✙ ☞ ✒▲✍ ☞ ✒ ✤ ☞ ✒ ❚ ☞ ✒✝✙⑨☞✎✏✫✍ ✒ ✟ ✮ ✵ ✟ ❷ ✟ ❷ ✟ ❷ ✟ ❷ , . It follows that ✟ ❷■❸❺❹✹❻ and the reverse inequality is obvious.

✑

✡✂

☛

✑

✁

✁

☛

✞✝

✁

✠

☛✡

✠✟

✆

✁

✁

✞

✂

☛

✠

✑

✑

✑

✠

✝

☛

✝

✞

✑

☎

✂ ✌☞

✑

✑

✞

✑

✑

✂ ✂ ✌☞ ✌☞

✁

☛

✑

✁

✑

☛

✂ ✌☞ ✎✍ ✏✒✑ ✁

✁

☛

✑

✑

✑

4. This really follows from Exercise 3 and the observation that if is a rectangle whose interior includes ❷ , then the outer area of ❷ plus the inner area of ➆❾❷ equals the area of , and likewise with ❷ replaced

4.3. Multiple Integrals and Iterated Integrals

27

by ❷ . The inner area of ➆✦❷ is the same as the inner area of its interior ❸❺❹❜❻✬➆ ❷ , which ☞ is✒s✄ the same ☞ ✒ as the inner area of ➆ ❷ since the boundary of has zero content; it follows that ✟ ❷ ✟ ❷ . Alternatively, one can redo the argument of Exercise 3 by considering a partition of and shrinking the rectangles that do not meet ❷ slightly to produce rectangles that do not meet ❷ . ✑

✑

✑

✑

✑

5. Given any partition of a rectangle whose interior contains ❷ , the subrectangles of the partition fall into three classes: (i) those contained in ❷ ❸❺❹✹❻ , (ii) those ✒ intersect ❽❴❷ , and (iii) those✹that ☞ that ✟ do not intersect ✹✟✤☞ ✒ ❷ . (Any rectangle that intersects both ❷❁❸❺❹✹❻ and ❷ ✆ also intersects ❽❾❷ ; otherwise ❷❁❸❺❹✹❻ and ❷ ✆ would be a disconnection of ☞ .) The ✒ sum of☞ the✒ areas☞ of✒ the rectangles in class ☞ (i)✒❙✡ (resp.☞ class ✒☛✄ (ii),☞ classes ✒ (i) and (ii)) approximates ✟ ❷ ❸❺❹❜❻ ☞ (resp. ✟ ❽❴❷ , ✟ ❷ ). It follows that ✟ ❷ ❸❺❹❜❻ ✟ ❽❾❷ ✟ ❷ , and ✒❄✡ ☞ ✒❁✄ ☞ ✒ ✟ ❽❴❷ ✟ ❷ . hence by Exercises 3 and 4 that ✟ ❷ ✑

✑

✑

✑

✁

6. (a)

is a union of open rectangles, so it is open (any point is an interior point).

☛ P by stopping at the thP stage, with P the rectangles of length (b) to obtained ✙ ✎✠☛ Let✁ ☛ be the approximation ☛ ✎ ✎✠☛ P P P ☞✎✏✬✍ ✙ ✎✠☛❍of✒ area ✪ , two rectangles of area ✓ ☞ ,✁ . . .✒ , ✕ P rectangles☞ ✁ of✒✂area . is the union of 1 rectangle . q ✆ ✕ ☛ ☛ ✟ ✟ ✟ , and there is some overlap, so ✟ ; in fact ✟ The sum of these areas is ✆ ✙ with independent of . If is any partition of the unit square, the union of the (closed) subrectangles ✙ ✁ ✁ of that are contained in is a compact set, and the open rectangles out of which is built are an P in other P all these subrectangles are open cover of it, so by Heine-Borel there is a finiteq subcover; ✆ ✕ ☞ ✁ ✒✝✕words, ✁ ☛ ✟ ✟ contained in for some . It follows that ☛✌☞✁ , so ✟ . ✁

✁

✟

✟

✂

☞❊➇❄➂❨➁❬✒❤✐

✰✮✕✥✚ ➁ ✕✌✏ ➇ ☞ ✁ ✒❁✄ ☞ and✒❁✄✳has ✏ a terminating base-2 decimal expansion; ✟ by Exercise 4. ☞ ✁ ✒❁✄✳✏❝✍⑤✏✤✄✖✰ ☞ ✒❁✄ ☞ ✒■✍ ☞ ✁ ✒ ✙✌✏✯✍ ✟P ✄ ✟P

✟

✁

(c) contains every such that ☞❊➇❄➂❨➁❬✒ is dense. Hence ✟ the set of all such (d) We have ✟

☞ ✒✿✄ ☎

✑

☞ ✒✿✍ ✑

✟

✟

✑

and ✟

☎

✟

✑

✟

.

✍✂☞ ✗✒ ✄ ✰ . With ☞❊➇❫✒✑✄ ✝ ❵ ☞❊❛❜✒ ❛ , ✝✠✞ ❵ ☞❊➇❫✒✚✍❑☞❊➇❫✒ ➇⑤✄ ☞❊➇❴✒✚✍✂☞❊➇❴✒✗ ✞ ✍ ✝✠✞ ☞❊➇❫✒✚✍ ☞❊➇❫✒ ➇⑤✄ ✍ ☞❊➇❫✒✚✍ ☞❊➇❫✒ ➇ ✍ ✤❞✰ , we can apply the mean value theorem ☞✳✱✫✒ ✄ ✍✂☞ ✗✒♣✄❉✰ since . Since ✝✟✞ ✍ ☞❊➇❫✒✚✍ ☞❊➇❫✒ ➇❈✄❉✍ ☞ ✱✒ ✍ ☞❊➇❴✒ ➇t✄ ☞ ✗✒✚✍✂☞✳✱✫✒ ✄ ✍✂☞✳✱✫✒ ✆ ☞❊➇❫✒ ➇ , the claimed result to get ✝✠✞ ✝✟✞ ✝ ❵ ✂✍ ☞ ✗✒✉✄ ✰ ✄ ☞❊➇❫✒ ✍✂☞❊➇❴✒☛✡ ✄✭➉ ➇✕✄ ✍❑☞✳✱✫✒ ✆ ☞❊➇❫✒ ➇★✡ ✝ ❵ for the case . Moreover, for any constant , ✝ ✞ ❵ ➈ ✂☞ ✻✒✑✄✕ ✍ ❶ ✰ ✄ ☞❊➇❫✒ ➇❦✄ ✂✍ ☞✳✱✫✒❄✡ ✄✦➉ ✆ ☞❊➇❴✒ ➇ ✡ ✄ ☞❊➇❴✒ ➇ ➈ ✂✍ ☞❊➇❴✒■✍ ✍✂☞ ✗✝ ✒ ❵ ✄✌✄✏✍❑☞ ✗✒ ✝✠✞ ❵ . Finally, if , we apply this result with ✂✍ ☞❊✝✟➇❴✞✒ ❵ ✆ replaced by and to get the desired conclusion. ✗

7. Suppose

☎

✁

✁

✠

✁

✁

✠

✠

✗

✁

✠

✠

✌

✌

✌

✌

✁

✝✆

✁

✠

✗

✝✆

✁

✠

✌

✠

✌

✌

✠

✠

✠

✗

✠

✗

✌

✗

✌

✌

✌

✌

4.3 Multiple Integrals and Iterated Integrals 1.

P P ✎ ✖ ☞❊➇✕✡ ❊☞ ➇✖✡ ✞▲➁ ✜ ✒ ✄ P ✎ (a) P ✝ ✝ ✍ ☞✎✏❝✍✪➇ ✟ ✒ ✜ ✒ ✟ ✡ ☛✜ ✟ ☞❊➇ ✍ ✟ ➇ ✝ ✜ ✡ ✝ P ➇ ✒ ➉ P✎ P ✄ ✪ ✪ ✜ ➈ ✜ ✄ ✟ ✗ ✟ ✗ ☞❊➇ ✟ ✍ ✛ ➁❬✒ ➇ ✁ ☞❊➇ ✟ ✍ ✛ ➁✫✒ (b) ✝✝ ✟ ✝ ✝ ✖ ➁ ✄✒ ✟ ✒ ➁✉✄ ✟ ➁ ✪ ✍ ✪ ➁ ✄✒ ✟ ✍ ✟ P P ➁ ✡ ✟ ➁ ✒ ✟ ➉ ✟ ✄ ➈✜ P ✎ P ❅▲➇❍➁❫☞✎✏♣✍✚➇✥✍⑨➁❬✒ ➁ ➇ ✄ ✄ ☎ P ✝ ➇ ✟ ✍✇✝ ➇ ✜☛✡ ✝ ✜ ➇✢✪❝✍ P ➇ ➉ P ✄ ✟ P ✪ . ➈✟ ✁

☎

✌

✌

✌

✄

✄

✏

✏

✄

✏

✌

☎

2.

✌

✄

✄

✄

✌

✌

✄

3. (a) and (b): See answers in back of text.

P ➇ ✛ ✏❝✍✇➇➃✟★✡ ✜ ☞✎✏⑩✍ ➇ ✟ ✒ ✟ ➉ ➇ ✄ ✪ ✝✎ P➈ . ➁❢✄ ✟ P ➇ ✜✯✍✪➇ ✛ ➁✾➉ ✗✟ ✗ ➁♣✄ ✟ ☞ ➁ ✜✤✍⑩✙▲➁ ✜ ✒ ✟ ✍ P ➁✞✓☛✡ ✜ ✝ ➈✜ ✝ ✜ ✖ ✜ ✟ ☞ ❂✘✍ ✛ ✙✾✒ ✜ . P ✞▲➇●☞✎✏❢✍✌➇❫✒ ➁ ✟ ✍r✙▲➇➃➁ ✜ ➉ P ✎ ➇ ✄ P ➇■☞✎✏♣✍⑨➇❴✫✒ ✜ ➇ ✄ ➈ ✝ ✞▲➁ ✜ ✒ ➁ ➇

☎

✄

✄

✌

✌

✞

✌

✌

✄

☎

✌

✌

Chapter 4. Integral Calculus

28

☞✎✍✤✞✈➂✹✴✜✒ ☞✎✏✠➂✱✏✓✒ ➇❢✄✌❆ ✛ ➁ ➇⑥✄✳✍✤✙✤❆ ✛ ✗✏ ✰✑✍✇➁ and , andP they✗ are given by and P ✗ ✎✠✪ ✎ ✖ ☞❊➇✬➂❨➁✫✒ ➁ ➇⑥✄ ☞❊➇❄➂❨➁❬✒ ➇ ➁❬✡ P ✎ ✟ ✗ ✎ ☞❊➇✬➂❨➁✫✒ ➁ ➇❴✡ ✄ ✎ ✗ ❵ ✎ ❵ Thus P ✎ ✟ ❵ P ✟ ✎ ✗ ☞❊➇✬✎✢✜➂❨➁✫✒ ✖ ➁ ➇ ❵ . ✎ ✟✎ P ✎ ✗ ❵ (c) The parabolas intersect at P

☛

☎

☎

☎

✓

☎

✁

☎

☎

☎

✝

✌

✝

✝

✌

✝

✌

✝

✌

✝

✌

✝

✌

✝

✌

☛

☎

☎

✝

✌

✝

✌

4. (a) and (b): See answers in back of text. ✁ ✟ ✟ ☞❊➇✬➂❨➁✫✒ ➇ ➁ . (c) ❹ ✂☎✄ ❵

P ✗ P ✜ P ➁ ✟ ➇ ➁✉✄ P ✜ ✟ ☞❊➁ ✟ ✍ P P ✻✸ ✺✜✼✩☞❊➁ ✜ ✡ ✏✓✒ ➁ ★ ➇ ✄ P (b) P ☞♦✼✄✔ ✕❝✙✦✍⑩✼✄✔ ✕✘✏✓✒ ✜ . ✟ ✓P P ✒ ➁ ✗ ➁ ➇❢✄ P✓P ✒ P✓✟ ✒✛✗ ➁ P ✟ (c) ✝

✝

✌

✟ ➁❬✒ ➁✉✄ P ➁ ✟ ✗ ✍ P ✟ ✗ ✍ P ✟ ➁ ✟ ➉ ✜P ✄ ✄ ✯✍ ✜ ✜ ✟ ✞ ✞ ✞ . ✪ ➈✪ ✖ ✸✻✺✜✼✓☞❊➁ ✜ ✡ ✓ ✏ ✒ ➁ ➇❢✄ P ➁ ✟ ✸✻✺✜✓✼ ☞❊➁ ✜ ✡✥✓✏ ✒ ➁♣✄ P ✄✼ ✔ ✕❙☞❊➁ ✜ ✡⑤✏✓✒✗ P ✄

✌

✗

☎

✂

5. (a)

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✜

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✌

✌ ✝

P P ✟ ✗ ✍ ✱➁✾➉ P ✄ P ✟ ✍ ✗ P✓✒ ✟ ✟ ✓P ✒ ☞ ✟ ✍ ▲✒ ➁✉✄ ✟➈ ✟ . ➁✕✄ ❋ ➇ ✡⑦✏ ➁⑨✄ ✙▲➇ ✟ The region of integration is bounded above by ✗ , ✒ below by ✗ ✒ on the left by P ✛ ✟ ☞❊➁✫✒ ➇ ➁♣✡ P ✟ ,✗ ✛ and ✟ ☞❊➁❬✒ ➇ ➁t✄ ➁ ✎ P ❵ theP -axis. Reversing the order of integration gives ❵ ☞❊➁❬✒ ✆ ➁✫✽✠✙ ➁ ✡ P ✟ ☞❊➁✫✒✻☞ ✆ ➁✫✽✠✙✘✍✪➁✘✡✖✏✓✒ ➁ ❵ ❵ . ☎ ☞❊➇❫✒☛✄ ✚ ☞❊❛❜✒ ➁ ❛❁✄ ☞❊➇❢✍✇❛❜✒ ☞❊❛❜✒ ❛ ✂

➇ ➁ ✄

✗

☎

☎

✂

✂

✂

✂

✂

✂

✂

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6.

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✝

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☎

7. Reversing the order of integration gives

✝

☎

☎

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✠

✌

✌

✝

✠

✌

.

8. See answer in back of text.

➇❍➁

9. (a) The region is bounded below by⑧✄the ➇❢of ✄✳integration ✏❝✍✪➁ ✟ ❈➁ region in the first quadrant of the and above by the plane . the parabola (b) and (c): See answers in back of text.

P✎

-plane under

P ✗ ✆▼☞✎✏✑✍ ✒ ✟ ➇❈✄ P ✱ ✗ ✆ ▼☞✎✏ ✍ ✍ ✗ ✒ ➁ ➇❈✄ ➇ ✟ P 10. The volume is . The -moment is ✗ P ☞ P ✎ ☞ ✒ ✌ ✌ ✆✮➇●☞✎✏✂✍ ✟ ✟ ✍ ✒ ➁ ➇⑥✄ ✟ ✗ ✆ ■➇ ☞✎✏❑✍ ✒ P ➇★✗ ✄ ✟ ✪ P ✱ ✆ ✗ ✆ , so the ➇ -coordinate of the center P✱ ➁ of mass is ✪ . By symmetry, the and coordinates are ✪ and ✪ . ✟ ✟ ✟ ➁ ➇ ➁ ✟ ➇ ✟ ➁ ➁ ✟ ✄ ✄ ★ ➇ . The -moment is 11. The mass is ✟ ✟ ✟ ➇❍➁ ➇ ➁ ♣✄ ★ ✟ ✟ ✟ ➁✟ ➇ ➁ ➁ ✄ ✜✟ ✜ , and likewise the -moment ; the is ✪ ➂ ✪ ✒ ✜✟ ☞✎✏✠➂ -moment ✜ ✜ . is ✜ . So the center of mass is ✟P ✟ ✎ P ✙ ✟➇ ✍⑩✞ ✕t➇♣✍⑤✏ ✍✑✏✹✕⑤➇ ✕t✙ ➁ ➇❋✄❈✙ ✟ P ☞❊➇❢✍⑤✏✓✒ ✟ ✍ ✎ ➈ when , so the net charge P is ✎ 12. ✖ ✎✢✜ ☞❊➇ ✟ ✍⑩✞✜✒ ✟ ➉ ➇❢✄✌✙ ✟ P ☞✎✍✯➇ ✪❑✣ ✡ ✠✩➇ ✟ ✍ ✙▲➇♣✍ ★✜✒ ➇★✄ ✍ ✄✟ ☞

✠

☞ ✒

✌ ✌ ✆

✠

☎

✠

☎

☎

✠

✠

✓

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✠

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✌

✞

✠

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✠

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.

☞❊➇❄➂✹✙▲➇❫✒✤✄❞☞ ✙▲➇❴✒ ✎ ✟ ✣ ✥

✌

➇ ✣ ✰

(❵ as✏ ). However, 13. (a) ❵ is not ➁ integrable ☞❊➇❄➂❨➁ ✒ on ❷ because it is✰✣unbounded ➂✱✏✮➉ ✰ ➁ ➈ for fixed , ❵ is continuous on except at the three points , , and and is in ➁ ✎ ✟ ✰✣➂✱✏✮➉ ☞❊➇ ➂❨➁✫✒ ✰✣➂✱bounded ✏✮➉ ; hence it is integrable on . Likewise is integrable on . absolute value by ➈ ❵ ➈ P P P ✗ P P P P (b)P ✝

✝

✝

☞❊➇✬➂❨➁✫✒ ➇ ♠ ➁ ✄ P ❊☞ ➇❄❵➂❨➁❬✒ ➁ ➇★✄ P ✍ ❵ ➈ ✌

✝

✌

✌

✌

➈

✝

☎

✝

➁ ✎ ✟ ➇ ✍ ✗ ➇ ✎ ✟ ❍➇ ➉ ❖ ➁ ✄ P ➇ ✎ ✟ ➁ ✡ ✛ ➁ ✎ ✟ ✾➁ ➉ ⑥ ➇ ✄ P ✯ ✍ ➇ ➈ ✌

✌

✝

✌

✝

✟

P ✟✞✡✠☞☛ ✁ ✁

✟✝✆

✄ ✝

☛

✝

✑

✌

✾✵ ✌

✌

☎

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✝

✌

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✌

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✝

4.4 Change of Variables for Multiple Integrals 1.

➁ ✎ ❖ ✍ ☞✎✍✑✏✲✡r➁ ✎ ✒ ➉ ➁❖✄ ✏ ➈P , ✎ ✡✕☞✎✍✑✏✂✡◗➇ ✎ P ✒ ➉ ➇❦✳ ✄ ✍ ✏

✄ ✝

✟✝✆ P ✎☞ ✏❑✡❲✸✻✺✜✼ ✒ ✟ ✄ ✜ ❃ ✟ ✵ ✾✵ ✟ ✌

.

✌

.

and

.

4.4. Change of Variables for Multiple Integrals 29 ✒ P P P P P P ✄ P ❃ ✆ ✟ ✁ ✁ ✄❖❃ ✁❬☞✎✏✦✍ ✁✾✒ ✁ ✝ ✄ ❃ ✁ ✟ ✍ ✁ ✜ ➉ ➇ ✒ ✟ ✆ ✟ ✜ ✎ ☎ ✵ ➈ 2. The✒ volume is . The -moment is ✒ P P P P P P ✄ ✆ ✟ ✆ ✟ ✟ ✁ ✻ ✸ ✜ ✺ ✼ ✁ ✄ ✼ ✙ ✔ ✕ ➉ ✁ ✭ ✜ ✍ ✩ ✁ ✪ ➉ ➁ ✒ ✒ ✎ ✆ ✟ . The P -moment by symmetry. ✵ ✒ ➈ ✵ ✎ ✆ ✟ ➈✜ P P ✪ P ☎✵ P P P P vanishes ✆ ✟ ✁ ✁ ✄✖❃ ❃ ☞✎✏❬✍ ✁ ✟ ✒ ✁ ✁ ✖ ✄ ❃ ✁ ✟ ✍ ✁ ✪✮➉ ✄ ➇❢✄ ✏✓✽▲❃ ✒ ✟ ✪ ✪ . Thus , The -moment is ✎ ✆ ✟ ✾✵ ➈✟ ➁✉✄❈✰ ❧✄ ✜ ✪ . , and ✟✆ P ✙ ✛ ✝ ✄ ✥✄❉❆ ✛ ✲✍ ✁ ✟ ✍ ✁ ✟ ✁ ✁ ✄ 3. The top and bottom hemispheres , so ☎ ☎ ✵ ✒ P ✒ are given by ✍ ✪ ❃❁☞ ✍ ✁ ✟ ✒ ✜ ✟ ✄ ✪ ❃❁☞ ★✑✍⑩✞ ✜ ✟ ✒ ✜ ✜ . P P ➇ ✟ ✡r➁ ✟ ✄ ✙▲➇ ✁✕✄ ✙■✸✻✺✜✼ ✍ ❃ q q ❃ ✄ ✟ P ). Hence ✒ ☎ ✵ ( ✟ ✵ 4. The✒ equation of the cylinder is ✒ , or ✞✡✠☞☛ ✆ ✟ ✆ ✟ ✆ ✟ ✟ ✄ ✙ ✡★✼✄✔✖✕❝✙ ✍ ✞ ☞♦✼✄✔ ✕ ✍ ✼✄✔ ✕ ✜ ✒ ➉ ✒ ✄ ☞ ✙☛✍ ✁✾✒ ✁ ✁ ✄ ✍ ✞ ✸✻✺✜✼ ✜ ✒ ✒ ✒ ☞ ❁✸✻✺✜✼ ✟ ✆ ✟ ✜ ✜ ✜ ✎ ✆ ✟ ✵ ✾✵ ➈ ✵ ✵ ✵ ✵ ✎ ✆ ✟ ✾✵ ✵ ✙✠❃★✍ ✜ ✟ . ✠ ✟✝✆ ✆ ✁ ✁ ✄ P ✟ P coordinates P with ☎✵ 5. In ✆▼☞ ✙✠cylindrical ❃❄✒✻☞ ✒✻☞ ☎ ✟ ✒☛✄ ✆✻❃ ✟ ☎ ✟ the origin at the center of the base, the mass is ✟ ✟ ✟ . P P ✄ ✟✝✆ ✜ ☞ ✛ ✑✍ ✁▼✟✣✍s✏✓✒ ✁ ✄ ✁ ✄✌✙✠❃ ✍ ☞ ❙✍ ✁ ✟ ✒ ✜ ✒ ✟ ✍ ✁ ✟ ➉ ✜ ✄ ❃ ✟ ✜ ✜ ✾ ✵ ➈ 6. In cylindrical coordinates, ☎ . Al✄ ✏ ✁ ✄ ✼ ✸ ✍ ✄ ✒ ☎ ✒ is given by , so ternatively, coordinates, the plane P ✄ ✆ ✟✝✆ ✆ ✒ ✜ ☛✂✟ ✁ ✞☎in✄ ✁ spherical ✆ ✜ ✜ ✟ ✟ ✟ ✄✼ ✔ ✕ ✍ ✁ ✍ ✟ ✄ ❃ ☞ ★☛✍♣✼ ✸ ✜ ✍■✒✈✄✼ ✔ ✕ ✍ ✍✇✄ ❃ ✤ ✍ ★■✸✻✺✜✼ ✍ ✍ ✼ ✸ ✍❫➉ ✄ ❃ ✟ ✜ ✜ ✜ ➈ . ✾✵ P P P ✟✝✆ ✆ ✄ ✆✠☞✎✏✤✍ ✁✾✒ ✁ ✟ ✄✼ ✔✖✕ ✍ ✁ ✍ ✄ ✆▼☞ ✙✠❃❄✒✻☞ ✙✜✒✻☞ ✜❤✍ ✪✓✒❁✄ ❃ ✆ ✪ ✜ ✪ ✜ . 7. ✾✵ ✓

✝

✝

✌

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✌

✌

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☎

☎

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✁

✁

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✌

✁

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✌

✒ ✒ 8. By symmetry the coordinates of the centroid P areP all equal,✒ soP it sufficesP to calculate . The -moment P ✪ ✆ ✟ ✆ ✟ ☞✂✁✿✸✻✺✜✼ ✍●✒ ✁ ✟ ✼✄✔✖✕ ✍ ✁ ✍ ✄ ☞ ❃❄✒ ✼✙✔ ✕ ✟ ✍❫➉ ✆ ✟ ✁✩✪✮➉ P ✄ P ❃ ☞ ❃❄✒❝✄ ✟ ➈✟ ✞ ✜ ✪ isP ❃ ✾ ✵ , and the volume is ➈ ⑧✄ ✜ ✞ . , so P P ✄❈➇✦✍♣✞▲➁ ✄✕✙▲➇❝✡❢➁ ☞ ➂ ✒❨✽ ☞❊➇✬➂❨➁✫✒❤✄ ✠ ➇★✄ ☞ ✡★✞ ✒ ➁s✄ ☞✎✍❀✙ ✡ ✒ ✏ ✁ ✏ 9. Let ✁ ❽ , and , ✙, ➇❍➁ ; then ❽ ✁ ✰sq ✁ q✌✏✗.✰ The ✰sq given q P ✄ P by P given P -plane becomes the rectangle in the ✁ -plane P✟✄ ✁ parallelogram in the , ✜ ✏✱❂ ✙ ☞✎✏✗✰✾✒✻☞✎✏✱❂✜✒ ➇ ➇ ➇ ➁❦✄ ☞ ✡⑤✞ ✒ ✄ ✪ ✪ ✄ , . So the area of is ✏ ; the is ✁ P✄ P P -moment ✄ ✄✟✄ ✎✁ ✔ ☞✎✍✤✙ ✡ ✒ ✄✝✍✌✄ ✜ ✏ ➁ ➁ ➇ ➁⑥✄ ➇❋✄ P ➁ ✄✝✍ P ✪ ✪ ✪ and ✪ . ✁ ✎✁ ✔ and the -moment is , so ✙ P✄ ✄✟✄ ✄ under the map (Shortcut: ☞❊➇❄➂❨➁❬✭ ✒ ✣ ☞ The ➂ ✒ centroids of and are ☞ their ❂✈➂ ✒ geometric centers,✙ which ☞ ➂✱correspond ✍ P ✒ ✟ , so the center of is P ✪ ✪ .) ✁ . The center of is clearly ✄ ➇⑩✡♠➁ ✄ ➇❲✍⑨➁ ☞ ➂ ✒❨✽ ☞❊➇✬➂❨➁✫✒t✄ ✍✤✙ ☞❊➇⑩✡♠➁✫✒ ✪ ☞❊➇✥✍⑨➁❬✒ ✎ ✄ ✄ ✁ ❽ ✁ ❽ ☛✟ 10. Let and ; then , so P P P P P ✄ ✄ P ✜ ✪P ✄ ➉ ✍ ✪ ✎ ✎✠✪ ➉ ✜ P ✄ ✟ ✎ P P ✁ ✟ ➈✄ ✁ ✎ P ➈ ✪ ✞ . ✔✁ ✎ ✓

✝

✝

✌

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✓

✑

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✌

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✑

✑

✁

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✌

✄❈➇❑✡ ✙▲➁ ✄❈➇❝✍ ✙▲➁●✡ ✄✌✛ ✞ ☞ ➂ ➂ ✒❨✽ ☞❊➇❄➂❨➁❾➂✁☎✒✯✄✳✍ ✈✛ ✞ , ,✍ ; then ❽ ✁ ✍ ❽ , and☞✎the 11. Let➇❍➁ ✁ ✏✓✽✁❾✛given ✞✾✒✻☞ ✜❃ellipsoid ❄✽✠✞✜✒❑✄ -space becomes the unit ball in in ✁ ✔✍ -space. Thus the volume of the ellipsoid is ❃❄✽✠✞✜✛ ✞ . ☞❊➇✬➂❨➁✫✒ ✏ ✆ ✏✓✽ ✏ ✍ ✆ ✽ ✜ ➇r✄ ✆ ✽ ➁✖✄ ✛ ✄ ✄ ❽ ✁ ✁ ✆ ✽ ✆ ✽ ✙ ✙ . Thus the ✁ ✁ , so ☞ ➂ ✒ 12. We have and ❽ ✁ ✁ ✁ P ✄ ✞ ✺✍✓ ➉ ✪ P ✄ ✜ ✺✍✓ ➇ ✄ P ✪ P ✪ ☞✎✏✓✽✠✙ ✒ P ✪ P ✪ ✆ ✽ ☞✎✏✓✽✠✙ ✒ ✟ ✝ ✎✁ ✎ ➈✟ ✝ ✁ ✒ ✎✁ ✔ P area , the -moment is P ✟ is✒ ✟ P ✓P ✒ P ✓ ✒ ✟ ✜ ✟ ➉ ✪P ✙ ✟ ➉ ✪P ✄ ✪ ➉ ✪ P ✍❀✙ ✎ ✟ ➉ ✪ P ✄ ✏ ➁ ☞✎✏✓✽✠✙ ✒ ✄ ✜ P✪ P✪ ✛ ✟ ➈✜ ✁ ✟ ➈✜ ✁ P ✜ ✜ . ✁ ✔✁ ✔ ➈ the -moment is ✞✆ ➇⑥✄ ➈ ✠✝✪ ✆ ➁s✄ ,✟✟ ✠✝and ✁ ✁ ✪ and ✪ . Thus ✂

✞

✁

✝

✠

✝

✌

✌

✝

✝

✝

✌

✌

✝

✌

✌

30

✞

13.

✄

✞✂ ✂

✁

✎ ✂

✟

✁

✝

.

✗

✎ ☎

✟ ✗ ✑ ✄✳✍✤✙✫☞❊➇ ✟ ✡✦➁ ✟ ✒ , so ☎

☞❊➇ ✟ ✡✦➁ ✟ ✒ ✁

✝

✄ ✟P

☛✟ ✌

✝

Chapter 4. Integral Calculus ✁ ✝

✄✂ ✂

✁

✝ ✝

✄ ✟P P ✜ P ✪

✟ ✌

✝

✄

✎✁ ✎ ✌

✌

☞❊➇❄➂❨➁❬✒ ✏✯✍ ✍ ✄⑨➇ ✡✥➁ ✄ ➇s✡✥➁ ✄ ➇❴✽✩➁❢✄ ✎ P ✍t✏ ❽ ☞ ➂ ✒ ✄ ✁ , so ✁ and ✁ . Also, ✁ and ✁ P ❽ ✁ ✄✕➁❬✽✣☞❊➇ ✡✥➁✫✒ ➇ ➁ ✄⑨✰ ✄♠✏ ✄ ✁ ☞❊➇s✡◗➁✫✒ ✎ and axes correspond to and . Therefore P ✪ ✎ P ✧ ; hence✄✕the ✟ ✞ P ✁ . ✁ ✎✁ ✎ ❥ ✪ ✁ ✟ ✡✍✂ ✟ ✄ ✑ ✟ , and the volume element is is given by In double polar coordinates, the unit sphere in P ☞✑ ✟ ✍ ✟ ✆ ✝✟ ✆ ✝ ✖ ✎ ✖ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ☎✵ ✍ . P Thus the volume ✍✇✄ ☞ ✙✠❃❄✒ ✟ ✟ P P ✾✵ ✂ ✟ ✒ ✂ ✂✘✄✕✙✠❃ ✟ ➈ ✟ ✑ ✟ ✂ ✟ ✍ ✪ ✂ ✪ ➉ ✄ of✟ the ❃ ✟ ✑ ball ✪ . is ✂

14. We have

✞

✁

✠

✝

✝

15.

✝

✝

✌

✝

✌

✌

☎

✌

✌

✌

✝

✌

✝

✝

✝

✌

✌

✌

✌

✝

✌

4.5 Functions Defi ned by Integrals

P

➇

➁

☞♦✰✣➂❨➁❬✒

✰

➇✖✄ ❶ ✰

of for each , and ❵ . For fixed , one☞❊➇❄studies 1. (a) ❵ is obviously ☞❊➇❄➂❨➁❬✒ ➁ as a function ✰✾✡ ➂❨➁❬✒❨✽✩➁ ❘ the ❘ P ✒✛pital) ✗ ❵ ✂ 2.1. ❵ behavior of as as in Exercise 9, First one verifies (by l’Hˆ o that ✰ ➁ ✰✾✡ ✒ ✖ ➁ ❈✰ ✙■❘ ✗ ☞❊➇❄➂❨➁❬✒❑✄ ✙✬❘✣☞❊➇❄➂❨➁ as for every and (by induction) that ❽ ❵ for where is ❘ ✥✤ ✰ ✰ ➁ ✰✾✡ ✗ ☞❊➇❄➂❨➁❬✒❨✽✩➁ ,❽ ❵ as , and hence a polynomial. induction again, it follows that for all ❘✗ ✆☛ P ☞❊➇❄➂❜By ✰✾✒ ❵ P exists and equals zero. ✒✛✗ P that ❽ ☞❊➇❫✒☛✄ ☞❊➇❄➂❨➁❬✒ ➁♣✄❈➇ ➇ ✟ ✖ ➉ ✄❈➇ ☞❊➇❫✒❁✄ ☞✎✏❀✍⑩✙▲➇ ✟ ✒ ✖ ☞♦✰✾✒☛✄ ✏ ✖ ❵ ➈ (b) , so and . But ✒✛✗ ☞❊➇✬➂❨➁✫✒✿✄♠☞ ✞▲➇ ✟ ✍⑩✙▲➇ ✩✒ ➁ ✟ ✖ ➁ ✰ ☞♦✰✣➂❨➁❬✒ ✰ ❽ ❵ for , so ❽ ❵ . ☞❊➇❴✒❁✄ P ✗ ✽✣☞✎✏✣✡ ➇ ✗ ✒ ➁✉✄ P ✂ ✏✓✽✣☞✎✏✈✡s➇ ✒ ✄t➇ P ✺✍❾✓ ☞✎✏✈✡s➇ ✒✗ ✂P ✄❈➇ P ✺✍❾✓ ☞❨☞✎✏✈✡ ✗➇❴✒❨✽✣☞✎✏✣✡ ➇❫✒❨✒ . 2. (a) ✁ ✔✁ ✝ ✁ ✝ ☞❊➇❫✒❁✄❈➇ ✟ ✸✻✺✜✼▲☞❊➇ ✄ ✒ ✧ ✙▲➇●✍ P ✖ ➁✂✼✙✔ ✕❫☞❊➇❍➁ ✟ ✒ ➁✉✄✕✙▲➇ P ✸✻✺✜✼✩☞❊➇ ✄ ✒✮✍ ☞ ✙▲➇❫✒ P ✁ ✼✄✔ ✕ ✄✌✙▲➇ P ✸✻✺✜✼✫➇ ✄ ✡ (b) ✁ ✔✁ ☞ ✙▲➇❴✒ P ☞♦✸✻✺✜✼❍➇ ✄ ✍✪✸✻✺✜✼❬➇❴✒ . ✗ ☞❊➇❫✒❁✄ ☞ ✞▲➇❫✒ P ✖ ✧✩✞❝✡ P ➁ ✄❈➇ P ✖ ✡ ➇ P ✗ ➉ ✗ ❱ P ✄t➇ P ☞ ✙ ✖ ✍ ✾✒ ➈ (c) . ✗ ☞❊➇❴✒❝✡ ☞❊➇◗✍✖➁⑥✡⑦✏✓✒ ✗ ☞❊➁❬✒ ➁❖✄ ☎ ☞❊➇❴✒❝✡ ☞❊➁❬✒ ➁ ☎ ☞❊➇❴✒✇✄ ☞❊➇◗✍✖➇❴✒ ☎ ☞❊➇❫✒✇✄ ✗ , so 3. ☞❊➇❴✒❄✡ ☞❊➇❫✒❙✡ ☞❊➁✫✒ ➁♣✄ ☎ ☞❊➇❴✒❄✡ ☞❊➇❴✒✬✡ ☎ ☞❊➇❫✒●✍ ☎ ☞❊➇❫✒ ➉ ☎ . ➈ P ✼✟ ✄✔ ✕✯✙✫☞❊➇◆✍✥➇❴✒ ☞❊➇❫✒☛✡ ✸✻✺✜✼❾✙✫☞❊➇❇✍✥➁❬✒ ☞❊➁❬✒ ➁✥✄ ✸✻✺✜✼❾✙✫☞❊➇❋✍✥➁❬✒ ☞❊➁❬✒ ➁ ☎ ☞❊➇❴✒⑧✄ ☎ ☞❊➇❴✒⑧✄ 4. ✸✻✺✜✼➃✙✫☞❊➇❢✍✇➇❴✒ ☞❊➇❴✒✿✍ ✙ , and so ✼✄✔ ✕❝✙✫☞❊➇❢✍✇➁✫✒ ☞❊➁✫✒ ➁♣✄ ☞❊➇❴✒✿✍ ☎ ☞❊➇❫✒ . ☞❊➇❴✒❁✄ ☞❊➇❄➂ ✍✂☞❊➇❴✒❨✒✚✍ ☞❊➇❴✒✿✍ ☞❊➇❄✒➂ ☎✦☞❊➇❴✒❨✒ ☎ ☞❊➇❫✒❙✡ ✂ ☞ ☞ ✌ ☞❊➇❄➂❨➁❬✒ ➁ ✌ ❽ ❵ . 5. ❵ ❵

✮✣ ✶

✄

✣

✂

✌

✝

✂

✁

✛✎

✎

✎

✂

☎

✎

✂

✁

✎

☎

✝

✌

✍✙ ✮✣

☎

✂

✂

✝

✁

✙

☎

☎

✂

☎

✎ ✩✎ ✮✣

✶

✜ ✎ ✎ ✪ ✢✎ ✎

✁

✣

✄

✎

✌

☎

☎

✁

✎

✄

✎

✂

✎

☎

☎

✝

✎ ✜ ✎ ✂

✁

✂

☎

✎

✂

✜

☎

✂

☎

✎

☎

✠

☎

☎

✂

✝

✎ ✜

☎

☎

✂

✎

☎

☎

✂

✎

☎

✌

✝

☎

✠

✌

✌

✌

✜

✂

☎

✂

☎

✎

☎

✝

✠

✌

✠

☎

✠

✎

☎

✂

✠

✠

✜

☎

☎

✝

✝

✂

✌

☎

✠

✝

✌

✠

✝

✌

☎

✝

✠

✠

✌

✠

✄

☎

✁

✙ ✏ P we have ☞ ❵ ☛ ✒ ☞❊➇❫✒s✄ ➈ ☞ ✍✌✏✓✒ P ✎ ➉ ✎ P ➈ ☞❊➇◆✍ ➇❴✒ ☛ ✎ P ❵ ☞❊P➇❴✒❁✡ ☞ ✍⑨✏✓✒✻☞❊➇✇✍ ➁❬✒ ☛ ✎ P ✟ ❵ ☞❊➁❬✒ ➁☎➉✭✄ For ☞ ✍⑩✙✜✒ ➉ ✎ ➈ ✎ P ☞❊➇❢✒ ✍◆➁✫P ✒ ☛ ✎ ✟ ❵ ☞❊✗ ➁❬✒ ✒ ➁s✚ ✄ ❵ ☛ ✎ ☞❊➇❴✒ .P✓✒ For P ✄✳✏ , ✗ ❵ ✒ ☞❊✚➇❫✒✿✄ ❵ ☞❊➁✫✒ ➁ , so ☞ ❵ ✒ ✄ ❵ . ☞ ✟ ✩✎ ☞ ✎ ✌ ✖ ✪ ☞❊➇★✍ ➁❬✒ ✟ ✽✁▼❛ ✟ ➉ ☞❊➁❬✒ ➁ ✚ ✄ ✍ ❛ ✎✢✜ ✟ ✟ P✓✒ P ✩✎ ☞ ✎ ✗ ✌ ✖ ✒ ✪ ✚ ❵ ☞❊➁❬✒ ➁⑧✡ ❛ ✎ (a) ❽ ✁ On the other ✟ ✄ ❛ ✎ ➈ P✓✒ ✟ P ✎ ☞ ✎ ✗ ✌❵ ✖ ✒ ✪ ✚ ☞❨☞❊➇◆. ✍❲ ✄ ❛ ✎ ✟ ✩✎ ✎ ✌ ✖ ✪ ✍ ☞❊➇❋✍❲➁❬✒❨✽✠✙▲❛ ➉ ➁ ➁✫✒❨✽✠✙▲❛❜✒ ✟ ✍ hand, , so ❽ ✁ ➈ ❽ ✁ ➈ ☞✎✏✓✽✠✙▲❛❨✒ ➉ ☞❊➁❬✒ ➁❢✄ ✚ ❵ ❽ ✄ ✁ .☞❊➇❢✍✇➁❬✒ ✟ ✡❲❛ ✟ ✄❈❛ P ☞✎✍✑✏✓✒ ✎ ✟ ✙✫☞❊➇ ✍✇➁❬✒ ☞❊➁❬✒ ➁ ✍ P ❵ (b) For short, let ✍ , so P . Then ❽ ✟ ✄❈❛ ✙ ✎✢✜ ❍☞❊➇ ✍◆➁❬✒ ✟ ☞❊➁✫✒ ➁✘✡❲❛ ☞✎✍ ✏✓✒ ✎ ✟ ✙ ☞❊➁❬✒ ➁ ❽ ✍ ❵ ✍ ❵ P P ✎ ✟ ❵ ☞❊➁✫✒ ✯➁ ✄ ✄ ★▲❛ ☞❊➇ ✍✇➁✫✒ ✟ ✎✢✜ ☞❊➁✫✒ ➁❧✍ ✙▲❛ ✍ ❵ ✍ ☎

☎

✝

✌

✝

☎

6.

✟

✟

✝

✝

✟

✟

✝

☎

✝

✌

✟

✝

✟

✂

✝

✌

✂

✌

✟

☎

✂

7.

✌

✝

✟

☎

✟

☎

✝

✌

☎

✂

☎

☎

☎

✝

✌

✝

✌

☎

✌

✝

✂

✂

☎

✌

✂

✌

✌

✂

✌

4.6. Improper Integrals

31

✚ ✄ P ✎ P ❊☞ ➁❬✒ ➁ ✡✥✙▲❛ ✟ P ✎☞ ✍✑✏✓✒ ✎ ✟ ☞❊➁✫✒ ➁ ✍ ❵ ✌ ✍ ❵ ✌ , so On the other hand, ❽ P P P ✟✚ ❈ ✎ ✟ ☞❊➁❬✒ ➁❧⑩ ✄ ❛ ☞✎✍✑✏✓✒ ✎ ✟ ✙▲❛ ➁ ✍ ▼❛ ✍ ✙▲❛ ✟ ✎☞ ✍❀✙✜✒ ✎✢✜ ☞ ✙▲❛❜✒ ☞❊➁✫✒ ➁ ❽ ✍ ✌ ✍ ❵ ✌ ✍ ❵ ✌ P P ✎✢✜ ☞❊➁✫✒ ➁⑧✍⑩❅▲❛ ✎ ✟ ❊☞ ➁✫✒ ➁✯✄ ✄ ★▲❛ ✜ ✍ ❵ ✌ ✍ ❵ ✌ P ✎ ✟ ☞❊➁❬✒ ➁ ✟☎ ✡ ✚ ✟ ✎✢✜ ★▲❛ ✎ ✟ Thus, in ❽ ❽ , the terms involving ✍ add up to ✍ ❵ ✌ , and the terms involving ✍ ✝

✝

✂

✂

✂

✂

✂

✝

add up to the negative of this quantity.

