Solutions Manual of Continuum Electromechanics by Melcher

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An Introduction to Analysis Table of Contents Chapter 1: The Real Number System 1.2 1.3 1.4 1.5 1.6

Ordered field axioms……………………………………………..1 The Completeness Axiom………………………………………..2 Mathematical Induction…………………………………………..4 Inverse Functions and Images……………………………………6 Countable and uncountable sets………………………………….8

Chapter 2: Sequences in R 2.1 2.2 2.3 2.4 2.5

Limits of Sequences……………………………………………..10 Limit Theorems………………………………………………….11 Bolzano-Weierstrass Theorem…………………………………..13 Cauchy Sequences……………………………………………….15 Limits Supremum and Infimum………………………………....16

Chapter 3: Functions on R 3.1 3.2 3.3 3.4

Two-Sided Limits………………………………………………..19 One-Sided Limits and Limits at Infinity…………………………20 Continuity………………………………………………………..22 Uniform Continuity……………………………………………...24

Chapter 4: Differentiability on R 4.1 4.2 4.3 4.4 4.5

The Derivative…………………………………………………...27 Differentiability Theorem………………………………………..28 The Mean Value Theorem……………………………………….30 Taylor’s Theorem and l’Hôpital’s Rule………………………....32 Inverse Function Theorems……………………………………...34

Chapter 5: Integrability on R 5.1 5.2 5.3 5.4 5.5 5.6

The Riemann Integral…………………………………………….37 Riemann Sums……………………………………………………40 The Fundamental Theorem of Calculus………………………….43 Improper Riemann Integration…………………………………...46 Functions of Bounded Variation…………………………………49 Convex Functions………………………………………………..51

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Chapter 6: Infinite Series of Real Numbers 6.1 6.2 6.3 6.4 6.5 6.6

Introduction……………………………………………………….53 Series with Nonnegative Terms…………………………………..55 Absolute Convergence……………………………………………57 Alternating Series………………………………………………...60 Estimation of Series……………………………………………....62 Additional Tests…………………………………………………..63

Chapter 7: Infinite Series of Functions 7.1 7.2 7.3 7.4 7.5

Uniform Convergence of Sequences……………………………..65 Uniform Convergence of Series………………………………….67 Power Series……………………………………………………...69 Analytic Functions……………………………………………….72 Applications……………………………………………………...74

Chapter 8: Euclidean Spaces 8.1 8.2 8.3 8.4

Algebraic Structure………………………………………………76 Planes and Linear Transformations……………………………...77 Topology of Rn…………………………………………………..79 Interior, Closure, and Boundary…………………………………80

Chapter 9: Convergence in Rn 9.1 9.2 9.3 9.4 9.5 9.6

Limits of Sequences……………………………………………...82 Heine-Borel Theorem…………………………………………….83 Limits of Functions……………………………………………….84 Continuous Functions…………………………………………….86 Compact Sets……………………………………………………..87 Applications………………………………………………………88

Chapter 10: Metric Spaces 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction………………………………………………………..90 Limits of Functions………………………………………………..91 Interior, Closure, and Boundary…………………………………..92 Compact Sets……………………………………………………...93 Connected Sets……………………………………………………94 Continuous Functions……………………………………………..96 Stone-Weierstrass Theorem……………………………………….97

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Chapter 11: Differentiability on Rn 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Partial Derivatives and Partial Integrals……………………………99 The Definition of Differentiability…………………………………102 Derivatives, Differentials, and Tangent Planes…………………….104 The Chain Rule……………………………………………………..107 The Mean Value Theorem and Taylor’s Formula………………….108 The Inverse Function Theorem……………………………………..111 Optimization………………………………………………………...114

Chapter 12: Integration on Rn 12.1 12.2 12.3 12.4 12.5 12.6

Jordan Regions………………………………………………………117 Riemann Integration on Jordan Regions…………………………….119 Iterated Integrals……………………………………………………..122 Change of Variables…………………………………………………125 Partitions of Unity…………………………………………………...130 The Gamma Function and Volume………………………………….131

Chapter 13: Fundamental Theorems of Vector Calculus 13.1 13.2 13.3 13.4 13.5 13.6

Curves………………………………………………………………..135 Oriented Curves……………………………………………………...137 Surfaces………………………………………………………………140 Oriented Surfaces…………………………………………………….143 Theorems of Green and Gauss……………………………………….147 Stokes’s Theorem…………………………………………………….150

Chapter 14: Fourier Series 14.1 14.2 14.3 14.4 14.5

Introduction…………………………………………………………..156 Summability of Fourier Series……………………………………….157 Growth of Fourier Coefficients…………………………………...…159 Convergence of Fourier Series……………………………………....160 Uniqueness…………………………………………………………...163

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10.5.2. a) It is relatively open in {(x, y) : y ≥ 0} because each of its points lies in a relative open ball which stays inside the set. It is relatively closed in {(x, y) : x2 + 2y 2 < 6} because the limit of any convergent sequence (in the SUBSPACE sense) in the set stays in the set. b) It is relatively open in B1 (0, 0) because each of its points lies in a relative open ball which stays inside the set. It is relatively closed in B√2 (2, 0