✁ ✐ ✂

P ☞❊✁✬✒

and hence is continuous by 8. We show that for each ✢ ✄⑦☞ ❯ ➂❜✰✣, ✆➂ ❽ ✄✆✄✆✄▲➂❜✰✾✒ exists and ✣ is✰ given by (4.48), ❯ ❯ ❯ Corollary where and each is ☞❊small so that the points ✁❢✡ ✢ ❯ 4.53a. Let ✁⑥✡✣✢ enough ✂ ❯ ➂✫✢■✒❁✍ ❵ ☞❊✁✿➂❬✢●✒✗❺✽✫ ❯ ✣✄ ❵ are all in a ball contained in . By the mean value theorem, ✘ ☞❊✁✿➂❨✢●✒ P ☞❊✁✪✡⑨❛☎✢ ❯ ➂❬✢■✒✗✦q ☞❊✁✪✡ ✢ ❯ ➂✫✢●✒❀✍ ☞❊✁✿➂❬✢●✒ ➉❊✽ ❯ ✣ ✢ ✌ ✣ ✥ , and ➈❺❵ for each as . ❽ ❵ ❵ ❽ ❵ Hence, the bounded which are ☞❊✁❢✡ by ✢ ✒✿✍ ☞❊✁✬✒ ➉❊✽ ❯ convergence theorem, ✧✗✧✗✧ ✁ ✘ the☞❊✁✿integrals ➂❨✢●✒ ☛✈✢ of these difference P quotients, ☞❊✁✬✒ ❯ , converge to , which is therefore ❽ . ➈ ❽ ❵ ✁

☎

☎

☎

☎

✄

☎

✁

☎

✁

✝

4.6 Improper Integrals

✝

☎

☎

✌

✁

➇✞✎✢✜ ✒ ✟ . ➇ ✎ P

1. (a) Converges by comparison to

. ✂ ✎ ☎ (c) Converges by comparison to

(b) Diverges by comparison to

➇

(for example), using the fact that

large . (d) ➇ The integrand is bounded in absolute value by ; hence converges absolutely. P

☞❊➇ ✟ ✍⑥➇⑧✍❦✙✜✒ ✎ P

✂☎

➇ ✟ ✕

and

✂✎☎

✕ ✩✂ ✎ ✟ ☎ ✖

➇✡✎ ✟

, which is comparable to

for

for large

➇ ✎ ❜✶ ✕✬☞♦✰✾✒❁✄✕✰ ❜✶ ✕ ☞♦✰✾✒❁✄ ✏ ❜✶ ✕❄☞ ✒ ☛ ✁ ✁ for small ✁ .) .( and , so ➇❴✽ ✛ ✏✯✍✇➇ ✟ ☛✳✏✓✽ ✆ ✙✫☞✎✏✤✍✇➇❴✒ ➇ for near 1. (a) Converges since P ❃❄✒ ✎ P ❊ ☞ ❢ ➇ ✍ ✼ ✔ ✕✂➇ ☛✝✏ ✄ ✻✸ ✺✜✼❬➇❋✄✚✼✄✔ ✕❙☞ ✟P ❃❦✍⑩➇❴✒ ☛ ✟P ❃❦✍✪➇ ➇ ✟ P❃ (b) Diverges by comparision to since and for near ✟ . ☞ ✞❴✍✘➇❫✒ ✛ ✏✯✍✪➇❬➉ ✎ P ✏✓✽✠✙ ✛ ✏❝✍✪➇ ➇ (c) The integrand equals ✓P ➈ ✒ , which is roughly ✄✄✒ for near 1; hence converges. ✓P ✒ ✄✄✒ ✏✓✽✩➇ ✟ ☞❊➇ ✟ ✡❲➇❴✒ ✜❀✄✳✏✓✽✩➇ ✓✾☞❊➇❧✡✕✏✓✒ ☛❈➇ ✎ ✓ ➇ for near 0. (d) Converges since ➇ ✏❝✍✪✸✻✺✜✼❬➇ ☛ ✟P ➇ ✟ ✼ ✔ ✕ ✜ ✙▲➇ ☛ ☞ ✙▲➇❴✒ ✜ ✄ ☛✳✏✓✽✈✏✱❅▲➇ ✂

✂

✂

(e) Diverges by comparision to

2.

(e) For

near 0,

and

➇✞✎✢✜ ✪

, so the integrand is

✒

; hence diverges.

✂✯✎ ☎

near✓P ✒ infinity; hence converges. P ✟ ❈➇ ✎ ✜ ☛ ☞✎✏❝✍✇➇❫✒ ✎ ✟ ✓✒ P ✟ diverges. near 0 and near 1; hence converges but (b) The integrand is P ✓✒ P ✟ ☎ ☎ ➇ ✂ ✍t✏ ☛✌➇ ❫ ➇ ✣ ✽ ☞ ✂ t ✍ ✓ ✏ ✒ ✌ ☛ ➇ ✎ ➇ ✛ (c) For near✒ 0, , so and converges. For large, the integrand is ✂ ✎ ☎ ✟ P ; hence ✒ converges. less than P✓✒ ✟ P ☛✳✍✑✏✓✽✩➇ ➇ ✓✒ P P ✟ and ✟ ✒ all converge, though.) (d) The integrand is for near 0; hence diverges. ( ✟ , ➇ ✎ ✓P ☎✒ ✄ ✄✼ ✔ ✕✂➇ ✎ P ☛q ➇ ✎ ✓P ☎✒ ✄ ➇ ➇✡✎ P✓✒☎✄ ✼✄✔ ✕❤➇ ✎ P ☛❞➇ ✎✔✓ ✒☎✄ ➇

3. (a) The integrand is comparable to P✓✒ ☛

near 0 and less than

✝

✝

✝

✝

✝

(e) verges. (f) For

➇✁ ✰

,

✂ ☎ ✽✣☞ ✂ ☎

for

✡✉➇ ✟ ✒☞✍ ✕ ✂

✝

✝

near 0 and

☎ , so

✝

✎ ✒

converges; but for

✝

for

➇✄✂ ✰

,

✂ ☎ ✽✣☞ ✂ ☎

near infinity; hence con-

✡✉➇ ✟ ✒ ✳ ☛ ✏

, so ✝

✒

diverges.

Chapter 4. Integral Calculus

32

☞ ✺✍✓☛➇❫✒ ✎ ✙✥➇ ✎

☞ ✺✍✓❑➇❴✽✩✁➇ ✒ ✒❁✄✖✰

✔✟✞

4. (a) This follows Corollary 2.12 (✝ ✄✂ , so ✝ P ☞ ✺✍✓❑➇❴from ✒ ✝ ➇ ✎ ✒ ✎ ⑥ ➇ ✄ ✎ ✙✌✏ ✝✆ ✝ ✝ ✒ ✟ ✁ ✎✁ converges if and only if . (b) ✝ ✟ ✒ ✆

☎

✆

✆

➇

✆

for large ).

✟

✠

✁

☞ ✺✍✓ ✺✍✓❁➇❴✒ ✎ ✓✙⑨☞ ✺✍✓☛➇❫✒ ✎ ➇ ✝ for large. ✎ ✙✌✏ . ✝ ✝✒ ✆ ✝✆ ✜ ✁ ✎✁ converges if and only if ❘ ☞❊➇❴✒ ➇ ✄❖✙ ✎ P ✡t✙ ✎ ✟ ✡⑨✧✗✧✗✧✾✡❈✙ P ✎ ❘ ✄ ✏✦✍❲✙ P ✎ ❘ ✤r✰ ✶⑩q ✗ q ✶⑧✡✚✏ , so since ❵ , for we have 6. (a) ✝ P ❵ ❘ ❘ ✏❝✍⑩✙ ✎ q ☞❊➇❫✒ ➇❋q⑨✏❝✍⑩✙ ✎ ☞❊➇❴✒ ➇❢✄✳✏ . It follows that ✝ ✒ ❵ . ✝✠✞ ❵ ☞❊➇❴✒✿✄✌✙ ❘ ☞❊➇❴✒❁✄✖✰ ➇◆✐ ✶❾➂ ✶✑✡✥✙ ✎ ✟ ❘ ➉ ✶✉✄✳✏✠➂✹✙✈➂✹✞✈➂✆✄✆✄✆✄ ➈ for ( ) and ❵ elsewhere. (b) One possibility: ❵ ✄✌✼✜✛✣✢ ✍✂☞❊➇❴✒❝➄✾➇ ✤✷✱❍➅ ✍ ✙t✰ ➇ ➀ 7. Let (which exists since is bounded). Given , there exists such ✍✂☞❊➇ ✒ ✙ ✍ ✍ ✍ ✏ ✕ ✍✂☞❊➇❴✒✭q ✍✂☞❊➇❴✒☛✍ ✕ that . Since is increasing, we have and hence for ➇ ✤✥➇ ✔✞ ✍✂☞❊➇❴✒☛✄ all . It follows that ✝ ✄✂ . ➇✛✎ ✒ P ✔✼✙✔ ✕❤➇■ ➇ ✤ ✆ P ➇ ✎ P ➇◆✄ ✥ P ✒ way to carry out the hint is as 8. Granted the hint, we have ✝ ☞ ☛ P ☛ ✌ ➇ ✎ P ✔✼✙✔ ✕❤➇■ ➇ ✝ ✙ ✒ ☞ ☛ ☛ P ✌ ☞ ✡✖✏✓. ✒✔One ❃❾➉ ✎ P ✔✼✄✔ ✕❤➇✿ ➇★✄✕✙✜✽✣☞ ✡✖✏✓✒✔❃ follows. On the one . On P ✌ ➇✛✎ ✝ P ☛ ➇ ✕ ☞ ☛ ☛ P ✌ ❃❾➉ ✎ P ➇❇✝ ☛ ✄⑦✏✓✽ ➈ ☞ ☛ ☛ hand, ✙ ✽✣☞ ✡✌✏✓✒ ✤✳✏ ✤ ✏ . Since for all , we the other ✆✯hand, ✝☛ ➈ ✄ ✏✓✽▲❃✝ ☛ can take . ☞❊➇❴✒ ☞❊➇❴✒ ➇✳✄ ☞❊➇❫✒ ☞❊➇❴✒✗ ✍ ☞❊➇❫✒ ☞❊➇❴✒ ➇ ✗ ✣ ✥ ☞ ✗✗✒ 9. Integrating by ☞ parts, ✝✟✰ ✞ ❵ ☞ ✗✗✒ ☞ ✗✗☞ ✞ ✝✠✞ ✗✗✒ ✣ ✒ ✣ ✰ q❈.✰ As ☞❊➇❴✒ ☞❊➇❴✒✗,❍q ✄♣ remains ☞❊➇❴✒✗✈✄ ✕ bounded and , so . On the other hand, since , ✍ ☞❊➇❴✒ ➇❲✄ ☞✳✱✫✒❑✍ ✔✟✞ ✂ ☞ ✗✗✒✘✄ ☞✳✱✫✒ ☞❊➇❫✒ ☞❊➇❴✒ ➇ ✍ ✄ ☞❊➇❫✒ , and ✝ ✒ ✝ . Hence ✝ ✒ is absolutely ✒ ✞ convergent. ✏✓✽✩➇●☞❊➇✉✡✥✙✜✒ ✟P ✺✍✓✦ ➇❫✽✣☞❊➇ ✡ ✙✜✒✗ is ✝ , so 10. The antiderivative of P P ✎ ➇ ✏ ✂ ✺✍✓ ➇ ➇ ✙ ✄ ✄ ✄ ✔✟✞ ✡ ✍ ✺ ✓ ✡ ✡ ✡✡ ➇❧✡✥✙ ✡✡ ☛ ☎ ✝ ✎ P ➇●☞❊➇s✡✥✙✜✒ ✝✂✝ ✙ ✝ ✡✡✡ ➇ ✡✥✙ ✡✡✡ ✎ P ✡✡ ✡✡ ☎✄ ✏ ✏ ✄ ✔✟✞ ☛ ✙ ✍✺ ✓ ✍ ✡✥✙ ✍ ✍✺ ✓❀✏❑✡ ✍✺ ✓ ✞ ✍ ✍✺ ✓ ✡✥✙ ✝ ✝ ✝ ✝ ✝✂✝ ✏ ✥ ✡ ✙ ✏ ✄ ✍ ✙ ✺✍✓✂✞❝✡ ✔✟✞ ☛ ✺✍✓ ✍ ✡✥✙ ✄r✍ ✙ ✺✍✓❑✞ ✄ ✝ ✝ ✝ ✝✂ ✝ P ✍✂☞❊➇❴✒❑✄ ✍✂☞♦✰✾✒❙✡ ✍ ☞♦✰✾✒ ➇ ✡ ✡ ✑s☞❊➇❴✒ ✑s☞❊➇❫✒✗✣q ✄✦➇ ✜ for ➇■✫q✌✏ , so ✟ ✍ ☞♦✰✾✒ ➇ ✟ ✍ 11. We have where P ✍✂☞❊➇❴✒ P ➇ P ➇ ✍ ☞♦✰✾✒ P ➇ P ✑s☞❊➇❴✒ ✙ ✄ ✄ ➇⑥✄ ✍✂☞♦✰✾✒✬✧ ✙ ✄ ✄ ✡✧✍ ☞♦✰✾✒✬✧ ✙ ✄ ✄ ✡ ✙ ✧✙ ✄ ✄ ✡ ➇ ✄ ☎ ☎ ☎ ☎ ✎ P ➇ ✜ ✎ P ➇ ✜ ✎ P ➇❾✟ ✎ P ➇ ✎ P ➇ ✜ P ➇ ✎ ✟ ➇⑥✄ ✙ ✄ ✄ P ✎ The but ☎ ✝ ✥ last integral is proper, and the first and third P.V. integrals exist (and✍ are☞♦✰✾zero), ✒ ✌

✌

5. (a) As in the preceding exercise, ✝ ✝ ☞❊➇ ✺✍✂✓ ➇❴✒ ✎ P ☞ ✺✍✓ ✺✍☛✓ ➇❫✒ ✎ ➇❢✄ (b) ✝ ✒✜ ✝ ✝ ✝

✆

✆

✆

✠

✟

✠

✁

✁

✌

✌

✌

✌

✌

✁

✁

✁

✁

✁

✁

☎

✌

✌

✆

✆

✆

✆

✌

✆

✟

✌

✟

✆

✆

✆

✌

✟

✌

✁

✟

✟

✟

✟

✁

✠

✠

✁

✠

✠

✌

✠

✠

✌

✁

✁

✠

✠

✠

✠

✠

✁

✠

✠

✠

✠

✌

✠

✠

✠

✠

✌

✂

✂

✂

✌

✞

✠

✂

✂

✂

✂

✌

✌

✂

✌

✌

✌

✌

; hence the original P.V. integral exists if and only if the coefficient

4.7 Improper Multiple Integrals 1. Spherical coordinates turn ✝ ✝ ✝ ✞ . ✙⑤Likewise, the integral over ✞ . ✎

✟

☎

❭❫P ✁❑ ✎ ✜✻✁ ✁❑ ✙ ✏ ✆

✎

✌

✜❃

into becomes

of this term vanishes.

P✁✎ ☛ ✟ ✁ ✕ ✝ ☛ , which converges precisely when ✜❃ P ✁ ✎ ✟ ✁ ✝ ✒ , which converges precisely when ✆

✟

✌

✆

✌

4.7. Improper Multiple Integrals

33

✒ ✁ ✟ ✜✁ ✽✣☞✎✏✭✡ ✁ ✟ ✒ , which diverges since the integrand ✒ ✆ ✟ ✁ ✾✁ ✽✣☞✎✏✂✡ ✁ ✟ ✒ ✟ ✄✳✍ P ❃❁☞✎✏❑✡ ✁ ✟ ✒ ✎ P ✄ P ❃ ✪ (b) In polar coordinates, the integral is ✒ ✾✵ P ✒ ✪ . ✟ ✆ ✆ ✿✁ ✸✻✺✜✼ ✟ ✍✲✼✄✔ ✕ ✍ ✁ ✍ ✝ ✄r✙✠❃ P ✸✻✺✜✼ ✜ ✍❙➉ ✆ ✟P ✁ ✟ ➉ P ✄ (c) In spherical coordinates, the integral is ✾✵ ➈✜ ➈ ✜❃

2. (a) In spherical coordinates, the integral is tends to 1 at infinity. ✝

✙✠❃❄✽✠✞

➇ ✩✎

☎

✂

✖

✎

✄

✗

✖

☛✟

☎

✝

✝

✌

✝

. (d) ✁ ✝

✌

✒ ➇ ✩✎ ✂

✝

➇

☎ ✖

✌

✝

✝

✌

✌

✌

P P ✒✎ P ✎ P ➁ ✄ ➈ ✍ ✟ ✩✎ ➉ ✒ ✛ ❃❦✄ ✟ ✛ ❃ . ✟ ✆ ✒ ✁ ✎ ✸✻✺✜✼ ✟ ✁ ✝ ✁ ✵ ☎✵ ; the -integral diverges. ✗

✂

✌

(e) In polar coordinates, the integral is

✌

✝

✝

✝

☎

✂

✖

✖

✌

✌

❙q

✝

✌

✁

that 3. By ☞❊✁❁the ✍✯✢■✒☛extreme ✄✖✰ value ✢✂☎✤ theorem, ✂ is bounded, say ✂ ✍✂☞❊➇❴✒ . For a given ☞, ✦let✽✁✜❃❄✒ be big❭ enough ✢❤ P so ✢✇ ✄ ✂ for . Then the integral defining is dominated by ✄

✑

✎

✁ ✁✓✕✺✥

✄

✝

✎

✑

✄

✝

✝

✎

✜

✌

✝

✝

.

✌

✒ quadrant P P ✔✸of 4. (a) Since the first is contained in the square ❷ , using polar coordinates we have ✆ ✟ ✟ unit ✤ ✻✺✜✼the ✍⑩✼✙✔ disc ✕ ✟ could also be done by a P ❵ ☛✟ ✵ ✵ ✾✵ ; the -integral diverges. ✄✕✙ (This ☞❊➇✬➂❨➁✫✒ ➁ ➇ .) calculation like that in part (b) below,P using the fact that ❵ ☛✟ ❵

✁✎

✁

✝

✝

✌

✝

✁

✝

✌

✁

✌

☎

☞❊➇❄➂❨➁❬✒ ✄ ☞❊➇ ✟ ✡✌➁ ✟ ✒ ✎ ❈ ✍ ✙▲➁ ✟ ☞❊➇ ✟ ✡✕➁ ✟ ✒ ✎ ✟ ➁ ✙ ✰ ➇✚✄ P ✸✻✺✜✼❾✙ (b) have ❵ ☞❊➇❄➂❨➁❬✒ ➇ ➁ ❜✶ We ✕ ➁ ✎ P , so☞✎✏ for ✍⑤✙■a ✸✻fixed ✺✜✼ ✟ ✒ ✄ the ✍ ➁✢✎ substitution ✯ ✄ P P P into ✁ turns the indefinite integral ❵ ✁ ✎✁ ✁ ✔✁ P ☞❊➇❄➂❨➁❬✒ ➇★✄✳✍✑✏✓✽✣☞✎✏❑✡ ➁ ✟ ✒ ☞❊➇❄➂❨➁❬✒ ➇ ➁✉✄ ✍✯➁ ✎ ✼✙✔ ✕ ✸✻✺✜✼ ✄✳✍❝➇❴✽✣☞❊➇ ✟ ✡ ➁ ✟ ✒ ✁ ✁ . Hence ❵ and so ❵ P P P P ✍ ❃ ☞❊➇❄➂❨➁❬✒ ➁ ➇★✄ ❃ ✪ . Since ❵ ☞❊➇❄➂❨➁❬✒❁✄r✍ ❵ ☞❊➁❾➂❨➇❴✒ , it follows that ✪ . ❵ ✁

✝

✝

✌

✝

✝

✌

✌

✂

✝

✝

✌

✝

✝

✌

✝

✝

✝

✌

✌

✌

✝

✌

✌

✌

Chapter 5

Line and Surface Integrals; Vector Analysis 5.1 Arc Length and Line Integrals

✟✝✆ ✟

1. (a) ✝

(b)

P✂

(c)

✝

(d)

✝

☛

❊☞ ❛❨✒✗ ❛❑✄ ✟✝✆ ✆ ✱ ✟ ☞♦✼✄✔ ✕ ✟ ❛❙✡❲✸✻✺✜✼ ✟ ❛❜✒❫✞ ✡ ✗ ✟ ❛❁✄ ☞❊❛❜✒✗ ❛☛✄ ✟ ✆ ☞❊❛✎✟✯✍ ✏✓✒✔✟☛✡ ▼❛✎✟ ❛❁✄ ✟ ☞❊❛ ✟ ✡✖✏✓✒ ☞❊❛❨✒✗ ❛❁✄ P ✂ ✛ ❛ ✎ ✟❁✡ ✭✡ ▼❛✎✟ ❛☛✄ P ✂ ☞❊❛ ✎ P ✡✥✙▲❛❜✒ ☞❊❛❜✒✗ ❛☛✄ ✟ ✛ ▼✞ ❅❀✡✥✞▼❅▲❛❄✡✥✞▼❅▲❛❄✡✥✞▼❅▲❛✔✟ ❛✿✄✌❅ ✟ ✌

☛

✝

✌

☛

✟

✝

✌

☛

✌

✌

✝

✌ ✝

✌

✝

✝

✌ ✝

✴✱ ✖✱

✝

✟✝✆ ✛ ✱ ✟ ✡✞✗ ✟ ❛☛✄✕✙✠❃ ✛ ✱ ✟ ✡✞✗ ✟ . ❛☛✄ P ✪ ✜. ❛☛✄✳✏✂✡ ✟ ✍✥✏❀✄ ✟ . ☞✎✏❑✡◗❛❜✒ ❛☛✄✕✙ . ➁

✌

✌

✂

✂

✌

✌

✝

2. (a) ➇❢ With origin and the major axis on the axis, the ellipse is described parametrically ✄ ✗➃the ✸✻✺✜✼❍center ❛ ➁✉✄ at❑✼✙the ✔ ✕✂❛ ◗✄ ✒ ✒ length is 4 times the length in the first quadrant. Hence , , and the whole by ✆ ✟ ✆ ✆ ✟ ✆ ✈✟✯✍⑤☞ ✈✟✯✥ ✟ ✄✳✏❬✍ ☞ ✗✗✽ ✣✒ ✟ ✗✮✟❫✼✄✔ ✕ ✟ ❛❫✡ ☎✟❫✸✻✺✜✼❜✟❴❛ ❛❁✄ ✍ ✗✮✟✗✒✈✙✼ ✔ ✕ ✟ ❛ ❛✿✄✡ ☞ ❬✒ where . ➇ ✟ ✡⑨☞❊➁s✍⑤✏✓✒ ✟ ✄❖✏ ❍ ➇ ➁ (b) The base of the cylinder is the circle ➇✪✄ ✸✻✺✜✼❍❛ ➁❦✄ ✏✯✡⑤✼✄✔ ✕❤❛ in ✍ the✟P ❃ q⑨-plane; ❛✦q ✟P the ❃ semicircle where both coordinates are positive is given by , , . On the sphere we then ✲✄ ✆ ✍✇➇ ✟ ✍✪➁ ✟ ✄ ✛ ✙✑✍⑩✙■✼✄✔ ✕❤❛ . Hence the arc length is have

✱

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✳✱

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✯✱ ✶

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✟

✸✻✺✜✼ ✟❴❛ ✼✙✔ ✕ ✟ ❛❫✡❲✸✻✺✜✼ ✟ ❛❫✡ ☞ ✙✑✍⑩ ✙■✼✄✔ ✕❤❛❜✒ ❛❁✄ ✎ ✆✒✟ ✒ P P P ✆ ✟ ✂✲✄ ✟ ☞ ✟ ❃❦✍⑩❛❨✒ , so ❛❤✄ ✟ ❃❦✍◗✙✄✂ . Then ✎ ✆ ✒ ✟ ✧✗✧✗✧ ❛❝✄✳✙ Let ❁✼✄✔ ✕ ✟ ✂ ✸✻✺✜✼ ✟ ❛ ✄ ✄✼ ✔ ✕ ✟ ✄✙ ✂ ✄ ❁✼✄✔ ✕ ✟ ✂✬✸✻✺✜✼ ✟ ✂ , ✒ , and ✙ ✆ ✟ ✆ ✙✘✍✪✄✼ ✔✖✕ ✟ ✂ ✂✦✄✕✙ ✜ ✒ ✟ ☞ ✙ ✎ ✓P ✒ ✟ ✒ ◗✄

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4.

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✟ ✧✗✧✗✧

✙ ◗ ✍ ✙■✄✼ ✔ ✕✯❛✯✄✳✙ ✍◗✙■✸✻✺✜✼❾✄✙ ✂⑧✄ ✄ ✙ ✆ ✒ ✟ ✛ ✏❑✡❲✸✻✺✜✼ ✟ ✂ ✂ ✄ ✂

✝

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so

;

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.

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try. ✂✪✄

✟ ✆ ❛ ✛ ❑✏ ✡◗❛ ✟ ❛♣✄ ✝ ☞❊❛❜✒✗ ❛ ✄ ✆ ❁✼✄✔ ✕ ✟ ❛❙✡ ❁✸✻✺✜✼ ✟ ❛❫✡ ▼❛ ✟ ♣ ❛ ✄ ✙ ✛ ❑✏ ✡◗❛ ✟ ❛ ✂ ✄ ✛ ✇ , so ✁ P ✎☞ ✏❑✡ ✜❃ ✟ ✫✒ ✜ ✒ ✟ ✍⑤✮✏ ➉ ✜➈ . ●☞❊❛❜✒✦✄❞☞❊❛✮➂❨❛ ➂❨❛❜✒ ✰❇qr❛✑q ✏ ☞ ●☞❊❛❜✒❨✒✦✄ ☞❊❛ ✟ ➂❨❛ ✟ ➂❨❛ ✟ ✒ ☞❊❛❜✒❀✄ ✎☞ ✏✠➂✱✏✠➂✱✏✓✒ (a) Parametrize , ; then and , so ✧ ✁❇✄ P ✞▲❛ ✟ by❛❁✄✳✏ ☛

✌

✌

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✄

5.

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✒

✸✻✺✜✼ ✟➃❛ ✏❑✡ ☞ ✙✑✍⑩ ✙■✼✄✔ ✕❤❛❜✒ ❛

✆ ❑✏ ✡❲✼✄✔ ✕✂ ✟ ➇ ❋ ➇ ✄⑨✸✻✺✜✼✄❝➇ ➇ . Thus➁ the✂ arc✄ lengthP or✸✻✺✜“mass” ✙■✼✄✔ ✕✁✘✏ ➁ ✼✄ ✟ ➇ ➇ of the ✄ P ✎ ✎ , and the -moment is ➇ ✄ ✟ ☞ ✙ ✡⑨✼✄✔ ✕✁✯✙✜✒ ➁❈✄ ☞ ✙❧✡✚✼✄✔ ✕✁✯✙✜✒❨✽✁❁✼✄✔ ✕✂✑✏ ➇r✄ ✰ P ☞✎✏✑✡⑨✸✻✺✜✼✄✤✙▲➇❫✒ ✳ . Thus , and by symme-

3. The element P of✸✻arc ✺✜✼✄✤➇length ➇ is✄ P curve P P is P

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.

34

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5.2. Green’s Theorem

35

✄ ■☛ ☞❊❛❜✒✂✄✝☞❊❛✮➂❨❛ ✟ ➂❨❛ ✜ ✒ ✰♣P q❈❛❝P qr✏ ☞ ☛●☞❊❛❜✒❨✒❤✄❖☞❊❛ ➂❨❛ ✟ ➂❨❛ ✪ ✒ ☛ ☞❊❛❜✒❑✄✝☞✎✏✠➂✹✙▲❛ ➂✹✞▲❛ ✟ ✒ P (b) Parametrize by , ; then and , ✟ ✜ ✜ ✥✧ ✁❋✄ ☞❊❛ ✡✥✙▲❛✫✜☛✡ ✞▲❛ ✓✒ ❛❁✄ ✡ ✟ ✡ ✄ ✟ P . so ✝✁ ✝ ✄ ☛■☞❊❛❨✒❝✄⑦☞♦✼✄✔✖✕❤❛ ➂❜✸✻✺✜✼➃❛❨✒ ✰❢q✖❛❀q✕✙✠❃ ✄ (c)☞ ☛■Parametrize clockwise!); ☞❊❛❨✒❨✒ ✄ ☞♦✼✄✔ ✕❤by❛❲✍ ✸✻✺✜✼✫❛✮➂❍✼✄✔✖✕❤❛✇✡ ✸✻,✺✜✼❬❛❨✒ ☛ (remember ☞❊❛❜✒ ✄ ☞♦✸✻that ✺✜✼➃❛ ➂✱✍❦is✼✄✔ oriented ✕✂❛❨✒ ✧ ✁ then✄ and , so ✁ ✝ ✟✝✆ ☞✎✍❦✄✼ ✔ ✕ ✟ ❛■✍✪✸✻✺✜✼ ✟ ❛❜✒ ❛☛✄✳✍✤✙✠❃ . ✝ ✄ ■☛ ☞❊❛❨✒✭✄ ☞❊❛ ➂❨❛ ✟ ✒ ✍✤✙★q✚❛✑q✳✙ (d) by✄ ✟ ☞❊❛✫✠✪ ➂❨❛ ✒❫✧✩☞✎, ✏✠➂✹✙▲❛❨✒ ❛❴✡ ✟ , ☞and ▼❛ ✟ ➂✱the ✍✑✏✱straight ❅▲❛✫▲✜ ✒❾✧▲☞✎✍✑portion ✏✠➂❜✰✾✒ ❛❑by ✄ ●☛ ☞❊❛❜✒☛Parametrize ✄♠☞✎✍✯❛ ➂✚☎✒ the ✍❀✙❧parabolic q✥❛✂qt✙ portion of⑩✧ ✁❋ , . ThenP ✁✝ ✝ ✎ ✟ ✝ ✎ ✟ ✟ ☞❊❛✫✪❑✡❲✙▲❛ ✍ ▼❛ ✟ ✒ ❛☛✄✕✙✫☞ ✜ ✟ ✡ ✟ ✪ ✍ ✜ ✟ ✒❁✄ ✜ ✪ . ✝ ✎ ✟ ✄

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P ☞❊➇ ✎ ✄✖ ✡ ☞♦✼✄✔ ✕✯❃❴➇❴✒✎✙▲➇❫✒ ➇ ✄ ❊☞ ➇ ✩✎ ➇❲✡ ✼✙✔ ✕✯❃❴➇ ➁✫✒✳✄ ✍ ✟P ✎ ✄✖ ✍ ☞ ✙✜✽▲❃❄✒ ➇❀✸✻✺✜✼❴❃❴➇✚✡ ✝ (a) ✝ P ☞ ✙✜✽▲❃ ✟ ✒✈✼✄✔ ✕✯❃❴➇ ✄ ✟P ☞✎✏❝✍ ✎ P ✒❫✡✌☞ ✙✜✽▲❃❄✒ . ☞❊➁ ➇✭✡ ➁✯✡❋➇❍➁ ☎✒❁✄ ✟✝✆ ☞✎✍★✼✙✔ ✕ ✟ ❛✣✡❋❛✣✸✻✺✜✼✣❛✣✡❇✼✄✔✖✕❑❛✣✸✻✺✜✼❬❛❜✒ ❛❁✄ P ✸✻✺✜✼❍✙▲❛❾✍ ✟P ✡❋❛✣✼✄✔ ✕✂❛✣✡❇✸✻✺✜✼❬❛✣✡ ✪ (b) ✝ P ✼✄✔ ✝ ✕ ✟ ❛ ✝✟ ✆ ✄✳✍❝❃ ✟ . ➁❦✄ ✰ ➁❋✄♠✰ (c) On the segment and ; on the segment ➇⑥✄✳ ✏ P from ➇⑥✄✖(0,0) ✰ P to (1,0) we have ➁ ✄❈➇ from➁✉✄(1,0)➇ to (1,1) and , and on the segment from (1,1) to (0,0) we have and . Hence we have ✰ ➇ ✡ ☞✎✍❀✙✜✒ ➁ ✡ P ☞❊➇ ✟ ✍ ✙▲➇❴✒ ➇❦✄✳✍❀✙❝✡ ✟ ✄✳✍ ✪ ✜ ✜ . the integral is ✝ ✝ ✝ ✄ ✁✳✄ ☛●☞❊❛❜✒ ✱✕q ❛★q ✗ ✄ ✲☞ ☛●☞❊❛❜✒❨✒✗ ☛ ☞❊❛❨✒✗ ❛ q by , . Then we have ✡ ✝✁ (a) Parametrize ✡ ✡ ✝ ✞ ✡ ✡ ✡ ✡ ✡ ☞ ☛■☞❊❛❨✒❨✒✗✗ ☛ ☞❊❛❜✒✗ ❛❑✄ ✲ ✝ ✞ ✝✁ . ☛ ❈✧ ✁ ✄ ☞ ✲☞ ☛●☞❊❛❜✒❨✒✂☎✧ ☛ ☞❊❛❜✒ ❛ q ✲☞ ☛●☞❊❛❜✒❨✒✂☎✧ ☛ ☞❊❛❜✒✗ ❛❧q ✁ ✝ ✝ ✝ (b) With as in part (a), we have ✡ ✡ ✡ ✞ ✡ ✲☞ ☛●☞❊❛❜✒❨✒✗✗ ☛ ☞❊❛❨✒✗ ❛❑✄ ✲ ✡ ✡ ✡ ✡ , where the next-to-last step uses Cauchy’s inequality. ✝✁ ✝✁ ✙ ✙ ✙ ✱➃✙➂ ✗✹➉ As noted ✆in the text, if is a partition of ✞ ➈ ☞ ✄ and is the partition obtained from by adding ✒ ✤ ✞ ☞✄ ✒ ✞✭☞ ✄ ✒❧✄ ✜✼ ✛✣✢ ✞ ☞ ✄ in✒ , so in the point if it is not already there,✙ then ✆ ✙ ✄ ❛ ▲➂✆computing ✄✆✄✆✓ ✄ ➂❨❛ ➅ it is enough to consider partitions that contain . If is such a ✱➃partition with ➀ ✆✦✄⑨❛ ✙☛P❝✄ ❛ ➂✆✄✆✄✆✓✄ ➂❨❛ ❬➅ ✙ ✟ ✄ ❛ ❍➂✆✄✆✄✆✱✄ ➂❨❛ ➅ ✙☛P ✙ ✟ ➂✆➉ ✆✩➂✙✗✹➉ ➀ ➀ ➈ ➈ , let and ; then and are partitions of and , ✙✂P ✙ ✟ ✱❍➂ ✆ ➉ ✆▲✙➂ ✗✹➉ and are partitions of ➈ ✞ and can concatenate respectively. Conversely, if ✘ ☞ ✄✤P✮✒✈✡ ✞ ✙ ✱➃✙➂ ✗✹➉ ☞ ✄ ✒❁➈ ✄ ✞ , we ☞ ✄ ✟ ✒ them to ➈ obtain the partition of . In these cases we clearly have . It follows ✖ ✞ ☞ ✄ ✒✂q ✞✭☞ ✄✤P✮✒✈✡ ✞✭☞ ✄ ✟ ✒ ✙ ✞✭☞ ✄ ✒✂q ✞✭☞ ✄✦P✮✒✣✡ ✞✭☞ ✄ ✟ ✒ , and ✙taking supremum over all gives . On that P the ✙ ✟ ✞ ✕ ☞ ✄ ❯ ✒ ✤ ✞❀☞ ✄ ❯ ✒❄✍ ✌❧✄✳✏✠➂✹✙ ✞✭☞ ✄ ✒ ✤ ✙ ✰ , choose and so that for ; then the other hand, given ✞ ☞ ✄ ✒☛✄ ✞ ✘ ☞ ✄✤P✮✒❙✡ ✞ ☞ ✄ ✟ ✒ ✤ ✞✭☞ ✄✤P✹✒❄✡ ✞❀☞ ✄ ✟ ✒■✍ ✙ ✞❀☞ ✄✲✒ ✤ ✞✭☞ ✄✭P ✒❙✡ ✞❀☞ ✄ ✟ ✒ . Since is arbitrary, , ✖ ✌

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6.

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8.

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☛

and we are done.

☞❊❛ ✒ ✟ ✡

✁

☞❊❛ ✒ ✟ ☞❊❛ ✍◗❛ P ✒

❯ ❯ ✎ for some ❯ ❯ 9. By the mean value theorem, the displayed expression equals ❛ ❯ ➂❨❛ ❯ ✐ ❛ ❯ ✎ P✗➂❨❛ ❯ ➉ ✌❢✄ ✏ ✌❢✄ points . By Exercise 9, ✂ 4.1, the sum of these quantities from to , which ➈ ✆ ☞❊❛❜✒✔✟☛✡ ☎ ☞❊❛❨✒✔✟ ❛❋✄ ✞ ☞✄ ✒ ☛ ☞❊❛❨✒✗ ❛ ✙ is , can be made as close to ✝✠✞ as we wish by taking ✝✟✞ ✞❀☞ ✄✲✒ sufficiently fine. It follows that this integral equals . ☎

✠

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5.2 Green’s Theorem

✌

✥✧ ✁◆✄♠✍

1. (a) Let be the unit disc; then ✝ circle has the wrong orientation). ☞❊➁ ✟ ➇♣✍ ✙▲➇ ➁❬✒☛✄ P ☞✎✍❀✙✘✍⑩✙▲➁❬✒ ✝ ✝ (b) ✝

✙ ➇ ➁♣✄♠✍✤✙✠❃ (the minus sign is there because the ➁ ➇★✄ P ☞✎✍✤✙▲➇ ✍✇➇ ✟ ✒ ⑥ ➇ ✄✳✍ ✪

✁

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✝ ✝✄✂

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✌

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✜ .

Chapter 5. Line and Surface Integrals; Vector Analysis

36

✟ ✟ ✎☞ ✏✗✰✠➇❫✡✘❂▲➁❬✒✮✍ ☞✎✏✗✰✠➇❴✡✘✙▲➁❬✒ ➉ ➇ ➁⑥✄ ➈ ☞ ✞▲➇ ✟ ✼✄✔✖✕❤➁ ✟ ✒

✟ ➇

☞❊➇ ✟ ✡ ✏✗✰✠➇❍➁❬✡✭➁ ✟ ✒ ➇❴✡ ☞ ❂▲➇ ✟ ✡✘❂▲➇❍➁❬✒ ➁☎➉❴✄

(c) ✙✦✧✩❅✑✁ ✄✳➈ ✏✱✙

✄

✌

✝

✌

☞ ✙▲➇ ✜✻➁✂✸✻✺✜✼❍➁ ✟ ✒❑✄⑨❅▲➇ ✟ ➁✂✸✻✺✜✼✫➁ ✟ ✄

(d) ❽ theorem vanishes.

❽

☎

✵

2. Directly, the polar ✟✝✆ ✙■✸✻✺✜✼ ✟ by✼✄✔ using ✕ ✟ ✄ P ✟ ✄ ✟✝✆ angle ✼✄✔ ✕ ✟ ✙ ✵ ☎✙✠✵ ❃❢✧ P ✄ ✄ P ✄ ❃ ✟✝✆ P ✟ ✁✩✜ ✵ ✁ ✄✕ ✟ . ✪ ☎✵ ✝

✝

✝

✌

✌

✝

✌

✟ ✞▲➁ ➁s✄

✌ ✝

✌ ✝

, so the integrand of the double integral in Green’s

✟✝✆ ✞▼✙■✸✻✺✜✼ ✟ ✄✼ ✔ ✕ ✟ ✍ ✵ ✵ ☎ ✵ ☞❊➁ ✟ ✡t➇ ✟ ✒ ➇ ⑩ ➁ ✄

as theP ✄ parameter for both circles: ✄ ✟ ❃ . By using Green’s theorem: ✾✵ ✝

✵

✝

✌

✗

✝

✌

✌

✁

✝

✌

✌

✞✫☞✎✏✑✍❲➇ ✟ ✍❲➁ ✟ ✒ ➇ ➁

✌

➁ ✜ ➇⑥✡✳☞ ✞▲➇❋✍❲➇ ✜ ✒ ➁✾➉❝✄

✝

3. If ❷ is the region inside , ✁ ➈ . The integrand is positive inside the unit disc and negative outside, so the integral is maximized by taking ❷ to be the unit disc and to be the unit circle. ✄

✌

✝

✁

✌

✝

✌

✝

✌

✄

✰sq✥❛❤q⑤✙✠❃

✰✣➂✹✙✠❃❾➉

: the region under it is bounded on the bottom by the segment ➈ 4. Take the ➇ arch given by➁❋✄❖✰ of the -axis (where ) and ✟ ☞✎✏✯✍✇✸✻from ➇ ➁s✄ ✟✝✆ ✟ ☞❊❛■✍◆✼✄✔ ✕❤❛❜✒✈✼✄✔ ✕❤❛ on❛ top✍ by the ➁ cycloid ➇⑥✄ ✟✝✆ (traversed ✺✜✼❍❛❨✒ ✟ right ❛ to left). Thus the area is or ✂ ; both integrals are equal to ✞✠✁❃ ✟ . ✌

✑

✑

✁

✁

✝

✌

✝

✌

✝

✌

✝

✌

✑

➇

5. The oriented of ❷ consists of two vertical line segments, a segment of the -axis, ➁ ✄ ☞❊boundary ➇❴✒ ✍ and➁ the➇ curve ➇★✄✖❵ ✰ , traversed from right to left. ➇ The vertical segments contribute nothing to ✁ ➁ ✄✖✰ since on them, and on it. The integral ✍ ☞❊➇❴✒ the➇❢segment ✄ ☞❊➇❫of✒ the➇ -axis contributes nothing since . over the curve is ❵ ❵ ✟ ✡✕☞ ❯ ✒✻☞ ❯ ✒ ☞ ✽ ✒☛✄ ✥✂✧ ✄ ❈✄ ✂ ✄ ☞ ✒❁✄ ❽ ❵ , and ❽ ❯ ❯ ❽ ❯ ❵❫❽ ❯ ❵❫❽ ❯ ❽ ❵ ❽ . The result 6. We have ❵ ❽ where therefore follows from Corollary 5.17. ✁

✝

✌

✠

✌

✠

✝

✌

✝

✞

✌

✞

✁

✁

✁

✠

✟

✠

✁

✠

✠

✠

❥

✄✂

7. (a) The image of under the map ☞♦is✰✣➂ a✥ connected ✒ ☞✎✍ ✥✖subset ➂❜✰✾✒ of , by Theorem 1.26. It does not or ; otherwise it would be disconnected. contain 0, hence must be contained in either ✂

✁

✝

✕✄ ☞ ➂ ✒ ✄ ☞ ➂ ✒ ➇✌✄ ☞ ➂ ✒ ➁⑤✄ ☞ ➂ ✒ ➁ ✄ ✡ , ✗☞ , , and ✁ ✡ , so ✒ ✁ ✁ ❽ ✎✁ ❽ ✔ . Thus ➁❢✄ ✟ ❽ ✎✁ ❽ ✔ , where ❽❾❷ is given the positive orientation with ☞❊➇❄➂❨➁❬respect ✒✝✣ ☞ to➂ ❷✒ ✁ and ❽ is given the orientation induced from the one on ❽❾❷ by the change of variable . This may or may not be the positive orientation of ❽ with respect to , so in applying ✄ ❆ Green’s☞ theorem ✒✭✍ ❽ there will be an ambiguity of sign. The result is ✟ ❺ ➈ ❽ ✈ ❽ to ☞ the integral over ✒➉ ✄✌❆ ☞ ✍ ✒ ✄✌❆ ❽ ✈❽ ✔✁ ✎ ❽ ✈❽ ❽ ✈❽ ✎✁ ✎ ✄✂ ✟ . The easiest✡ way to determine ✟ is positive; hence the sign must be ✞✂ ✙ ✰ which sign is right is to observe that the area if ✍ ✕ ✰ (b,c) ✄ Let ➇

☛

☎

☎

☎

☎

✝

✁

✠

✠

☎

✌

✌

✌

☎

✁

✝

✁

✝

✌

✝

✠

✌

✌

✠

✂

✂

✂

☎

✂

✝

✁

✝

✝

✠

✠

✂

☎

☎

☎

✁

✝

✝

✝

✁

✁

✠

✌

✌

✝

✝

✁

✠

✠

✌

✌

✝

✠

✝

✝

✌

✠

✂

✞✂

✂

and

if

✁

✁

✝

.

✝

5.3 Surface Area and Surface Integrals 1.

✟✝✆

✄ ✟ ❃ ☞✎✏❤✡ ✱ ✟ ✒ ✜ ✒ ✟ ✍⑤✏✮➉ ✜ ➈ . P ✒ ✄ ☛ ❳ ✆ ✏❑✡ ▼➇ ✟ ✡ ▼➁ ✟ ➇ ➁✉✄ ✟✝✆ ✛ ✏☛✡ ✁ ✟ ✁ ✁ ✾✵ ✄ ❃ ➈ ☞✎✏❑✡ ✯✱ ✟ ✒✫✜ ✟ ✍ ✏✮➉ . ✟ ☞✎✍❁➂ ✒✭✄ ☞❨☞ ✗❤✡✘✱✂✸✻✺✜✼ ✍■✒✈✸✻✺✜✼ ➂❄☞ ✗✂✡✷✱✂✸✻✺✜✼ ✍■✒✈✼✄✔ ✕ ➂✠❑✱ ✼✄✔✖✕ ✍●✒ ✄❞✍ ✱✂✼✄✔ ✕ ✍✲✼✄✔ ✕ ❫✍ With ✱❑✼✙✔ ✕ ✍✲✂ ✄✼ ✔ ✕ ✵ ●✡ ✱✂✸✻✺✜✼ ✍ ✄✳✵ ✍ ☞ ✗❙✡ ✱❑✸✻✺✜✼ ✍●✒✈✼✙✔ ✕ ✵ ▼✡ ☞ ✗❙✡ ✱❑, ✸✻we ✺✜✼ ✍■have ✒✈✸✻✺✜✼ ❽ ✂ ☞ ✒■②❇☞ ✵ ✒❁✄ ✵ ❽ ✂ ✵ ✵ ❽ ✂ ❽ ✂ and , so ✍ ✱❴☞ ✗❲✡ ✱✂✸✻✺✜✼ ✒✈✸✻✺✜✼ ✍✲✸✻✺✜✼ ♣✡ ✱❴☞ ✗✥✡ ✱❑✸✻✺✜✼ ✍●✒✈✸✻✺✜✼ ✍✲✼✄✔ ✕ ✌✍ ✱❾☞ ✗✥✡ ✱✂✸✻✺✜✼ ✍■✒✈✄✼ ✔ ✕ ✍ ✵ ✵ ✵ and hence ✟

✄

✆ ✏❑✡◗➁ ✟ ✡◗➇ ✟ ➇ ➁s✄

❳

☛

✗

✠

✛ ✏☛✡ ✁ ✟ ✁ ✁

✠

✾✵

☎

✝

✝

✌

✖

✖

✝

✌

✝

✌

✌

✖

✠

2.

✠

✗

☎

✝

✝

✓

✖

✖

✌

✖

✌

✝

✝

✌

✌

✄

3.

✂

☎

✂

✑

✂

✄

✑

☎

5.3. Surface Area and Surface Integrals

♥☞ ❽

✒✘②✖☞

✰✮✕✷✱ ❽ ✕

✄

✂

Since

37

✒✗ ✟ ✄ ✱ ✟ ☞ ✗✤✡ ✱✂✸✻✺✜✼ ✍■✒ ✟ ☞♦✸✻✺✜✼ ✟ ✍ ✸✻✺✜✼ ✟ ✡✌✸✻✺✜✼ ✟ ✲ ✍ ✼✄✔ ✕ ✟ ✡✌✼✄✔ ✕ ✟ ✍■✒✉✄ ✱ ✟ ☞ ✗✤✡ ❑ ✱ ✸✻✺✜✼ ● ✍ ✒✟ ✵ ✵ ✟ ✝ ✆ ✟ ✝ ✆ ✗ ■ ✗ ✡ ✱❑✸✻✺✜✼ ✍ ✄ ❾✱ ☞ ✗✿✡ ✂ ✱ ✸✻✺✜✼ ✍■✒ ✍ ✄ ✜❃ ✟ ✱ ✗ . ✾✵ , is always positive, so finally ✟ .

✑

✂

✝

✝

✌

✌

4. The integral as in Example 1 (p. 232). First way: The upper half of the ✟ ❧✄ can☞ ✗✗✽ be ✱✣✒ ✆ set✱ ✟ up✍✇in➇ ✟ two ✍✇➁ ways, , so the area is surface is

✙ ✂

✏☛✡ ✗✮✟ ✟ ✧ ✟ ❾➇ ✟☛✡◗✟ ➁✣✟ ✟ ➇ ➁ ✄ ✜❃ ✱ ✱ ✇ ✍ ➇ ✍✇➁

✂

☎

❳

☛ ✗✖ ✖

✖ ✠

✌

✱ ✪ ✕ ✡ ☞ ✗✮✟✯✍ ✈✱ ✟✓✒ ✁✠✟ ✁ ✁ ✄ ✱ ✟ ☞✳✱ ✟ ✍ ✁ ✟ ✒ ✠

✂

✌

❇✄ ✛ ✱✈✟✯✍ ✁▼✟

✌

☞ ✜❃❄✽ ✱✫✒ ✆ ✗ ✟ ✱✈✟❁✡✕☞✳✱✈✟✯✥ ✍ ✗✮✟✓✒ ✓✟ The substitution simplifies this to , and the substiP ✄ ✠✽ ✱ ✜❃ ✱ ✆ ✗✮✟❁✡✕☞✳✱☎✟✯✍ ✗ ✟✓✒ ✟ tution ✁ turns this into ➇ ✄ ✱✂✄✼ ✔ ✕ ✁ ✍✲✎✸✻✁✺✜✼. Second ➁ ✄ ✱✂way: ✼✄✔ ✕ ✍✲Use ✼✄✔ ✕ modified ✪✄ spherical ✗❾✸✻✺✜✼ ✍ , coordinates to parametrize the ellipsoid. With ✙✠❃ ✱ ✆ ✄✼ ✔✖✕ ✍ ✆ ✗ ✵ ✟❴, ✼✄✔✖✕ ✟ ✍ ✡✖✱☎✟❫✸✻✺✜✼ ✟ ✵ ✍ , and ✍ the formula (5.20) for area yields the integral , and the sub✄ ✸✻✺✜✼ ✍ ✙✠❃ ✱ P P ✆ ✗ ✟❁✡✕☞✳✱✈✟✯✍✥✗✮✟✓✒ ✟ ✎ ✁ ✔ ✁ stitution ✁ turns this into . This is the same as the integral P ✄ ✙ P P even functions. Finally, one uses a trig substitution obtained in the first way since ✎ ✗ ✙ ✱ ✄ ☞ ✗ ❜✶ ✕❝❛❜✒❨✽ ✛ ✱✈✟❤✍✥✗✮✟ ✱✺✙✡✗ ✄ ☞ ✗❾✼✄for ✔ ✕❝❛❨✒❨✽ ✛ ✗✮✟✯✍ ✱☎✟ if , if (✁ ✁ ✟ ✥ ✟ ✒❨✽ integrals ✙✠❃ ✱ ✟ ✡⑨✙✠❃ ✱ ✗ ✟ ✭✽✣✆ ✱ ✟ ✥ ✍ ✗✟ ✄ ✍✺ ✓❴☞❨✳☞ ✱♣) ✡ or ✛ a ✱ table ✍ ✗ of ✗✗✒ ✱ ✙ to ✗ eval✝ uate✄ the integral as , where if ✶✠✷✹✸✗✙✼ ✔ ✕❙☞ ✛ ✗ ✟ ✍ ✱ ✟ ✽ ✗✗✒ ✱ ✕ ✗ ✜❃ ✱ ✟ and ✠

✂

✂

✂

✝

✌

✂

✝

✌

✌

✝

✌

✝

✝

✝

✂

✄

✄

✄

✣ ✱

✗

if

. (Both expressions for the area have the limiting value

.)

➇ ✄ ➁ ✄ ✰ by symmetry. ✒ ✟ ✻✸ ✺✜✼✯✍✲✼✄✔ ✕ ✍ ✇ ✍ ✄✖❃✑✼✄✔ ✕ ✟ ✍❤ ✆ ✟ ✄✖❃

5. Clearly✒

✙✠❃

✆

✝

✌

✟✝✆

6.

✆

✝

as

✒

✜ ☞ ❁✼✄✔✖✕ ✟ ✍●✒✻☞ ❁✼✙✔ ✕

✝

The

-moment, in spherical coordinates, is P

, and the area is

✙✠❃

⑧✄ ✟

; hence

✍■✒ ✍ ✄✕✞▼✙✠❃ P ✻✸ ✺✜✼✲✜ ✍❇✍✪✸✻✺✜✼✯❙✍ ➉ ✆ ✒ ✜ ✄ ✟ ❃ ✜ ✾✵ ➈✜ ✟ ➇ ✟ ➁✟ ✌

✄

☛✟ ✁

✝

✝ ✌

.

.

✌

7. By ➇ ✟ ✡◗symmetry, ➁ ✟ ✍⑩✙ ✟ the integrals of , , and over the unit sphere are equal, so the integral of ✙✠❃ ✆ ☞♦✼✄✔ ✕ ✟ ✍❇✍⑩vanishes. ✙■✸✻✺✜✼ ✟ ✍■✒✈✄✼ ✔✖✕ The ✍ ✍✇integral ✄✕✙✠❃ ✆ is☞✎✏❝also ✍✪✞■✸✻easily ✺✜✼ ✟ ✍●✒✈computed ✼✙✔ ✕ ✍ ✍✪✄✌in✙✠❃ spherical ✸✻✺✜✼ ✍❇✍✪✸✻coordinates: ✺✜✼✲✜ ✍ ✆ ✄✖✰ it is . P P ✟ ✄ ✄ ☞✎✍❝➁ ✩✍❧➇ ✂❍✡ ☎❄✒ ➁ ➇ ❢✧ ✄ ✄ ☞✎✍✯➇ ✟ ➁ ✟ ✍❧➇❍➁❬✒ ➁ ➇★✄ ☞✎✍ ✞ ➇ ✟ ✍s✙▲➇❫✒ ➇★✄ ✜ ✟ 8. (a) , so ✍ ✞ ✍✥☛✟✏❀✄ ✍ P ✏ . ☎

✝

✌

✝

✝

✌

✁

✁

✌

✌

✌

✝

✌

✝

✝

✌

✝

✌

✌

✝

(b) Since lines through perpendicular unit ✁❈✐ the center of a sphere are✁✥ ✄❖➇ ❴✡⑤➁ ✂❝✡ to✆☎the sphere, the❈ ✧ ✄ normal ✄ to the ➇ ✜ unit ☛✟ ✟ , sphere ❷ at a point ➇ ✜ ❷ is simply the vector . Hence ✄ ✄ is an odd function. Alternatively, in spherical coordinates one finds ✟ which vanishes since ✆ ✝ ✟ ✆ ☞♦✼✄✔ ✕ ✍✲✸✻✺✜✼ ✈✡✥✼✄✔ ✕ ✍✲✼✄✔ ✕ ✂☛✡❲✸✻✺✜✼ ✍ ☎❄✒✈✄✼ ✔✖✕ ✍ ✍ ✥✧✂✄ ✄ ✼✄✔ ✕ ✪ ✍✲✸✻✺✜✼ ✜ ✍✇✄✖✰ . ✵ ✵ ✾✵ , so ☛✟ ✵ ☎✵ ➇♣✡⑤➁✲✡ ⑥✄✝✙ ➇✬➂❨➁ ✄ ✄ ☞ ➃✡ ✂ ✡ ❝ (c) taking ✟ ✟ ✎ ☞❊➇❍➁✤✡⑩✙❝✍★➇❧; ✍★ ✟ ☞ P ☞❊➇ ✍ ✏✓✒✻☞we ❄☎ ✒ The ➁ ➇ triangle lies ⑩✧ ✄ in the✄ plane ➁❬✒ ➁ ➇⑥✄ as parameters, ✙✤✍❦have ➇❴✒ ✟ ✡⑤☛☞ ✙❝✟ ✍❋➇❴✒ ✟ ✒ ➇★✄ ✟ , so ☛ ✟ P ✟ ✟ ✟ ☞❊➇ ✜✯✍ ✞▲➇ ✡ ☎✒ ➇⑥✄✕✙ . ✁

✁

✝

✝

✁

✌

✝

✌

✝

✌

✁

✁

✌

✌

✝

✝

✌

✝

✝

✌

✌

✌

☎

✁

✁

✌

✝

✌

✝

✝

✌

✝

✝

✌

✌

✝

✌

✌

(d) so ✟ ✄✛✗ ✟ ✄ on✄ the☎ vertical side of the ✖✄ The✗ normal is horizontal ✖✄ cylinder, ✱ ( ✥✧✂✄ ) we have ; on the bottom ( ), we have ✄✖❃❁☞ ✗ ✟ ✍ ✱ ✟ ✒ , ☛✟ . ✁

✧

✄ ✄ ✰ ✟ ✄ ✱✟

there. On the top ✄ ✄ ✍ ☎ , . Hence

✁

✁

✝

✝

(e) ⑧✄❈❷ ➇

✄

P✡✟ ✟ P ⑧✄ ✆ ✙✘✍◆➇➃✟✯✍✇➁✣✟ ✟ and the paraboloid ✟ ✡ ❷ ➁ ✟ ❷ where ➇ ✟ ✡◗➁ ❷ ✟ and q ✏ ❷ are the portions of the sphere ✌ with , oriented with the normal pointing up and down, respectively. As in (b), ✁ ✁✬✽ ✛ ✙ ✧ ✄✇✄❖ ✁❑ ✟ ✽ ✛ ✙✲✄ ✛ ✙ ✌

✁

the normal at a point

on the sphere is

, so

. Also, the element of area in

38

Chapter 5. Line and Surface Integrals; Vector Analysis

✍ ✍

✟ ✼✄✔✖✕

✍ ✍

✘

✍ ✍

✒

✄ ■ ✙ ✙✼ ✔ ✕ ✧ ✄ ✄ ✛ ✙ ✧▲✙✠❃ ✆ ✪ ✙■✄✼ ✔✖✕ ✥✄ ✟ ✾✵ ✾✵ , so spherical coordinates is ✄✒ ❃❁☞ ✙ ✟ ✍ ☎✒ ➇ ➂❨➁ ✬ ✄ ✄ ☞ ✙▲➇ ❍✡✥✙▲➁ ✭✍ ❄✒ ❖ . On the paraboloid, with as parameters, we have ☛✟ ◆✧ ✄ ✄ ✡ ✙▲➁ ✟ ✍✪☞❊➇ ✟ ✡❋➁ ✟ (remember ✒❨✒ ➁ ➇⑥✄ ❳❫P ☞ ✙▲➇ ✟ ◆ ✟ that the -component must be negative), so ✄✒ P P ✙✠❃ ✁ ✜ ✁✲✄ ✥✂✧ ✄ ✄✖❃❁☞ ✙ ✟ ✍ ❃ ✒ ✟ . Hence ✟ . ☛✟ ✁

✌

✌

✁

✌

✌

✝

✝

✁

✌

☛

✄

✁

✁

☎

✝

✝

✌

✝

✝

✖

✌

✝

✂

☎

✌

✗

✌

✌

✖

✄

✖

✁

✏

✁

✝

✌

✝

✝

✌

5.4 Vector Derivatives 1. These are all simple computations. ✁ ✄ ♣ ✁❑ ☞✂✁✾✒✤✄ P 2. These are simple computations too. For (c), with and ☞ ✍✥✏✓✒ ✁ ✎ ☞✂✾✁ ✒❁✴ ✄ ✱❴☞✳✱✲✍ ✏✓✒ ✁ ✎ ✟ ✡✕☞ ✍ ✏✓✒✫✱ ✁ ✎ ✟ ✴ ✄ ✱❴☞✳✱✦✡ ✍⑩✙✜✒ ✁ ✠

✠

✁ ✎ ✠

✠

✟

✠

✟

✟

✠

✟ , one has .

✂

✟ ❵

☞❊✁✬✒❝✄

☞✂✁✜✒✬✡ ✠

3. The first two formulas are most easily obtained just by writing out the indicated products and taking the indicated derivatives. The last one follows from the first one by using (5.29). 4. (Also 5, 6, 7.) These are straightforward calculations, but the verifications of (5.25), (5.27), and (5.33) require a little masochism. It is less frustrating to start with the complicated expressions (on the right in (5.25) and (5.27), on the left in (5.33)) and work toward the simpler expressions on the other side. ✌② is skew-symmetric in and . 8. ✁

✁

✂

✂

9. This follows immediately from (5.29) and (5.30).

5.5 The Divergence Theorem t✄✕✙▲➇ ✙▲➇ ✙▲➇ , and the integral of over the ball vanishes by symmetry since is odd. P P unit P ✟✝✆ ☞ ✛ ✙✘✍ ✁✠✟✈✍ ✁ ✟ ✒ ✁ ✔ ❈✄✕✞ ✞ ✄✕❅✠❃ ✍ ☞ ✙❴✍ ✁ ✟ ✒✫✜ ✒ ✟ ✍ ✁✩✪✮➉ P ✄ ✪ (b) ✄✒ , so polar coordinates yield ☎✵ ➈ ✜ ❃ ✙ ✟ ✍ ➉ ✟ . ➈ ✆✔ ✕✄✳✙✫☞❊➇✉✡✥➁✲✡ ☎✒ ➇ ➁ (c) ❅ and ➇ ➇ ➁ ⑧✄,✌ ✩ ✞ ✱ ✪ the integrals of , , and over the cube are equal by symmetry, so we . get ✪ ❃ ✱ ✗ ✔ ✌✄ ✱✢✎ ✟ ✣ ✡ ✗ ✎ ✟ ✡ ✎ ✟ ✜ (d) , and the volume of the ellipsoid is (reduce it to the ✳☞ volume ✄⑦➇❫✽ ✱ ✄✝➁❬✽ ✗ ✄ ☎✽ ✱ ✎ ✟ ☞ ✡ ✗ of✎ ✟ the ✡ , , ), so the integral is unit sphere by the change of variable ✜❃❁☞ ✗ ✟ ✟ ✗ ✡ ✱ ✟ ✟ ✖ ✡ ✱ ✟ ✗ ✟ ✒❨✽✠✩✞ ✱ ✗ ✎ ✟ ✒ ✪ ❃ ✱ ✗ ❝✄ ✜ . ✆✔ t✄✌✙ ✄✴✱ ✟ ➉ ✟ P ✄✕✞ P✟ ✙ , so the integral is ✂✁ (e) ✟ ➈ ✟ . ⑤✧ ✄◗✄ ☞❊➇ ✟ ✡✥➁ ✟ ✡ ✟ ✒ ✟ ✽ ✱⑥✄ ✱ ✜ Directly: of the exercises), ✟ so ❷ (see the remark at the beginning ✱ ✜ ✜❃ on ✱ ✆✔ ✖✄✌❂✫☞❊➇ ✟ ✡◗➁ ✟ ✡ ✒ ❷ the integral is ❂ times the area of , i.e., . By the divergence theorem: , ✆ ✝✟ ✆ ✁ ✟ ✧ ✁ ✟ ✄ ✼ ✔✕ ✁ ✇✄ ✜❃ ✱ ☎✵ . so the integral is P P ✆✔ t✄✕✞ ✥✂✧ ✄ ✄ ✔ ✄ ✄ ✜ , so ✜ ✂ ☛✟ ☎ ☎ volume of . ⑨✄ ⑤✧ ✄ ✄ ✆✔ ⑨✄ ☞ ✒✯✄ ✡ Let ❵ ; then ❵ and ❽ ❵ ❵❫❽ ✈❽ ❵ , so the result follows from the divergence theorem. ✟ ☞ ✽ ✒ ✄ ✧ ✄ ✄ ✆✔ ☞ ✒ ✄ (a) ❽❾❵ ❽ ✟ ❵ ☛✟ ❵ ☎ ❵ ☎ . ✟ ✡ ✟ ✆✔ ☞ ✒✿✄♠ ❈✄ (b) By (5.28) we have ❵ ❵ ❵ ❵ ❵ ; apply the divergence theorem to ❵ ❵ .

1. (a)

✆✔

✁

✁

✝

✝

✌

✌

✂

✄

✏

✁

✠

✠

✝

✝

✠

✝

✌

✌

✌

✁

✆

✆

✆

✁

✆

✆

✆

✍

✆

✆

✁

✝

✝

✝

✌

✌

✁

2.

✍

✠

✝

✁

✝

3.

5.

✌

✌

✁

✝

✌

✝

✝

✌

✑

✝

✁

✌

✝

✠

✟

✌

✝

✌

☎

✝

✝

☎

✟

✠

✂

✝

✝

✁

☎

✠

✝

✁

✄

✁

✝

✁

4.

✝

✍ ✡

✄

☎

✠

✠

✂

✝

✌

✝

✝

✂

✌

✝

✝

✝

✌

✁

✂

✂

✂

✂

5.6. Some Applications to Physics 6. (a) ❽ ☎

✟ ✒ ✎✢✜ ✒ ✟ ✄✳✍✯➇❴✽✫ ✁❑ ✜

✍✯➇●☞❊➇ ✟ ✡◗➁ ✟ ✡ ✄

✠

39

➁

, and likewise for

(b) See Exercise 2c in 5.4.

✱ ③ ✄◗✄✚✁✬✽✫ ✁❑ (c) If ❷ is ☞ the ✽ sphere radius , so ✒ of✄✳ ✍ ✱ ✎ ✟ ✧ about ✜❃ ✱ ✟ ✳ ✄ , on ✍ ✜❃❷ we have ❽ ✠ ❽ ✟ ☛✟ P hence .

.

✧ ✄ ✄⑦✍✲ ✁❑ ✟ ✽✫ ✁❑ ✪✲✄⑦✍✤✱✠✎ ✟

✂

✠ ✂

☞❊✁✬✒❁✄✳✍✯✁✬✽✫ ✁❑ ✜ ✠

✂

and , so

;

✁

✝

✌

✝

(d) ✠ is not of class

✑

✄

on the region inside the sphere, so the divergence theorem doesn’t apply. ✟ ✄⑨✰ ✄ ✁◗➄❴ ✁❑❬q ➅ ➀ (e) Choose so small that the ball is ✰❲ contained in the interior of . Then ✟ ✄ ✄ ☞ ✽ ✒ ✠ ✍ ✁✄✂✁☎ on ➆ , so by the divergence theorem and part (c), ✠ ☎ ❽ ✠ ❽ ✟ ☛✟ ☞ ✽ ✒ ✄ ☞ ✽ ✒ ✡ ✜❃ ✂✆☎ ❽ ✠ ❽ ✟ ☛✟ . ❽ ✠ ❽ ✟ ✟ ✟ ✄ ✟ ✄♠✰ ✰ ✄ ☞ ✍ ✒❁✧ 7. (a) Since on , by (5.39) we have is the union ❵ ✁❑❍✄ ✁ ✠ ✁ ❵ ✁❑➃✄✠ ✠ ❵ ✟ ✟ . ❽ P of the outward (with the inward normal). On ✄ (with ✁✏✎ ✽ ✄ normal) ✍ ✁ ✎ ✟ and the sphere ✍ ✁❑➃✄the sphere ✁ ✎ ✎❱ ☞ we have ✠ and ❽ ✠ ❽ ✟ P (as in Exercise 6c), so by Exercise 5a, ❵ ✠ ✟ ✒❍✧ ✄ ✄✳☞ ✍ ✁ ✎ ✟ ✎ ✎ ✍ ✁ ✎ ✚ ✄ ☞ ✍ ✁ ✎ ✟ ✎ ✎ ✍ ✜❃ ✎ ✎❳ ❱ ❵ ✟ ❱ ❵ ☛✟ , which is times the ✠ ❵ ✟ ❵ ☎ ✁❑☎✄ ✁ ✁❑✣✄ mean value of ❵ on the sphere . Likewise, the integral over (with the inward normal) is ✜❃ times the mean value of ❵ on this sphere. The sum is zero, so the mean values are equal. ☞ ③❾✒ ✁❑✜✄ ✣ ✰ (b) Since ❵ is continuous, ❵ is the limit of the mean value of ❵ on the sphere as .

✑

✝

✝

✂

✂

✝

✝

✌

✂

✑

✝

✂

✝

✝

✝

✌

✝

✌

✑

✌

✂

✝

✂

✝

✌

✂

✝

✝

☎

✂

✂

✌

✝

✝

✌

✝

✝

✝

✌

☎

✝

☎

✌

☎

5.6 Some Applications to Physics

☞❊✁✬✒✭✄ ✍ ✎ ✝✒✎ ✞⑩✍ ✁❑ ✎ P ❱ 1. To evaluate the potential ☛✟ (we take the density to be 1) at a particular ✁ ✁ point , the key to proceeding efficiently is to rotate the coordinates so that is on the✁ positive -axis ☞❊✁✬✒ it suffices (or, equivalently, ✟ ✡✕ ✁◆✄ to ☞♦✰✣observe ➂❜✰✣✁➂ ☎✒ that✞❤☎✄ ✁ is spherically ✞❇✍✪symmetric ✁❑✜✄ ✆ ✟ P so✡ that ☞ ✜ ✍ ☎✒ to✟ take ✄ ✆ ✁ ✟ on✍⑩the ✙ positive ✜ ✡ ✟, ✟ -axis). For and we have so in spherical coodinates we have ✆ ✆ ✝✟ ✆ ✁ ✟ ✼✄✔ ✕ ✙✠❃ ✁ ✆ ✁ ✟ ✍⑩✙ ✯✁❁✸✻✺✜✼ ❢✡ ✟ ☞❊✁✬✒❁✄✳✍ ✄ ✍ ✾✵ ✆ ✁ ✟ ✍ ✙ ✁✿✸✻✺✜✼ ⑥✡ ✟ ✙✠❃ ✄✳✍ ✁❝✡ ➃✠✍t ✁✘✍ ➃ ✁

✝

✌

✂

✁

✟

✍

✂

✟

✟

✟

✍

✂

✍

✌

✌

✡

✁

✡

✍

✡

✡

✎

✄

✑

✰◆q ✪q ✁ ✍ ✜❃ ✁ ✁ ✍ ✜❃ ✁ ✟ ✽ ❋✄ ✍ ✜❃ ✁ ✟ ✽✫ ✁❑ For ✜❃ ✁ ✟ this is ; for it is ✍ ❊☞ ✁✬✒ .③ The latter ✁❑✡✕ is✁ the potential ✍ ✜❃ ✁ ✟ ✁✬✽✫for ✁❑ a✜ is for and mass✁❑✏✙✘✁ located at the origin. The corresponding field for . ✁ ✰ ✕✴✁✚✕ ✁ ) and thickness . For a 2. Think ✁ of the ball as the union ✁ ✙ ✁❑of thin spherical shells of radius ( ✁ ✕ ✁❑ , the shells with contribute nothing to the field, and the shells with contribute given ✍ ✜❃ ✁ ✟ ✁✬✽✫ ✁❑ ✜ ✰ ✔ ✕❫☞ ❧➂✩ ✁❑❺✒ ✍ ✪ ❃❴✁ ✁❑✏✕ ✍ ✪ ❃ ✜ ✁✬✽✫ ✁❑ ✜ ✜ . Integrating from to gives the field as for and ✟ ❃❁☞❜ ✁❑ ✟ ✜ ✍◗✞ ✟ ✒ ✁❑ ✙ ✁ ✜ for , as claimed. (The potential can also be found by integrating in ; it is for ✁❑✏✕ ✍ ✪ ❃ ✜✱✽✫ ✁❑ ✁❑ ✙ and ✜ for .) ✄✳✏ 3. (a) We take . The field is ➇ ❬✡◗➁ ✫✡✖✆☞ ✍ ❴✒ ➇ ✫✡◗➁ ✣✡❲❛ ★✄ ❛ ☞❊❛❁✄ ✍ ❴✒ ✟ ✄ ✒ ✒ ✒ ☞❊➇❾✟✿✡❲➁✈✟☛✡✕✆☞ ✲✍ ❴✒✔✟✓✒ ✜ ✟ ☞❊➇➃✟☛✡❲➁✈✟❁✡◗❛✎✟✱✒ ✜ ✒ ✟ ✤

✂

✑

✁

✟✑

✑

✞

✑

✑

✑

✑

✑

✌

✂

✑

✟

☎

☎

✂

✂

✎

✟

✟

✒

✌

✎

✟

✒

✌

✄

Chapter 5. Line and Surface Integrals; Vector Analysis

40

➇

➁

❛✱ ✎✢✜

❛✱✝✣

✥

The and❛ ✎ ✟ components of the integrand decay like as , while the component devanishes since its❛❦ integrand cays ☞✳like ✟ integral ✟ ✛ ✱✈✟☛✡ ❛✎✟ The ✄ -component ✱ ✟ ✡⑨❛ ✟ ,✒ ✎✢so✜ ✒ the ✄ ❛❜✽ ✱ converges. ✙✜✽ ✱ ✟ ✄ ✱ ❜✶ ✕ is odd. Since ✒ ✒✎ ✵ ), we obtain (via the substitution ✎ ✄✕ ✒ ✙✫☞❊➇ ❬✡◗➁ ✩✒❨✽✣☞❊➇ ✟ ✡◗➁ ✟ ✒ . ✒ ☞❊✁✬✒ ☞❊➇ ✟ ✡ ➁ ✟ ✡⑨☞ ❢✍ ✈✒ ✟ ✒ ✎ P✓✒ ✟ ■ ✎ P ■✏✣ ✥ should be ✒✎ , but the integrand decays like as , (b) so the integral diverges.✒ ✺✍➃✓ ☞ ✛ ✱✈✟☛✡◗❛✎✟✿✡❲❛❨✒ ✏✓✽ ✛ ✱✈✟❁✡◗❛✔✟ is an antiderivative of , (c) Since ✝ ✟

☎

✝

✂

✌

✝

✂

✟

✟

✁

✟

✝

☛

☞❊✁✬✒❁✄

✓

✎ ✓

✌

Multiplying top and bottom of the fraction by

✆ ➇❾✟☛✡◗➁✣✟✿✡✌☞✆ ✍

☞❨✆ ➇❾✟☛✡◗➁✣✟✿✡✌☞✆✭✡✍✑ ✒✔✟❁✡ ✦✡✍✑ ✒✻☞❨✆ ➇➃✟☛✡❲➁✈✟❁✡✕☞✆✲✍ ➇➃✟☛✡ ➁✣✟ ☞ ✆ ➇ ✟ ✡◗➁ ✟ ✡✌✆☞ ✭✡✍✑ ✒ ✟ ✡ ✦✡✍✑ ✒✻☞ ✆ ➇ ✟ ✡❲➁ ✟ ✡✕☞✆✲✍ ✑✟

✺✍✓

✄ ✺✍✓ ✝

✑

✟

✆ ➇❾✟☛✡◗➁✈✟❁✡✕☞✆✦✡✍✑ ✔✒ ✟☛✡ ✭✡ ✆ ➇ ✟ ✡◗➁ ✟ ✡✕✆☞ ✲✍ ✑ ✒ ✟ ✡ ✍

❛ ✄ ✺✍✓ ✆ ➇➃✟☛✡❲➁✈✟❁✡◗❛✎✟ ✝

✂

✁

✝

✟

✌

✑

✒✔✟❤✍ ✦✡✍✑

✑

✑✂✁ ✄

turns this into

✑

✒✔✟❤✍ ✭✡✍✑ ✒

✑

✒ ✟ ✍ ✭✡✍✑ ✒ ✡✥✙ ✺✍✓ ✕ ✑ ✍ ✺✍✓➃☞❊➇ ✟ ✡◗➁ ✟ ✒ ✄

✁ ✁

✑

✝

✝

✙ ✺✍✓ ✺✍✓ ⑧✍ ✺✍✓❴☞❊➇ ✟ ✡❲➁ ✟ ✒ ✝ ✝ ✝ , so subtracting ✺✍off the yields , ✓ . (Of course the ✝ can be discarded too.) ✺✍✓✂✢❇✄❈✢●✽✫ ✢❤ ✟ 4. The argument essentially ✟ ✺✍✦ ✓ ✢✂❾✄is♠ ✰ ✢ ✄❖ ❶ the③ same as the proof of Theorem 5.46, using the facts that ✝ and ✝ for (Exercise 2d, ✂ 5.4). The two-dimensional analogue of Green’s formula (5.39) is easily obtained from Exercise 6, ✂ 5.2, and it yields the following analogue of (5.47): ✣ ✥

As , the first ✍✤✙✫term ☞❊➇ ❬✡◗approaches ➁ ▲✒❨✽✣☞❊➇ ✟ ✡ ➁✝ whose gradient is ✂

✺✍✓ ✟✒

✂

✂

✟ ☞❊✁✬✒❁✄ ✂

✔✟✞

✝✂

✁

❨☞ ✺✍✓✘ ✢✂❺✒

✂

✎

✎

✝

❱

✝

✂

✄

where is the unit inward normal to the circle, namely, term on the right becomes

☞ ✺✍✓✘ ✢✂❺✒

✂

✡

✡

✎

✎

✡

✝

✡

❱

✝

☞❊✁❢✡◗✢●✒■✧ ✂

and from the second term, since

✢✂✜✄

✏

✔✟✞

✝✂

✎

✆✸ ✛✫✷ ✖✧ ✄ ✄ ☞ ✬✍ ✤✡ ❄✒❑✧❬☞✎✍ ✭✡ ❄✒ ➇ ➁ ✝ ➇ ✟ ✡◗ ➁ ✟ q✌☛✏ ✟ ✙✠❃ , i.e., . ✆✸ ✛✫✷ ✘✧ ✄ ✄ ☞✎✍ ✜✍ ❄✒ ✧ ☞ ▲✡ ❄✒ ➇ ➁❢✄ ✍✤✙ ➇ ✝ ✟ ☛ ✟ ➇ ✡✪➁ ✟ ✡⑤☞✳✱✑✍❦➁✫✒ ✟ ✄ ✱ ✟ ✙▲➇ ✟ ✽ ✱ ✟ ✡ ❍☞❊➁ ✍ by , or ✍✤✙ ✁

2.

✂

✎

The integral is

☎

✂

✌

✌

✌

✂

✌

✌

☎

✌

❱

✄ ✙ ➇ ➁

☎

✌

✂

✌

✝

as

✣ ✰✣➂

☞❊✁ ✡◗✢●✒ ✘ ✂ ✄✕✙✠❃ ☞❊✁✬✒ ✄

✂

5.7 Stokes’s Theorem ☎

. The estimate (5.48) for the first

on the circle, one obtains

✝

✂

✌

✡

✁

✌

✧✂✄ ✜✂ ➂

✡

✟ ☞❊✁✬✒❁✄ ✂

✁

✄

✡

✂

✂

q ✉ ✺✍✓ ✙✠❃ ✣ ✰ ✡

✌

✡

✡

1.

✂

✄

✂

☞❊✁ ◗ ✡ ✢■✒ ✢ ✢✂ ✟ ✄ ✄ ✍✯✢●✽ ✕

☞❊✁⑥✡◗✢●✒■✍ ✂

✌

✂

, so the integral is twice the area of the disc

➁P ➇❍➁ .✱✫The curve lies over the curve in the -plane given ✟ ✟ ✟ ✒ ✽ ✱ ✄✳✏ , an ellipse with semiaxes ✱✫✽✾✛ ✙ and✟ ✱✫✽✠✙ . ☞✎✍❀✙✜✒✔❃❁✳☞ ✱✣✽ ✛ ✙✜✒✻☞✳✱✫✽✠✙✜✒☛✄✳✍✤❃ ✱ ✽ ✛ ✙ ✄

✌

times the area of the region inside the ellipse, i.e.,

.

5.8. Integrating Vector Derivatives

41

➇ ❧✄✏✗ ➁✂✡ ✆ ✸✆✛✫✷ 3. The equation of a nonvertical plane parallel to the -axis has the form . Thus ☞✎✍✯➇ ✣✡⑩➁ ✂❁✡❲✙ ☎❄✒❄✧✜☞✎✍ ✗ ✂❁✡ ☎❄✒ ➇ ➁✉✄ ☞✎✍ ✗ ➁✭✡❲✙✜✒ ➇ ➁ ➇ ✝✟ . The integral of this over the disc ✙✠❃ ✱ ✟ ✗ ➁ is twice the area of the disc, i.e., (the integral of vanishes by symmetry).

✌ ✌

4.

✌ ✌

✱

❽❾❷

➇❍➁

◆✧ ✄ ✄ ☛✟ ✌ ✟ ✡⑩➁ ✄ ✱ ✟ ✁

is the circle of radius about First ✱ ➇❍➁ the origin in ✸✆the ✫✛ ✷ t-plane. ✧✄ ✄ method: ✸✆✛❬✷ ✖✧ ❽❴☎ ❷ is also ✄ the ➇➃boundary ➁ ✜ ➇ ➁✪of✄ ✝ ☛✟ ✂ the ✰ disc of radius in the -plane, so ➇ ✄ ✱❑✸✻✂ ✺✜✼ ✝ ➁ ✄ ☛✱❑✟ ✼✄✔✖✕ ✄⑦✄ ✰ , by symmetry. Second method: parametrize by , , . Then ❴ ❽ ❷ ✵ ✵ ✸✆✛✫✷ ✥✧✂✄ ✄ ✧ ✁❇✄ ✟✝✆ ☞✎✍★✼✄✔✖✕ ✟ ✡❲✸✻✺✜✼ ✟ ✒ ✄✖✰ ✝ ☛✟ ✵ ✵ ☎✵ . ✝

✁

✝

✁

✝

✌

✁

✝

✝

✌

✁

✁

✝

✌

✁

✌

✁

✝

✌

✝

✝

✝

✝

✌

✁

✌

✝

✌ ✌

✝

☞❊➇ ✟ ✽✁☎✒✾✡ ☞❊➁ ✟ ✽✠✴✜✒❁✄✳✏ ➇➃➁ in the ✄✖✰ ✸✆✛✫✷ -plane. ✇✧ ☎❇If✄✖✰ ✁ ✌ ❋✧ ✄ since✄ ✝ ✁ ❇✧ ✁❋✄

5. Like Exercise 4, this can be done two ways. ✸✆❽❴✛✫❷ ✷ is✇the ✧ ✄ ellipse ✄ ✸✆✛✫✷ ✇✧✄☎ is the region inside this ellipse, we➇❢have ✝ ✸✆☛✛✫✟ ✷ ✄✌✙■✸✻✺✜✼ ➁s✝ ✄✕✞■✼✄✔ ✕ ✟ ⑧✄✖✄✂ ✰ ❾ ❽ ❷ ✝ ✵ ✵ Alternatively, parametrize by , , ; then ✟✝✆ ✏✗✰✿✸✻✺✜✼ ✼✄✔✖✕ ✄✖✰ ✵ ✵ ✾✵ .

✁

✝

✝

✁

✝

✁

✌

✝

✝

✁

✌

✁

☛✟

✝

✝

6. (a) This is a simple calculation.

➇◗✄ ❑✱ ✸✻✺✜✼ ➁✪✄ ✱❑✼✙✔ ✕ ❋✄ ✆ ❖✄ ☞✎✍ ☞♦✼✄✔ ✕ ✒ ❾✡✳☞♦✸✻✺✜✼ ✒ ✂▲✒❨✽ ✵✥ , ✧ ✁❇✄ ✵ ✵ ✟✝✵✆ , ✕ ✄ ✙✠;❃ then ✵ P ☎✌ ✵ , so ✁ ✌ ✌✾✵ .

(b) Parametrize ✱❾☞✎✍✲☞♦✄ ✼ ✔ ✕ ✒ ✣✡✕☞♦✸✻✺✜✼ by✒ ✩✂ ✒ ✄

✵

✝

✄

✄

(c) is not of class on any surface bounded by Stokes’s theorem doesn’t apply. in between the✸✆circles ✚✧be ✁ the✍ annulus ✛✫✷ ✚✧ ✄ ✘ ⑨✧ ✁♠✄❉❆ ✁ ✌ ✌ ✄✂☎ ✸✆✛✫✷ ✥✧ ✄❇✄✕✞ ❃☛ ✁ ✟ ✝ ✍ ✏✾ ✄

✟

7. Let

✁

✝

✁

✝

✁

✝

✝

✁

✌ ✁t✄

and

✁

✝

✁

✱

✁

.

✝

, and the area of ✟

8. Use (5.26) and (5.30):

✸✆✛✫✷ ☞ ✝

❵

✠ ✂

✝

✒❁✄

is

(such a surface must intersect the -axis), so

❀P

➇

✄✥✄

and in the -plane, oriented so ✍that ✁✷✕ ✡ ✁✘ ✙❏✏ if and if ☛✟ , the sign being ✥✧ ✁❇✄✖❂✤✡✥✞✠❃❁☞✂✁ ✟ ✍ ✏✓✒ . It follows that ✁ . ✄

✌

✌

✁

✝

✸✆✛✫✷ ☞ ❵

✝

✠ ✂

✒✩✡

✂

②

❵

✠ ✂

✄

②

❵ ✂

✠ ✂

✂

.✏ Then . But

. Now apply Stokes’s theorem.

5.8 Integrating Vector Derivatives

☞❊➇❄➂❨➁❬✒✯✄ ☞ ✙▲➇❍➁⑧✡ ➇ ✟ ✒ ➇✪✄r➇ ✟ ➁✲✡ P ➇ ✜❤✡ ✍✂☞❊➁❬✒ ✗ ☞❊➇✬➂❨➁✫✒✯✄✳➇ ✟ ✡ ✍ ☞❊➁❬✒❝✄r➇ ✟ ✍ ➁ ✟ ✍✂☞❊➁✫✒❝✄ ✜ ;❽ ❵ , so ❵➁ ✜ ✡ ✌ ✜ . ✗✣☞ ✞▲➁ ✟ ✡✥❂▲➇ ✪ ➁✫✒✿✄✌❅▲➁✑✡ ❂▲➇ ✪ ✄✕ ❶ ❂▲➇ ✪ ✍⑩❅▲➁ ✄ ☎ ☞❊➇ ✍ ❅▲➇❍➁❬✒ ❽ . (b) ❽ ☞❊➇❄➂❨➁❬✒✿✄ ☞ ✙ ✂ ✟ ☎ ✼✄✔✖✕❤➁✦✍❇✞▲➁❀✡✇❂✜✒ ➇❦✄✄✂ ✟ ☎ ✼✄✔ ✕✂➁✘✍❋✞▲➇➃➁❀✡✇❂▲➇ ✡ ✍✂☞❊➁✫✒ ✗ ☞❊➇❄➂❨➁❬✒✿✄✄✂ ✟ ☎ ✸✻✺✜✼❬➁✘✍❋✞▲➇✑✡ ❵ ✌ ;❽ ❵ (c) ✍ ☞❊➁❬✒❁✄✄✂ ✟ ☎ ✸✻✺✜✼✫➁❧✍⑩✞▲➇ ✍✂☞❊➁✫✒❁✄ , so . ☞❊➇✬➂❨➁➃✁➂ ✈✒⑥✄ ☞❊➁ ⑥✍⑤➁❤✄✼ ✔ ✕❤➇➃➁✫✒ ➇✳✄❏➇➃➁ s✡⑨✸✻✺✜✼❬➇➃➁ ✡ ✍❑☞❊➁❾➂✁☎✒ ✗ ☞❊➇✬➂❨➁➃➂✁☎✒⑥✄ ➇ ❢✍❈➇❀✼✄✔ ✕❤➇➃➁♣✡ ❵ ❽ ➂❨➁❾❵ ➂✁☎✒❁✄❈➇➃➁➃✡✦➁❑✸✻✺✜✼❍➁ ✣✡ ☎ ☞✆✈✒❁✄ (d) ; ☞❊➇❄ ✗ ✍✂☞❊➁➃➂✁☎✒❑✄❈➇ ➃✍❝➇✭✄✼ ✔ ✕❤➇❍➁➃✡ ❁✸✻✺✜✼❬➁ ✌ ✍✂☞❊➁➃✁➂ ☎✒❑✄✖✼✄✔ ✕✂➁ ✈✡ ☎✭☞✆☎✒ ; ❽ ✱❵ ➇❍❽ ➁✑✡◗➁❤✸✻✺✜✼❬➁ ☎✦☞✆☎✒☛✄ . , so , so ☞❊➁ ✍ ☎✒❁✄ ✍✑✏s✄✳ ❶ ✏✤✄ ☎ ☞❊➇ ✍✪➁✫✒ ❽ . (e) ❽ ☞❊➇✬➂❨➁➃✁➂ ☎✒❑✄ ✙▲➇❍➁ ➇❦✄✖➇ ✟ ➁✑✬ ✡ ✍❑☞❊➁❾➂✁☎✒ ✗ ☞❊➇✬➂❨➁➃➂✁☎✒❑✄✖➇ ✟ ✡ ✗ ✍✂☞❊➁➃✁➂ ☎✒❑✄✖➇ ✟ ✡ ✍✺ ✓ ✍✂☞❊➁❾➂✁☎✒❑✄ , so (f) ✝ ➁ ✍✺ ❵ ✓ ❀✡ ☎✦✆☞ ☎✒ ☞❊➇❄➂❨✌ ➁❾✁➂ ☎✒❁✄ ☞❊➁❬✽ ☎✒❙✡ ☎ ✆☞ ;☎✒❽ ❁✄ ❵ ☞❊➁ ✡✥✙✜✒❨✽ ✭✆☞ ❽ ✈✒☛✄✕✙ ✍✺ ✓ ❀✡ ☎ ✝ ✝ ; ❽ ✱❵ , so . ☞❊➇❄➂❨➁❾✁➂ ✫➂ ✒❈✄ ☞❊➇ ✟ ✡♠➁ ✒ ➇ ✄ ✟P ➇ ✟ ✟ ✡♠➇➃P ➁ ✡ ✍✂☞❊➁❾➂✁✫➂ ✒ ✗ ☞❊➇✬➂❨➁➃➂✁☎✒t✄ ➇ ✓ ✡ ✗ ☛ ✌ ✗ ☛ ❵ ❽ ❵ ; (g) ✟ ✟ ✟ ✟ ✟ ✗ ✍✂☞❊➁➃➂✁✫➂ ✒❧✄⑦➇ ✡❈➁ ✍ ✙ ✂ ✍❑☞❊➁❾➂✁✫➂ ✒✲✄ ➁ ✡✍ ✂ ✆✡ ☎✦✆☞ ✫➂ ✒ ; ❽ ✱❵ ☞❊➇❄➂❨➁❾➂✁✫➂ ✒✲✄ ➇❍❽ ➁ ✡t➁ ✟ ✡ ✍ ✂ ✟ ✗ ☛ ✡ ☎✭✆☞ ✫➂ ✒ ✄⑦➇➃, ➁ so ✡t➁ ✟ s✍ ✂ ✟ ✗ ☛ ✟ ✍ ✄✼ ✔✖✕ ☎✭✆☞ ❬➂ ✒⑧✄⑦✸✻✺✜✼ ✡ ☞ ✒ ; ❽ ➇❍➁ ✍ ❁✼✄✔ ✕ ☞❊➇❄➂❨➁❾➂✁✫➂ ✒❑✄❈➇ ✟ ✡◗ ✡ ☞ ✒❁✄❈➇❍➁ ✭✡ ➇ ✟ ✍ ❑✼✄,✔ ✕ so ☞ ✒☛✄ ✝❽ ✆■❵ , so .

1. (a) ✍ P

✝

✄

✄

✝

✄

✝

✓

✄

✓

✝

✄

✓

✍

✍

✝

✍

✍

✍

✍

✓

✍

✍

✓

✍

✍

✓

✍

✓

✍

✓

✍

✓

✍

✍

✍

✍

✍

✍

✄

✍

✍

✍

☞

✍

✍

✍

☞

✍

☞

✍

Chapter 5. Line and Surface Integrals; Vector Analysis

42

✄✕✞▲➇ ✟ ✡✕☞✎✏❝✍⑩✞▲➇ ✟ ✒❙✡❲✰✲✄r✏s✄✕ ❶ ✰ . ✄ P ❫✡P ✟ ✸✆✛✫✷ ✄❉✍ ✟ ✡ ❽✔✓ P✂✡✳☞ ❽ P ✟ ✍ ❽ ✗ P✮✒ . Then ✍ ❽✙✓ ✟ ✄ ➇➃➁❧✡ , , so (b) Take ✝ ❽ ✓ ✟ ✟ ✄ ✍✯➇➃➁ ❧✍ ✟ ✡✡✍❑☞❊➇✬➂❨➁✫✒ , and ❽ ✓ P ✄♠➇ , so P ✄ ✟ ➇ ✟ ✡ ☎✭☞❊➇❄P ➂❨➁❬✒ . Hence, ❽ ✟ ✍ ❽ ✗ P✑✄ so ✟ ✍✯➁ ❀✡ ✍❑☞❊➇✬➂❨➁✫✒✿✍ ✗ ☎✭☞❊➇✬➂❨➁✫✒❁✄ ✍✯➁ ✍✇➇ ✍✂☞❊➇✬➂❨➁✫✒❑✄✳✍ ✟ ➇ and✚ ☎✭☞❊➇✬➂❨➁✫✒❑✄✖✰ . ❽ ❽ . One solution is ✍ ✄⑨➇ ✎ ✄✖ ✓ ✖ ✍ ❅▲➇ ✟ ✄⑦✍❝➇ ✓ ✩✎ ✄✖ ✖ ❛❙✡t❅▲➇ ✘✡✬✍✂✚ ☞❊➇✬➂❨➁✫✒ , (c) Proceeding as in (b), we have ❽ ✓ ✟ P◗✄ ❂▲➁❦✡ ✙ P◗✄ ❂▲➁ ❢✡ ✟ ✡ ☎✦☞❊➇❄➂❨➁❬✒ , so ✟ ✍ ❽ ✗ P❲✄ ✍ ✓ ✎ ✖ ✖ ❛❝✡ and ❽ ✓ , so . Hence, ❽ ✚ ✙▲➇ ✟ ✓ ❛ ✟ ✩✎ ✖ ✖ ❛❴✡⑩❅ ❝✡ ✍✂☞❊➇❄➂❨➁❬✒✬✍❇❂ ✦✍ ✗ ☎✭☞❊➇✬➂❨➁✫✒❑✄ ✦✍ ✎ ✖ ✓ ✖ hopeful, but ✙▲➇ ✟ ❽ ✓ ❛ ✟ ✩✎ ✖ ✚ ✖ ❛❁✄✳❽ ✍ ✩✎ ✖ ✓ ✖ ✡ ✓ ✎ ✖ ✚ ✖ ❛ . This does not look ✍◆✄✆☎ ✄✖✰ , so we can take . by integration by parts, ✄ ✆✔ ✁ ✂ ✟ By outlined before Theorem 5.64 we can find ❵ such that ✆✔ ☞ the ✍ procedure ✸❵ ✛✫✷ ✄ .✍ ✂ Then ✆ ✂ ✒■✄ ✆✔ ✁ ✍ ✂ ✟ ✄✕✰ ❵ ❵ , so by Theorem 5.63 there exists such that ✝ ❵ . ✰ ✕ ✁ ✕ ✳✧ ✁ ✍ ✳✧ ✁❞✄ (a) If , let ✟ be the annulus between and . Then ✄✂ ☞ P ✍ ✟ P✮✒ ✄✕✰ ✄ ✥✧ ✁ ✁ by Green’s theorem. Thus is independent of . ✄✂☎ ✂ ❽ ✟ ❽ ✟ ✁✁ P ✙✕✰ ✁❧✄ ✟✰t✄ . (b) Let be the minimum distance from points in the compact set to the origin, and let a region Then ☞ P the✟ ✍ curve P✮✒ and✄ the circle ✧ ✁★✍ together ✥✧ ✁❇bound ✄ ✧ ✁❦✍ , so by Green’s theorem again, ❽ ❽ ✟ ☛✟ . ❙✧ (oriented counterclockwise) we have (c) From Example 1 and part (b), for any closed curve in ❷ ✁❇✄✌✙✠❃ ☞ t✍ ☞ ✽✠✙✠❃❄✒ ✒✬✧ ✁❇✄✖✰ ✍t☞ ✽✠✙✠❃❄✒ ✆✔

2. (a)

✂

✁

✂

✁

✁

☎

✁

✁

✁

✁

✁

✁

☎

✁

✁

✁

✁

✁

☎

☎

✂

☎

☎

✂

✁

✁

✝

✌

✂

✁

✁

✁

☎

✁

☎

✌

✝

✂

☎

☎

✂

☎

✝

✌

☎

✂

✂

✝

3.

☎

☎

✂

✌

✝

✌

✂

✂

4.

✄

✂

✄

✁

✁

✌

✝

✌

✝

✁

✁

✁

✝

✌

✝

✝

✌

✄

✄

✄

✑

✁

✁

✝

✁

✁

✁

✌

✌

✝

✌

✝

✝

✌

✝

✁

✝

✁

✌

and hence

✝

✁

✝

✁

✌

. By Proposition 5.60,

✁

✝

is a gradient.

Chapter 6

Infi nite Series 6.1 Defi nitions and Examples

✙✫☞❊➇■✡⑥✏✓✒ ✙✫☞❊➇■✡⑥✏✓✒ ✜ ✙❬ ➇■✡❢✏✾ ✜ ✕✌✏ and , ✙✫☞❊➇ ✡❈ ✏✓✒❨✽ratio ✏✭✍⑩✙✫☞❊➇s✡❈✏✓✒ ;✜✻it➉ converges for . ➈ ✏✗✰✠➇ ✎ ✟ ✙▲➇ ✎ ✟ ✙❬ ➇■ ✎ ✟ ✕ ✏ (b) This is a geometric series with initial term and ratio ; it converges for , i.e., ➇ ✕⑨✍❀✛ ✙ ➇ ✙❈✛ ✙ ✏✗✰✠✡➇ ✎ ✟ ✽✣☞✎✏✤✍⑩✙▲➇✛✎ ✟ ✒❁✄✳✏✗✰✾✽✣☞❊➇ ✟ ✍⑩✙✜✒ or , to the sum . ☞✎✏✘✍◗➇❫✒❨✽✣☞✎✏❀✡⑤➇❴✒ . The is ➇❴less (c) This is a geometric series with intial term 1 and ratio ➇ ✍ ✏ ➇ ✙ ✰ ✏✓✽ ✏❾ratio ✍⑥☞✎✏➃✍⑧ ✒❨✽✣☞✎✏✫than ✡✉➇❴✒ 1➉✿in ✄ absolute . The sum is ➈ ☞✎✏❑✡◗➇❫✒❨✽✠value ✙▲➇ if and only if is closer to 1 than to , i.e., . ✍✺ ✓❑➇ ✍✺ ✓☛➇ ✍✺ ✓❑➇■✞✕ ✏ P and ratio ; it converges for , i.e., (d) This is a geometric series with intial term ✝ ✝ ✝ ✂ ✎ ✕✥➇ ✡ ✕ ✂ ✺ ✓❁➇❫✽✣☞✎✏✤✍ ✍✺ ✓❁➇❴✒ ✍ ✝ , to the sum ✝ . P ✡✥✙ ✎✠☛ , which does not tend to 0; the series diverges. P (a) The th term is ✟ ✏✯✍⑤☞ ✡✖✏✓✒ ✎

P✓✒ ✜ ✕✥➇ ✕⑨ ✒ 1. (a) This series with ✍✑✏✯is✍⑩a✙ ✎geometric ✍✑✏☛✡✥ ✙ ✎ P✓initial ✜ term , to the sum i.e.,

2.

✟

(b) This is a telescoping series; the sum of the first (c) This is a telescoping series; the sum of the first

✟

✟

terms is

terms is

✡✖✏✤✍ ✏

✛

, so the full sum is 1.

✟

✟

, so the series diverges.

(d) The odd-numbered terms do not tend to zero; the series diverges. ☞ ❘✆☛ P ✌ ☞❊➇❴✒❑✄♠☞✎✍✑✏✓✒ ❘ ✶ ✽✣☞✎✏❤✡◗➇❴✒ ❘✆☛ P ☞❊➇❫✒☛✄ ✺✍➃✓ ☞✎✏✂✡◗➇❫✒ ❘☎☞❊➇❴✒❑✄ 3. If reads ☞✎✍✑❵ ✏✓✒ ❘ ➇ ❘✆☛ P ✝ ✽✣☞ ✶❦✡⑦✏✓✒✻☞✎,✏⑧then ✡ ✱✆ ✒❵ ❘✆☛ P ✆ ➇ , so Lagrange’s ➇ ✙ ✰ formula ⑧ ✏ ✡ ✆ ✙ ✏ P ✕✳have where , so ❘✈☞❊➇❫P ✒✗■q✳➇ ❘✆☛ P ✽✣☞ ✶ ✡r✏✓✒ ➇tq ✏ is between 0 and ✶ .✣ For ✥ ✍ we ➇✷✕♠✰ ✟ ,❘✈and . (For we still ➇■ ✕ ✟ ✕ ✏✦✡ ✆ ☞❊➇❫✒✗❀qfor❏✏✓✽✣☞ ✶⑥✡✝this ✏✓✚ ✒ vanishes ✣ ✰ as ❘✣☞❊➇❴have ✒❢✄ , so .) The integral formula (2.56) gives ❘ ✆☛ P P ❘ ❘ P ☞✎✍✑✏✓✒ ➇ ☞✎✏❤✍◆❛❨✒ ☛❴☞✎✏✿✡ ❛✔➇❴✒ ✎ ✎ ❛ ❘☎☞❊➇❫✒❁✄ , and then gives ☞✎✍✑✏✓✒ ❘ ➇ ❘✆☛ P ☞✎✏✾✍✁❍✒ ☛ ☞✎✏✠✂ ✡ ✣➇❫✒ ✎✠☛ ✎ P ⑥✐the✰✣mean ➂✱✏✮➉ value theorem ✍✑✏ ✕✥➇ for✕ integrals ✰ ✰sq✌ ✏✾✁ ✍ ⑥q✌✏✠✡✂✣➇ weq have ✏❑✡✄✣➇ ✙r✏✂✡✥➇ ❘✈☞❊➇❫✒✗for ☎✄ some ➇■ ☛ ☛ P ☞✎✏✂➈ ☎ ✡ ✣➇❴. ✒ Now, ✎ P ☞✎✏✤for ✆ ✍ ❍✒❨✽✣☞✎✏❤✡☎✣➇❴✒ ➉,☛★ ➇■ ☛ ☛ P ✽✣☞✎✏❤✡❲➇❴✒ and ➈ , which ✶ ✣ ✥ , so . vanishes as ✑

✆

✎

✑

✆

✑

✑

✆

✌

✝

✑

4. (a)✶✮ Note ✣ ✥ that if as . (b) Assume all ❚ Conversely, if

✄✖ ❶ ✰

✙

✱☛

✑

✆

✆

✆

then all

✱☛

are nonzero. Let

✝ ✙■❘ ❘ are positive. With as in (a), if ✺✍✓ ✱ ☛ ✄ ✙●❘✘✄ ●✢ ☞ ❚ P ✱ ☛ ✒✭✣ ✝ ❷ , then ✒

✁

43

✱ P ✱ ✟ ✧✗✧✗✧ ✱✈❘

✙✿❘✘✄

✒P

✂

✱☛ ✄ ✁

.

✙

✱✈❘✘✄ ●✙ ❘✠✽ ✙●❘ ✎ P ✣ ✙❀✽ ✳ ✙ ✄ ✏ ❚ ❘P ✺✍✓✝✱ ☛ ✄ ✺✍✓ ✙●✮ ❘ ✣ ✺✍✓ ✙ , then ✝ ✝ ✝ . ; then

Chapter 6. Infi nite Series

44

6.2 Series with Nonnegative Terms 1.

✱ ☛✁

✟

✒

✎✢✜ ✟

❚

; converges by comparison to

✎✢✜

✟ ✒

✟

.

2. Practically any test you can think of will work on✙ ✠ ☛ one! (Ratio test, root test, integral test using ❚ ✎ this Corollary 2.12, comparison to geometric series , ...) 3.

❚ ✎ ✟✒✜ . ✱ P✹✽ ✱ ☛ ✄ ☞ ✥ ✡ ✙✜✒❨✽✣☞ ✡✖✏✓✒ ✟ ✣ ✰ Converges by ratio test: ☛ . P✱ ✓✒ ☛ ✄ ☞ ✙ ✡✖✏✓✫✒ ✜✓✽✣☞ ✞ ✡✕✏✓✒ ✟ ✣✦✥ . Diverges by root test: ☛ ✱ ☛ P✹✽ ✱ ☛ ✄ ☞ ✙ ✡❲✞✜✒ ✟ ✽✠✞✫☞ ✙ ✡✕✏✓✒✻☞ ✙ ✡✥✙✜✒✰✣ P ✜ Converges by ratio test: ✱ P ✽ ✱ ☛ ✄ ☞ ✡✖✏✓✒❨✽✈✏✗✰ ✣✦✥ . Diverges by ratio test: ☛ P ❚ ✎ Diverges by comparison to , using (2.13). ✱ ☛ P ✽ ✱ ☛ ✄ ☞ ✙ ✡❲✞✜✒❨✽✣☞ ✞ ✡ ❂✜✰ ✒ ✣ ✟ ✜ . Converges by ratio test: ✟ ✱ P ✽ ✱ ✄♠☞ ✡✖✏✓✒ ✽✣☞ ✙ ✡✖✏✓✒✻☞ ✙ ✡❲✙✜✒✭✣ P ✱ ☛

✟✒✜

✎

✟

; diverges by comparison to

✟

☛

4.

✟

✟

✞

5.

✟

✟

✟

☛

6.

✟

✟

.

✟

☛

7. 8.

✟

✟

☛

9.

✟

10. Converges by ratio test: ☛

✱ 11. Diverges by ratio test: ☛

☛

☛

✟

✪ .

P ✽ ✱ ☛ ✄✚✞ ✣✽ ☞ ✡✕✏✓✒ ➉ ☛ ✣ ✞✜✽ ✂✓✙r✏ ➈ . (See Exercise 7 in

☛

✟

✟

✟

✟

✟

➇★✄

2.1, with ✂

✎ ✟

P .

The root test is a little easier if you know Stirling’s formula.)

P✓✒ ✱ ☛ ☛ 12. Converges by root test:

✽✣☞ ✡✖✏✓✒ ➉ ☛✓✣ ✓✏ ✽ ✂ ➇❢✄ ✎ P ➈ . (See Exercise 7 in 2.1, with .) ✟ ✏ ✍t✻✸ ✺✜✼✩☞✎✏✓✽ ✒ ➉ ✣ P ✟ ; series converges by comparison to ✎ ✟ opital’s rule or Taylor’s theorem, ➈ ❚ By l’Hˆ

13.

✄

✟

✟

✂

✟

.

✟

✼✄✔✖✕

15.

➈

✟

✟

✽✣☞ ✟ ✡✥✞✜✒ ➉

✟

✡✖✏▼✡ ✛

✒✛

✟

✽✣☞ ✟ ✡✥✞✜✒

✟

✟

✄✳✏✓✽✣☞ ✛

✱

14. By rationalizing the numerator, ☛ ❚ ✎ P . to

✏✓✽

✟

✟

✡✥✙

✏✓✽✠✙

✟

; series diverges by comparison ✟

❚

; series diverges by comparison to ✟

✟

P✓✒ ✱ ☛ ☛ ✄

16. ✏✓Converges ✒❨✽✠❂✮✕ ✴ by the extended root test (part (a) of Theorem 6.14): for large .

✎ ✟

P .

✟ ☛ ❝ ❃ ✡❋☞✎✍ ✏✓✒ ☛✾➉❊✽✠❂ q ➈

✟

✒

✄

✟ ✒ ☛❾☞♦❃❝✡ ✟

P ✽ ✱ ☛ ✄ ☞ ✙ ✡✥✏✓✒❨✽✣☞ ✙ ✡ ☎✒ ✏☛✍⑩☞✳✱ ☛ P✮✽ ✱ ☛ ✒ ➉❴✄✕✞ ✽✣☞ ✙ ✡ ☎✒✰✣ ✜✟ , so ☛➈ . P ✱ P ✽ ✱ ☛ ✄ ☞ ✙ ✡✥✙✜✒❨✽✣☞ ✙ ✡✥✞✜✒ ✏✯✍t✳☞ ✱ ☛ P✮✽ ✱ ☛ ✒ ➉❴✄ ✽✣☞ ✙ ✡✥✞✜✒✰✣ ✟ 18. Diverges by Raabe’s test: ☛ , so ☛➈ . ✱ ☛ ✣ ✰ ✱ q ✏ ✱ ✕ ✱ ☛ ✖✙ ✏ ❚ ✱☛ , so ✱ ☛ for large . For such we have ☛ for , so 19. If ✱ ❚ ☛ converges, then ❚ ☛ converges by comparison to . ➇❴✽✩➇■☞ ✍✺ ☛✓ ➇❫✒ ★✄ ☞ ✺✍✓☛➇❴✒ P ✎ ✾✽✣☞✎✏✦✍ ❴✒ ✌✄ ❶ ✏ ➇❴✽✩➇ ✺✍✓❑➇ ✄ ✺✍✓ ✺✍✓❁➇ for and ✌ , 20. Use ☞ the ✺✍✓❑integral ➇❴✒ P ✎ test: ☞✌ ➇ ✣✦✥ ✧ ✙✌✏ and remains bounded as precisely when . ➇❴✽✩➇■☞ ✍✺ ☛✓ ➇❫✒✻☞ ✺✍✓ ✺✍✓❁➇❴✒ ✄ ☞ ✺✍✓ ✺✍✓☛➇❴✒ P ✎ ✾✽✣☞✎✏✥✍ ❴✒ ✄❶ ✏ for and 21. Use➇❫✽✩the ➇●☞ ✍✺ integral ✓✂➇❴✒✻☞ ✍ ✺ ✓ test: ✺ ✓❑➇❴✒✇☞✄ ✌ ✍✺ ✓ ✍✺ ✓ ✍✺ ✓❑➇ ✍ ✄ ✺✍✓ ✺✍✓❁➇ ✌ by ✙⑨ the substitution , so as in the preceding ✏ ✟

✱

17. Converges by Raabe’s test: ☛

☛

☛

✟

✟

✟

✟

☛

✟

☛

✟

✟

✟

✟

✟

✆

✟

✟

✟

✆

✆

✝

✆

✝

✟

✟

✝

✝

✝

✝

✟

✆

✝

✆

✝

✝

✝

✝

✝

✝

✝

✝

✝

✝

✝

.

✟

✝

✁

✟

problem, the series converges if and only if

✆

✝

✝

✝

✟

✝

6.3. Absolute and Conditional Convergence 22.

45

❚ ✟P ✁✄✂ ✎ P ✏✓✽ ✺✍✓ ✤ ✟ P ✁✄✂ ➇❴✽✩➇ ✺✍✓☛➇ ✤ ❚ ✜P ✁✄✂ ✏✓✽ ✺✍✓ ✝ ✝ ✝ From Theorem . The integral is P ✄✂ 6.7, ✺✍✓ ✺✍☛✓ ➇ ✟ ✄ ✺✍✓ ✜✰●✡ ✍✺ ✓ ✺✍✓❀✏✗✰❑✍ ✺✍✓ ✺✍✓❤✙ ✣✄ ★ ★▼✴ ✏✓✽✈✏✗✰✩✪ ✺✍✓➃☞✎✏✗✩✰ ✪ ✒ ✣✄ ★ ★▼✴✮✕ , and is negligible, so ✝ P ✝ ✄✂ ✝ ✝ ✝ ✝ ✝ ✝ ❚ ✟ ✏✓✽ ✺✍✓ ✕ ✣✄ ★ ★▼✴♣✡ ☞✎✏✓✽✠✙ ✺✍✓❤✙✜✒ ❂ ✄ ❅✣✏▼✏ ❚ P ✄✂ ✏✓✽ ☞ ✺✍✓ ✒ ✟ P ✝ ✝ ✝ . Also from Theorem 6.7, ✒ ✟ ✰☎✄ ✏▼✏ P ✄✂ ➇❴✽✩➇■☞ ✍✺ ❑✓ ➇❫✒ ✄❉✍ ☞ ✍✺ ✓❑➇❴✒ ✎ ✒P ✄✂ ✄❉✏✓✽✁✜✰ ✺✍✓✦✏✗✰ ✟

✟

✌

✝

✟

✟

☛

✡

☛

✡

✝

✒

✟

☛

✟

✟

✝

✌

✝

☛

✝

✡

✟

. (The error in adding or removing the

initial term in the sum is negligible.) ✡

➇❴✽✣☞❊➇ ✟ ✡✪✏✓✒ ✟ ☞✎✏❄✍❢✞▲➇ ✟ ✒❨✽✣☞❊➇ ✟ ✡⑩✏✓✒ ✜ ➇ ✤✌✏✓✽☎✛ ✞ . By ➇ is➇❴✽✣☞❊➇ ✟ ✡♣✏✓✒ ✟ ✤ ❚ ✪ ,✽✣which ☞ ✟ ✡♣✏✓is✒ ✟ nonpositive for ✍✑✏✓✽✠✙✫☞❊➇ ✟ ✡♣ ✏✓✒ ✒Theorem ✄ ✄ ✰✜❂ ✒✜ ✟ ✕✏✄ ✰✜❂❀✡✥✞✜✽✣☞ ✞ ✟ ✡✌✏✓✒ ✟ ✒ ✄ ✄ ✰✯★ . The integral is , ✜ . Adding on the first two terms, namely .25 and ✟ ✡✕✏✓✒ ✟ ✕ ✄ ❬✏

23. The derivative ✽✣☞ ✟ of✡♣✏✓✒ ✟ ✤ ❚ 6.7,✄ ✰✜❂ ✜✕✒ ❚ ✽✣☞ ✟ ✡✕✏✓✒ ✜ so ✒ ✄ ✞ ★✓✕❈❚ P ✽✣☞ .08, gives ✒ ✆ ✄ ❚ ☛ P ✏✓✽✩✦ ✶ ✍ P ☛ ➇❴✽✩➇ ☛ P 24. ☛ ✟

✝

✟

✟

✟

✟

✡

✡

✟

✟

.

✟

☛ P❫✍ ✆ ☛ ✄✳✏✓✽✣☞ ✡◗✏✓✒➃✍ ✝ ✺✍✓➃☞ ✡◗✏✓✒✣✡ ✝ ✺✍✓ ✄ ✆ ➅ ✏✓✽✣☞ ✡✖✏✓✒■✍ ☛ ➇❴✽✩➇ ✕✌✏✓✽✣☞ ✡✖✏✓✒■✍ ☛ ➇❴✽✣☞ ✡✖✏✓✒☛✄✖✰ ☛ ☛ , so ➀ ☛ is decreasing. P✓✒ ✁❦✄ ☞✎✏✤✡ ✞✂✒❨✽✠✙ ✞ ✕ ✏ ✁ ✕ ✏ ✱ ☛ ☛ ✕ ✁ P✓✒ ✙ for✏ all but finitely many ; it follows . If then , and from Let ✞ ✙♠✏ ✱ ☛ ☛ ♠ ✱ ☛ ✙❖✏ ❚ ✱☛ Theorem 6.14a that converges.✱ If then for infinitely many and so for ❚ ☛ diverges. infinitely many ; it follows that ✝

✟

25.

✌

✝

✌

✌

✙ ✰

as in Theorem☛ 6.7. P Also, ✟

✝

✌

✆ ☛

✟

✟

✟

✟

✟

✟

✟

6.3 Absolute and Conditional Convergence 1. For (a) and (b), just use ❚ the ➇■ ☛ fact that ❚ and the convergent series

✔✸✻✺✜✼ ❾q♠✏ ✎ ✟ ❙✵

✔✼✄✔ ✕

✟

✡ ✥

❾q❖✏

❙✵ and for all . For (c), use the ratio test. ✟

✟

✟

to get a comparison with

, add up the positive terms until the sum exceeds 1; then put 2. To get a rearrangement whose sum is in one negative term; then add more positive terms until the sum exceeds 2; then put in another negative ✍✑✏ term, etc. Since the original series converges, only finitely ✡ ✥ . ✡⑨✏ many negative terms can be less than After they are all used up, once the sum exceeds ✢ it never drops below ✢ , so it tends to as more terms are added. 3.

✰✂✡ ✟P ✪ ✡ ✰❀P ✍ P ✡✇P ✰✂✡ P P ✡✥✧✗✧✗✧ ☞ ✙ ✦⑦✍ ✏✓✒ ☞ ✙ ✦❋P ✒ ✪ ✍ ✡ ✍⑩✧✗✧✗✧✮✒ , the th and th sums both coincide with For ✦ the series ☞ ✎ ☛ ✏ ✺✍✓✂partial ✙ ✟ ✟ ✟ P P ✜ ✝ the th partial sum of ; hence they converge to . When this series is added to ✏❬✍ ✟ ✡ ✍s✧✗✧✗✧✈✄ ✍✺ ✂✓ ✙ ✜ ✶✥✤ ✰ ☞ ✙ ✶✭✝ P ✡ ✙✜✒ , the odd-numbered (positive) terms of the☞ latter ✶❬✒ series survive unchanged, and for P P , the th terms of the two series cancel and the th terms add up to give the negative all ✍ ✍ ✍ terms ✟ , ✪ , , . . . . ✜ After omitting the resulting zero terms one obtains the stated rearranged series, ✺ ✓✂✙ ✍ ✟ . whose sum is therefore ✝ ❘ ❘ ✂✩❘✭✄ ❚ ✱ ☛ ❛✎❘✦✄ ❚ ✗ ☛ . Then ❛❨❘✭✄✟✂✩❘ if ✶ is odd, and ❛❨❘✭✄✟✂✩❘❁✧✍ ✱✈❘☞☛ P ✍✧✱☎❘ if ✶ is even. Since Let ✱☎❘ ✣ ✰ ✔ ✞ ✂✓❘✦✄ and✔✟✞❲❛✎❘ ✝ ,✝ . ✱ ☛ ❫q ✟P ❚ ✱☛ ✕ ✥ (a) Suppose ✺ ✓❴☞✎✏✤✺ ✍ ✡ ✱ ☛. ✒✗After ✞✕ ✄ throwing out finitely many terms we ✍✺ ✓➃may ☞✎✏✤✡ assume ✱ ☛ ✒✘✄ ✱ ☛ ✡ ✑s✳☞ ✱ ☛ for ✒ all , for all . By Taylor’s theorem, where in✑swhich case ✝ ✟ ✝ ✳☞ ✱ ☛ ✒✗✣q ♣ ✱ ☛ q ✉ ✱ ☛ ❚ ✍✺ ✓❾☞✎✏☛✡ ✱ ☛ ✒ ❚ ✱☛ ✝ , so is the sum of the two absolutely convergent series ✗ ✡ ✱☛✒ ✱ ✄ ☎ ✍ ✏ ❚ ✑s✳☞ ✱ ☛ ✒ ❚ ✗ ☛ ✏✕ ✥ ☛ ✄ ✍✺ ✓❾☞✎✏✂✗ and . Conversely, suppose have ☛ ✱ , and ✱ ☛ ✏ ✄ ✗ ☛ ✡ ✁❬☞ ✗ ☛ ✒ where✁❬☞ ✗ ☛ ✒✗✣q ✝ ♣ ✗ ☛ ✟ q .✉We ✗ ❚ ☛ , so ☛ is the sum of where Taylor’s theorem again gives ❚ ✗☛ ❚ ✁✫☞ ✗ ☛ ✒ ✓

✓

4.

5.

✟

✌✠

✟

✄

✄

✂

✞

✄

the two absolutely convergent series

and

.

✄

Chapter 6. Infi nite Series

46

❚ ✱ ☛ ❚ P ✟ test, but not absolutely convergent since (b) is convergent ✺✍by the alternating series ✓❾☞✎✏❝✡✷✱ ✒✭✄ ✱ ✍ ☛ ✟ ✱ ☛ ✡ s☞✳✱ ☛ ✒ whereP ✟ s☞✳✱ ☛ ✒✗❄q ♣ ✱ ☛ ✜❧✄ By Taylor’s theorem, s☞✳✱ ✒ ☛ ❚ ☛ is absolutely convergent; but ❚ ✟ ✱ ☛ ✄ ❚ ✏✓✽✠✙ diverges. converges as above;

✝

✑

✑

✄

✟

✄

✟

✎ ✓P ✒ ✟ ✄ ✥ ✢✎ ✜ ✒ ✟ ❚ ✱ ☛ . .

✑

✟

6.4 More Convergence Tests

➇♣✡ ❬ ✙ ✕ ✏ ✍❀✞✚✕✕➇ ✕♠✍✑✏ 1. By the the✄ series diverges for ✟ ✡✌ ➇♣✡⑤ ✙ ratio ❬ ✙❖✏ test, ➇✇ ✍✤✞ converges ➇✇✄ ✍✑✏ absolutely for ✽✣☞ ✟ ✡✌✏✓✒ ✏✓✒ ❚ ✎☞ ✍✑✏✓✒ ,☛✫i.e., ❚ ✏✓✽✣,☞ and . At or the series becomes or , both of ❚ ✏✓✽ ✟ which are absolutely convergent by comparison to . ✙▲➇★✍t✏✾ ✕ ✏ ✰✧✕✖➇ ✕ ✏ , and 2. By the ratio test absolutely for ☞✎✍ ✏✓✒ ☛ ✜ , i.e., ✜ ✙▲➇❢or ✍❈✏✾the✙ root ✏ test,➇◆the ✄✳✰ series ➇✇✄converges ✏ ❚ ❚ . At or the series becomes or , both of which diverges for ✜❧✣ ❶ ✰ diverge since . ➇ ✱ P ✽ ✱ ☛ ✜✄✖➇ ✟ ✽✣☞ ✙ ✡✥✞✜✰ ✒ ✣ ✰ ✣✦✥ as . 3. By the ratio test, the series converges absolutely for all : ☛ ➇❴✽✠❂❬ ✕ ✏ ✍✤✗ ❂ ✕ ➇ ✕ ❂ , i.e., , and diverges for 4. By ratio ➇❴✽✠❂❬the ✭✙❞ ✏ test, ➇✥the ✄❞❂ series converges ❂ ✽✣☞ ✡✳✏✓for ✒✟ ❚ ✙▼absolutely ❚ ✏✓✽ . At the series is , which diverges by comparison with . At ➇⑥✄ ✍✤❂ ❚ ✙▼❂ ☞✎✍✑✏✓✒ ☛ ✽✣☞ ✡✖✏✓✒ ✟ it is , which converges (conditionally) by the alternating series test. ➇❇✍ ➃ ✕ ✙ ✙ ✕⑦➇ ✕ ❅ ➇❇✍ ➃ ✙ ✙ P ✍ ✞✜✒ ✺✍for ,☛⑥i.e., diverges , by 5. The series converges absolutely ✱ ☛ P✮✽ ✱ for ❄✄ ❏ ❋ ➇ ✍ ➃♥☞ ✙ ✍ ✜ ✞ ✒ ✍✓❫☞ ✡✕✞✜✒❨,✽✣and ✺ ☞ ✙ ☛ ✓❫☞ ✡ ☎✒ ☛ P P . We have the ratio test. (In detail: ☞ ✙ ☛ ✍✕✞✜✒❨✽✣☞ ✙ ☛ ✍✖✞✜✒❋✄ ☞✎✏❧✍✕✞♣✧✫✙ ✎✠☛ ✒❨✽✣☞ ✙❢✍✌✞♣✧✣✙ ✎✠☛ ✍ ✒ ✣ ✺✍✓❴☞ ✡r✞✜✒❨✽ ✺✍✓❫☞ ✡ ☎✒ ➉☛✍✳✏ ✄ ✟ , and ➈ ✺✍✓ ☞ ✡★✞✜✒❨✽✣☞ ✡ ☎✒ ➉❊✽ ✺✍✓❴☞ ✡ ☎✒✔✣ ✰✾✽ ✥ ✄✖✰ ➇❢✄✕✙ ❚ ✙ ☛❬✽✣☞ ✙ ☛❤✍ ✞✜✒ ✺✍✓➃☞ ✡❦✞✜✒ ➈ ✡❢✞✜✒ ❚ ✏✓✽ .) At ➇❢✄✕❅ the series is ❚ ☞✎✍✑✏✓✒ ☛✈✙ ☛❬✽✣☞ ✙ ☛❑✍s✞✜✒ ✺✍✓❾☞ , which . At the series is , diverges by comparison to (for example) which converges (conditionally) by the alternating series test. ♥☞❊➇❑✍s✏✓✒❨✽✣☞❊➇✿✡❢✏✓✒✗ 6. By✕⑨ the ratio test or the root test, the series converges absolutely or diverges according as ✏ ✙✌✏ ➇ ✙ ✰ ➇ ✕⑤✰ ➇⑥✄✖✰ ❚ ☞✎✍✑✏✓✒ ☛✣✽ ✛ is or , i.e., or . At the series is , which converges (conditionally) by the alternating series test. P ➇❦✍ ✞❬✞✕ ✏ ✕❖➇✺✕ ★ P the ratio test, the series converges absolutely forP ✟ 7. By , i.e., , and diverges for ☞ ➇ ✍ ✞✜✒ ☛ ➇❢✍⑩✞❬✸✙✚✏ ✟ ✟ is clearly bigger than the denominator, so ➇⑥✄ ➇⑥. ✄ The ★ numerator of the coefficient of at or the terms of the series do not tend to zero and the series diverges. ➇❢✡r✏✾ ✕ ✏ ✍✤✙ ✕✳➇ ✕ ✰ , i.e., , and diverges for 8. By the ratio test, the series converges absolutely for ➇✿✡★✏✾✸✙✌✏ ➇❢✄✳✍❀✙ ➇⑥✄✖✰ ❚ ☞✎✍✑✏✓✒ ☛❬✽✣☞ ✞ ✡✉✙✜✒ . At both and the series is , which converges (conditionally) by the alternating series test. ✱ P ✽ ✱ ❫✄ ☞ ✙ ✡✖✞✜✒✗ ➇■❺✽✣☞ ✞ ✡t❂✜✒ ✣ ✟ ➇■ ✜ 9. We have ➇■☛ ✕ ✜ ☛ converges abso✜✟ by the ratio P ✽ ✱ ☛ the✄ series ➇✿ ✙ ✜ ➇❈✄ , so ✱ ☛ test, ☞ ❅ ✡✕ ✴✜✒❨✽✣☞ ❅ ✡✳ ✏✗✰✾✒ ✟ ✟ P . ✏✗At , we have , lutely for ✏✘✍✌☞ ❅ ✡✖✴✜and ✒❨✽✣☞ ❅ diverges ✡r✏✗✰✾✒ ➉❀for ✄ ✽✣☞ ❅ ✡r ✰✾✹ ✒ ✣ and ☛➈ , so the series✓P ✒ diverges by Raabe’s test. How✎ , and ever, test, ➇⑥ the ☞ ❅ ✡sby ✴✜✒❨✽✣the ☞ ❅ proof ✡⑥✏✗✰✾✒ of✕✌Raabe’s ✏ ✄r✍ th✜ term is comparable to ☞✎✍✑the ✏✓✒ ☛ terms decrease since ✟ , where there is an extra factor of . Hence, at , the series converges conditionally by the alternating series test. ✺✍✓ ✏ ✡ ☞✎✏✓✽ ✒ ➉♣✄ ☞✎✏✓✽ ✒✯✡ ✁ ☛ ✁ ☛ ✘q ✎ ✟ ❚ ☞✎✍✑✏✓✒ ☛❬✽ where . The series 10. By Taylor’s theorem, ➈ ❚ ☞✎✍ ✏✓✒ ☛ ✁ ☛ converges conditionally, while converges absolutely; hence the original series converges conditionally. ✟

✟

✟

✟

✟

☛

✟

✟

✟

✟

☛

✝

✟

✟

☛ ✪

✥ ✝

✟

✧

✟

✟

✟

✌

✝

☛

✝

✟

✝

✟

✧

✝

✟

✝

✟

✟

✟

✟

✝

✟

✟

✟

☛

✟

☛

✟

✟

✟

✓

✟

✟

✟

✟

✟

✓

✟

✟

✟

✟

✝

✄

✟

✟

✟

✟

6.5. Double Series; Products of Series

47

P ✺✍✓➃☞❊➇✇✡✑✠▼✒ ➇❴✽✩➇ ☛ ☛ P ➇❫✽ ✄ ✏✓✽ ✱ ☛ ✄ ✱ ☛ ✙ ☛ ☛ ✝ ✌ ✍✺ ✓➃. ☞❊➇❢Then ✌ 11. Let ✡ ✠▼✒❨✽✩➇ ➇✺✙♠✰ , so the series is not ✱ ☛ absolutely ✱ ✕ is decreasing for , and hence so is , and ☛ convergent. ✍✑✏✓✒ ☛ ✱ ✝ ☛ ✺✍❾✓ ☞ ✡ ✠▼✒❨✽ On✣ the ✰ other ❚ ☞✎hand, ✝ , so converges by the alternating series test. ✓P ✒ ☛✓✣ ✏✉✄✖ ❶ ✰ ☛ ☛

✝

✟

12.

✟

✟

, so the series diverges.

✟

13. By Taylor’s ✺✍❾✓ ☞✎✏✑ ✡ ✁ ✒⑥✄ theorem, ✁ ✡

☛

✝

☛

❚

by comparison to

✑

✼✄✔ ✕ ✎ P ✄

✝

✟

.

✟

✟

✟

☎✢

✑ ☛

✈q

❚ ☞✎✍✑✏✓✒ ☛ ✎ P ✂✩✽✠✙ comparison to

➈ ✄

✟

✁ ☛ . Hence, ✁ ☛ ✟

✄✄✂

✠

✒

✁

✢

❚

❄✍ ✙ ✏ ✡

✁ ☛

✞

✟

✟

❫q

✁ ☛

✎✢✜ ✄

➉✟ q

✁ ☛

✟

✎

✄

. Finally,

✡✖✏ ☛ ✄ ✂ ✍ ✙ ✟

✂❀✍ ✞

✟

✠

☞✎✍✑✏✓✒ ☛✈➇ ✟ ☛✣✽✣☞ ✙

✁ ☛ ✟

✟

✑ ✡

➂ ☛ ✠

✟

✂ ✟

✍

✁ ☛

✂✑

☛ ✄

❚ ☞✎✍✑✏✓✒ ☛ ☎✂ ☞ ✁ ✡✌✑ ✒ ✟

is absolutely convergent by

✟

❚

✟

✞

✠

✟

✺✍✓❴☞❨☞ ✡⑨✏✓✒❨✽ ✒ ☛ ✄

✄✄✂ ❙✏❝✍ ✙ ✏ ✡

☛ ☛ is ✎ conditionally convergent, and ✟ , so the original series is conditionally convergent. ✟

, so ✝

✟

✟

✟

✟

✟

where

✟

✟

✟

, and hence ✝ Hence the series converges absolutely

✄

P ✍ P ✎ ✟ ✡✖✁ ☛ ✟

✎

✺✍✓➃☞ ✼✙✔ ✕ ✎ P ✒♠✄

✟

✎

✄

✟

✟

✟

✍✲☞ ✙ ✒ ✎ P ✡

✟

✄

✞

✠

where

✟

❄✏✯✍ ✙ ✏ ✡

✒

✁

✏ ✡ ✁☛ ✁ ☛ ❲q where ✟ ✭q ✁ q ✎ ✪ ✠ ☛ ☛ .

P ✒❝✄

✎

P ✡

✟

✡✖✏ ☛ ✄

✞

✑

☛ ✎ ✟ where ✟

✺ ✓❴☞✎✏❤✡ ✍ P ✒☛✄✳✏❝✍⑤✝ ☞ ✙ ✒ ✎ 14. By✺✍❾ ✓ Taylor’s ☞✎✏✂✡ ✎ theorem, ✟

✟

✝

✡ ✏✓✒

➇✛✎ P ✼✄✔ ✕❤➇

✎

➇

for is an alternating 15. The power series ✒ ➇ ☞ ✡♠series ✏✓✒ (for any , since the th term to the th is ➇ ✟ ✽✣powers ☞ ✙ ✡ ✙✜of ✒✻☞ ✙ are ✡⑤✞✜all ✒ even), ➇■❾qand ✕❃ the absolute value of the ratio ✤ ✏ of the . For this is less than 1 when , so the terms decrease ➇ ✽✠✴ q ❃ ✽✠✴ ✄ ✙▼❅✣✏after the first . one. Hence the error is smaller than the first neglected term, namely ✟

✞ ✎ ✞ ✎☛ ✟

✟

✲✄ ✗

✟

✟

❚ ✱

✔✟✞✞✗

✗

✄ ❚ ✱ ☞ ✗ ✍✣✗✻✒●✡

✗

❚ ✱

✟

✗

✍✞✗

☛ . First Method: We have ☛ ☛ ☛ ☛ ☛ . Since ☛ ✝ 16. Let decreases ❚ ✱☛ converge Dirichlet’s test gives the convergence of to sums of ☞ ✗ ☛ the ✍ ✗✗partial ✒ ✄ ✱ to✡✌the ✧✗✧✗✧▼full ✡✘✱ sum, ❚ ❘0✱ ☛ and ☛ and ✗ ☛ ✄ ✗ ☛ ✍ ✗ ☛ ✎ P . Then by Lemma 6.23, ❘ ✟ ☛ . Second Method: Let ❘ ✗ ❘✮✣ ☞ ❚ ✱ ☛ ✒ ✗ ❚ ✱ ☛ ✗ ☛ ✄ ❘ ✗✮❘✤✡ ❚ P ☛ ✎ P✙✗ ☛ ❚ ☛✎ P✗☛ . We have , and the series is absolutely ✟ ✟ ✟ ✟ P ✗ ☛ ✈q ❚ ✗ ☛ ✾✄ ❚ ☞✎✍ ✗ ☛ ✒❁✄ s☞ ✗ ✍✥✗✗✒ ❚ ✟ ☛ ✎ convergent since . ✄

❚

✄

✎ ✩✱

✄

✄

✱

☛ follows from Dirichlet’s test. (Take ☛ 17. The convergence ✗ ✱ ☛ of ✙✌✏ ✱ ✣ ☛ , since ☛ relabel as .) Absolute convergence is guaranteed for ✆

✟

✎ ✟

✟

❚

➇✿ ☛

✰ ✆

in Theorem ✱ 6.25, ✈q then and hence ☛ . ✄

➇✿ ✕✌✏ ✄ ❶ (for ☞ ✙ ✶❝any ✡◗✏✓✵ ✒✔❃ ).

18. The series absolutely✄✕ to the geometric ➇❢ series when ➇⑥✄✳converges ✏ ❶ by✙ ✶✣comparison ❃ ✄ ✍✑✏ it converges☞✎for by Corollary 6.27. When it converges for ✵ When ✵ ✍ ✏✓✒ ☛ ✸✻✺✜✼ ✄✖✸✻✺✜✼ ☞ ✍⑩❃❄✒ ❙✵ ✵ by Corollary 6.27, since . ✟

✟

6.5 Double Series; Products of Series

☞✎✏❝✍✇➇❫✒ ✎ P ☞✎✏✤✍✇➇❴✒ ✎ P ✄ ☞ ❚ ➇ ☛ ✒✻☞ ❚ ➇ ✮ ✒✿✄ ❚ ❯ ❱ ☞ ❚ ☛ ☛ ❱✫❯ ✏✓✒ ➇ ❯ ✄ ✒ ✒ ✒ ✮ ☞✎✏◗✍ ➇❫✒ ✎ P ☞✎✏◗✍❞➇❫✒ ✎ ✟ ✄ ☞ ❚ ➇✢☛✫✒✻☞ ❚ ☞ ✦ ✡ ✏✓✒ ➇ ✮ ✒ ✄ ❚ ❯✮❱ (b) P ✒ ❚ ✟ ☞ ✌✦✡✖✏✓✒✻☞ ✌✘✡✥✙✜✒ ➇ ❯ ❚ ☛ ✒ ☞ ✦✝✡✖✏✓✒❁✄✳ ✏✂✡✥✙❝✡❈✧✗✧✗✧✠✡✖☞ ✌✘✡✖✏✓✒☛✒ ✄ since ✮ ☛ ❱✫❯

1. (a)

❚ ✮❯ ❱ ☞ ✌✦✡✖✏✓✒ ➇ ❯ . ✒ ☞ ❚ ☛ ☛ ❱✫❯ ☞ ✦ ✡ ✓✏ ✒❨✒ ➇ ❯ ✄ P ✮ ✟ ☞ ✌✭✡✕✏✓✒✻☞ ✌✦✡ ✙✜✒ .

Chapter 6. Infi nite Series

48 2.

☞❊➇❴✒ ❊☞ ➁❬✒✿✄ ☞ ❚ ➇ ☛ ✽ ✒✻☞ ❚ ➁ ✮ ✽✁✦ ✒☛✄ ❚ ❯✮❱ ❚ ☛ ☛ ❱✫❯ ➇ ☛ ➁ ✮ ✽ ✦ ❵ ✒ ✒ ✒ ❚ ☛ ☛ ❱✫❯ ➇ ☛ ➁ ✮ ✽ ✦ ✄ ❚ ❯☛ ❱ ➇ ☛ ➁ ❯ ✎✠☛ ✽ ☞ ✌❤✍ ✒ ✄ ✮ ☞❊➇✯✡⑥➁✫✒ ❯ ✽ ✌ , so ❵ ✮ ✡◗➁❬✒ ☞❊➇❧ ❵ . ☞ ☛✍✌❜☞❊➇❫✒✝✄ ☞❊➇❴✒⑦✄ ☞✎✏❇✍✓▼➇❴✒ ✎ P✓✒ ✟ ☞✎✍ ✟P ✒✻☞✎✍ ✜✟ ✒➃✧✗✧✗✧▲☞✎✍ ✡ If ❵ ☞ P✓✒ ✟ ✌ ❵ ✏❾✧ ✞✂✧✗✧✗✧▼☞ ✙ ✍⑥✏✓✒✎✙ ☛ ☞✎✏❴✍ ▼➇❫✒ ✎✠,☛ ✎ then ✏❾✧ ✞✂✧✗✧✗✧▲☞ ✙ ✍⑥✏✓✒❁✄ ☞ ✙ ✒ . Moreover, ➇ ☛ ☞ ✙ ✒ ✽✣☞ ✒ ✟ ❵

☞❊➇❴✒ ❊☞ ➁❬✒✿✄ ❚ ❯ ❱ ☞❊➇❝✡❢➁❬✒ ❯ ✽ ✌ ✄ ❵ ✒ P ✍ ▼➇❫✒ ✎✠☛ ✎ ☞ P✓✒ ✟ ✌❖✄ ✟ ✒✻☞✎✍ ☎✒ ☛❫☞✎✏❇✑ ✽✠✙✬✔✧ ✯✧✗✧✗✧✜☞ ✙ ✒❁✄ ☞ ✙ ✒ ✽✠✙ ☛ ➇✿✏✕ . ItP ✪ in the Taylor series is , so that series converges✆ for follows that the coefficient of ❚ ❯➇❯ by the ratio test. The product of ❯ this series with itself is the series where ✆❯ ✄ ❚ ☞ ✙ ✒ ☞ ✙ ✦❋✒ ✽✣☞ ✦ ✒ ✟ ✄ ❚ ☛ ❱ ☞ ✙ ✒ ☞ ✙ ✌❢✍t✙ ✒ ✽ ☞ ✌❢✍ ✒ ➉ ✟ ☛ . On the other hand, the ➈ ✮ ☛ ❱✫❯ ☞ ▼➇❫✒ ❯ ☞✎✏❁✍ ▼➇❴✒ ✎ P ❚ sum of the latter series is , whose Taylor series is the geometric series ❯✮✒ ❱ . Equating ❯ ➇❯ ✆ ❯ ✄ ❘ coefficients of in these two series gives the P asserted formula . (The justification for ✆❘ ☛ P last ♥☞✎✏✯✍ ▼➇❴✒ ✎ ✍ ❚ ✆ ❯ ➇ ❯ ✜✄♠ ➇✿ ☞❘ ☛ P ❚ ❘✆☛ P ✆ ❯ ➇ ❯ ✎ ❘ ✎ P ✣q ✉ ➇✿ the for step is P contained in❘ Theorem 2.77: ✒ ➇■✣q ✶ ☞✎✏❝✍ ▼➇❫✒ ✎ P ❚ ✆❯➇ ❯ , say, so is the th Taylor polynomial of . See also Corollary 7.22.) ❚ ☞✎✍✑✏✓✒ ☛ ☞ ✡✚✏✓✒ ✎ ✓P ✒ ✟ is convergent by the alternating but not absolutely convergent. The ✒ ✆❯ ✄ ❚ ❯ ☞ ✡✌✏✓✒✻☞ ✌ ✍ ✡✕✏✓✒ ➉ ✎ ✓P ✒ ✟ ❚ ❯ ❱ ☞✎✍✑✏✓✒ ❯ ✆ ❯ series test, ❱ ☛ where . Cauchy product of the series with itself is ➈ ✒ P ✌✦✡❈✏✓✒ ✎ P ✄ ☞ ✌✦✡✖✏✓✒❨✽✣☞ P ✌✦✡❈✏✓✒✘✣ ❶ ✰ ✌ ✣✦✥ ❯ ✆ ❯ ✤t❚ ❯ ☞ ❚ ✎ ☞ ✍ ✓ ✏ ✒ ✆❯ ✟ ☛ ❱ ✟ By the hint, as ; hence diverges. ✄ ❚ ☛❱ ✱ ☛ ✁❘ ✦ , and for each , let Let ❷ ✄ ✒ ✮ , which we think of as the limit of the square partial sums ✮ ✱ ✄ ❚ ☛❱ ❚ ❷ ✮ ✮ ☛ . The claim is that ❷ ✒ ✒ ❷ ✮ , whether these quantities are finite or not. ✕✺✥ ✂ q ❚ q ❚ ✙⑤✰ ❚ ❱ given Case I: ❷ ✮ ❚ . Clearly ❷ ✮ . On the other ✙ ❚ ✍ ✮ ❷ ✮ , so ❷ ✱ ☛ ✙ ✍ ☞ , we ✡⑥✏✓can ✒✎ P ❚ ☛ ❱ hand, ❷ ✮ ✮ ❷ ✮ find such that . We can then find such that ✒ ❷ ✮ ✦ ✄r✰✣✆➂ ✄✆✄✆✓✄ ➂ ✄ ✞s✓✶ ✒❙☞ ➂ ✒ ✂ ✤ ❚ ✍❲✙ ❚ ☛❱ ✱ ☛ ✤ ❚ ✮ for . Let ✢ ; then . It follows ✒ ❷ ✮ ✮ ❱ ✤ ❚ ✄ ❚ ❷ ✮ and hence ❷ ❷ ✮ . that ❷ ✕✴✥ ✦ ❚ ✄ ✥ ✙t✰ ❚ ✙ ✮ ✮ for each , but . Given , we can find such that Case II: ❷ ✮ ❷ ❷ P ✍ ☞ ✡★✏✓✒ ✎ ✦ ✄✖✰✣➂✆✄✆✄✆✩✄ ➂ ✄ ✞s✓✶ ✒❄☞ ➂ ✒ . ✡ ❚ ☛❱ ✱ ☛ ✙ We can then find such that . Let ✢ ; ❷ ✮ ✤ ❚ ✄ ✥for ❚ ☛ ❱ ✱ ☛ ✤ ✖✍⑤✮ ✏ ❱ ✮ then . It follows that ❷ . ✮ ✦ ✄ ✦ ✙♠✰ ❚ ✱ ☛ ✙ ✄ ✥ for some✙ . Given . Let Case III: ✮ ✄ ✞s ✡ ✓✶ ✒❄❷ ☞ ✦ ✮ ➂ ✒ ✄ ✥ , we can find such that ✢ ; then . It follows that ❷ . ✱ ✬✕ ✥ ✙ ✰ ❚ , it is approximated by its square partial sums, so given we have Since ✮✒ ☛ ❱ ✱ ✮ ☛ ✘✕ ❚ ☞ ☛ for ✢ sufficiently large. This is a double series (it’s obtained from ✱ ✞s✓ ✶ ✒❙☞ ✦✇➂ ✒❢q ❚ ✄✆☎✞☛ ✝ ❱ ✮ ☛✍✱ ✌ ☛ ✮ ✮ by replacing ✮ ☛ by 0 when ✢ ), so by Exercise 5, it can be summed ✮✒ ✦ ✦ ✦ q , then in . Summing over only those such that✱ gives as an iterated series first in ✢ ✦ ✄ ❚ ❱ ❚ ☛ ❱ ☛ P ✱ ☛ ✓✕ ❚ ☛❱ ☛ ✮ ✮ , that is, the sum of the tail ends of the series over ✒ ✒ ✮ ✰✣✆➂ ✄✆✄✆✩✄ ➂ ✁ ✍ ❚ ❱ ❷ ✮ ✕ . But also ✢ is less than . With notation as in Exercise 5, this implies that ✮ ✍ ✟ ✈q ❚ ✱ ☛ ✸✕ ✍ ❚ ❱ ✸✕⑤✙ ✄ ❚ ❱ ✄✆☎✞✝ ☞ ✮ ☛✍✌ ✮ , so ❷ . It follows that ❷ ❷ ❷ ✮ ❷ ✮ . ✮ ✮✒ ❘ ❘ ❘ P ✟❘✠ ✄ ❚ ❯✮❱ P ❚ ✙t✮ ✙ ☛ ☛ ❱✫❯ ☞ ✦ ✡ ✒ ✎ ✄ ❚ ❯✮❱ P ✌❧✧ ✌ ✎ ✄ ❚ ❯ ❱ P ✌ ✎ . This has a finite limit as ✶ ✣ ✥ if ✟

✟

3.

✎

✎

✎

✎

✟

✟

✎

✟

✎

✎

✎

. But by the binomial theorem,

✎

✎

✟

✟

✟

✟

✟

✟

✎

✎

✟

✎

✎

✟

✎

✎

✟

✟

✎

✟

✟

✎

✟

✎

✎

✎

✟

✎

✟

✎

✄

✞

4.

✟

✟

✟

✂

5.

✆

✂

☎

✞

✁

✁

☎

☎

✆

✞

✂

✁

✁

☎

✆

☛

✄

✄

✁

☎

✞

✁

✁

✁

✆

✆

✞

✂

✄

☎

✞

☛

✄

✄

✂

✆

✂

✄

✆

☛

6.

✆

✆

✆

☛

✟

✟

☛

✂

☛

☛

✂

✆

☛

☛

☛

☛

✂

7.

✆

✆

✆

✟

✟

and only if

.

❚

✱

✡❧✏

✦

✄

✍✑✏

✦

✄

✡✕✏

❚

✄✕✰

✱

☛ ✮ ☛ are 8. The only nonzero terms in ✮✒ ❱ at ✱ ☛ at✡❧✏ ✄ ✦ , so ✍✑✮✒ ✏ ❱ ✮ ✄ ✦ ✍ for✏ ❚ ☛ ❱ and terms in ✒ ✮ are ✦ at and ✦ ✄⑨ at ✰ , all . On the other hand, the only nonzero ✦ ✄✚✰ ✙✖✰ and the latter term is missing when . Hence the sum is 0 for and 1 for . It follows that the sum of the first iterated series is 0, whereas the sum of the second one is 1. ✟

✟

✟

✟

✟

Chapter 7

Functions Defi ned by Series and Integrals 7.1 Sequences and Series of Functions

❘✈☞❊➇❫✒❁✄✖✰

✔✟✞

✰sq✥➇ ✕⑨✏

✔✟✞

❘✈☞❊➇❴✒❙✍❦✰❍✣q⑨☞✎✏✂✍ ✒ ❘ ✣ ✰

❘☎☞✎✏✓✒❑✄✳✏

➇❋✐ ✰✣➂✱✏✂✍ ➉

❵ ❵ ✵ ➈ ✵ , 1. (a) ✝ if ,✝ . We have ❵ for so the convergence is uniform there. ❘✈☞❊➇❴✒✿✍⑤✏✾✾✄ ✏✤✍ P✓✒ ❘ ✣ ✰ ➇◆✐ ➂✱✏✮➉ ✔✟✞ ❘☎☞♦✰✾✒✂✄✕✰ ✔✟✞ ❘☎☞❊➇❫✒☛✄♠✏ ✰ ✕⑤➇❇q✚✏ ✵ ➈ ✵ , so ❵ ❵ ,✝ if . We have ❵ for (b) ✝ the convergence is uniform there. P ✔✟✞ ❘☎☞❊➇❫✒☛✄✌✰ ➇✇✐ P ✰✣➂❜❃❾➉ ✟ ❃●➅ ✔✟✞ ❘✈☞ ✟P ❃❄✒☛✄ ✏ ❘✣☞❊➇❴✒■✍✪✰❍❬q⑤✼✄✔✖✕ ❘ ☞ ✟P ❃★✍ ✒ ✣ ✰ P . We have ❵ ✵ for (c) ➇❇✐ ✝ ✰✣➂ ❵ ✟ ❃⑥✍ ➉ if➇❇✐ ✟ ➈ ❃✉✡ ➆❀➂✱✏✮➀ ➉ , ✝ ❵ ➈ ❘✈☞❊➇❫✒✗✣q✷✵ ✶ ✎ orP ➈ ➇ ✵ ❘ ✣ , so✰ the convergence❥ is uniform there. (d) ❵ ✔✟✞ ❘☎☞❊➇❫✒✭✄✝✰ for all ➇ , so ✐ ❵ ✰✣➂ ✥ ✒ uniformly on . ❘ ✯✎ P ➇ ✄ ✶ ✎ P (e) ✝ ❵ for all ➈ , but the ❘✣maximum (at ✶✬✤ ☞❊➇❴✒❤✍ ✰❍☛q of ✶ ❵ on ➇ ✤ is ✎ ❘✁ this interval ✎ P ), so ✁ is not uniform. However, ❵ ❘ ✵ ✵ ✵ the convergence for provided , and ✔✟✞ ❘ ✶ ✩✎ ✄✖✰ ➂✥ ✒ ✝ ✵ ; hence the convergence is uniform on ➈ ✵ ➇◗✐ ✰✣➂ ✥ ✒ ➇✪✄ ✶ ✔✟✞ ✒ ❘☎☞❊➇❴✒❀✄ ✰ ❘ . ✎ P ➈ ❵ ❵ (f) ✝ for all , but ❘✈the maximum of on this interval is (at ), so the ☞❊➇❴✒❁✍ ✰❍❾q ✗✗✽✩✶ ➇⑩q ✗ for , so the convergence is uniform on convergence is not uniform. However, ❵ ✰✣➂✙✹✗ ➉ ✗ ➈ ✔ ✞ for ❘☎any . ☞❊➇❫✒❁✄✖✰ ➇❲✄r ❶ ✏ ❘☎☞❊➇❴✒☞✥ ✕ ➇ ❘ ➇ ✕✌✏ ❘☎☞❊➇❫✒ ✕✥✛➇ ✎ ❘ ➇ ✙⑨✏ ✔✟✞ ❘☎☞✎✏✓✒❁✄ ✟P ❘ since for and for , and (g) ✝ ❵ ✙♠✰ for ❵ ❵ ✝ ❵ ❘✈☞❊➇❴✒✂✍✥✰❍❁q❞☞✎✏✘✍ ✒ ➇✖✐ ✰✣➂✱✏✑✍ ➉ ❘✣☞❊➇❴✒✂✍✥✰❍■q ☞✎✏✤✡ ✒ ✎ ❘ . ✵ for ➈ ✵ and ❵ ✵ For for ➇❇✐ any ✏☛✡ ✵ ➂ ✥⑤✒ we have ❵ , so the convergence is uniform on these intervals. ➈ ✵ ✂

✂

✂

✂

✂

➇ ✙✖✰

✏✓✽✣☞✎✏❀✍

✾✒

✎

2. (a) The series is a geometric convergent to the sum ✄ ✩✎✠☛ . The convergence is ➂ ✥ series, ✒ ✙✖✰ for ☛ for any , by the M-test with . The sum is continuous absolute and uniform on ➈ ✵ ✵ ☞♦✰✣➂ ✥⑤✒ on . ✟ ✙⑤✰ ✍✑✏✠➂✱✏✮➉ ☛ ✄✳✏✓✽ ➈ (b) The series is absolutely and uniformly✣ ❶ convergent on by the M-test with ( ); ✰ ✍✑✏✠➂✱✏✮➉ . The sum is continuous on ➈ . it diverges elsewhere since the th term ✙✚✰ P is☞✎✏✯absolutely P ✒ ☛ ❚ and uniformly convergent on ➈ ✍✤✙✦✡ ✵ ➂❴✙ ✍ ✵ ➉ for ➇■any ✵ (c) The series by the M-test ✄ ✍ ✣✤⑤✙ ☛ ☛ converges by the ratio test). It diverges for ✟ ( since the th term with ✵ ✣❶ ✰ ☞✎✍✤✙✈➂✹✙✜✒ . The sum is continuous on . ❥ ✄❖✏✓✽ ✜ ; the sum is (d) The series is absolutely and uniformly convergent on by the M-test with ☛ everywhere continuous. ❥ ✄⑦✏✓✽ ✟ (e) The series is absolutely and uniformly convergent on by the M-test with ☛ ; the sum is everywhere continuous. ✂

☎

✂

✁

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49

Chapter 7. Functions Defi ned by Series and Integrals

50

✏✤✡ ➂ ✥ ✒ ✙❖✰ P ➈ ✵ ✵ (f) The series is absoutely and uniformly convergent on for any by the M-test with ☛ ✄ ✎ ✎ , and it diverges for ➇❇q✌✏ (Theorem 6.9). The sum is continuous on ☞✎✏✠➂ ✥ ✒ . ☞❊➇❴✒✗ ✰✣➂✱✏✮➉ ✙✝✰ ✙❖✰ ☞❊➇❫✒✗✰✕ ❘ ❘ be➇✥the value of on ➈ . Given for Let ✏✦✍ q✚ q maximum ✏ ✶ ☞✎✏✦✍ ✒ ✕ ,✽ choose ✵ ❘✣so☞❊➇❴that ✒✗❄q ➇ ✕ ✵ ✵ ❵ . Then if is large enough so that we have ➇ q ✏❀✍ ❘✈☞❊➇❫✒✗❙q ☞❊➇❫✒✗✡✕ ➇ ✙ ✏✭✍ ✘ ❘✈☞❊➇❴✒✗✡✕ ➇ ✐ ✰✣➂✱✏✮➉ ✶ for for ✰✣➂✱✏✮➉ for all when is ✵ and ❵ ✵ . That is, ❵ ➈ ❘ ✣ ✰ ✬ sufficiently large, so ❵ uniformly on ➈ . ✙r✰ ✠P✑✄ ✍✑✏✯✡ ➂✱✏✘✍ ➉ ✶ ✤ ✙ ✱❘♣✄ ✶ ✍❈✏✯✡ ➂ ✶✉✍ ➉ ✶ P ✵ ➈ ✵ ✵ ➈ ✵ ✵ Given , let , and for let . For a given ✟ ✟ ✟ ✄ ✞s✶✓✒ ✁ ➇ ✍ ✎ ✕ ✥ ✽ ✎ ✣ ✏ ✣ ✥ ✕ , ✥ let ❚ ☛ ✏ ☛ ☛ ☛ ✂ . Then for all , and . ➇❋✐ ✜❘ as ✍❝➇❋✐ ✗, ❘ so or . The M-test therefore gives the uniform convergence of the series for ✏✓✽✣☞❊➇ ✟ ✡ ✒ ❚ ✏✓✽ The series fails to converge absolutely by comparison to . However, decreases to 0 ✣ ✥ ➇ ➇ for each , so✶ by the alternating series test, the series converges , and as ✏✓✽✣☞❊➇ ✟ ✺ ✡ for ❧ ✶ each ✡⑨✏✓✒✲q✝ ✏✓✽✣☞ ✶ the✡✚absolute ✏✓✒ sum and the full sum is at most . The difference between the th partial ➇ ✶✮✣ ✥ latter quantity is independent of and tends to zero as , so the convergence is uniform. ❚ ☛ ☛ ❙q , so for ➇✥✐ ➈ ✍ ✱❍➂✲✾✱ ➉ (✱ ✕⑦✏ ) we have ☛ ➇ ☛ ✽✣☞✎✏✘✍⑩➇ ☛ ✒✗❄q (a)✱ ☛✣Since we ❚ ➇☛ ☛✫✽✣have ✢ ☞✎✏❝✍✇➇ ☛✣✒ ✽✣☞✎✏❝✍ ✱✣✒ converges ✍ ✱➃✲➂ ✜✱ ➉ . Hence converges absolutely and uniformly on ➈ by the M-test. ➇ ☛ ✽✣☞✎✏✘✍◗➇ ☛ ✒✦✄ ❚ ✽✣☞✎✏✘✍◗➇ ☛ ✒☛✍ ❚ ❚ ☛ ☛ ☛ (b) By the hint, . The first series on the right converges ➇✿ ✤ ✗ ✙ ✏ ☛ ✽✣☞✎✏❧✍t✢➇ ☛✫✒✗✭q ✭✽✣☞ ✗ ☛★✍✳✏✓✒ , and by the M-text, since absolutely and uniformly for ➇ ❚ ✏✓✽✣☞ ✗ ☛❧✍⑤✏✓✒ ❚ ✏✓✽ ✗ ☛ converges by comparison to . The second one is independent of , so it does not affect uniform convergence, but it decides the issue of absolute convergence. ❘✈☞❊➇❴✒✜✍ ☞❊➇❫✒✗✫q ➇❇✐ ✶ ✣✦✥ ❘ ✣ ❘✮ ❘✮ ✣ ✰ If ❵ ❵ ❷ ✮ , where ❵ uniformly on ❷ ✮ , weP have ❵ for as P ❘☎☞❊➇❫✒■✍ ☞❊➇❴✒✗✈q✬✞s✓✶ ✒❄☞ ❘ ✆➂ ✄✆✄✆✱✄ ➂ ❘ ✒ ➇❇✐ ✞s✶✓✒❙☞ ❘ ✆ ➂ ✄✆✄✆✓✄ ➂ ❘ ✔ ✒ ✣ ✰ ✶ ✣✦✥ . But P ✑ ❷ ✮ , and for as . then ❵ ❵ ❘✾➅ ✱❍✙➂ ✗✹➉ is uniformly But By Theorem✳☞ ✱➃7.7, ➂ ✗✗ ✒ the point is✙✖to ✙ ✰ show that ➀✩❵ ☞❊➇❫✒✿Cauchy ✍ ❘☎☞❊➇❴✒✗on ✢✕ ➈ . ➇✪ ✐◗✳☞ it✱➃✙➂ is✗✗✒ uniformly ❯ ❵ , so given❘ there exists ✢ ✱➃such that ❵ whenever ✌▼Cauchy ➂ ✶ ✙ on ✙➂ ✗✹➉ ➇ for ✣ ✱➃✙➂ ✗ are continuous on , we can take the limit as to conclude ✢ . Since ❵ ❯ and ❵ ➈ ➇❇✐ ✱➃✙➂ ✗❜➉ ✌▼➂ ✶ ✙ ❘▼➅ ✱➃✙➂ ✗✹that ➉ ❯❵ ☞❊➇❫✒■✍ ❵ ❘✈☞❊➇❫✒✗✣q ➈ ✢ , and hence ➀✩❵ for whenever is uniformly Cauchy on ➈ . ✍ ❘ ❘ ❘ ❵ is continuous, Since ❵ the ✕ set✧✗✧✗❷ ✧ in the hint is closed and hence❘ compact. Moreover, since ❵ ❘ P P increases ❷ ✟ ➇ to ❵ , we have ❷ ☞❊➇❫. ✒✿By ✍ Exercise ❘☎☞❊➇❫✒ ✤ 5 in ✂ 1.6,✶ if each ❷ is nonempty ✔✟then ✞ ❘✈so ☞❊➇❴is ✒✯✄ ✒ ☞❊❷ ➇❴✒ . But if is in the latter set we have ❵ ❘ for all , which . ✶✚❵ ✙ ☞❊➇❴✒➃is ✍ false ❘✾☞❊➇❫✒✗since ✸✕ ✝ ❵ ➇❋✐ ❵ ❷ (and hence every ❷ with ✢ ) is empty, i.e., ❵ ❵ for all ❷ when Hence some ✶✥✤ ❘ ✣ ✬ ✢ . Thus ❵ ❵ uniformly on ❷ . ✁

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3.

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✂

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6.

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✝

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✆☎

☎

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✠

☛

7.2 Integrals and Derivatives of Sequences and Series ❥ ✄ ✎ ✟ 1. The series defining ❵ converges absolutely and uniformly on ✒ ✟ by the M-test with ☛ ✒ ✟ . Hence ☞❊➇❴✒ ➇◗✄ ❚ P ✎ ✟ ✼✙✔ ✕ ➇ ➇◗✄ integration is permissible: ❵ is continuous, and✒ termwise ❵ ✒ P ✸✻✺✜✼ ❃ ☞✎✍✑✏✓✒ ☛ ✒ ✟ ❚ P ✍ ✎✢✜ ✸✻✺✜✼ ➇❍➉ ✟ ✟ . Now, is 0 when and✙✈➂✹❅✈➂✱✏✗✰✣➂✆✄✆✄✆✄ when ✰ is even;✄ hence ✒ ➈ ✎✢✜ ✙ ✎✢✜ is odd ✄✕ ✫➂ ★✈➂✱✏✱the ✙✈➂✆✄✆✄✆th ✄ when is odd, when , and when . term of the last series is ✟ ✰✣➂ ✥ ✒ ☛ ✄ ✎ . P P 2. The series defining ❵ converges absolutely ☞❊➇❫✒ ➇❋and ✄ ❚ uniformly ☞❊➇s✡ on ➈✒ ✎ ✟ ➇❋by ✄ ❚ theP M-test ☞ ✡✌✏✓✒ ✎ P ➉ ✎ P ✍twith P ❵ . This Hence ❵ is continuous there, and ✒ ➈ ✏❝✍t☞ ✒ ✡✖✏✓✒ ✎ P is a telescoping series; the th partial sum is , so the full sum is 1. ✁

✟

✆

✆

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7.3. Power Series

51

✉❘ ✂ ✶✠✷✹✸ ❜✶ ✕✬✶✈➇ ✟P ❃ ➇ ✙ ✰ ✍ ✟P ❃ ➇ ✕ ✰ ✰ ➇❢✄✕✰ is if , if , and if . ☞❊➇❴✒✯✄ ✶✠✷✹✸ ❜✶ ✒ ✕ ✶✈➇✉✡✘✈✶ ➇❫✽✣☞✎✏❝✡✘✶ ✟ ➇ ✟ ✒ ➇t✄r ❶ ✰ ✏✓✽✩✶ ❘ P P . If the second term tends to zero like (b) We have ❵ ✔✟♣ ✞ ❘ ✂ ☞❊➇❫✒ ✟ ❃ ✰ ✶ ✣ ✥ ➇⑤✄❞✰ ✍ ✟❃ ➇✴✙⑦as✰ ❘ ❵✔ is➇✕✄ , ✰ , or it vanishes to begin with. Hence ✝ for , ➇✖✄ ✰ , and ➇ ✕ if ✰ ✒ ✔✟✞

3. (a) Just observe that ✝

✂

✂

, or discontinuous.

respectively. The convergence cannot be uniform near

because the limit is

4. In each case it is a matter of using the M-test to establish the uniform convergence of the derived series on compact subsets of the interval of convergence. In what follows we write down the derived series and the constants ☛ in the -test. ✁

✁

✄ ✍ ❚ ➇ ✤ ✎ ☛ ✎✠☛ ✵. ; ☛ for P ➇✿✣q✌✏❝✍ ❚ ✢➇ ☛ ✎ ✽✣☞ ✟ ✡ ✡✖✏✓✒ ☛ ✄ ☞✎✏❝✍ ✒ ☛ ✎ P ; for (b) ✵ P ✵. P ☛ P ❚ ✟ ➇✢☛ ✎ ✽✠✙ ☛ ✜ ☛ ✄ ✟ ☞✎✏❝✍ ✟ ✒ ☛ ✎ ➇✿✈q⑤✙✘✍ ; for (c) ✵ ✵. ✍ ❚ ☞♦✄✼ ✔ ✕ ➇❫✒❨✽ ✟ ☛ ✄ ✏✓✽ ✟ (d) ; . ➇✿✣q ❚ ☞✎✍✤✙▲➇❴✒❨✽✣☞❊➇ ✟ ✡ ✟ ✒ ✟ ☛ ✄✕✙ ✽ ✪ ; for (e) ✢ . ✢ P ✍ ❚ ✎ ✍✺ ✓ ✄ ✍✓ ✺ ✿ ➇ ☎✤ ✕ ❑ ✏ ✡ ✎ ✎ ✝ ✝ ✵. (f) ; ☛ for ✙▲➇❫✽✣☞❊➇ ✟ ✍ ✟ ✒✇✄ ☞❊➇ ✍ ✒ ✎ P ✡ ☞❊➇✇✡ ✒ ✎ P , we see that the derived seUsing the observation that ✍ ❚ P ☞❊➇❇✍ ✒ ✎ ✟ ✡✳☞❊➇❢✡ ✒ ✎ ✟ ➉ . This series converges uniformly on compact subsets ries ❥ of❆❧❵ ✏✠➂ is❆✘✙✈✆➂ ✄✆✄✆❪✄ ✒➅ ➈ ❚ ✎ ✟ of ➆✦➀ by the same argument as in Exercise 4, ✂ 7.1 (basically, a comparison to ), so ✂

✂

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(a)

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✁

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✟

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✁

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5.

✟

✟

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the termwise differentiation is justified.

6.

✰ ✰sq✷✶ ☞ ✶✈➇❫✒✂q ✶❫☞ ✶✈➇❫✒ ✎ P ✎ q ✎ P✎ ✶ ✎ ✣ ✰ ✶ ✣ ✥ ,P as . (a) For ✵ ❵ ✵ ❘ ➁s✄ ✶✈➇ ✶ ☞ ✶✈➇❫✒ ➇★✄ ☞❊➁❬✒ ➁ ✣ ☞❊➁❬✒ ➁✉✄✴✱ ✒ ❵ , . (b) With ❵ ❵ P ❘✣☞❊➇❴✒ ☞❊➇❫✒ ➇ ✙♠✰ ✙♠✰ ☞❊➇❫✒❤✍ ☞♦✰✾✒✗✔✕ ✽✠✩✞ ✱ ✰✇q ➇tq (c) Given , pick ✵ so that whenP ✵ . Then ❵ ❘✈☞❊➇❴✒ ☞❊➇❫✒✂✍ ☞♦✰✾✒ ➉ ➇ ❘☎☞❊➇❫✒ ☞❊➇❴✒✂✍ ☞♦✰✾✒ ➉ ➇ ☞♦✰✾✒ P ❘☎☞❊➇❴✒ ➇ ➈ ✶ ❵ ➈ ❵ ✽✠✞ ✱ , ☞♦✰✾✒ ❵ is the sum of , and . By part (b), is suffuciently large. The absolute value of the second the first term is within ☞ ✽✠✩✞ ✱✣✒ of❘☎☞❊➇❫✒ ➇❖provided q ✽✠✞ . Finally, sinceP integrable functions are bounded we have term is at most ❵ ☞❊➇❴✒❁✍ ☞♦✰✾✒✗✫q ❘☎☞❊➇❴✒ ➇ ✽✠✞ ✶ , so the term is no bigger than , which is less than for large P ❘☎third ❵ ☞❊➇❫✒ ☞❊➇❴✒ ➇★✍ ✱ ☞♦✰✾✒✗✸✕ ✶ by part (a). In short, for sufficiently large, and we are done. ❵ ➇

✙

✤

✄

✝

✌

✄

✝

✌

✠

✠

✌

✝

✝

✠

✠

✝

✌

✝

✠

✌

✠

✝

✠

✠

✌

✌

✠

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✌

✄

✠

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✝

✠

✌

✌

✠

7.3 Power Series ☛

P

✱ ☛ P❜➇ ☛ ✽ ✱ ☛ ➇ ☛ ✾✄⑦ ✱ ☛ ☛ P✮✽ ✱ ☛ ✗ ➇■ ✣ ✦ ✞ ➇■ ✞✭ ➇■✢✕r✏ ❚ ✱☛➇ ☛ 1. (a) ☛ , so by the ratio test, converges when ✞✦ ➇■ ✙✌✏ and diverges . ✓✒ when P ✱ ☛ ➇ ☛❴ ☛ ✣ ✭ ✞ ➇✿ ✞✦ ➇■ ✕ ✏ ❚ ✱☛➇ ☛ , so by the root test, converges when and diverges when (b) ✞ ➇✿✏✙✌✏ ✭ .

✣q

✱ 2. If ☛

✄

✱ ☛ ➇✢☛❫ P✓✒ ☛⑥q , then

✄

P✓✒ ☛❫ ➇■

. If

➇✿✏✕r✏

✄

then

P✓✒ ☛❴ ➇✿✸✕✚✏

by the root test (Theorem 6.14a).

❚

3.

✱ ➇ ❘☛ P✓✒ ✒ ❘ ☛

✑

.

converges when

➇ ❘ ✕ ✑

and diverges when

for large , so ✟

➇ ❘ ✙

❚ ✱☛➇ ☛

converges

✑

, so the radius of convergence is

Chapter 7. Functions Defi ned by Series and Integrals

52

✏✓✽

✄

✼✜✛✣✢ ✱ ✓P ✒ ☛

✔✟✞

➇■✠✕

✁⑧✄✝☞ t✡✚ ➇■❺✒❨✽✠✙

➇ ☛ ✓P ✒ ☛ ✕ ➇■❺✽ ✁ ✕r✏

✱

☛ ☛ 4. Let ✱ ➇✢☛ ✝ . If , let ➇■. Then ✡✙ ✱ ☛ ➇✢☛❴ P✓✒ ☛ ✙⑦✏ for large , ❚ ☛ , then for infinitely so ✱ ☛ ➇ ☛ test (Theorem 6.14a). If ✱ converges ➇ ☛ ✣ ❶ ✰ by the ❚ root many , so ☛ and diverges. ☞✎✍✯❛ ✟ ✒ ☛ ✽ ❛✂✐⑥❥ ☞✎✍✑✏✓✒ ☛ ➇ ✟ ☛ ☛ P ✽ ☞ ✙ ✡✥✏✓✒ ✎ ✚ ✖ ❛❁✄ ❚ ✎ ✚✖ ✄ ❚ 5. (a) for . By Theorem 7.18, for ✒ ✒ ➇❇✐★❥ . ✸✻✺✜✼✫❛ ✟ ✄ ❚ ☞✎✍✑✏✓✒ ☛ ☞❊❛ ✟ ✒ ✟ ☛ ✽✣☞ ✙ ✒ ❛ ✐ ❥ ✸✻✺✜✼❬❛ ✟ ❛ ✄ (b) . By Theorem 7.18, ❚ ☞✎✍✑✏✓✒ ☛✈➇✢✪ ☛ ☛ P ✽✣☞ ✙ ✒ ✒ ☞ ✡✖✏✓✒ ➇❇✐⑥❥ for for . ✒ P ❛ ✎ ✺✍❾✓ ☞✎✏▼✡❧✙▲❛❜✒❁✄ ❚ P ☞✎✍✑✏✓✒ ☛ ✎ P ✙ ☛☎❛ ☛ ✎ P ✽ ✍✑✏ ✕⑤✙▲❛✯q✌✏ P P ☞ ✙▲➇❴✒ ☛ ✽ ✟ (c) for with the ✝ ✒ ❛ ✎ ✺✍✓❾☞✎✏❝✡✖✙▲❛❜✒✗ ✚ ❱ ✄ ✙ ❛ ✓❴☞✎✏❝✡✖✙▲❛❨3✒ in❛✘✂ ✄ 6.1), ✎ ☞ ✑ ✍ ✓ ✏ ✒ ✎ P ✺✍(Exercise ☛ ✎ understand❚ P ✝ ing that . By Theorem 7.18, ✒ ➇✿✏✕P ✟P ✝ ➇■✈q ✟P for . However, the integral on the left is continuous for (it is improper ➇⑥✄ ✍ ✟ ✙▲➇■✫q✌✏ but convergent at P ), and the series on the right converges absolutely and uniformly for by comparison to ➇■▼✄ ✟ ❚ ✎ ✟ . Thus the equality persists for . ✑

✑

✑

✟

✑

✟

☎

✂

✂

✟

✎

✌

✝

✟

✎

✟

☎

✟

✟

✎

✝

✎

✌

✟

✟

☎

✌

✝

✟

❚

✟

☞✎✍ ✏✓✒ ☛ ✽

☞✙

✡

✏✓✒

6. Each☞✎✍✑of✏✓✒ ☛❬the , ✒ ✽✣☞ ✙ series ✒ ☞ ✡✕for ✏✓✒ the ❚ three ☞✎✍ ✏✓integrals ✒ ☛ ✎ P ✽ ✟ in question, namely, ❚ P , and ✒ , is an alternating series whose terms decrease mono✒ tonically to zero in absolute value, so the full sum lies in between any two successive partial sums. One simply computes the✄ partial sums (with a calculator or otherwise) until the desired accuracy is attained. ✄✌✠✁✾❅✎✠✠✙ ✄✆✄✆☛ ✄ ✄ ❚ ☞✎✍✑✏✓✒ ☛❬✽ ☞ ✙ ✡r✏✓✒✓✕ P ✎ ✚ ✖ ❛✓✕ ❚ ☞✎✍✑✏✓✒ ☛❬✽ ☞ ✙ ✡r✏✓✒ ✄ ✄✌✠✁✾❅ ★▼✞ ✄✆✄✆✄ (a) , so the answer to three decimal places is .747. P ✸✻✺✜✼✫❛ ✟ ❛ ✕✌✏➃✍ ☞✎✏✓✽✠✙ ❂✜✒✩✡❦☞✎✏✓✁✽ ✴✜✒❀✄ ✄ ✴✠✰ ✾❅▼✙ ✄✆✄✆✄ ✄ ✴✠✰ ✾❂▼✙ ✄✆✄✆❬ ✄ ✄ ✏❍✍❢☞✎✏✓✽✠✙ ❂✜✒✓✡❋☞✎✏✓✽✁ ✴✜✒▼✍❢☞✎✏✩✽✠❅ ✏✱✞✾✒ ✕ , (b) so the answer to three decimal places is .905. (c) This series, alas, converges much more slowly. One has to go to the 123rd partial sum to be sure that the✄ ★▼answer decimal places is .822 and not .823. I used Maple to find that the 123rd partial ✙▼✙ ✾✴▼✴ ★ to✄✆✄✆three ✄ ✄ ★▼✙▼✙ ✾✞ ✠ ✄✆✄✆✄ and the 124th partial sum is ; the full sum is in between. (It is actually sum is ❃ ✟ ✽✈✏✱✙✲✄ ✄ ★▼✙▼✙ ✾❅✎✠▲✰ ✄✆✄✆✄ .) ✟

✟

✎

✟

✎

✟

✟

✓

✂

✟

✎

✟

✝

✎

☞❊➇❴✒✦✄

☞✎✍✯➇❫✒

✎

✌

✎

✟

✝

✎

✟

✌

✎

✎

✱ ☛ ✄ ☞✎✍✑✏✓✒ ☛ ✱ ☛ ❚ ✱ ➇ ☛ ✄ ❚ ☞✎✍ ✓✏ ✒ ☛ ✱ ☛ ➇ ☛ for all and ☞❊➇❫✒ ✄ ✍ , so☞✎✍✯by➇❫✒ Corollary ✱ ☛ ✄ 7.22, ☞✎✍ ✏✓✒ ☛ ☛ P ✱ ☛ then for all and hence ❵ ❵

☛ 7. If ❵ then ✱ ☛ ✄❵ ✰ for odd. Likewise, if hence ✱ ☛ ✄✖✰ for even. ✟

✟

✟

☞ ✡r✏✓✒

✟

8. (a) The absolute value th term to the th term ✣✦of ✥ the ratio➇ of the ➇ is for all . which vanishes as ☛ ❘ for all ; hence the series ☛ ❘ converges P ✟

☞✎✍✑✏✓✒ ☛✾➇ ✟ ☛ ✟ ✄ ✒ ✙✠✟ ☛ ☛❴❘ ☞ ✡✘✶❬✒

✟

➇ ✟ ✽✁❍☞ ✡r✏✓✒✻☞ ✡✴✶ ✡r✏✓✒ ✟

✟

,

☞✎✍✑✏✓✒ ☛✈➇ ✟ ☛ ✟ ✎ ✄✖➇ ❘ ❘ ✎ P ☞❊➇❫✒ ✄ P ➇ ✠ ✙ ✟ ☞ ✖ ✡ ❧ ✶ ✍ ✓ ✏ ✒ (b) ☛ ✎ ☛ ❱ ☛ ❱ ☞✎✍✑✏✓✒ ☛☎➇ ✟ ☛ ☞✎✍✑✏✓✒ ☛✈➇ ✟ ☛ ✎ P ☞✎✍ ✏✓✒ ✮ ➇ ✟ ✮ ☛ P ✒ ✒ ✒ ✄ ✄ ✙ ✟ ☛ ☛❴❘ ☞ ✡✘✶❬✒ ✙ ✟ ☛ ☛❴❘ ✎ P ☞ ✍ ✏✓✒ ☞ ✡✖✶❬✒ ✙ ✟ ✮ ☛❴❘☞☛ P ✦ ☞ ✦✝✘ ✡ ✶ ✡✖✏✓✒ ✄ (c) ➇ P ☛ ❱ ☛ ❱ ✮ ❱ ✆☛ P ☞❊➇❴✒ ❘ ✍ ➇ ❘ ✦ ✄ ✍ ✏ . (For the second equality, .) ☞✎✍✑✏✓✒ ☛ ☞ ✙ ✡ ✶❬✒✻☞ ✙ ✡ ✶✇✍❖✏✓✒✭✡ ☞ ✙ ✡ ✶❬✒✑✍ ✶ ✟ ✁ ➇ ✟ ☛ ☛❴❘ ✄ ➇ ✟ ✡ ➇ ✍ ✶✟ ✄ ✒ ☞ ✖ ✡ ✶❬✒ ✙✄✂ (d) ☛ ❱ ☛❴❘ ☞✎✍✑✏✓✒ ☛ ➇ ✟☛ ☞✎✍✑✏✓✒ ✮ ☛ P ➇ ✟ ✮ ☛❴✆❘ ☛ ✟ ✄ ✍❝➇ ✟ ❘✣☞❊➇❴✒ ✒ ✒ ✄ ❴ ☛ ❘ ☞ ✍ ✏✓✒ ☞ ✡✖✶❧✍ ✏✓✒ ✙ ✟ ☛ ✎ ✟ ✦ ☞ ✦✝✖ ✡ ❬✶ ✒ ✙ ✟ ✮ ☛❴❘ . (In the third equalP ☛ ❱ ✦ ✄ ✮ ❱ ✍ ✏ ✟

✒

✖

✌

☛❴❘

✖

✌

✟

✎

✟

✁

✎

✟

✖

✎

✟

✎

✖

✖

✌

✌

✟

✎

✟

✎

✟

✎

✟

✎

✎

✎

✁

✟

✖

☎

✁

✁

✝

✁

✟

✟

✖

✎

✟

✟

✟

✎

✖

✁

✟

ity,

✎

✟

✟

✎

.)

✎

✎

7.4. The Complex Exponential and Trig Functions

➇

9. The ☞ ✟ ✽ series ➇ ✟ ✒✛✆ ☛ converges ➇ ✜ ☛ ✄ for ✆ all ☛ ✎ P❨➇ ✜ ✄❈➇ ❚ ✆ ➇ ✄❈➇ ☞❊➇❫✒ ❵ . ✒

✌ ✌

53

✒ ✄ Let ☞ ✎ by✟ the ratio☞❊➇❴test. , so ❵ ☛

✮

✆

✙⑥✧❍✞❢✧❍❂❢✧➃❅✂✧✗✧✗✧✩☞ ✞ ✍♠✏✓✒✻☞ ✞ ✒ ➉ ✎ P ➈ ✟ ✽ ➇ ✟ ✒ ❚ ✆ ☛ ➇ ✜ ☛ ✄ ➇ ❚ P ✆ ☛ ✎ ❜P ➇ ;✜ ☞ ☛ then ✎ P✌

☛ ✄

✌ ✌

✟

✒

✒

P

✮

❚ P ➇ ☛ ✽✣☞ ✡❉✏✓✒ ✄ ➇■☞ ✽ ➇❴✒ ❚ ➇ ☛ ✽✣☞ ✡ ✏✓✒ ✄ ➇●☞ ✽ ✌✄ ➇❴✒ ✒ ➇✛✎ P ☞ ✂ ☎ ✍❇✏✓✒ ✄✄✂ ☎ ✍✉➇✛✌ ✎ P ✌ ☞ ✂ ☎ ✍❇✏✓✒ ✒ since ☞ ✂ ☎✘✍ ✏✓✒ ❚ ✌ ✌ ➇ ☛ ✽ ✍ ❚ ➇ ☛ ✽✣☞ ✡✖✏✓✒ ✄✄✂ ☎✑✍◆➇ ✎ .P Alternatively, . ✒ ✒ ☎❚ ✄ ❚ ☞✎✍ ✏✓✒ ☛ ➇ ✟ ☛ P ✽✣☞ ✙ ✡ ✏✓✒ ✧✚☞ ✙ ✡ ✙✜✒ P (b)☎ ❛ ✎ ✟ ✒ ☎ ✟ ✟ ❛❨✽✣☞ ✙ ❋✒ ✄ ❛ ✎ ☞✎✏❝✍✪✸✻✺✜✼❍❛❨✒ ❛ ❚ P ☞✎✍ ✏✓✒ ✎ ❛ ✌ ✌ . ✄ ✒ P ✄ ➇ ✎ ❚ ➇ ☛ P ✽✣☞ ✡ ✏✓✒ ✟ ❚ ➇✢☛✣✽✣☞ ✡ ✏✓✒ ✟ (c) ✒ ➇ ✎ P ☎ ✒ ❛ ✎ P ☞ ✂ ✍ ✏✓✒ ❛ (see part (a)). ✌ ❚ ☞✎✍✑✏✓✒ ☛❴☞ ✙ ✡♠✏✓✒ ➇ ✟ ☛✫✽✣☞ ✙ ✒ ✄ ☞ ✽ ➇❴✒ ❚ ☞✎✍✑✏✓✒ ☛☎➇ ✟ ☛ P ✽✣☞ ✙ (d) ✌ ✌ ✒ ➇❀✄✼ ✔✖✕❤➇ ✒

10. ➇● (a) ☞ ✽

✟

✟

✎

☎

✟

✌ ➇❴☞ ✒ ✡✪➇ ✎ ✏✓✒❬✍❋❚ ✏ ✒ ☎

✎

✝

✟

✟

✎

✟

✟

☛

✟

✎

✒

✝

✦

✮

✮

✝

✎

✝

☛

✟

✎

✟

✟

❚

.

☎

11. (a) Integrate by parts:

✎

✶✠✷✹✸ ❜✶ ✤ ✕ ❛ ❛✿✄❈❛✣✶✠✷✹✸ ❜✶ ✤ ✕ ❛✗ ☎ ✍

✌

✂

✝

✂

✎

✟

✡

✎

✙✜✒

✟

✄ ✎

❛✫☛ ❛❜✽✣☞

✌

✒

✝

✡ ✏✓✒ ✟

✄ ✎

✒ ✄ ☞ ✽ ➇❫✒✻☞❊➇❀✸✻✺✜✼❾➇❴✒ ✄❉✸✻✺✜✼❬➇✇✍

☛

✟

✝

✟

✌

☎

➇ ✎ P

✎

✝

✟

☛

✟

✚

✟

➇ ☛ P ✣✽ ☞ ✡❉✏✓✒ ✄ ❚ P ➇✢☛✈✽✣☞ ✡✪✏✓✒ ✄ , ✒

☞✎✍✑✏✓✒ ☛ ❛ ✟ ☛ ❛❜✽✣☞ ✙

✎

✟

✟

✟

✌ ✌

✎

☎ ❛ ✌ ❛❜✽✣☞❊❛ ✟ ✡✕✏✓✒❁✄❈➇❀✶✠✷ ✸ ❜✶ ❝➇♣✍ ✶✠✷✹✸ ❜✶ ❝➇✚✄ ❚ ☞✎✍✑✏✓✒ ☛✾➇ ✟ ☛ P ✽✣☞ ✙

P ✟ ✟ ✍✺ ✓➃☞❊➇ ✡✖✏✓✒ . ✡♠✏✓✒ ➇ ✐ ♠

✕

✂

✝

✕

✝

☛

(b) for ✒ ✍✑✏✠From ➂✱✏✮➉ Example 2 and Abel’s theorem, we have ❆❧✏ , the series converging uniformly on that interval. (The convergence at comes from the ➈ ✍✑✏✠➂✱✏✮➉ alternating series then gives the uniformity on ➈ and the validity ☞❊of ➇⑤✄❞test, ❆❧✏ and Abel’s ✶✠✷ ✸ ❜✶ theorem ✕ ➇❫✒✑the ✄ , since is continuous.) It then follows from Theorem 7.13a that expansion at ❵ ❚ ☞✎✍✑✏✓✒ ☛ ➇ ✟ ☛ ✟ ✽✣☞ ✙ ✡✖✏✓✒✻☞ ✙ ✡ ✙✜✒ ➇❋✐ ✍✑✏✠➂✱✏✮➉ ➈ P . for ✒ P ❃⑥ ➇⑥✄♠✏ ✍ ✺✍✓✂✙ ✄ ☞✎✏✓✒❁✄ ❚ ☞✎✍✑✏✓✒ ☛ ✽✣☞ ✙ ✡✖✏✓✒✻☞ ✙ ✡✥✙✜✒ ✪ ☞ ✙ ✟✡✖✏✓✒ ✎ P ✍⑤❵ ☞ ✙ ✡✥✙✜✒ ✎ ✒ P (c) Setting in (b) , and ☞ ✙ and ✡✖✏✓using ✒✻☞ ✙ (a) ✡✥✙✜gives ✒ ➉ ✎ P ✄♠ then yields the desired result. the observation that ➈ ✂

✟

✂

☛

✟

✟

✝

✟

✟

✟

✟

✟

✟

7.4 The Complex Exponential and Trig Functions

✼✄✔ ✕✁✁ ➇⑨✄ ✟P ☞ ✂ ❩ ✉ ☎ ✍ (a)

✂ ✎ ❩ ☎✜✒♣✄✂✾✼

✸✻✺✜✼✄✄ ➇⑨✄ ✟P ☞ ✂ ❩ ❧ ☎✡ and

✂ ✎ ❩ ☎✾✒♣✄ ✸✻✺✜✼✫➇ 1. by (7.30) and (7.32). Alternatively, one can examine the power series expansions. ✂✆☎ and ✂✝☎ , multiply out the right (b) First method: express the sinh and cosh of and in terms of ✯➇

✙✔ ✕

✍

sides of the asserted formulas, and simplify. Second method: set part (a).

✲✄✞ ➇

✄✟ ➁

✓

✆

and ✍

, and use (7.34) and

✼✄✔ ✕✂❙☞❊➇✲✡✠ ➁❬✒❁✄✖✙✼ ✔ ✕✁✂➇❀✸✻✺✜✼✄✡ ➁✭✡ ✸✻✺✜✼✄✯➇✭✼✄✔ ✕✁✁ ➁✉✄✖✼✙✔ ✕✁✂➇❀✸✻✺✜✼❍➁✭✠ ✡ ☎✸✻✺✜✼✄❝➇✭✼✄✔ ✕❤➁ ✸✻✺✜✼✄✬☞❊➇✲✡☛ ➁❬✒✿✄ , and ➁ ✡❲✙✼ ✔ ✕✁✂➇❀✄✼ ✔✖✕✁✁ ➁✉✄✖✸✻✺✜✼✄✯➇✭✸✻✺✜✼❬➁✑✡☞✾✼✄✔ ✕✂❤➇❀✼✄✔ ✕✂➁ . ❩ ✆✯✄ ✱✬✌ ✡ ✓✗ ✱❍➂✙✗✭✐⑥❥ ☞ ✽ ➇❴✒ ✂ ☞ ✌ ☎ ✄ ☞ ✽ ➇❴✒ ✂ ☎ ☞♦✸✻✺✜✼ ✗ ➇❑✡✍☎✼✄✔ ✕ ✗ ➇❴✒ ➉❫✄✴✱ ✂ ☎ ♦☞ ✸✻✺✜✼ ✗✹➇❑✡✌✾✼✙✔ ✕ ✗ ➇❴✒▲✡ with , If ✌ ✌ ✗ ✂ ✎ ☎❍☞✎✍❦✄ ✼ ✔ ✕ ✗ ➇❧✡☞☎✸✻✺✜✼ ✗ ➇❴✒❁✄♠✳☞ ✱✘✡✎ ✗✻✒ ✂ ☎➃☞♦✸✻✺✜✼ ✗ ✌ ➇❧✌ ✡☞✾✼✙✔ ✕ ✗ ➇❴✒❁✄ ✆ ✂ ☎ ✆ . ❩ ✂ ☎ ✸✻✺✜✼ ✗ ➇ ➇ ✡✎ ✂ ☎ ✼✄✔ ✕ ✗ ➇ ➇⑥✄ ✂ ☞ ✌ ☎ ➇ (c) ✸✻✺✜✄✼ ❝➇✭✸✻✺✜✼✄✄

✠

2.

✠

✠

✠

✞

✠

3.

☛

✠

✝

✂☞

❩ ✌☎ ✄ ✱✘✡☞✓✗ ✠

☛

✞

✌

✠

✂ ☎❍☞♦✸✻✺✜✼ ✠

✝

✌

✠

☛

✝

✞

➇ ✡✏✾✼✄✔ ✕ ✗ ❴➇ ✒ ✧ ✱✲✍✠ ✗ ✄ ❧ ✱✦✡✏✓✗ ✱ ✍✠ ✗ ✲ ✗

✌

. By Exercise 2, this is

✂ ☎✫☞✳✱✂✸✻✺✜✼ ✹➇ ✠

✗

✡✞❾✗ ✼✄✔ ✕ ✗ ❴➇ ✒❫✡✏ ✂ ❍☎ ☞✳✱❑✼✄✔✖✕ ✗ ♣ ➇ ✍✥✗❾✸✻✺✜✼ ✗ ❫➇ ✒ ✱✈✟☛✡ ✗ ✟ ✠

✄

The asserted formulas follow by taking real and imaginary parts. (Of course some constants of integration are being suppressed here.)

Chapter 7. Functions Defi ned by Series and Integrals

54

7.5 Functions Defi ned by Improper Integrals

➇◗✐

2.

✆✒

✟

✒

✂

☞❊➇✬we ➂❨❛❨✒ have ❛✗✫q

✆✒

✠

✠

✠

✆✒

✝

✟

✒

✣

✠

✠

✆✒

✠

✂

✒✆ ✳✱ ✆✒

✎ ✒ ☛ ✎ ☛ ✩✎ ✎ ✎ ✒ ✒ ☛ ✎ ☛ ✠✎ ☛ ✎ ✎ ✛✎ ✛✎ ✒ ✒ ☛✎ ✠✎ ☛ ✒ ✥✵ ✙✵ ✎✠☛ ✜ ☛ ✎✢✜ ✵✎ ☛ ✎ ☛✎ ✎ ✎ ☛ ✒ ☛ ☛✎ ✒ ✎ ✎ ✎ ✥ ✒ ✩✎ ✒ ✩✎ ✳✱ ✒ ✎ ✎ ✩✎ ✥ ✒✙ ✩✎ ✵ ✵ ✒ ✎ ✠✱ ✒ ✎ ✎ ✒ ✎ ✥ ✎ ✳✱ ✒ ✎ ☎✙ ✒ ✕ ✒ ✵ ✙ ✒ ✒ ✒ ✎ ✎ ✣ ✥ ✦ ✒ ✵ ✵ ✒ ✎ ✥ ✒ ✙ ✚

3.

✒✆ ✣✦✥ ✕ ✥ ✱

☞❊➇✬➂❨❛❨✒✗ ❛ q ☞❊❛❨✒ ❛ , so the integral converges absolutely. Like❵ ❊☞ ❛❜✒ ❛ ✌ ✰ ✝ ✌ ❵ ✌ ✝ ✌ as ✌ , so the convergence is uniform. ☞❊➇❫✒ ✄ ❊ ☞ ✬ ➇ ❨ ➂ ❨ ❛ ✒ ❛ ✌ ✐ ☎ Let ☎✝ ☞❊➁❬✒ ✌ ➁◆✄ ✝ ✝ ❽ ☎ ✝ ❵ ☎ ❽ ☎ ❵ ☞❊➁❾✌ ➂❨,❛❜✒ and ➁ pick ❛❀✄ a point☞❊➇❄➂❨❛❜✒ ❛❑✍. By Theorem ☞ ➃➂❨❛❜✒ ❛ 7.39 (with ❵ replaced by ❽ ❵ ), ✌ ✌ ✝ ❵ ☞❊➇❫✒☛✄ ☞ ✌ ✽ ➇❴✝ ✒ ❵ ☞❊➇❄➂❨❛❜✌ ✒ . ❛ The last integral is a constant, so by the fundamental theorem of calculus, ✌ ✌ ✝ ❵ ✌ . ✂ ☎ ❛ ✄ ✍✯➇ P ✂ ☎ ✄ ➇ P ✂ ☎ ✌ ❛ times gives ✌❛ ✂ ☎ ❛ . Formally differentiating➂ ✝ ✒ ✝ ☞✎✍✑✏✓✒ ✡✡ P ✄ ✝ ✌☞❊❛❨✒⑥.✄❉The❛ ✂ latter integral is uniformly convergent on ➈ for any ☞ ❞✽ ✰ ,➇❴✒by✣➇ Theo, so the differentiation is justified. On the other hand, ✌ ✌ rem ☞✎✍✑✏✓✒ 7.38➇ withP ; the result follows. P ☞❊❛ ✟ ✡⑤➇❴✒ P ❛✦✄✝➇ ✓P ✒ ✟ ✶✠✷✹✸ ❜✶ ✕✬☞❊❛❜✽✩➇ ✓P ✒ ✟ ✒ ✄ ✟P ❃❴➇ ✓P ✒ ✟ ✝ ✍✕✏ ✌ ☞✎✍✑✏✓✒ P ☞ ✍✕✏✓✒ ☞❊❛ ✟ ✡✡ ✡t➇❴✒ ❛ . Formally differentiating ✝ ☞❊❛ ✟ ✡⑤➇❴✒ ✌ ❛ ✝ ➂ ⑤✒ times gives ✰ ☞❊❛❜✒❁✄ ☞❊❛ ✌ ✟ .✡ The ✒ latter integral is uniformly convergent on P P ✓P ✒ for any by Theorem 7.38 with ➈ ☞ ✽ ➇❴✒ ➇ ✟ ✄ ☞✎✍ ✟ ✒✻☞✎✍ ✟ ✒➃✧✗✧✗✧✠☞✎✍ ✟ ✟ ✒ ➇ ☞ ✟ , Pso✌ ✒ ✟ the differentiation is justified. On the other hand, ✌ ✌ . Hence ❛ ❃ ✯ ✏ ✓ ✧ ✂ ✞ ✗ ✧ ✗ ✧ ▲ ✧ ☞ ✙ ✍ ✜ ✞ ✒ ❃ ❤ ✏ ✧✩✞✂✧✗✧✗✧✠☞ ✙ ✍⑩✞✜✒ ➇ ☞ P ✟ ✍✌ ✒ ✟ ✄ ☞P ✟ ✌✒✟ ✄ ✄ ✧ ➇ ✧ ✌ P ☞❊❛✔✟☛✡❲➇❴✒ ✙ ✙ ☞ ✍⑤✏✓✒ ✙ ✙✭✙✧ ✯✧✗✧✗✧✠☞ ✙ ✍⑩✙✜✒ ➇ P ☞ ✂ ☎ ✍ ✂ ☎ ✒ ➇⑨✄ ✍ ✂ ☎ ✌ ❛ ✌ ➇ . Since ✝ ✂ ☎ ✌ ❛ is uniformly convergent onP any ✝ ✌ ✝ ✝ ✞ ✞ ☞♦✰✣➂ ✒ ✍ ➇ ➇⑥✄ (see Exercise 3), we can reverse the order of integration to get compact interval in ✟ ✝ ✞ ✌ ✺ ❾✓ ☞ ✫✽ ✗✗✒ ✍ ✝ . ➇ P ☞ ✂ ☎ ✍ ✂ ☎ ✒✈✸✻✺✜✼➃➇ ➇❢✄✳✍ ✂ ☎ ✸✻✺✜✼✫➇ ✌ ❛ ✌ ➇ . The integral ✝ ✂ ☎ ✸✻✺✜✼❬➇ ✌ ➇ is uniformly ✝ ✝ ✰ ✝ ✞ ✞ ❛❤✐ ➂ ✒ ✌ ☞❊➇❫✒❁✄ ✄ ✂ ☎ can reverse the order convergent for ➈ ✍ for any ✂ ✿✸✻✺✜✼❍➇ by➇ Theorem ☎ ❛❝✄✝✍ 7.38❛ with ❛❨✽✣☞❊❛ ✟ ✡✌✏✓✒✯✄✝✍ ✟P , ✺✍so✓➃☞❊we ❛ ✟ ✡⑨ ✏✓✒ ✌ ✌ ✌ ✝ of integration to get ✝✟✞ ✝ ✝✟✞ ✡✡ ✞ . (For the first equality, use the result of Exercise 3, ✂ 7.4, or integrate by parts twice.) ✂ ☎✠➇ P ☞✎✏ ✍⑨✸✻✺✜✼ ✾➇❫✒ ✌ ➇⑦✄ ✝ ✝ ✂ ☎✿✄✼ ✔ ✕❤❛✔➇ ✌ ❛ ✌ ➇ . The integral ✝ ✂ ☎❁✄✼ ✔ ✕✂❛✔➇ ✌ ➇ is uniformly ✝ ❛❤✐ ✰✣➂ ✒ ☞❊➇❫✒☛P ✄✄✂ ☎ 7.38 with so✏✓we convergent for✂ ☎✿✼✄✔✖➈ ✕❤❛ ➇ ➇ by❛☛Theorem ✄ ❛ ❛❜✽✣☞❊❛ ✟ ✡✖✏✓✒☛✄ ✟ ✍✺ ✓❾☞ ✟ , ✡✖ ✒ can reverse the order of integration ✌ ✌ ✝ ✌ ✝ to get ✝ ✝ . (For the first equality, use the result of Exercise 3, ✂ 7.4, or integrate by parts twice.) ➇ ✰ ❈✄ ➇❍❛ P ❛❜✽✩❛✪✄ ✠✽ ✼✄✔✖✕❤➇❍❛ ❛❜✽✩❛✇✄ ✼✄✔✖✕ ▲✽ ❈✄ ✟P ❃ , set ; then , so . It follows If ✌ ✌ ✝ ✌ ✝ ✌ ✼ ✔✖✕❤➇❍❛ ❛❜✽✩❛◆✄ ✍ ✟ ❃ ✄ ➇ ✰ ✌ ✰ that ✝ for since the sine function is✰✕odd, ➇✳✄❏ ✐ and of course the integrand . The convergence cannot be uniform on if by Theorem vanishes if ➇ ✤ 7.39, since ➇✕q the ✍ resulting or ✰ function is discontinuous at 0. However, the convergence is uniform for ( ✄✼ ✔ ✕❤). see 3, ✂ ➇■4.6,✤ or Example 3, ✂ 7.5: we have ➇➃❛ To❛❨✽✩❛✉ ✄ this, ☞✎✍❦✸✻✺✜use ✼❬➇❍❛❜integration ✒❨✽✩➇❍❛ ✍ by parts ✸✻✺✜✼❬➇➃as ❛ ❛❨in✽✩➇➃Example ❛✟ P ✗ P ✡ ✌ ✌ , and for this is bounded in absolute ✝ ✝ ✡ ❛❜✽✩❛ ✟ ➉❾✄✌✡ ✞ ✙✜✽ ✗ ✞ ✗ ✞value by ➈ ✝ ✌ , which vanishes as . ☞❊➇❴✒☛✄ ✼ ✔ ✕ ✟ ➇❍✞ ❛ ❛❜✽✩❛ ✟ ➇ ✤ ✰ ✄ ✼ ✔ ✕ ✟ ➇❍❛❤q ✞ ✔ ✕❙☞❨☞❊➇➃❛❨✒ ✟ ➂✱✏✓✒ ✄ ✰sq✥➇❋q ✄ , for Let ✝ ✌ for . Since ✞ ✔ ✕❫☞ ✄ ✟ ➂❨❛ ✟ ✒ ✰✣the ➂ ✄✭➉ integrand✄ is ➈ less than , whose integral is finite. Hence the convergence is uniform on ✰✣➂ ⑤✒ ☞❊➇❫✒✯✄ P ✙■✄✼ ✔ ✕❤➇➃❛✈✸✻✺✜✼❍➇❍❛ ❛❜✽✩❛❤✄ ✼P ✙✔ ✕✯for✙▲➇❍any ❛❜✽✩❛ ❛ , . Next, formally ☞❊➇❫✒❤✄ and so is continuous on ➈ ✝❃ ✌ ☞❊➇❫✒❤✝ ✄ ❃❴➇s✡ ✆ ✌ . ➇ t✰ ✟ ✟ for . Hence , and By Exercise 8, the differentiation is justified and ✆✤✄ ☞♦✰✾✒❁✄✕✰

1. For all wise, ✝

✠

✚

✎ ✥✵ ✒ ✵ ✙

✚

✚

☛ ✎

✟

✚

✠

✟

✂

4.

✟

✟

✎

✒

✎

✎

✠

✎ ☛

✂

✟

✟

✎

✠

5.

✚

✟

✚

✠

✠

6.

✟

✠

✎ ✒ ✎

✚

✠

✠

✎

✚

✚

✠

✠

✠

✒ ✎

✠

7.

✠

✠

✠

✂

8.

✂

✂

✂

9.

✒

✂

✂

✂

✵

✂

✵

✁

✁

✁

✁

✁

.

✁

✒

✵

7.5. Functions Defi ned by Improper Integrals

55

P ➇ ✎ ✟ ✌➇

✏✠➂ ✥ ✒

✰✣➂✱✏✮➉

P ☞✳✱ ✟ ✍ ✗ ✟ ✒ to ✝ ✒ 10. (a) The integral over☞♦✸✻✺✜➈ ✼ ✗ ➇ ✍ converges , and the integral over ➈ is ✔✟✞ ✸✻✺✜✼✢✱☎➇❴✒❨✽✩➇ P ✟by✄ comparison ✄ ✂ ✟ P . One is to proper since ✝ ✸✻✺✜✼ ✗ ➇ ✍❇✸✻✺✜✼ ✱✾➇⑥✄✕✙■✼✄✔ ✕ ☞✳✱❀✡ ✗✗✒ ➇❀✼✄✔✖✕ ☞✳✱✑ ✍ ✗✗✒ ➇way to obtain the uniformity ✔✼✄✔✖✕ ✈q ✞ easily ✔✖✕❙☞❜ ➂✱✏✓✒ ✟ ✟ P ✁ ✁ use the✔✸✻identity and the estimate ✺✜✼ ✗ ➇✲✍❦✸✻✺✜✼ ✱✾➇✿❺✽✩➇ ✟ q ✞ ✔ ✕❙☞ ✟ ✱ ✟ ✍ ✗ ✟ ➂❬➇✛✎ ✟ ✒ ✱❍➂✙✗ to get , which for in a bounded set is less than a fixed ✟ ✞ ✔ ✕❙☞ ✄✦➂❨➇✡✎ ✒ . integrable function P P P

☎

❾☞✳✱❍✙➂ ✗✗✒❁✄✳✍

P ❃❁☞✳✱⑧✍✥✗✗✒ (b)

✝ ✒ ✝✠✞

✂

✟

✠

➇✛✎ ✼✙✔ ✕❤❛ ➇ ❛ ➇

✌ ✌

.

➃☞✳✱➃✙➂ ✗✻✒

. By Exercise 8, this is

✱

✍

➇ ✎ ✼✄✔✖✕❑❛✔➇ ➇ ❛❁✄ ✍

✝✠✞ ✝ ✒ ✠

✌

✠

P ✒ ✟ ❁❃ P ✳☞ ❁❃ ✱★☞❜ ✍ ✱❴✾✍✗✗✌ persists for ✗✠❺✒ ✟ for any

➃☞✳✱➃➂✙✗✗✒⑥✄ by ✱ part (a), the formula ✗ ➃☞✳✱➃➂✙✗✗✒✤✄ ❾☞❜ ✱❴ ➂✩ ✗▼❺✒✭✄

✗

❃ ❛❁✄ ✝✟✞ ✟

✌ ✌

is continuous (c) ✱❍➂✙✗ Since ✤⑨✰ ❾☞✳✱❍➂✙✗✗✒ in and . Also, clearly is even in both and , so ✱❍➂✙✗ . ✍ ✒ ❛ ✩✎ ✚ ✖❴✼✄✔ ✕✂➇➃❛ ❛ ❥ 11. (a) The differentiated integral is ✚ ✝ , which converges absolutely and uniformly ☞❊❛❜✒❀✄ ❛ ✎ ✖ ☞❊➇❴✒❀✄ ✍ ❛ ✎ ✚ ✖❴✼✄✔✖✕❤➇❍❛ on❛✤✄ by 7.38 with . Hence,P by integration by parts, ✝ ✒ P ✎ Theorem P➇ ✚ ✚ ✖❾✄✼ ✔✖✕❤➇❍❛ ✍ ✖❾✸✻✺✜✼❬➇❍❛ ❛☛✄r✍ ➇ ❊ ☞ ❴ ➇ ✒ ✎ ✟ ✟ ✝ ✒ ✟ . ✡✒ P P ✂

✂

✂

✂

✂

✂

✠

✡

✂

✂

✂

✌

✌

✂

✌

✁

✁

✽

✺✍✓✭ ☞❊➇❫✒✗❴✄❞✍ ➇ ✟ ✡ ✺✍✓✦ ✄♣

✄ ✍ ➇

✟ , so ✝ ✪ ✝ , hence (b) Solving the ✒ differential equation from (a), ☞❊➇❴✒☛✄ ✄ ✎ ✖ ✪ ✄✌✄ ☞♦✰✾✒☛✄ ✩✎ ✚ ✖ ❛❁✄ ✟P ✛ ❃ ✒ , and (Proposition 4.66). ✝ ☞❊➇❫✒❀✄ ❛ ✎ ✚ ✖❾✸✻✺✜✼❬➇➃❛ ❛✭✄❞✍ ✟P ✎ ✚ ✖❾✸✻✺✜✼❬➇➃❛ ✒ ✍ ✟P ➇ ✎ ✚ ✖❴✼✄✔ ✕❤➇❍❛ ❛❀✄ ✟P ☞✎✏✦✍ ✒ ✒ 12. As in Exercise 11, ✝ ✝ ✡ P P ➇ ☞❊➇❫✒❨✒ ☞❊➇❴✒✿✡ ✟ ➇ ☞❊➇❴✒✲✄ ✟ ✡ . Write this equation as and multiply P ✒ ✒ ✒ ✒ ✚ ✒ through by the integrating factor ✄✖ ✪ ☞ ✄✖ ✪ ✒ ✄ ✟ ✄✖ ✪ ✄✖ ✪ ☞❊➇❴✒❑✄ ✖ ✪ ❛❫✡ ✄ ✄✌✄ ☞♦✰✾✒☛✄✖✰ to get . It follows that , and . ✝ ✚ ✟ ☞❊➇✬➂❨❛❨✒✲✄ ☞✎✏✘✍ ✎ ✖✗✒❨✽✩❛ ☞❊➇❴✒✲✄ ☞❊➇❄➂❨❛❜✒ ❛ ☞❊➇✬➂❨❛❨✒ ➇ ✤✝✰ . is continuous on the region , 13. Let P ✝ ☞❊➇✬✒ ➂❨❛❨✒ ❛ ❛✩✤❖✰ ☞❊➇✬➂❜✰✾✒ ✄⑦➇ and ➇ ❊ ☞ ❄ ➇ ❨ ➂ ❜ ❛ ✒ ❛ P ✒ if we define is☞❊➇❫continuous in , and➇ so is✰ ✝ ☞❊➇❄. ➂❨Hence ❛❜✒✗■qr❛ ✝ ✎ ✟ ✤ ✒ ☞❊➇❴since ✒✦✄ ✚ ). Thus is continuous for . We have the convergence is uniform ( ➇✷✙✳✰ ✎ ✖ ❛ ✝ ✒ , the differentiation being justified since the latter integral converges uniformly ➇ ✤ ✙❈for ✰ ✑✄⑨❛ ✛ ➇ ☞❊➇❫✒❑✄ ✒ ✩✎ ✖ ✠✽ ✛ ➇❦✄ . Hence, by the substitution and Proposition 4.66, for ✝ P ✆ ❃❄✽✩➇ ✵ ❴ ❃ ❧ ➇ ✡ ✄ ➇ ✥ ✙ ✰ ⑨ ✄ ✄ ✔✟✞ ❊ ☞ ❴ ➇ ❁ ✒ ✄ ♦ ☞ ✾ ✰ ☛ ✒ ✖ ✄ ✰ ❊ ☞ ❫ ➇ ❁ ✒ ✄ ✛ ✄ ✂ ✟ ✝ . Thus for , and . ✚ ➇ ✎ ✖ ✒ ✚ 14. (a)➇ The integrand is at most for all , so the integral to a continuous function ✍✤✙▲➇ converges ✩✎ ✚ ✖ ✎ ☞ ✖ uniformly ✖ ✌ ❛❜✽✩❛ ✟ ✝ ✒ ; this integral is still conof . Formal differentiation of the integral ✒ yields ✚ ➇✳✄ ❶ ✰ ✎ ✄✖ ✖ ✏✓✽✩❛ ✟ ❛✘✄ ✰ vergent for ➇✿✣✤ because ✙t✰ the factor ➇✥✄✕ ❶ ,✰ and➇ the✙tconvergence ✰ ✄✕➇❫✽✩is❛ ☞❊➇❴✒✂✄ ✍❀✙▲➇ kills the ❛✟ ✎ ✚ ✖ factor ✎ ☞ ✖ ✒ ✚ ✖ ✌ ❛❨✽✩near ✒ ✵✟ ✁ uniform for . Hence for . If , let ✝ ✄✳✍✯➇ ❛❜✽✩❛ ☞❊➇❴✒❁✄✌✙ ✄ ✍✤✙ ☞❊➇❴✒ ➇ ✕ ✰ ✄✳✍✯➇❴✽✩❛ , ✎ ☞ ✖✒ ✖✌✎ ✖ ✎✁ so ✎✁ ,☞❊➇❫ and . If , the substitution ✁ ✒❁✄✕✙ ☞❊➇❫✒ ✝ ✒ likewise yields . ☞❊➇❫✒✯✄ ✄ ✁ ✟ ❆✭➇ ✙✖✰ P ✚ for P give (b) The differential equations for at 0, we ✄ ✄ ☞♦✰✾✒☛✄ ☞❊➇❴✒☛✄ . Since ✎ ✖ ❛❁✄ ✟ ✛ ❃ ✛ ❃ ✎ is✟ continuous ✟ ✝ ✒ have by ✚ Proposition 4.66, so . ✒ ✚ ✒ ✚

✂ ☎

✁

✂☎

✂ ✌

✁

✂

✂☎

✂☎

✡✂ ☎

✁

✁

✂

✌

✌

✠

✂ ☎ ✌

✌

✁

✠

✌

✠

✁

✁

✂

☎

✂ ☎

✌

✌

✁

✁

✁

☎

✁ (c) P By the ✟substitution ✂ ✟ ✆ ❃❄✽ ✎ .

✂

✂ ☎

✁

✝

✝

✁

✂ ✌ ✄

✁

✌

☎

☎

✟

✂✎

✖

✆

✎

☞✄✂

☎

✆

✂

✖ ✌

✁

☎

✂

✌

✂ ☎

✌ ❛✉✄

✁

✟

✂ ☎

✌ ✂ ✄☎ ✂ ☎

✂ ☎ ✎

✝ ✒

☞✎✍✑✏✓✒ ☛

✂✎

✁

✝

✖

✎

☞ ✂ ✆

✖ ✌

✌✔✁ ✽ ✛

✕✄

✎

✟

☞ ✛ ✆☎✠✒❨✽ ✛ ✖✄ ✟

✁

✟

✂ ✎ ☎ ➇✢☛ ❵ ☞❊➇❴✒ ✌ ➇ . The convergence of ✂ ✎ ☎✠➇ ☛ ❵ ☞❊➇❫✒✗✣q✖✱ ✂ ✎ ☎❍☞✎✏✿✡✪➇❴✒ ☛ , ✂ ✂ ☎ ❵ ☞❊➇❫✒ ✌ ➇★✄✳✍ ❵ ☞♦✰✾✒❴✡✍✂✄✞ ➈❺❵ ➉ ☞ ✂▲✒ ; the

yields ✙ ✰ 15. (a) Formal differentiation of the integral times ✝ ✒ ✤✣✗●✡ ✵ (✵ the latter integral, for any , is uniform for ) since so the differentiation is justified. ✎ ☞❊➇❫✒ ➇★✄ ✩✎ ☞❊➇❴✒ ✒ ✡ ✒ ✩✎ (b) Integrate by parts: ✝ ✒ ❵ ☞❊➇❫✒✔❵ ✣ ✰ ✡✡ ➇✚✝ ✣ ✥ ✎ ❵ as . assumed estimate on ❵ guarantees that ✟

✌

✁

✁

✛ ✤❛ , ✝ ✒

✁

✁

✌

✂ ✌✂

✠

✂

✂

✌

✠

✁

✟

✂

☎✂

✂☎

✠

✠

✁

☛

✞

Chapter 7. Functions Defi ned by Series and Integrals

56

7.6 The Gamma Function 1.

✙✟☛ ✛ ☞✙

P

✒ ☞ ✡ ✟P ✒❁✄✌✙ ✟ ☛ ✎ P ✏■✧✹✙✂✧✗✧✗✧✩☞ ✍⑩✏✓✒➃✧ ✟P ✧ ✜✟ ✧✗✧✗✧▲☞ ✍ ✟P ✒❁✄✕✙✂✚✧ ✯✧✗✧✗✧✩☞ ✙ ✍❦✙✜✒➃✧✱✏■✧✹✞✂✧✗✧✗✧▲☞ ✙ ✍✪✏✓✒✿✄ ✄ ☞✙ ✒ . ❛✑✄ ✎ ❛✑✄ ✍ ✎ ✥ ❛ P ✁✺✍✓❾✄ ☞✎✏✓✽✩✝ ❛❜✺✍✒ ✓➃➉ ☞✎✏✓✎ ✽✩P ❛❜❛ ✒ ,✎ so P ❛❁✄ ✎✁ ✗ ,✎ and☞✳✱✣✁ ✒ goes from Let , ✎ P ✎ to 0 as goes from 0 to 1. Thus ✄✏ ✒ ✁ by (7.51). ➈✝ ✎✁ P ✄ P P ➇ ✪ ✎ ✄✖ ➇⑥✄ ✟ ☞ ✟ ✒✿✄ ✟ ✧ ✜✟ ✧ ✟ ✛ ❃❦✄ ✜✞ ✛ ❃ (a) By (7.52), ✒ . ✎✢✜ ✛ ➇ ➇❢✄✕✞ ✎✢✜ ✒ ✟ ☞ ✜ ✒❁✄✕✞ ✎✢✜ ✒ ✟ P ✛ ❃❦✄ P ✆ ❃❄✽✠✙✎✠ ✟ ✟ ✟ (b) By (7.51), ✒ . ✄✖➇ ✪ ✄ ▼➇ ✜ ➇ ✡ ➇ ✎ ➇⑥✄ P ✄ P ☞ ✄✟ ✒❁✄ P ✧ ✜✟ ✧ ✟P ✛ ❃❦✄ P ✜ ✛ ❃ ✜✒✟ ✎ ✪ ✒ ✁ ✪ ✪ ✎✁ , ✎✁ ; then ✒ . (c) Let ✁ ✄✳✏✯✍✪❛ ☞❊➇✬➂❨➁✫✒ ☞❊➁❾➂❨➇❴✒ ✁ (a) The substitution in (7.53) into . P turns ☞❊➇❄➂✱✏✓✒☛✄ P ❛ ✎ P ❛❁✄❈➇ ✎ P ❛ ✄❈➇ ✎ P . (b) ❛ ✫☞✎✏✯✍❇❛❨✒ ✗ ✎ P ✡⑩❛ ✎ P ☞✎✏✯✍❇❛❜✒ ✗ ✄❈❛ ✎ P ☞✎✏❤✍❋❛❜✒ ✗ ✎ P ❛❴✡✖☞✎✏✂✍❇❛❜✒ ➉❴✄❈❛ ✎ P ☞✎✏✂✍❇❛❜✒ ✗ ✎ P ➈ (c) ; the result follows by integrating from 0 to 1. ❛✿✄ ✏✓✽✣☞ ✡✕✏✓✒ ☞❊➇❄➂❨➁❬✒❁✄ ✒ ☞✎✏❑✡ ✒ ✎ ✎ ✗ ✗ ✎ P turns (7.53) into (d) The substitution ✁ ✁ ✁ ✔✁ . ✰ ✘ P P P ✝ ☛ P ✒ P P P ☞ ✄ ➇ ➇✳✄✦✗ ✎ ➇ ☞✎✏❧✍t➇ ✒ ➇♠✄ ✗ ✎ ✒ ✄ ✎ ✌ ✎ ☞✎✏❧✍ P ✁ ✁ ✎ ✁ ✆ ✁ ✁ ✆ ✎✁ Let ; then , so ✠✟ ✗ ✎ ☞❨✳☞ ✱✘✡✖✏✓✒❨✽ ✗✩✏➂ ☛✆ ✡❈✏✓✒ ; use Theorem 7.55. ☞ ✙▲➇❴✒❁✄ ☞ ✙▲➇❤✍✉✏✓✒✻☞ ✙▲➇❤✍✲✙✜✒ ☞ ✙▲➇✂✍⑧✙✜✒ ☞❊➇❫✒❁✄ ☞❊➇❤✍✉✏✓✒ ☞❊➇❤✍✉✏✓✒ ☞❊➇❁✡ ✟P ✒✿✄ ☞❊➇✂✍ ✟P ✒ ☞❊➇❤✍ ✟P ✒ We have , , and . P ✓P ✒ P ☞ ✙▲➇❫✒❁✄✖❃ ✎ ✟ ✙ ✟ ✎ ☞❊➇❴✒ ☞❊➇ ✡ ✟ ✒ Hence, given that , we have P ✓P ✒ P ❃ ✎ ✟ ✙ ✟ ✎ ☞❊➇ ✍ ✏✓✒ ☞❊➇❢✍ ✏✓✒●✧✾☞❊➇❢✍ ✟ ✒ ☞❊➇ ✍ ✟P ✒ ☞ ✙▲➇❢✍⑩✙✜✒❁✄ ✄✕❃ ✎ ✓P ✒ ✟ ✙ ✟ ☞ ✎ P ✌✳✎ P ☞❊➇❢✍✥✏✓✒ ☞❊➇⑥✍ ✟P ✒ ➂ ☞ ✙▲➇❢✍ ✏✓✒✻☞ ✙▲➇⑥✍⑩✙✜✒ ✎

✁✍ ✏✓✒ ☞ ❃

✟

✟

✟

✟

✎

✟

✟

✟

✝

✝

✂

2.

✟

✂

✠

✠

✠

✌

✌

✝

✂

✝

✌

✝

✌

✞

✞

✂

3.

☎

✌

✝

✂

☎

✌

✝

✝

✂

☎

✂

✓

✌

4.

✌

✝

☎

✝

✌

✌

☎

✝

✌

✡

✡

☎

☎

☎

☎

☎

✝

✌

✠

5. 6.

✌

✌

✞

✠

✌

✝

✞

✌

✝

✞

✞

☎

☎

☎

➇ ✍⑩✏

so the✍ duplication formula is valid for . Thus, if the formula is valid for ➇ ✙⑨ ✍ ✏ ➇ , and by induction, it is therefore valid for all .

P P P ✟ ✼✄✔ ✕ ❘ ➇ ➇★✄ P ☞ P ☞ ✶ ✖ ✟ ✟ ✡ ✏✓✒ ➂ ✟ ☛✒ ✄ ✟ ☞ ✟ ☞ ✶ ✡✕✏✓✒❨✒ ✶ ✍ P ✒ ✛ ❃❢✧ ✛ ❃ ✏ ✧ ✟P ✧ ✜✟ ✧✈☞ ✟P ❧ ✟ ✙ ✏❝✧✓✂ ✙ ✧✗✧✗✧▲☞ ✟P ✶❬✒

✒

7.

✌

✝

✶

✟

it is also valid for

☞ ✟P ❨✒ ✽ ☞ ✟P ✶✑✡✖✏✓✒ ✶ . If is even, this is ✏✓✒ ✧ ❃ ✄ ✄ ✏✯✧✩✞✂✙✘✧✗✆✧ ✧✗✯✧✧✩✗☞ ✶✧✗✧ ✍⑤ ✶ ✙

✟

✆

➇ ✙✌✍

✏ ✧✓✙✂✧✗✧✗✧✠☞ ✟P ✶ ✍ ✟P ✒✬✧ ✛ ❃ ✏ ✧ ✯ ✙✦✧✆✯✧✗✧✗✧▲☞ ✶s✍✥✏✓✒ ✄ ✄ P P ✙ ✏❤✧✩✞✂✧✗✧✗✧ ✶ ✟ ✧ ✜✟ ✧✗✧✗✧▲☞ ✟ ✶❬✒ ✛ ❃ (There ✶s✄ ✏is one more factor in☞✎the ✏✓✒ denominator ☞ ✟P ✒❨✽✠✙ ☞ ✜✟ ✒❁✄✳than ✏ in the numerator, which absorbs the extra factor of 2. If , we simply have .) ✒ ✒ ✒ ☛ P ✟ ✼✄✔ ✕ ✟ ☛ ➇ ➇ ✕ ✟ ✄✼ ✔ ✕ ✟ ☛ ➇ ➇ ✕ ✟ ✄✼ ✔ ✕ ✟ ☛ ✎ P ➇ ➇ If

is odd, it is

✆

8. Since

✝

✆

✌

✆

✌ ✝

✌ ✝

, by Exercise 7 we have

✙ ✧✙✯✧✗✧✗✧▲☞ ✙ ✒ ✕ ❤ ✭ ✏ ✧▲✞✂✧✗✧✗✧▲☞ ✙ ✍ ✏✓✒ ✧ ❃ ✕ ✙✭✧✙✯✧✗✧✗✧▲☞ ✙ ✍ ✙✜✒ ✏✯✧✓✞✂✧✗✧✗✧✠☞ ✙ ✡✖✓✏ ✒ ✙ ✧✆✯✧✗✧✗✧✠☞ ✙ ✒ ✦ ✙ ✏ ✧✩✞✂✧✗✧✗✧▲☞ ✙ ✍⑤✏✓✒ ❤ ✟

✟

✟

✟

✟

✟

✄

7.7. Stirling’s Formula

57

✙ ✧✣✯✧✗✧✗✧✩☞ ✙ ❨✒ ✽✈✏s✧❍✞✂✧✗✧✗✧▲☞ ✙ ✍ ✏✓✒ ✆ ✕ ✟P ❃ ✕ ☞ ✙ ✡⑦✏✓✒✛✆ ☛ ✽✠✙ ☛ gives . Since ✙ ✆ ☛ ✎ P , the sequence ➀ P ✆ ☛ ➅ is increasing, and we have just seen that ☛ ☛✎ P❃ it is ☞ ✙ ✡❋✏✓✒✛✆ ☛ ✽✠✙ ➅ ✞⑤q ❃ ✟ ✟ P P P ➀ bounded above by , so it converges to a limit . The sequence also converges ✞ ✙ ❃ ✞✬✤ ✟ ❃ . In short, ✞◗✄ ✟ ❃ . to , but its terms are ✟ , so ➉ ☞❊➇❴✒ ✙✖✰ ☞ ✽ ➇❫✒ ☛ P ➉ ☞❊➇❴✒❤✄ ☞ ✡✌✏✓✒ ✎ P ☞❊➇❢✍⑩➇❴✒ ☞❊➇❫✒❙✡ ☞❊➇⑥✍✪❛❜✒ ✎ P ☞❊❛❨✒ ❛ ➉❄✄ ➈❺❵ ✝ ➈ ❵ ✝ ❵ (a) For ✝ ☞ ✡✝,✏✓✒❋✄ ☞ ✒ ➈❺❵ ✕ ✏ since ✝ ✎ ☞❊➇❋then ➉ ☞❊➇❫✒ ✝ ✝ . Actually there is more to be❛✲said ✄ ➇ when ✝ ☞ ✽ ➇❫✒ , since ✍✥❛❜✒ the ☞❊integral ❛❨✒ ❛✲✄ defining ➈❺❵ . But ✎ is☞❊➇❁improper: ☞❊➇❁✍ ✒✮✡ ✍✭❛❨✒ ✎ P ☞❊the ❛❜✒ integrand ❛ ➇ blows up at ✰✣➂ ✦➉ ☞❊➇❫✒✗✈q ❵ ✕✺✥ ❵ ☞❊➇❁✍ ✒✗✈q✝ ❵ ✎ ☞❊➇❁✍✭.❛❜For ✒ ✎ P in☞❊❛❜a✒ finite ❛✩✍ interval, ☞❊➇❁✍✭❛❨say ✒ ✎ P ➈ ☞❊❛❜✒ , we ❛ q have ❵ ☞❊➇❁✍❀❛❜✒ ✎ P ❛❁, so✄ ✎ ❵ ✝ ❵ ✝ ❵ and ✝ ☞ ✽ ➇❫✒ ✎ ☞❊➇❦✍◗❛❜✒ ☞❊❛❨✒ ✬ ❛ ✣ ☞❊➇★✍◗❛❜✒ ✎ P ☞❊❛❨✒ ❛ ➇ ✐ ✰✣➂ ✦➉ . Hence uniformly for , so the ✝ ❵ ➈ ✣ ✰ ❵ differentiation in the limit as is justified. ✚ ✁ ➉ ➉ ☞❊➇❴✒⑥✄ ☞ ✒ ☞ ✞●✒ ➉ ✎ P ☞❊➇✪✍❈❛❜✒ ✎ P ☞❊❛✤✍ ✂▲✒ ✎ P ☞ ✂▲✒ ✂ ❛ ✚ order of in. Reversing the (b) ➈ ➈❺❵ ➈ ✝➂ ✞ ✕ ✏ ❵ ✧✗✧✗✧ ✂ ❛ ✝ tegration (OK even if since the integral is absolutely convergent) turns into ✧✗✧✗✧ ❛ ✂ ✄ ☞❊❛☛✍✡✠✂ ✒❨✽✣☞❊➇❦✍✍✂▲✒ . Now, in the inner integral, make the substitution ✁ , so that ✁ goes ❛ ✂ ➇ ❛☎✍ ✂❀✄♠☞❊➇✤✍ ✂▲✒ ➇✤✍❧❛✿✄ ☞❊➇✤✍ ✂✠✒✻☞✎✏❴✍ ✒ ❛☛✄ ☞❊➇❀✍ ✂▲✒ Multiplying ✟ ✒❨✽✣☞ by ✟ ✍✖✏✓✒ ✆ ✄ ✆ P✩☞ through ✟

✟

✟

✟

✟

✟

✟

✟

✑

✑

☎

✑

✑

✂

9.

✌

✂

✌

✌

✝

✑

☎

✑

✂

✑

✑

✌

✌

✝

✌

☎

✄

✁

✑

✝

✑

✌

✑

✑

☎

✑

☎

☎

✁

✁

☎

✌

✝

✡

✑

✌

✝

✌

✝

✡

✑

✑

☎

☎

✡

✡

✄

✁

✌

✌

✝

✌

✝

✌

✑

☎

✑

✂

✂

✝

✝

✌

✌

☎

✝

☎

✝

✌

✝

✌

☎

✝

✌

✌

✁ ,

from 0 to 1 when goes from to . Then so

P ➉ ➉ ☞❊➇❴✒❁✄ ☞ ✒ ✏ ☞ ✞●✒ ➈ ❺➈ ❵ ✁ ✝ ✄ ☞ ☞ ✝ ✒ ➂ ✞●☞ ✞●✒ ✒ ☞❊➇ ✍ ✂✠✒ ☛✂ ✎ P ❵ ✝ ☞ ✟ ✒❨✽ ☞ P ✒ ➉ ❚ ✜ 10. (a) The series is ➈ ✜ ✒ ☞ series diverges. P ☞ ✄ ✒✫❚ ☞ r ✡ ✏✓✒❨✽ ✁

✑

✂

✎ P ☞✎✏❝✍

✑

✂

✂

☎

✑

✂

✌

✒

✜

❃ ✎ ✆✒✟ ❚

11. The series is ✒ converges if and only if

✟

☞ ✡

✟

✟

☛✂

✪✒

✟

✜ ✒.

➈❺❵

➉ ☞❊➇❴✒ ✄

✎ ✓P ✒ ✜

✄

The th term is asymptotic to , so the ✎ ✄✄✒ ✪ , so the series . The th term is asymptotic to ✟

✟

✟

☞ ✡ P ✒❨✽ ☞ ✡♠✏✓✒ ➉ ✆ ➈ ✙⑤✙ ✟ ✟

✑

✂

✌

✡ ✪ ✒❨✽ ☞ ✡

✟

✪

✌

☎

✑

✪

✌

✑

✌

✂

(b) The series is converges.

✔✁ ,

✌

✒ ✎ P ❊☞ ➇ ✍ ✂✠✒ ☛✂ ✎ P ☞ ✂✠✒ ❛ ✂ ✁ ❵ ❊☞ ➇ ✍ ✂▲✒ ☛✂ ✎ P ☞ ✂▲✒ ✭ ☞ ✂✠✒ ✂✦✄ ☞ ✏ ✡✍✞●✒ ✂ ✄ ❵ ✝

☎

✂

✁ , and

✟

. The th term is asymptotic to . For Raabe’s test, we have ✟

✟

✆ ✄ ✏✯✍ ❙✏✤✍ ✙ ✏ ✡✥✙ ☛ ☞✎✏❁✡⑩❛❨✒ ✆ ☛✳✏❁☎ ✡ ✟❍❛ ❛ ✟❴✽✣☞ ✙ ✡◗✙✜✒ for large, and the limit is ✟❴✽✠✙ . Since for small, this is✟ approximately ⑤✙ ✙ ✟ ✕⑤✙ but is indecisive for ✟⑥✄✕✙ . and divergence for Hence Raabe’s test gives convergence for ☞✳✱❤✡ ✒❨✽ ☞✝✆❙✡ ✒ ☛ ✎ ☞ ✗❫✡ ✒❨✽ ✄ ☞ ✗❫✡ ✒❨✽ ☞ ✡❲✏✓✒ ☛ ✎ P ☛ 12. ✎ large, ✎ ✎ P ✆ and ✱❑✡ ✗❙✍ ✆❄for✍◆✏ ✕⑨ ✍✑✏ so the✱❑✡ th✗ term ✕ ✆ ✏✯✍ ✱ ✱☛ ☛ P ✄

✏❝✍

✂

✟

✞

✙ ✡✖✏ ✙ ✡✥✙

✆

✎ ✆ ✒ ✟ , so the series ✟

✂

✟

✟

✄

✂

✞

✠

✠

✟

✟

✟

✟

✟

✟

✠

✠

✟

✟

is asymptotic to

✟

✞

✆

✟

✟

✟

✟

✟

✟

✟

✟

✞

. Hence the series converges if and only if

7.7 Stirling’s Formula

, i.e.,

.

✺✍✓❴☞✎✏■✡⑩❛❜✒ P ✍ ☞✎✏■✡⑩❛❜✒ ✎ ✟ P ✒ ✺✍✓➃☞✎✏✦✡ remainder ✍ P ✟ ❛is✟ q ✑s☞❊❛❨✒❢q , ✰ Lagrange’s ❛ ✤ ✰ form of the ❢✍✚ ✱ ☞✳✱s✡ ✍ immediately ✟ for ✝ ✒☎✑s✳☞ ✱✫✽ ✒ ✰♣q ✱★qr.✏ Thus, ✟ ✟ P , and for this is, in absolute value, at most ✏✓✽✠✙ ✖ ✡ ☞ ✡ ✟ ❨✒ ✽✠✙ ✟ ✄✳✏✓✽ ✡✖✏✓✁✽ ✟ q⑤❂✜✁✽ .

1. Since the ✺✍✓❴☞✎second ✏✭✡❈❛❜✒ derivative ✄ P ❛☛✡ ✑s☞❊of❛❜✒ ✝ gives ☞✳✱✣✽ ✒❨✝✒❤✄❖✍ ☞✳✱❧✍ ✒✻☞✳✱✫✽ ✒❁✍t✳☞ ✱ where ✡ ✍ ✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

Chapter 7. Functions Defi ned by Series and Integrals

58 2.

☞ ✙ ✒ ✽✣☞ ✒ ✟ ✙ ✟ ☛ ✟

✎

✟

☞✙ ✒✟☛

✣✦✥

✎

☛ ☞

P✓✒ ✟ ✎ ✟ ☛ ✛ ✙✠❃❄✽ ✟ ☛ ☛ P ✎ ✟ ☛ ☞ ✙✠❃❄✒✎✙ ✟ ☛ ✄✳✏✓✽ ✛ ❃ ✂

✌

✂

✟

approaches 1 as

✙✂✧✚✯✧✗✧✗✧✮✙ ✟

✟

, where ✟

.

✄✕✙ ☛ ☞✎✏■✧✹✙✂✧✗✧✗✧ ✒✿✄✕✙ ☛

✏■✧✹✞✂✧✗✧✗✧✠☞ ✙ ✪ ✍ ✏✓✒❁✄ ☞ ✙ ✒ ✽✠✙✂✚✧ ✯✧✗✧✗✧✗✙ ✄ ☞ ✙ ✒ ✽✠✙ ☛ ✙ ☛ ➉✟ ☞ ✙ ✒ ➉ ✟ ☞ ✙ ✡✌✏✓✒❨✽ ✙ ☛ ➉ ✟ . and ➈ , so ➈ ➈ ✙ ☛ ➉✪ ➉✪ ☞ ✙ ✒ ✪ ☛ ☛ P ✩✎✠✪ ☛ ✧ ➂ ✄ ✧ ➈ ➈ ☞ ✙ ✒ ➉ ✟ ☞ ✙ ✡✖✏✓✒ ☞✙ ✒ ➉✟ ✙✫☞ ✙ ✖ ✡ ✏✓✒ ✪ ☛ ☛ ✟ ✎✠✪ ☛ ➈ ➈ P P P ✟ ✄ ❃ ✪ ✧ ☞✎ ✟ ✧ ✯ ✣ ✥ ✟ , or ◗✄ ✛ ✙✠❃ . ✪ as . Thus ✪

3. Note that , and Thus the numerator and denominator of Wallis’s fraction are the whole fraction is ✟

✟

✟

✎

✟

✟

✟

✎

✎

✟

✟

✂

✟

✎

✟

✎

✟

✟

✂

✟

✞

which tends to

means that the ratio

✎

✟

✞

✟

✟

✞

✟

✞

✎

✟

✎

✟

✟

✟

✎

✎

✟

✎

Chapter 8

Fourier Series 8.1 Periodic Functions and Fourier Series ✱ ☛ ✄r✰ ✗ ✼✄✔✖✕ ✄⑦✍ ☞ ✙✜✽ ❃❄✒✈✸✻✺✜✼ ✄ ☛ ✄ ☞✎✏✓✽▲❃❄✒ ✎ ❵ ☞ ✵ ✒✈✼✄✔ ✕ ❙✵ ✾✵ ✄ ☞ ✙✜✽▲❃❄✒ ✵ ☎✍◗ ✵ ✏ ❙✵ 1. ❵ ☞ ✙✜is ☎✽ ❃ ✄✕✙ ❙✝ ✽ odd, ❃ ✒ ✏❤so ❄ ✍◗☞✎✍✑✏✓✒ ☛☎➉ and if is odd, say ; thus the Fourier series ☞ ☎✽▲❃❄✒ ➈ ❚ P ✼✄✔ ✕❙☞ ✙ . This ✍ ✏✓is ✒ 0➉❊✽✣if☞ ✙ ❞is✍ even ✏✓✒ and . is ✒ ➈ ✵ P ☞✎✏❝✍✪✸✻✺✜✼➃✙ ✒ ✼ ✔✖✕ ✟ ✄ ✄ ✟ ✵ , and the thing on the right is a Fourier series! ✵ 2. By the double angle formula, ✗ ✚ ✄ ✰ ✱ ✭✄ ✝ ☞ ✜ ✙ ▲ ✽ ❄ ❃ ✒ ✼ ✄ ✔✕ ✄ ☎✽ ▲ ❃ ☞ ✒✈✸✻✺✜✼ ✄ ✙❈✰ ✱ ☛ ✄✝☞✎✏✓✽▲❃❄✒ ✎ P❵ ✵ , , and for , 3. ❵ is even, so ☛ ✵ ☎✵ ❙✵ P ✾✵ ☞ ✙✜✽▲❃❄✒ ✼ ✔ ✕ ✸✻✺✜✼ ✄ ✄ ☞✎✏✓✽▲❃❄✒ ✄✼ ✔✖✕❙☞ ✡❋✏✓✒ ✍❧✄ ✼ ✔ ✕❙☞ ✍ ✏✓✒ ➉ ✄ ☞✎✏✓✽▲❃❄✒ ☞ ✡❋✏✓✒ ✎ ✍⑥☞ ✍❢✏✓✒ ✎ ➉ ✏❾✍ ✵ ✾☞✎✍ ✵ ✏✓✒ ☛ ✎ P ➉❊✽✣☞ ✟ ✍✕ ➈ ✏✓✒✔❃ ✵ ✵ ✾✵ ➈ ☞✎✍✑✏✓✒ ☛ ✎ P ➉❤✄ ✵ ✍✤✙ ✏✘❙✍✌ ✍ ☎✽✣☞ ➈ ✟ ✍✕✏✓✒✔❃ P . This is 0 if is odd and if is even, ➈ ✟ ✍ say ✄✕✙ ✱ ✡ ❚ P ✱ ☛ ✸✻✺✜✼ ✄ ☞ ✙✜✽▲❃❄✒❁✍⑤☞ ☎✽▲❃❄✒ ❚ P ☞♦✸✻✺✜✼❾✙ ✒❨✽✣☞ ✏✓✒ ❙✵ ✵ ; thus the Fourier series is ✟ . ✒ ✒ P ✟ ✟ ✸✻✺✜✼ ✱ ✄ ☞✎✏✓✽▲❃❄✒ ✄ ✗ ❖ ✄ ✰ ✱ ✄ ☞ ✜ ✙ ▲ ✽ ❄ ❃ ✒ ✟ and ☛ ✵ 4. ❃❵ ✟ is✽✠✞ even, so✙ ☛ ✰ ✟ ✸✻❙✺✜✵✼ ✾✵ . The ✄ constant ☞ ✙✜✽ ✟ ✒ ✸✻term ✺✜✼ is✡ ☞ ✟ ✽ ✒▼✍♣☞ ✙✜✽ ✜✓✒ ✵ ➉✓✼✄✔ ✾✕ ✵ by✄ parts ❙✵ ✾✵ ✵ ❙✵ ➈ ✵ ❙✵ , ✟ ✒✔❃❁☞✎✍✑✏✓✒ ☛✉ ✱ . ✄ For☞ ✙✜✽▲❃❄✒✻☞ ✙✜,✽ integration ❍☞✎✍ gives ✏✓✒ ☛✫✽ ✟ ✵ so ☛ . ✆ ✄ ☞✎✏✓✽✠✙✠❃❄✒ ✩✎ ❩ ☛ ✄ ✎ 5. Here ❩ it is easier to use the exponential form of the series: ☛ ✾ ✵ ☞ ✎ ☛✍✌ ✽✠✙✠❃❁☞ ✗✯✍ ✒✿✄ ☞✎✍✑✏✓✒ ☛ ✍ ✩✎ ➉❊✽✠✙✠❃❁☞ ✗✯✍ ✒❁✄♠☞✎✍✑✏✓✒ ☛❾☞♦✼✄✔✖✕✁ ✗✮❃❄✒❨✽▲❃❁☞ ✗✯✍ ✒ . ➈ ✎ ✱ ☛ ✄ ✰ ✗ ☞ ✙✜✽▲❃❄✒ ☞♦❃⑦✍ ✒✈✄✼ ✔ ✕ ☛ ✄ 6. ❵ is☞♦❃✥odd, ✍ ✒✈✙✼so✔ ✕ ✄ ☞✎✏✓and ✽ ✒ ☞♦❃ ✍ ✒✈✸✻✺✜✼ ✡ ✵ ☞✎✏✓✽ ✟ ✒✻☞♦✵ ❃✥✍⑨✙ ❙✵ ✒✈✾✼✙✔ ✵ ✕ . Integration ✍❖☞ ✙✜✽ ✜ ✒✈✸✻✺✜by ✼ parts ✗ gives ☛ ✄ ✵ ✵ ❙ ✵ ✾ ✵ ✵ ✵ ❙ ✵ ✵ ❙ ✵ ❙ ✵ , so ☞ ✙✜✽▲❃❄✒✻☞ ✙✜✽ ✜▲✒ ✏✭✍t☞✎✍✑✏✓✒ ☛☎➉ ★✜✽▲❃ ✜ ✄✚✙ ✍⑤✏ ➈ is 0 when is even and when is odd, say ; hence ☞ ★✜✽▲❃❄.✒ ❚This ✍⑤✏✓✒ P ☞ ✙ ❞✍ ✏✓✒ ✎✢✜❫✄✼ ✔ ✕❙☞ ✙ the Fourier series is ✒ ✵ . P ✗ ✄ ✰ ✱ ✍❝❃■➂❜❃❾➉ ✟ ❵ ➈ 7. ❵ is even, so ☛ .✙ The constant term is the mean value of on , ✄ which con✰ ✱ ☛ ✄ ☞ ✙✜✽▲❃ ✱✫✒ ✸✻✺✜✼ ✍ ✙✜✽▲❃❁☞♦❃❦✍ ✱✣✒ ➉ ✸✻✺✜✼ ☞ ✙✜✽▲❃ ✱is 0✒✈✼✄by ✔✕ ✱✫✡ struction ➈ ❙✵ ☎✵ ✙✜✽▲❃❁☞♦❃❦✍ of✱✫✒ ❵ .➉✓For ✼✄✔ ✕ ✱ ✄ ,✙✜✽ ✱❴☞♦❃❦✍ ✱✫✒ ➉✓✼✄✔ ✕ ✱ ❙✵ ☎✵ ➈ ➈ . P ✗ ✄ ✰ ✱ ✍❝❃■➂❜❃❾➉ ✏✓✽✠✙✠❃ ☛ ✟ so . The constant term is the mean value of on , namely . For 8. ❵ is✙ even, ❵ ➈ ✰ ✱ ☛ ✄ ☞ ✙✜✽▲❃ ✱ ✟ ✒ ✳☞ ✱◆✍ ✒✈✸✻✺✜✼ ✄ ☞ ✙✜✽▲❃ ✱ ✟ ✒✻☞✳✱❇✍ ✒✈✄✼ ✔ ✕ ✍❖☞ ✙✜✽▲❃ ✱ ✟ ✟ ✒✈✸✻✺✜✼ ✄ ✵ ❙ ✵ ☎ ✵ ✵ ❙ ✵ ❙ ✵ , ✙✫☞✎✏✤✍✪✸✻✺✜✼ ✱✣✒❨✽▲❃ ✱ ✟ ✟ . ❘ ☞ ✶s✍t✏✓✒✛✙ q ✱ ✕ ✶ ✙ ✄ ✡ ❘ ☞❊➇❴✒ ➇ 9. First method: Suppose . Then (the integrand is ❵ in ❘ P all integrals). By periodicity of ❵ , the second integral on the right equals ☞ ✎ ✌ , so adding it to the ✆

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59

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☞✳✱✫✒✯✄

❘

Chapter 8. Fourier Series

❘✎ P ☛

first integral gives

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❵ method: Let . ❵ is discontinuous we have points where ☞ ✫✒☛✄ ☞♦✰✾✒ .

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. Second ✱ is☞✳✱✫a✒❁continuous ✄ ☞✳✱❝✡ ⑧function ✒❫✍ ☞✳✣✱ ✒✿of✄✖✰ , and except at the finitely many ❵ ❵ . It follows that is constant, so ☛

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8.2 Convergence of Fourier Series 1. Let ❵ be the sawtooth wave of P Example 1.

♦☞ ❃⑥✍ ✙ ✒✿✄ ❚ P ☞✎✍✑✏✓✒ ☛ ☛ P ✽ ➉✓✼✄✔ ✕❙☞ ❃⑥✍ ✙ ✒✿✄ ❚ P ☞✎✏✓✽ ✒✈✼✄✔✖✕✤✙ ✟ ✵ ❙✵ ❙✵ . (a) The function depicted is ❵ ✒ ➈ ✒ ✏✭✡✝☞✎✏✓✽▲❃❄✒ ☞ ✙ ✍t❃❄✒♣✄ ✏✦✡♠☞ ✙✜✽▲❃❄✒ ❚ P ☞✎✍✑✏✓✒ ☛ ☛ P ✽ ➉✓✼✄✔ ✕❙☞ ✙ ✍ ❃❄✒♣✄ ❵ ✵ ❙✵ (b) depicted is ✒ ➈ ✏❝✍ The ❚ P function ✒ ☞ ✙✜✽▲❃ ✒✈✙✼ ✔ ✕✤✙ ❙✵ . P❃ ☞ ✒ P ❃✘right P ❃✦✼✄✔✖✕ ❵ ☞ ✵ ✒✘✄ The ☞♦❃ ✟ ✽✠function ✞✜✒❄✡ ❚ ❵ P ✵ ☞✎✍✑here ✏✓✒ ☛ ✽ is ✟ the ➉✓✸✻✺✜function ✼ ☞ ✍ P of❃❄✒ Exercise ✸✻✺✜✼ 4, ☞ ✂ 8.1, ✍ P shifted ❃❄✒❁✄✖✸✻✺✜to✼ the ✸✻✺✜✼ by✡❲✪ ✼✄✔ .✕ Hence ✪ ✪ ✵ ✪ , and ✵ ✪ ❙✵ ❙✵ . ✒ P➈ ✏✪✡ ☞ ✒❨✒ ☞ ✒ ✄ P ✡✕❵ ☞ ✙✜☞ ✵ ✽▲✒❃❄✒ ❚ ✄ P ✟ ☞ ✙ ☞✎✦❞ ✵ P ✄✼ ✔ ✕❙where (a) is the square wave of Exercise 1, ✂ 8.1; hence ❵ ✵ ✍ ✓ ✏ ✒ ✎ ☞ ✙ ✦ ✍ ✏✓✒ ✟ ✒ ✵. ☞ ✒✕✄ ✟P ☞❜✔✼✄✔✖✕ ❴✡⑦✄✼ ✔ ✕ ✒ P ✕ ❃ , so by Exercise 3, ✂ 8.1, we have ❵ ☞ ✵ ✒✌✄ ☞✎✏✓✽▲❃❄✒✮✍ (b) ☞ ✙✜✽▲❃❄❵ ✒ ❚ ✵ P ☞♦✸✻✺✜✼❾✙ ✦ ✒❨✵ ✽✣☞ ✦ ✟ ✍ ✵ ✏✓✒❄for ✡ ✟ ✼✄✵ ✔ ✕ ✵ ✵. ✒ ☞ ✒❤✄ ☞♦❃❇✍ ✱✫✒❨✽✠✙✠❃❾➉ ☞ ✒✿✡✚☞♦❃❦✍ ✱✣✒ ✎ P ➉●✄ ☞✎✏✓✽✠✙✠❃❄✒■✡ ☞♦❃❇✍ ✱✫✒❨✽✠✙✠❃❾➉ ☞ ✒ (c) ❵ ✵ ➈ ✵ where is the function of ☞ ✒■➈ ✄ ✵ ☞✎✏✓✽✠✙✠❃❄✒●✡✖☞✎✏✓✽▲❃❄✒ ❚ P ☞♦✄✼ ✔ ✕ ✱✫✒❨✽ ➈ ✱✾➉✓✸✻✺✜✼ ❙✵ . Exercise 7, ✂ 8.1,P so ❵ ✵ ✒ ➈ ☞ ✒ ✄ ✟ ☞ ✍ ✩✎ ✒ ✕ ❃ ✗ ✄ ✏ ☞ ✒ ✄ 5, ✂ 8.1,☞✎✍✑with (d) ☞♦✄✼ ✔ ✕✁❵ ✯❃❄✵ ✒❨✽✠✙✠❃❾➉ ❚ ✎ ☞✎✍✑✏✓✒ ☛❬✽✣☞✎✏❴for ✍ ✒ ✵ ➉ ❩ ☛ ✍ ✎ , ❩ ☛ so➉❴✄ by ☞♦✄✼ Exercise ✔✖✕✁✯❃❄✒❨✽▲❃❾➉ ❚ ✓ ✏ ✒ ❬ ☛ ✣ ✽ ✎ ☞ ❴ ✏ ✍ ✒ ➉ ✾✼✄, ✔✖✕ ❵ ✵ ✎ P P ➈ ✒✒ ➈ ➈ ➈ ✒ ➈ ☞✎✍ ✒ ☞✎✍✑✏✓✒ ☛ ☞✎✏❉✍ ✒ ✎ ✒ ✍ ☞✎✏❞✡ ✒ ✎ ➉ ✾✼✄✔ ✕ ❙✵ . The ✄ P ➈ ❙ ✵ sum of the th and th terms is ☞✎✍✑✏✓✒ ☛ ✎ ✙ ✽✣☞✎✏✂✡ ✟ ✒ ➉✓✼✄✔ ✕ ➈ ❙✵ . ✄✳✰ ☞ ✙✜✽▲❃❄✒❁✍✖☞ ☎✽▲❃❄✒ ❚ P ✏✓✽✣☞ ✦ ✟ P ✍⑤✏✓✒✤✄r✰ ❚ P ✏✓✽✣☞ ✦ ✟ ✍⑤✏✓✒❝✄ ✟P (a) Setting ✵ gives result also ☞ ✦ ✟ ✒ ✍ ✏✓✒ ✎ P ✄ ✟ ☞ ✙ ✦❈✍♣✏✓✒ or ✎ P ✍❢☞ ✒ ✙ ✦❲✡❦✏✓✒ ✎ P ➉ , so that the, a series P ➈ obtainable from the observation that ❃ ✏ ✄ ☞ ✙✜✽▲❃❄✒⑦✍ ☞ ☎✽▲❃❄✒ ❚ ☞✎✍✑✏✓✒ ✮ ✽✣☞ ✦ ✟ ✍ ✏✓✒ tele✄ ✟ ☛ Setting P ✵ gives , or scopes. ❚ P ☞✎✍✑✏✓✒ ✮ ✽✣☞ ✦ ✟ ✍ ✏✓✒❁✄ ☞♦❃❦✍⑩✙✜✒❨✽✁ . ✒ ✄ ❃ ❃ ✟ ✄ ☞♦❃ ✟ ✽✠✞✜✒❝✡ ❚ P ✏✓✽ ✟ ❚ P ✏✓✽ ✟ ✄ ❃ ✟ ✽✠❅ ✄ ✰ (b) ; setting ✵ gives ✰✲✄ Setting ☞♦❃ ✟ ✽✠✞✜✒❙✵ ✌ ✡ ❚ P gives ☞✎✍✑✏✓✒ ☛✫✽ ✟ ❚ P ☞✎✍✑✏✓✒ ☛ ☛ P ✽ ✒ ✟ ✄❈❃ ✟ or✽✈✏✱✙ ✒ or ✒ . ✒ ✄❉✰ ✏✇✄ ☞♦✙✼ ✔ ✕✁✯❃ ✗✗✒❨✽▲❃❾➉ ❚ ✎ ☞✎✍✑✏✓✒ ☛❬✽✣☞ ✗✑✟ ✍ ✒ ✄ ✰ ✏✓✽ ✗ gives . The term is (c)✙⑦ Setting ✵ ➈ ✒ ✰ ☞✎✍ ✒ ✙ ✒ ✗✠☞✎✍✑✏✓✒ ☛ ✽✣☞ ✗ ✟ ✡ ✟ ✒ ✏★✄ ☞♦✙✼ ✔ ✕✁✯❃ ✗✗✒❨✽▲❃❾➉ , ☞✎and ✏✓✽ ✗✗✒❑for ✡ thus ✙ ✗ ❚ P the ☞✎✍✑✏✓sum ✒ ☛❬✽✣☞ ✗ of✟ ✡ the ✟ ✒ th➉ and❚ P ☞✎P ✍✑th✏✓✒ terms ❬✽✣☞ ✗ ✟ is✡ ✟ ✒★✄ ☞♦❃ ✗❾✸✗✼❜✸ ✭❃ ✗✑; ✍✳ ☛ ✏✓✒❨✽✠✙ ✗ ✟ ➈ ✄ ➈❃ . Setting ✵ gives ☞♦✄✼ ✔ ✕✁✯❃✒ ✗✗✒❨✽▲❃❾➉ ❚ ✎ ✏✓✽✣☞ ✗❀,✍☞or ✒✦✄ ✒ ✟ ☞ ✡ ✎ ✒✭✄✝✸✻✺✜✄✼ ✤❃ ✗ ➈ . (The function represented by the series ✒✒ ✄ ❃ ✵ , so the sum of the series is the average and right limits!) is discontinuous at ✟✒ ✄ ✰ ✏✓✽ ✗ ✙r✰ the sum of the th and of☞✎✍ the✒ thleftterms ✙ ✗✗✽✣☞ hand ✗ ✟ ✡ Again the term is , and for is , so ☞✎✏✓✽ ✗✗✒❄✡ ❚ P ✙ ✗✗✽✣☞ ✗ ✟ ✡ ✟ ✒✿✄✖❃ ✸✻✺ ✄✤❃ ✗ ❚ P ✏✓✽✣☞ ✗ ✟ ✡ ✟ ✒✿✄ ☞♦❃ ✗❾✸✻✺ ✄✭❃ ✗✂✍ ✏✓✒❨✽✠✙ ✗ ✟ hence☛ P ✒ . ✒ ✄ P❃ P ❃ ✟ ✄ ☞ ★✜✽▲❃❄✒ ❚ and ☛ P ✎ ☞ ✑ ✍ ✓ ✏ ✒ ✣ ✽ ☞ ✙ ✦✖✍❢✏✓✒✫✜ ✎ ☞ ✑ ✍ ✓ ✏ ✒ ✣ ✽ ☞ ✙ ✦❈✍❢✏✓✒✫✜✤✄✖❃ ✜✓✽✠✞▼✙ ❚ P P ✮ ✮ ✟ gives ✪ or ✒ . (d) Setting ✵ ✒ ✐ ❥ ✙ ✰ ✙❏✰ ☞ ✡ ✍■✒✭✍ ☞ ✡ ✒✗ ✕ ✽✠✞ Given and ☞ ✡ ✍■,✒❤choose enough ✰ ✕✒✍ ✵ ✕ ✗ ✍ ☞ ✍❀✵ ✒✗✭✕ ✽✠small ✞ ✍ ✕ so✍ that ✕ ✰ ❵ ✵ ✄❞★✼ ✛✣❵ ✢ ✵ ✎ ☞ ✒✗when ✵ and ❵ ✁ ✵ ❵ ✰ ✵ q ▼☞✎✍■✒sq when ✵ ❵ ✵ . By . Let ✽✠❅✠❃ ✍ ✐ ✍❝❃■➂✱✍ ➉ (8.23), there exists such that for ➈ ✵ ❿❲➈ ✵ ➂❜❃❾➉ when ✁ ✕ ✁ ✕ ✏ . ✟

✟

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8.3. Derivatives, Integrals, and Uniform Convergence

✁

✕

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61

☞ ✡ ■✍ ✒✛✙ ▼☞✎✍■✒ ✂ ✍ ✂q

☞ ✽✠❅✠❃

❵ ✵ Assume now that . We have ✎ ☞ ✡ ✍■✒✛✙ ✾☞✎✍■✒ ✍❤ ✕ P ✙ ✜☞✎✍●✒ ✍✚✄ ✙ ✜☞✎✍■✒ ✎ ❵ ✵ . Moreover, likewise P ✍ ✽✠❅ ✙ ▼☞✎✍■✒ ✍ ✟ , and this last integral is between 0 and . Now, we have ✆

✁

✻✒ ☞♦❃⑩✍ ✒ ✕ P ✵ , and ✍❲✍ ✙ ▼☞✎■ ✍ ✒ ✍r✄ ✁

✓

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☞ ✧ ✡ ✍■✒✛✙ ☞✎✍●✒ ✍❇✍ ✙ ✏ ☞ ✡✑✒✿✄ ✖❵ ✵ ❵ ✵

✏ ✄ ☞ ✧ ✡ ✍●✒✬✍ ☞ ✡ ✒ ➉ ✙ ☞✎✍●✒ ✍⑥✡ ☞ ✡✑✒ ✙ ☞✎✍●✒ ✍◆✍ ✙ ◆➈❺❵ ✵ ❵ ✵ ❵ ✵ ☞ ✽✠✞✜✒ ✙ ▼☞✎✍■✒ ✍✥✄ P The first term on the right is at most P P and the second term is ☞ ✽✠❅ ✒❁✄ ☞ ✡ ✍■✛✒ ✙ ✜☞✎✍●✒ ✍♣✍ in✟P absolute ☞ ✍✭✒✗ ✕ value, ✜ . Adding up these results, at most ❵ ✵ ☞ ✡✬✍●✒✛✙ ✜☞.✎Similarly, ✍■✒ ✍ ☞ ▲✒ ✎ ❵ ✵ ✁ abbreviating ❵ ✵ as we have ✎ ☞✁▲✒✿✍ ✙ ✏ ☞ ✡✑✒❫✡ ☞ ✍✭✒ ➉ q ☞✁▲✒ ✡ ☞✁▲✒■✍ ✙ ✏ ☞ ✡✑✒ ✡ ☞✁▲✒ ☞✁▲✒✿✍ ✙ ✏ ☞ ✍❀✒ ✡ ❺ ➈ ❵ ✵ ❵ ✵ ❵ ✵ ❵ ✵ ✎ ✎ ✎ ✕ ✡ ✡ ✡ ❅ ✞ ✞ ❅ ✄ ✂

✂

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✁

for

✆

✕✖✁✓✕✌✏

, as desired.

✙✌✰

✙✌✰

☞ ✡ ●✍ ✒☛✍

continuous, given we can choose ✵ small enough so that ❵ ✵ 6. Since ☞ ✒✗✏✕❵ is✽✠uniformly ✞ ✍✂✠✕ ❵ ☞ ✵ ✡✑✒✿✄ ☞ for ✵ ✵ any when . The argument of the preceding exercise (slightly ☞ ✒✬✍ ☞ ✒✗ ✕ ✁ ✕✖ ✁✓✕✌✏ simplified, since ✍❀✒■✄ ☞ ✒ ❵ ✵ ❵ ✵ ❵ ✵ ❵ ✵ ) shows that ✟ ❵ ✵ for all ✵ when .

8.3 Derivatives, Integrals, and Uniform Convergence

✄ ❃ ✎ P ☞ ✒✈✸✻✺✜✼ ✄ ❃ ✎ P ☞ ✒✈✸✻✺✜✼ ✍✥❃ ✎ P ☞ ✒✻☞✎✍ ✼✄✔ ✕ ✒ ✄ ✰ ✡ ✗ ☛ ✎ ✔ , and ❵ ☞ ✵ ✒✈✄✼ ✔ ✕ ❙✵ ☎✵ ✄✖❃ ✎ P ❵☞ ✒✈✵ ✼✄✔✖✕ ❙✵ ✎ ✍✪❃ ✎ P ✎ ☞ ❵ ✒✻✵ ☞ ✸✻✺✜✼ ✒ ❙✵ ✄✖ ☎✵ ✰✘✍ ✱ ☛ ✄✖❃ ✎ P ✎ ❵ ✵ ✎ . ❙✵ ☎✵ ❵ ✵ ❙✵ ✎ ❵ ✵ ❙✵ ✾✵ ☞ ✒ ✍ ✏❤✡r✳☞ ✱✲✡ ✒❨✽✣☞✳✱s✍◗❃❄✒ ✍✤❃ q ✕✝✍ ✱ ✱ ✎ P ❫q ✱ ✏❀✍✖☞ ✍ ✱✫✒❨✽✣☞♦❃❇✍✗✱✫✒ ✵ ✵ (a) for , for ✱✚✕ ✵ q is❃ ☞✎✵ ✍❝for ❃■➂❜✰✾✵ ✒ ☞✎✍ ✱➃➂✱, ✍✑and ✏✓✒ ☞✳✱➃➂✱✏✓✒ ✵ ☞♦❃■➂❜✰✾✒ ✵ . The graph is the broken line joining the points , , , and . ✙✜✽ ✱❾☞♦❃❋✍ ✱✣✒ ➉ ❚ P ☞♦✼✙✔ ✕ ✱❑✼✙✔ ✕ ✒❨✽ ✟ ❙✵ (b) Termwise integration of the series in Exercise 7, ✂ 8.1 gives ➈ . ✒ ✞ ✟ ✍t❃ ✟ ✄ ✏✱✙❁❚ P ☞✎✍✑✏✓✒ ☛ ✽ ✟ ➉✓✸✻✺✜✼ ✂q❞❃ ❙✵ ( ✵ we✏✱✙❁ have ), and termwise (a) From Exercise ✜● 4,✍⑥✂ ❃ 8.1, ✟ ✄✳ ❚ P ✵ ☞✎✍ ✏✓✒ ☛✫✽ ✜ ➉✓✼✄✔ ✕ ✒ ➈ ✜✬✍⑥❃ ✟ ✍❝❃■➂❜❃❾➉ integration yields ✵ ✵ ✒ ➈ ❙✵ . (✵ ✵ is odd, so its mean value on ➈ is 0.) ✪✕✍ ✙✠❃ ✟ ✟ ✄ ❙✡ P ☛ P of the result of (a) and multiplication by 4 gives ✵ (b) Integration ✵ ✟ ✟ ✟ ★ ❚ P ☞✎✍✑✏✓✒ ☛ ✽ ✪ ➉✓✸✻✺✜✼ ✄♠☞✎✏✓✽✠✙✠❃❄✒ ☞ ✪ ✍✉✙✠❃ ✒ ✄ ☞✎✏✓✽▲❃❄✒✻☞ ❃ ✍ ❃ ✒❁✄✳✍ P ❃ ✪ ✜ ✎ ✵ ❙✵ , where ✵ ✾✵ . ✒ ➈ ✄✖❃ ✍❝❃ ✪✤✄✳✍ P ❃ ✪❝✍ ★ ❚ P ☞✎✏✓✽ ✪✩✒ ❚ P ☞✎✏✓✽ ✪✩✒❁✄✕❃ ✪✱✽✠✴✠✰ in (b) gives , or ✒ . (c) Setting ✵ ✒ ☞ ✒■✄♠✔✼✄✔ ✕ ✈☞ ✒✈✸✻✺✜✼ The function ❵ ✵ ✵ is continuous and piecewise smooth, and its derivative is ✵ ☎☞ ✒✈✸✻✵ ✺✜where ✼ ✄ ✵ ✵ is wave (Exercise 1, ✂ 8.1). So by Corollary 8.27, we✸✻✺✜✼ have ✮ ✟ ☞ ★✜✽▲❃❄the✒ ❚ square ☞ ✼ ✙ ✔ ❝ ✕ ✙ ❨ ✒ ✣ ✽ ☞ s ✍ ✓ ✏ ✒ ✰ ✕ ✕ ❃ P ❙✵ , and in particular the series converges to ✮ ✒ ✔✼✄✔✖latter ✕ ✰ ✕ ✵ for✕ ❃ ✏✫✵ ✍ ✸✻✺✜✼ . On ✄ from 0 to gives, for the other hand, termwise integration of the series for ✵ ✵ ☞ ✙✜✽▲❃❄✒ ✍◗☞ ✙✜✽▲❃❄✒ ❚ P ☞♦✼✄✔✖✕❝✙ ✒❨✽ ☞ ✟ ✍✪✏✓✒ ✸✻✺✜✼ ✄✳✏❁✍◗☞ ✙✜✽▲❃❄✒ ✡✥☞ ✙✜✽▲❃❄✒ ❚ P ☞♦✼✙✔ ✕❝✵ ✙ ✒❨✽ , ☞ ✟ ✍✪✵ ✏✓✒ ✵ ❙✵ ☞✎✏✓✽ ☞ ✟ ✍⑤✏✓,✒❨or ✵ ❙✵ ✸✻✺✜✼ . ✒ ✽✣☞ ✒ ✟ P ✍ ✏✓✒❤✄✝ ✒●✡✌☞✎✏✓✵✽ ✒ of the two series for Now, since ✵ amounts ❃◆✍ ✄ ❚ P ☞♦✄✼ ✔ ✕❝✙ ✒❨✽ ✰ , the ✕ equality ✕♠❃ ✟ ✵ ❙ ✵ ✵ for , and one verifies this by substituting to the assertion that ✒ ❃★✍⑩✙ ✵ for ✵ in Example 1, ✂ 8.2. ✱

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Chapter 8. Fourier Series

62

☞ ✒ ❃ 5. ❵ ✵ is not continuous (it has jumps at the odd multiples of ), so Theorem does ☞ ✒ ☞ ✒✘✍ ☞ ✙■8.26 ✼✄✔ ✕✂✯❃❄ ✒ ✆ ☞ not ✒ apply. (For those who like distributions: The derivative of ❵ ✵ is really ❵ ✵ ✵ ✵ , where ❃ ❩ ✵ ✆ is the periodic delta-function with singularities at the odd multiples of . The Fourier series of ☞✎✏✓✽✠✙✠❃❄✒✫❚ ☞✎✍✑✏✓✒ ☛ ☛ ✄ ☛ but rather ☛ ✄ ☛ ✍ ✵ ☞✎✆✍✑✏✓is✒ ☛❫☞♦✼✄✔ ✕✁✯❃❄✒❨✽▲❃ ✒✎ , so the correct conclusion is not ☛ ,✒ which is true by Exercise 5, ✂ 8.1.) ❘ ✎ ✒☎✄ ✽✣☞✎✏✑✡ ✗✒ ✶✳q❉❅ ❚ 6. (a) The series ☛✁❱ converges if and only if , so the given series can be differentiated 6 times.❘ ❚ ✽✠✙ ☛ ✶ (b) The series ✒ converges for all , so the given series can be differentiated any number of times. ☛ ✄❞✙ ✎✠☛ ), so its sum is continuous, but the (c) The given series converges uniformly (M-test with ✍ ❚ ✄✼ ✔ ✕✯✙ ☛ differentiated series ✵ does not converge at most points (the terms do not tend to zero as ✣✦✥ ). ✑

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8.4 Fourier Series on Intervals ✏ 1. (a) The even periodic extension of ❵ is the constant function , which is its own Fourier series. The odd periodic extension is the square wave (Exercise 1, ✂ 8.1). ✔✼✄✔✖✕ ✼✄✔✖✕ ✵ (Exercise 3, ✂ 8.1). The odd periodic extension is ✵ , (b) The even periodic extension of ❵ is which is its own Fourier series. (c) The even ☞periodic extension✞✕✝ of ❃ ❵ is the☞ function ✒✘✄ ✒✑✄ ❃ ✍ of Exercise ☞♦❃◆✍⑨ ❺✒ 4, ✂ 8.1. The odd periodic extension ✵ ✵ for ✵ ✵ ✵ ✵ ; the Fourier series of the latter two is given by ❵ ✵ , so ❵ ✵ functions are given by Example 1 and Exercise 6, ✂ 8.1. P ☞✙ ✒ P 2, ✂ 8.1.P Then the even periodic extension of ❵ is ✟ ✵ , (d) Let be the triangle wave of Example ☞ ✡ ❃❄✒✑✍ ❃ ✟ ✟ . The Fourier series ☛ P of these are, respectively, and the odd periodic extension is ✵ ☞♦❃❄✽✁☎✒❤✍✌☞ ✙✜✽▲❃❄✒ ❚ P ☞♦✸✻✺✜✼✩☞ ✦ ✍ ✙✜✒ ✒❨✽✣☞ ✙ ✦❉✍✕✏✓✒ ✟ ☞ ☎✽▲❃❄✒ ❚ P ☞✎✍ ✏✓✒ ✮ ☞♦✼✄✔✖✕❙☞ ✙ ✦ ✍✕✏✓✒ ✒❨✽✣☞ ✙ ✦❉✍✕✏✓✒ ✟ P ✵ ✒ ✸✻✺✜✼▲☞ ✙ ✦❞✍ ✏✓✒ ✒✻☞ ✡ ❃❄✒❁✄ ☞✎✍ ✵ ✏✓✒ ✮ ✼✄✔✖✕❴☞ ✙ ✦ ✍ and ✏✓✒ ✟ (since ✵ ✵ ). ✈☞♦❃❴➇❴✒ is the square wave of Exercise 1, ✂ 8.1; its Fourier 2. (a) The odd ☞ ☎✽▲2-periodic ❃❄✒❬❚ P ☞♦✄✼ ✔ ✕❙extension ☞ ✙ ✦ ✍ ✏✓✒✔of ❃❴➇❴❵ ✒❨✽✣is☞ ✙ ✦ ✍ ✏✓where ✒ series is . ✒ P P ✈☞ ❃❴➇✯✡ ❃❄✒ ✟ ☛ P where is the function P wave, namely, P P ✪ is a square of (b) The even 4-periodic extension of ❵ ✼ ✔ ✕❙☞ ✙ ✦ ✄ ✍ ✏✓✒✻☞ ❃❴➇ ✡ ❃❄✒ ✄ ☞✎✍ ✏✓✒ ✮ ✸✻✺✜✼✓☞ ✙ ✦ ✍ ✏✓✒ ❃❴➇ ✄ ✟ P 8.1; P since P P ☛ ☛ P ✪ ☞ ☎✽▲❃❄✒ ❚ ✪ ✏✓✒ Exercise ☞✎✍✑✏✓✒ ✮ P ✸✻1, ✺✜✼✩☞ ✂ ✦❞ ✍ ✒✔❃❴➇ ✎ ☞ ✑ ✍ ✓ ✏ ✒ ♦ ☞ ✻ ✸ ✜ ✺ ▲ ✼ ☞ ✦ ✍ ✔ ✒ ❴ ❃ ❴ ➇ ❨ ✒ ✣ ✽ ☞ ✙ ✦ ✍ ✮ ✟ ✟ ✪ ✪ , its Fourier series is . ✒P ✙ ✠ ☞❊➇❴✒✇✄ ➇■✔☞ ✠✯✍✝ ➇■❺✒ ✍ ✠✔✜➂ ✠❪➉ ☞❊➇❴✒⑩✄ (c) The odd -periodic extension of ❵ is given by ❵ on ➈ , that is, ❵ ✔☞ ✠ ✽▲❃❄✒ ✟ ☞♦❃❴➇❴✽ ✠ ✒ of Exercise 6, ✂ 8.1; its Fourier series is ☞ ★ ✠ ✟ ✽▲❃ ✜✓✒✫❚ P ☞♦✄✼ ✔ ✕❙where ☞ ✙ ✦ ✍✥✏✓✒✔❃❴➇❴is✽ ✠ ✒❨the ✽✣☞ ✙ ✦ function ✍⑤✏✓✫✒ ✜ . ✒ ✓P ✒ ✟ ☞ ✙✠❃❴➇✤✍ ❃❄✒ ✗✯✄ P where is the function ✓P ✒ ❩ P (d) The 1-periodic extension of ❵ is of Exercise 5, ✂ 8.1, with ✟ ✟ ✝ ✆ ✆ ✏✓✽✠✙✠❃ ☞ ✼ ✔ ✕✁ ✒❨✽▲❃❾➉ ❚ ✄ ☞✎✍ ✏✓✒ ☛ ☛ ✎ ✽✣☞❨☞ ✙✠❃❄✒ ✎ ✍ ✒ ✄ ✟ ❩ series ✒ is ➈ ✒✎ ✆ ☛ ✾✽✣☞✎✏❀✍⑩✙✠❃ ☞ ✤✍⑤.✏✓✒ ❚ Its ✝✟ Fourier ✒ . ✒✎ ✒ ✱ ☛ ✄ ☞✎✏✓✽ ✠ ✒ ✄✟ ✂ ☞❊➇❴✒✈✸✻✺✜✼✩☞ ❃❴➇❫✽✠✙ ✠ ✒ ➇ ✗ ☛ . Replace ❵ ☞❊➇❫✒ by ❵ ☞ ✙ ✠✣✍⑥➇❴✒ , then set ❵ 3. We have and similarly for ➇⑥✄✌✙ ✠✫✍ ✸✻✺✜✼▲☞ ❃❁☞ ✙ ✠✫✍ ✒❨✽✠✙ ✠ ✒❑✄ ☞✎✍✑✏✓✒ ☛❤✸✻✺✜✼✩☞ ❃ ✽✠✙ ✠ ✒ ✼ ✔ ✕❙☞ ❃❁☞ ✙ ✠❍✍ ✒❨✽✠✙ ✠ ✒☛✄ ✄ facts that and ✁☛ P ✁ ☞✎✍✑✏✓✒ ☛ ☛ P ✄✼ ✁ ✔ ✕❫and ☞ ❃ use✽✠✙ the ✠✒ ✱ ☛ ✄❞☞✎✍ ✁ ✏✓✒ ☛ ✱ ☛ ✗ ✄ ✎ ☞ ✑ ✍ ✓ ✏ ✒ ✗ ☛ ☛ ☛ , whence ✱ ☛ ✄❖✰ for odd ✁ to deduce that and ✗ ✄✖✰ and ☛ for even. ✠

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8.5. Applications to Differential Equations

63

✰✣➂✹✙ ❪✠ ➉

✼✄❵ ✔✖✕❙☞as ❃❴suggested ➇❴✽✠✙ ✠ ✒

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in the hint and ✟✄expand ✗ ✄ ☞✎✏✓✽ ✠ ✒ ✂ ☞❊➇❫✒✈✼✄✔ ✕❙it☞ in❃❴➇❫a✽✠✙ Fourier ✠ ✒ ➇ sine series on ➈ ✗ ☛ ✄ : ✰ ❵ ❚ P ✗☛ ☛ where☞❊➇❴✒✈✼✄✔✖✕❙☞ ❃❴➇❴✽✠✙ ✠ ✒ ❵ for even, by Ex✒ ➇✕. ✄ Then ✠ ercise 3; and for odd, ❵ is symmetric about (as observed in the solution of ✰✣➂✹✙ ✠❪➉ ✰✣➂✜✠ ➉ ✄ ✙ ✦ ✍✖✏ is twice its integral thus yields Exercise 3), so its integral P over ➈ P ☞❊➇❴✒❁✄ ❚ P ✼✙✔ ✕❫☞ ✦❞✍ ✟ ✒✔❃❴➇❴✽ ✠ ✄♠☞ ✙✜✽ ✠ ✒ ✂ ☞❊➇❫over ✒✻☞♦✼✄✔ ✕❙➈ ☞ ✦ .✍ Setting ✟ ✒✔❃❴➇❫✽ ✠ ✒ ➇ . ❵ where ✮ ❵ ✒ ✮

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8.5 Applications to Differential Equations

☞❊➇❴✒ ✄ ➇ ✰✣➂✱✏✗✰▼✰▲➉ ❵ ➈ cosine series for on is ✟ ✒✫Fourier ✟ ❚ P ☞♦✸✻✺✜✼✩☞ ✙ ✦❏✍✌✏✓✒✔❃❴➇❴✽✈✏✗✰▼✰✾✒❨✽✣☞ ✙ ✦ ✍⑨✏✓✒ so the solution (8.35) of the heat equation is ☞❊➇✬➂❨❛❨✒❑✄✕❂✠✰✑✍⑤☞ ✜✰▼✒ ✰✾✽▲❃ ✟ ✒ ❚ P ✎ ☞✁ P❨P ✌ ☞ ✟ ✮ ✎ P ✌ ✖ ✖ ✚ ☞♦✸✻✺✜✼▲☞ ,✙ ✦❞ ✍ ✏✓✒✔❃❴➇❴✽✈✏✗✰▼✰✾✒❨✽✣☞ ✙ ✦ ✍ ✏✓✒ ✟ . ✒ ❛✿✄✌❅✠✰ ✦ ✄✌✙

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(b) When

✜✰▼✰ ✖ ✒ ❃❫✟

✍ ✏✓✒✻☞♦❃❴➇❴✽✈✏✗✰▼✰✾✒ q ✜✰▼✰ ✎ ☞✂ ✓✒✓ ✌ ✄ ✖ ✖ ✖ ✒ ✥ ✏ ☞ ✙ ✦❞✍⑤✏✓✒✔✟ ❫❃ ✟ ☞ ✙ ✦ ✍ ✏✓✒✔✟ ✜ ✜ ✜✰▼✰ ✎ P ✓ ✟✟✞ ☞✗✄ ✏✱✙▼✞✜✒ ☛ ✄ ✴ ★ ✄ ☛ ❃ ✟ ☞❊➇✬➂✹❅✠✰✾✒☛✄✕❂✠✰➃✍⑧☞ ✜✰▼✰✾✽▲❃ ✟ ✒ ✎ ✓✒✓ ✖❴✸✻✺✜✼✩☞♦❃❴➇❫✽✈✏✗✰▼✰✾✒✹✡ P ✎ ☞✂ ✓✒✓ ✌ ✖❫✸✻✺✜✼▲☞ ✞✠❃❴➇❴✽✈✏✗✰▼✰✾✒ ➉ ☛ ➈ To error, ✁ ❂✠✰❧within ✍✖☞ ✞✎✠ ✄ ✴✎this ✠▼✒✈✸✻✺✜✼✜☞♦❃❴➇❫✽✈✏✗✰▼✰✾✒✂✍✕☞ ✙ ✄ ❂✣✏✓✒✈✸✻✺✜✼✠☞ ✞✠❃❴➇❫✽✈✏✗✰▼✰✾✒ ➇⑩✄♠✰ ➇⑩✄ ✏✗✰ , which is about 10 when , 12 when ➇⑥✄ ✜✰ . and 40 when ❛ ✤ ✞▼❅✠✰▼✰ ☞❊➇❄➂❨❛❜✒✤✍✖❂✠✰❍❀q ☞ ✜✰▼✰✾✽▲❃ ✟ ✒ ✎ ☞✁ P❨P ✌ ✖ ☞ ✜✒✓ ✌ ❚ P ✏✓✽✣☞ ✙ ✦ ✍r✏✓✒ ✟ ✄❏❂✠✰ ✎ ☞✂ ✜ ✓ ✌ ✖ ☛ (c) , ✁ ✒ ✏ ✄ ✰▼✰✜For ✞✠ ✎ ☞❊➇✬➂❨❛❨✒❝✍⑤❂✠✰❍✂q . Almost good enough, but not quite! A slightly less crude ☞ ✜✰▼✰✾✽▲❃ ✟ ✒ ✎ ☞✂ ✜ ✓ ✌ ✖ ✡ ✎ ☞✂ ✜ ✓ ✌ ✖ ❚ ✟ ✏✓✽✣☞ ✙ ✦✪✍✑✏✓✒ ✟ ➉❴✄♠☞ ✜✰▼✰✾✽▲❃ ✟ ✒ ✎ ☞✁ estimate ✜ ✓ ✌ ✖ ✡ works: ✎ ☞ ✜ ✄ ✓ ✪ ✌ ✖✁ ☞❨☞♦❃ ✟ ✩✽ ★✜✒✮✍✑✏✓✒ ➉ ☛ ➈ ➈ ✒ ✄ ★✣✏ ✡

✡

✡

✎ ☞✂

✂

✌☞✟

✓✒✓

✎ P✌✖ ✖

✸✻✺✜✼✩☞ ✙

is

✮

✦

✡

✆

✆

✂

✡

✡

✡

✡

✡

✡

✂

✆

✆

✂

✂

✆

✆

✂

✆

✂

✆

✂

,

✆

✂

✆

✂

✂

.

2. One procedure as on p. 382 to find solutions of the form ✩✎ ❘ ✚ follows ☞ P ✸✻✺✜✼ ✛ the separation-of ✡ ✟ ✼✄✔ ✕ ✛ variables ✒ ✄ ✛ ❘ ✚ . The periodicity condition then forces , so the resulting ✵ ☞ ➂❨❛❜✒✲✄ ❚ ✲✵ ✎✠☛ ✖ ☞✳✱ ✸✻✺✜✼ ✡✏✗ ✼✄✔ ✕ ✒ ☛ ☛ ✵ ✗ ✼✄✔ ✕ ✒ ✒ ❙✵ ❙✵ ☞ .✒ To satisfy the initial condition one analog of (8.35) ❚ ☞✳✱ ☛ ✸✻✺✜✼ is ✁ ✡☞ be the Fourier series of looks a little neater in takes ✒ ❙✵ ☞ ➂❨❛❨✒☛☛ ✄ ❚ ❙✵ to ❵ ❚ ✵ . (The ❘ ✚☛ ❩☛ ✑ ❩ result ✆ ✩✎✠☛✍✖ ☞ ■ ✒ ✄ ✆ ☛ ✑ ☛ ☛ where ❵ ✵ .) exponential form: ✁ ✵ ✒✎ ✒✎

✝

✑

✂

✄

✝

✄

✝

✂

✟

✟

✟

✂

☞❊➇✬➂❨❛❨✒☛✄ ❚ P

✟

✟

☞❊❛❨✒✈✼✄✔✖✕❄☞ ❃❴➇❴✽ ✒✠ ✒

✗

✞

✂

✌✞

✚

✒

✄ ✶ ✟ ✡

☞❊➇✬➂❨❛❨✒❑✄ ❚ P

☞❊❛❜✒✈✼✄✔ ✕❙☞ ❃❴➇❫✽ ✠ ✒

3. If ✁ where , ✒ ✗ ☞❊☛ ❛❨✒✇✄ ✍✤✶❫☞ ❃❄✽ ✠ ✒ is✟ ✗ to ☞❊satisfy ✒ ☛ ❛❜✒❝✡ ❽ ☛ ☞❊✁ ❛❨✒ ❽ ✁ ☛ ☛ we must have , assuming that termwise differentiation of the se❘ ✒ ✚ ✒ ✚ ✚ ries is❘ ☞ ☛ justified. To solve this ordinary differential through by❘ the fac☞ ☛ ✒ ✂ ✌ ✖ ✚ multiply ☞ ✒ integrating ✂✌✖ ✂ ☞ ✽ ❛❨✒ ✗ ☛ ☞❊❛❨✒ ☞ ☛ ✂ ✌ ✖ ➉ ✄ ❘equation, ☛ ☞❊❛❜✒ , whence ✗ ☛ ☞❊❛❨✒ ☛ ✌ ✖ ✄ ✗ ☛ ☞♦✰✾✒✂✡ ➈ tor✚ ❘ ☞ ☛ ✒ ✂ ✌ to☞ ▲obtain ✒ ✖ P ☛ ✰✣✜➂ ✠ ➉ . For this ✟ are (more than)➇⑨sufficient: ☞♦✰✾✒ to ✄ work, ☞✔✠ ✒❧✄ the✰ following ☞❊➇✬➂❨❛❨conditions ✒ ✐ ✰✣➂✜✠❪➉ (1) ❵ is❛ on , and . (2) is as a function of of ☞♦class ➈ ❵ ❵ ➈ for each , ✟ ☞❊➇✬➂❨❛❨✒ ✰✣➂❨❛❜✒✉✄ ☞✔✠✔➂❨❛❜✒s✄ ✰ ☞❊➇❄➂❨❛❜✒ ☞❊➇❄➂❨❛❜✒ , ❽ , and ❽ are jointly continuous as functions of ➇⑩✐ ✰✣➂✜✠ ➉ ❛ ✤✖✰ , and conditions on and guarantee that their odd periodic extensions ➈ and P . The boundary ❵ ✟ ❽ are still at least , and ✗ that of is at least piecewise continuous. It follows that the ☞♦✰✾✒ ☞❊❛❜✒ Fourier sine co☛ ☛ )✎ are summable, and of (namely, ) are continuous efficients of ❵ (namely, ✟ absolutely ❛ ☞❊❛❜✒✗✣q ❛ ✰✣➂ those ✂ ➉ for in any finite interval ➈ . We then have in and satisfy ☛ ✟

✟

✞ ✂ ✂

✆

✂

✆

✂

✝

☎

✆

✞

✆

✂

✌

✂

✌

✟

✆

✂

✌

✄

✄

☎

☎

✞

✄

✗

✞

☛ ☞❊❛❨✒✗✣q ✂

✎ ❘ ☞☛

☎

✄

✚

✟

✒ ✆

✂

✌✖

✚

✗

☛ ☞♦✰✾✒✗✱✡

✎ ✄

✟

✟

✂

✂

✎ ❘ ☞☛

✒ ✆

✂

✌✖ ✌

✂

✂

q ✂

✎ ❘ ☞☛

✒ ✆

✂

✌✖

✚

✗

☛ ☞♦✰✾✒✗✓✡ ✶❫☞♦❃❄✽ ✠ ✔✒ ✟ ✪ ✄

✟

✄

Chapter 8. Fourier Series

64

☞❊➇❄➂❨❛❜✒ ☞❊➇❄for ➂❨❛❜✒

This the absolute and uniform convergence of the series defining ➇❇✐ ✰✣is➂✜✠ enough ➉ ❛✂✐ to ✰✣guarantee ➂ ➉ ➈ ✟ ☞❊➇❄and ➈ ✐ ✰✣, ➂✜as✠❪➉ well ❛✂as✐ the➂ absolute ➂❨❛❜✒ ➇❋ ➉ ✙ ✰ and uniform convergence of the series defining ❽ ➈ and ➈ and ❽ for ( ), so that all formal calculations are justified. ✁

✂

✚

✁

✂

☎

✁

☞❊➇✬➂❜✰✾✒

☞♦❃❴➇❴✽ ✠ ✒

✦

is where is as in Exercise 4. (a) The odd periodic extension of the initial displacement ✱★✄✳❃ ✗✗✽ ✠ 2, ✂ 8.3, with☞❊➇❄➂❨❛❜✒ , so its Fourier sine series can be read off from the answer to that exercise. The can then be read off from (8.37). series for ✁

✙ ✠ ✟ ✽▲❃ ✟ ✠✗ ☞✔✠❙✍ ✗✗ ✒❤✄r✙✠✰▼✰✾✽✠✙ ✜❃ ✟

✦✄⑦☞♦☎✰ ✄ ☎✒✗✠

✁

✗

✠

✠

✑✄ ★

✎

✟ ✼✄✔ ✕❙☞❨☞✗✄ ☎✒ ❃❄✒

✑✄ ✴▼❂✣✏

we have , and (b) ✍ ✄ ✰✜When ❅▼❂ ✍ ✄ ✰✜❂▼✴ ✄ ✏✠➂✹✙✈➂✹✞✈✚➂ ✫➂✹❂ ✦ , .147, , , so ☞♦the the overall factor of ) are ✍ ✄,✰✜0❂▼❂ when ✍ ✄ ✰✜❂✠✰ ✗❋✄ ✰☎✄ ✏✓✒✗first ✠ five coefficients ✙ ✠ ✟ ✽▲❃ ✟ ✗✠☞✔✠❁(up ✍ to ✗✗✒ ✄❉✙✠✰▼✰✾✽✠✴✠❃ ✟ ❏✙ ✄ ✙▼❂▼✙ , , 0. When we have , and .803, .124, ✎ ✟ ✼✄✔ ✕❙☞❨☞✗✄ ✏✓✒ ❃❄✒ ✄ ✞✠✰✜✴ ✄❞✏✠➂✹✙✈➂✹✞✈➂✚✫➂✹❂ , .147, .090, .059, .040 when , so the first five coefficients ☞✔✧ ➂❜✰✾✒ ✦❲✆ ✠ are ✦ ✞ ✟ ✽✠✞ ( times) .696, ✗ .331, .203, .133, .090. (Note: The norm of the initial✗ displacement is , independent of , so the total energy of these waves is independent of and a direct comparision of the coefficients is appropriate.) ☛

✟

☛

✟

✟

☛

☛

✟

✟

✟

✁

☞❊➇❄➂❨❛❜✒ ✄ ✍✂☞❊➇❫✒ ✭☞❊❛❜✒ ✍✂☞❊➇❴✒ ☞❊❛❜✒❁✡ ☞❊❛❜✒◆✄ ✆ ✟ ✍ ☞❊➇❴✒ to ✦☞❊be ❛❨✒ a solution ☞❊❛❜✒✯of ✡✳✙ the✦modified ☞❊❛❨✒ ➉❊✽ ✆ ✟ ✭☞❊wave ❛❜✒❇✄ equation, ✍ ☞❊➇❴✒❨✽ ✍✂☞❊we ➇❴✒✪must ✄ ✍ have ✵ ✵ , or ➈ ✄ ☞ ❃❄✽ ✠ ✒ ✟ ✍✂☞❊➇❴✒⑥✄❏✙✼ ✔ ✕❫☞ ❃❴➇❴✽ ✠ ✒ , a constant.☞❊❛❨✒✂As ✡ on 385, boundary conditions force ✟ ✡r✙ ❦✡⑦☞ and ✙ page ☞❊❛❜✒❑✡⑦ ☞ ❃ the ✆✱✽ ✠ ✒ ✟ ✭☞❊❛❜✒★✄ ✰ ❃ ✆✱✽ ✠ ✒ ✟ ✄ ✰ ✳✄ ✍ . Then ❆ ☛ are where ✵ ✵ ✄♠✆ ☞ ❃ ✆✱✽ ✠ ✒✔✟❝✍ ✟ ✭☞❊❛❜.✒☛✄ The✎ roots ☞ ✗ ☛ ✸✻of✺✜✼ ☛ ❛❙✡ ✵ ☛ ✙✼ ✔ ✕ ☛ ❛❜✒ ☛ ✵ , so . Taking linear combinations of ✄✳✏✠➂✹✙✈➂✹✞✈✆➂ ✄✆✄✆✄ ✕ ❃ ✆✱these ✽✠ . If solutions for q ✠ ✽▲gives ❃ ✆ the desired result analogous to (8.37). (This is assuming ✵

5. For ✙ ✍✂☞❊➇❫✒

☎

☎

✁

☎

☎

☎

☎

☎

✝

☎

✝

☎

✟

✟

☎

✚

✟

✟

☎

✂

✟

✟

not, the solutions for 6. (a)✄ If

P ✬P ✡

have pure exponential decay with no oscillation.)

✵

✟

P✇✄

✄ ✰

✟ solves the problem for and ✟ solves the problem in the general case.

✁

✠

✠

✟ ✁

P◆✄ ❵

solves the problem for

✄ ✰ ❵✟

, then

✼✄✔ ✕✁ ✆▼✝☞ ✞✳✍⑨➁❬✒❲✄ ✼✄✔ ✕✁ ✆ ✞✉✸✻✺✜✼ ✆✮➁✪✍✚✸✻✺✜✼ ✆ ✞✉✼✄✔ ✕✂ ✆ ➁ (Exercise ✆▼✝☞ ✞✲✍ ➁✫✒✱✡s✸✻✺ ✄ ✆ ✞♣✼✄✔ ✕✁ ✆✮➁ ✼✙✔ ✕✁ ✆✮➁ 1b, ✂ 7.4), ✸✻✺✜✼✄ ✆✮so➁ ; hence any linear combination of and ✼ ✔ ✕✁ ✆✮➁ ✄ ✼ ✔✖✕✁ ✆▼☞✝✞s✍❧➁✫✒ ✄ ☞❊➇✬➂❨➁✫✒☛✄ ✍✂☞❊➇❴✒ ✭☞❊➁❬✒ is also a linear combination of and☞❊➁❬✒❨✽ ✭☞❊➁❬✒☛✄✳✍ (and ✍ ☞❊➇❴vice ✒❨✽ ✍✂☞❊versa). ➇❫✒☛✄ Now, for to satisfy✍❑Laplace’s , ✍✂ and ☞♦✰✾✒⑧✄ ✍❑✔☞ equation, ✠ ✒⑧✄ ✰ we need ☞❊➇❫✒✲the✄ boundary ✼✄✔ ✕❙☞ ❃❴➇❴✽ conditions ✠✒ ✄ ☞ ❃❄✽ ✠ ✒ ✟ and become ☞❊➁✫✒ ✄ ☞ ❃❄✽. ✠ As ✒ ✟ ✭in☞❊➁❬✒ the text, this forces ✭☞❊➁❬✒ . Hence ✄ P , so (by the preceding remark) ✗ ✄✼ ✔✖✕✁❫☞ ❃❁✝☞ ✞⑨✍✕➁❬✒❨✽ ✠ ✒❝✌ ✡ ✗ ✟ ✄ ✼ ✔✖✕✁❫☞ ❃❴➁✫✽ ✠ ✒ ☛ we arrive at the general ☞❊➇❄➂❨➁❬✒✘✄ ❚ P ✼✄✔✖✕❴☞ ☛ ❃❴➇❫✽ ✠ ✒ ✗ ☛ P ✼✄✔✖✕✁❫☞. Taking ❃❁☞✝✞✥✍❲➁✫linear ✒❨✽ ✠ ✒■✡☞combinations, ✗ ✟ ✼✄✔ ✕✂❴☞ ❃❴➁❬✽ ✠ ✒ ➉ ☞❊➇✬➂❜✰✾✒✘so-✄ ☛ lution✗ P ✄✼ ✔✖✕✁❙☞ ❃ ✞✂✽ ✠ ✒✈✒ ✄✼ ✔ ✕❙☞ ❃❴➇❫✽ ✠ ✒ ➈ . WeP✱☞❊then ➇❫✒ have ☞❊➇✬➂ ✞✂✒✌✄ ❚ P ☛ , which must be the Fourier sine series of ❵ , and ✒ ☞❊➇❫✒ ❚ P ✗ ☛✟ ✄✼ ✔✖✕✁❙☞ ❃ ✞✂✽ ✠ ✒✈✄✼ ✔ ✕❙☞ ❃❴➇❫✽ ✠ ✒ ✟ , which must be the Fourier series of ❵ . ✒ ☞ ✁✩➂ ✒❁✄ ✍✂✂☞ ✁✾✒ ✭☞ ✒ ✂ ✁ ✟ ✍ ✂☞ ✁✾✒ ✦☞ ✒✾✍ ✡ ✁ ✍ ✂☞ ✁✾✒ ✭☞ ✒☎✡ (a) ✵ ✂☞ to✜✁ ✒✯satisfy polar Laplace ✵ ✵ ✍✂☞✂✜✁ For ✒ ☞ ✒❋✵ ✄ ✰ ✁ ✟ ✍ ✡ ✁ ✍ the ✂☞ ✁✜✒ ➉❊✽ ✍✂✂☞ ✁✾✒✇✄ ✍ equation, ☞ ✒❨✽ ✭☞ we ✒❇✄ need ❩ , or ➈ This differential ✵ ✵ ✵ ✄ ✟ , a constant. ✦☞ ✒✿✄ ✆ ☛ ☛ ✡ ✆ ✎✠☛ ✎ ❩ ☛ equation ✄⑨✰✣➂✱✏✠for ➂✹✙✈✆➂ ✄✆✄✆, ✄ together with the periodicity requirement, yields ✍ and ✵ (✁ ☛ ). The general solution of the differential equation for is then a linear combination of ✁ ✎✠☛ ✏ ✺ ✓✰✁ ✍ ✄❖✰ ✁ ✎✠☛ ✺ ✓✰✁ ✍ ❩ ✝ and (or and ✝ if ). But and blow up at the origin and must be discarded. ☞✂✁✩➂ ✒ ✄ ❚ ✎ ✆ ☛ ✁ ☛ ☛ ☞✎✏✠➂ ✒ ✄ ✵ ,➂ and the requirement that ✵ Hence ✒ ☞ ✒ we obtain the✆ ☛ general solution ☞ ✂ ▲ ✁ ❁ ✒ ✄ ☞ ✒ ❵ ✵ means that the ’s are the Fourier coefficients of✒ ❵ ; thus ✵ ✟ ❵ ✵ . (b) This follows immediately from part (a), (8.20), (8.22), and the observation that the Poisson kernel ✙ ✍ ✍ ✍ ✁

✁

✁

(b) we note that ✸✻✺✜✼✄ First ✆✮➁s✄✖✸✗✼❜✸ ✆ ✞✉✼✄✔✖✕✁

✂

☎

✁

☎

☎

✝

✝

☎

✟

✟

☎

☎

✟

✟

✟

✁

✟

✟

✟

✟

✟

✟

✟

✁

✁

☎

7.

☎

☎

✁

☎

☎

☎

✑

✑

✝

☎

☎

✝

✂

✟

✂

✟

✎

✎

✑

✟

✂

✁

✁

✁

is an even function, so that one can replace

by

in the integral.

8.6. The Infi nite-Dimensional Geometry of Fourier Series

65

8.6 The Infi nite-Dimensional Geometry of Fourier Series 1.

✸✻✺✜✼ ➇❀✸✻✺✜✼✫✦⑥➇ ✄ ✟P ✸✻✺✜✼✩☞ ✦ ✡ ✒ ➇✚ ✡❉✸✻✺✜✼▲☞ ✦ ✍ ✒ ➇❍➉ ✼✙✔ ✕ ➇❀✄✼ ✔✖✕ ✦⑥➇ ✄ PUse✸✻✺✜✼✩the ❩ ✒ identities ➈ ➇❀✸✻✺✜✼ ✦⑥➇⑨✄ P ☞ ❩ ☛ ✡ ✩✎ ❩ ☛ ✒✻☞ ❩ and ✻ ✸ ✜ ✺ ✼ ✡ ✎ ☞ ✦❏✍ ✒ ✇ ➇ ❈ ✍ ✻ ✸ ✜ ✺ ✓ ✼ ☞ ✦ ✡ ✒ ❬ ➇ ➉ ✮ ✮ ✟ ➈ ✪ , or , etc.,✙rto✰ see ✦ ✄❶ ❃❄✽✠✙ ✦ ✄ ✸✻✺✜✼ ➇✭✸✻✺✜✼✫✦⑥➇ ➇ ✼✄✔ ✕ ➇✭✼✄✔ ✕ ✦⑥➇ ➇ that ✝ and ✝ are 0 when and when (of ✄✖✰ ✸✻✺✜✼ ➇ ✼✄✔✖✕ ➇ ✆ ❃❄✽✠✙ ✙⑤✰ ✸✻✺✜✼ ✟ ➇ ➇⑥✄✖❃ course when ). The norm of or is for ; the norm of ✏❀✄✖✸✻✺✜✼➃✝ ✰✠➇ ✛ ❃ is . ✐ ✞ ✟ ☞♦✰✣➂❜❃❄✒ ✙✠❃ ❚ P ✗ ✼✄✔ ✕ , let ❵ be its odd -periodic extension and let ✒ ☛ be the Fourier series of If ❵ ✟ ✰✣➂❜❃❾➉ ☞ ❙✒❤✵ ✍✕ ❚ P ✗ ☛ ✼✄✔ ✕ ✄ to ➈ is the Fourier sine series of ❵ . Then ✝ ❵ P , whose☞ restriction ❵ ✵ ❙ ✵ ✾ ✵ ✟ ✟ ✝ ✎ ❵ ✵ ✒●✍ ❚ P ✗ ☛ ✄✼ ✔✖✕ ❙✵ ☎✵ ✣ ✰ as ✣✦✥ . Likewise for the cosine series. ➂ ✂P ✁s✄ P ☞❊➇★✡ ✱✫✒ ➇⑨✄ ✟P ✡ ✱ ➂ ✟ ✁s✄ P ☞❊➇ ✟ ✡ ✗✹➇★✡ ✆✱✒ ➇✌✄ P ✡ ✟P ✗❝✡ ✆ ✜ ❵ ➂ ❵ ✂P ✁❁✄ and ❵ ❵ P ✝✍ P ✗ P ✗❙✍ ✄ ✝ ➂ ✟ ✁❁✄✖✰ ✱ ✄✳✍ P ✆✤✄✳✍ P✻➂ ✟ ✁❁✄ P ☞❊➇✑✍ ✟P ✒✻☞❊➇ ✟ ,✡ so✗ ➇✑to✍ make ✟ ✟ ✟ ❵ ✄ P ❵ ➇ ✜❫✡❲ ❵ ☞ ✗✬✍ P we P ❵ ✒ ➇★ P ✗❴✡ P ➉ ➇★✜ ✄ P ✡ . Then P ☞ ✗✬✍ P ❵ ✒➃✍ ❵ P ☞ ✗❾✡ ✝ P ✒✈✡ P ✗❾✡ P ✄ P ☞ ✗❾✡◗✏✓✒ ✒ ➇ ✟ need ✍✪☞ ✗❴✡ P ✒ ➇✭✡ P and P✟ ✟ ✟ ✟ ✜ ✜ ✪ ✓ ✪ ✜ ✜ ✪ ✓ ✝ ➈ , ✗❝✄✳✍✑✏ ✆✤✄ ✟

✟

✟

☎

✂

✟

✟

✟

☎

✂

☎

✂

☎

✂

✟

✆

✆

✟

✌

✟

✌

✟

✟

✆

✟

2.

✌

✟

✟

✟

✟

✟

✆

✞

✟

✌

✆

✞

✆

✟

3.

✆

✌

✌

✌

✌

✌

☞❊➇❫✒❨✽▼✛ ✩✙ ➉ ✍ ☛ ☞❊➇❴✒❨✽▼✛ ✙✩➉ ✝✎ ➈ ➈ ✮ ✍ ✎ ☞❊➇❫✒❨✽ ✛ ✙✩➉ ✍ ✎ ☞❊➇❴✒❨✽ ✛ ✙✩➉ ✝✎ ➈ ☛ ➈ ✮ ✒ ☞ ✎ ✌ ☎ ☛ ➂✒☎ ✮ ✁☛✄ ✝ ☞ ✞ ✎ ✌ ✒ ✆ ✍ ☛ ☞✝✆✮➇s✡ ✆ ☎ ➂✒☎ ✁☛✄ P ✍ ☞❊➇ ✟ ✒ ✍ ☞❊➇ ✟ ✍

✂

4.

☛

✂

☛

✟

6.

✟

☛

✝

✮

☛

✮

☛ ➇✚✄ ✍ ☞❊➇❴✒ ✍ ☛ ☞❊➇❴✒ ☛ ✮ ✝ ➇❢✄ ✵ ✮ ☛ . Finally, ✝ ✎ ✒ ✍ ✝☞ ✆✮➇ ✡ ✒ ✆ ➇★✄ ✮ ✒❨✙▲➇ ➇★✄ P ✍ ☞ ✒ ✍

✌

✍

✂

✂

✠

.

✂

✌

✂

5.

✓

and hence

so we must have

✌

✌

✌

✌

☛

✝

✌

✮

✁

✂

➇r✄ ✵ ✮ ☛ since the integrand is even; likewise, ☛ ❊ ☞ ❴ ➇ ✒ ✍ ✎ ❊☞ ➇❫✒ ➇★✄❈✰ ☛ ✮

✍ ☞ ✒ ✍ ✝✟✞ ☛ ✁ ✁

✄

✔✁

✁

✵ ✮ ☛ (✁

✌

✱➃➂✙✗✹➉

☞ ✒ ✮

✠

☞ ✒

since the integrand is odd.

✌

✄

✔✁ ✌

✄✖➇ ✟

✵ ✮ ☛ (✁ ).

✄ ✆

➇s✡

). ✌

❲✐❞☞✳✱➃➂✙✗✗✒

P at P ✕ ✗ a ✍ jump . To ☞❊simplify 7. Suppose, to begin with, ✱✚✕ that ✰⑧✄ ❵ ✆ ✕ is ✗ continuous on ➈ P except for ✱✚✕✌ ☛ ➇❴✒ to be P P P P P P ☛ ☛ ❵ notation, we assume . For large enough so that and , define ☞❊➇❴✒ ✱ q✥➇❋q⑨✍ q ➇❋q ✗ ☞❊➇❴✒☛✄ ☞✎✍ ✒❫✡ ☛ ✟ ☞ ✒❄✍ ☞✎✍ ✒ ➉ ☞❊➇ ✡ ✒ ✍ ✕✥➇ ✕ P ☛ ☛ or ☛ ✱❍➂✙✗✹➉ ☛ if ☛ ☛ . ❵ if , and ❵ ➈❺❵ ☛ ➇✪✐ ❵ ✱➃✙➂ ✗✹➉ ☛ ☞❊➇❴❵ ✒✗❍q ☛ P ☛ ☞❊➇❴✒✗ , P if for all , the same is true of Then ❵ ☛ is continuous on ➈ ✍ P ➂ . P Moreover, ❵ ➈ ❵ ➉ ☞✎✍ ✒ ☞ ✒ ☞❊➇❫✒✂✍ ☞❊➇❫✒✗●q♠✙ ☛ and ❵ ☛ ; so ❵ ☛ ❵ since the values of ❵ ☛ on ➈ P✓☛ ✒ ☛ lie in between ❵ . Hence ✆

✟

✁

✁

☛ ☞❊➇❫✒✂✍ ☞❊➇❴✒✗ ✟ ➇ ✄ ☞❊➇❴✒✂✍ ☞❊➇❫✒✗ ✟ ➇✖q ★ ✟ ✽ ✣ ✰ ✣ ✥ ✎ P✓✒ ☛ ❵ ☛ as ✝✟✞ ❵ ☛ ❵ ✝ ❵ ✆✠P✱✆ ➂ ✄✆✄✆✓ ✄ ➂ ✆✻❘◆✐ ✱➃➂✙✗✹➉ ❊ ☞ ❫ ➇ ✒ ☞❊➇❴✒ . InP the general case ☛ P P P of ➈ , one likewise defines ❵ ☞✝✆ ✍ to be ☞✝✆ ❯ discontinuities ✍ ➂ ✆ ❯ ✡ ✒ at ✒ ❵ ☞✝except ✆ ✡ ✒ on the intervals ☛ ☛ , on which ❵ ☛ interpolates linearly between ❵ ❯ ☛ and ❵ ❯ ☛ , and one finds ☞❊➇❴✒✿✍ ☞❊➇❴✒✗ ✟ ➇❋✷ q ★✶ ✟✽ ✣ ✰ ✣ ✥ ☛ ❵ as . that ✝ ✞ ❵ ✟ ✄ ☞✝✆ ☛ ✍ ✆ ✎✠☛ ✒✗ ✟ ✄❞ ✆ ☛ ✟ ✡ ✱ ☛ ✟ ✄ ✆ ☛ ✡ ✆ ✎✠☛ ✟ ✄ ✆ ☛ ✟ ✡✚ ✆ ✎✠☛ ✟ ✡t✆ ✙ ☎ ✝☞ ✆ ☛ ✆ ✎✠☛ ✒ ✗ and ✄✖☛ ✰ We have ✟ ✟ ✟ ✟ ✟ ✆ ✍ ✙ ✆ ☎ ✝☞ ✆ ✆ ✒ ✱ ◆ ✡ ✗ ✕ ✄ ✫ ✙ ❜ ☞ ✆ ◆ ✡ ✆ ✒ ✱ ✄✌✙ ✆ ✗ ✄✖✰ ✎✠☛ ☛ ✎✠☛ , so ☛ ☛ ☛ ✎✠☛ . (When P we have and ✟ ✟ ✟ ✟ ✟ ✱ ✄✡➃ ✆ ❃☛ ✱ ✡✥❃ ❚ P ☞❜ ✱ ☛ ✡✌ ✗ ☛ ✒❑✄✕✙✠❃ ❚ ✎ ✆ ☛ ✟ ✄ , ✟ .) Hence and this☞ formula becomes ✒ ✒ ✒✗ ✟ ✒ ✝ ✎ ❵ ✵ ✾✵ . ✙✠❃ ✪ ✽✠✴✤✡✖✏✱❅ ❚ P ✏✓✽ ✪ ✄♠☞✎✏✓✽▲❃❄✒ ✄✌✙✠❃ ✪ ✽✠❂ ✪ ❚ P ✏✓✽ ✪ ✄✖❃ ✪ ✽✠✴✠✰ , or ✒ . (a) From Exercise 4, ✂ 8.1, ✒ ✝ ✎ ✵ ✾✵ ✟ ☞♦❃❋✍❈ ❺✒ ✟ ✄✝☞ ✙✜✽▲❃❄✒ ☞♦❃ ✟ ✟ ✍ ☞ ★✜✽▲❃❄✒ ✟ ❚ P ✏✓✽✣☞ ✙ ✍⑤✏✓✒ ✓ ✄❖☞✎✏✓✽▲❃❄✒ ✵ ☎✵ ✵ (b) From Exercise 6, ✂ 8.1, ✝✎ ✵ ✝ ✙✠❃ ✜☛✡ ✪✱✒ ✄✖❃ ✪✓✽✈✏✱❂ ❚ P ✏✓✽✣☞ ✙ ✒ ✍⑤✏✓✒ ✓✤✄✖❃ ✓✓✽✠✴▼❅✠✰ ✵ ✵ ✾✵ , or ✒ . ✙✫✡☞ ✠▲❃ ✪ ✽✈✏✱❂✜✒ ✟ ✡ ☞ ★✜✒ ✟ ❚ P ✏✓✽ ✞ ✄ ☞✎✏✓✽▲❃❄✒ ☞ ✪ ✍ ✙✠❃ ✟ ✟ ✒ ✟ ✄ ✵ ☎✵ (c) From Exercise 3b, ✂ 8.3, ☞ ✙✜✽▲❃❄✒ ☞ ✞ ✍ ✜❃ ✟ ✓❁✡ ✜❃ ✪ ✪✱✒ ✄✕✙✣☞✏ ✜❃ ✞ ✽✠✞✣✏✱❂ ❚ P ✒ ✏✓✽ ✞ ✄✕❃ ✞ ✽✠✴ ✾❂✠✰ ✝ ✎ ✵ ✝ ✵ ✵ ✵ ✾✵ , or ✒ . ✁

✠

✌

✌

✟

✟

✁

✠

✌

8.

✟

✟

✁

✁

✟

✆

✆

✌

✆

9.

✆

✟

✌

✟

✆

✆

✆

✟

✌

✌

✟

✆

✆

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Chapter 8. Fourier Series

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✰✮✕✘✱ ✕ ❃ ☎✽ ✱ ✟ ☞♦❃✿✍✤✱✣✒ ✟ ➉ ❚ P ☞♦✼✙✔ ✕ ✱✣✒ ✟ ✽ ✟ ✄ ☞ ✙✜✽▲❃❄✒ ✱ ✎ ✟ ✡ P P ➈ ➈ (d) ☞♦From Exercise 7, 8.1, for we have ✒ ✟ ✟ ✟ ❃❦✍ ✱✣✒ ✎ ➉❴✄❖☞ ✙✜✽▲❃❄✒ ✱✠✎ ✡✌☞♦❃★✍ ✱✫✒ ✎ ➉❴✄⑨✙✜✽ ✱❾☞♦❃❋✍ ✱✫✒ ✄ ✱❴☞♦❃★✍ ✱✫✒❨✽✠✙ ✾✵ ❚ P ☞♦✼✄✔ ✕ ✱✫✒ ✽ , or ✒ . This ☎✵ ➈ ✱❧✄✖✰ ✱ ✄✖❃ ❃ formula is still✱ valid when or (both sides vanish then), and the sum is clearly -periodic as a function of . P P ☞✎✍❝❃❄✒✦✄ ☞♦❃❄✒ ➂ ✦✄ ☞ ✒ ☞ ✒ ✄ ☞ ✟ ✒ ☞ ✒ ✄ ☞ ✒✟ ✄⑦✰ ✟ ✎ ❵ ✟ ❵ ✵ ❵ ✆ ➅ ✎ ❵ ✵ ✔ ❵ ✵ ☎✵ ✵ ✾✵ 10. First way: ❵ ✁ since ❵ . ✎ ✆ ➅❵ ☛ of ❵ , then the Fourier coefficients of ❵ are ➀ , and Second way: If ➀ ❩ ☛ are the Fourier coefficients ✆ ✎✠☛ ✄ ☞ ✒ ☛ ✄ ☞ ✒ ✎ ❩☛ ✄ ✆ ✍ ✆ ☛ ✒❁✄ ☛ . Hence, by (8.46), ❵ ➂ ❵ ☛✄✌✙✠❃ ❚ ✒✎ ✆ ☛ ☞✎✄ ✎ ❵ ✵ ✟ ✾✵ ✎✟ ❵ ✵ ☎✵ ✍✤✙✠❃ ❚ P ☞❜ ✆ ✒ ☛ ✍t ✆ ✎✠☛ ✒❁✄✕✰ . ✒ ✠

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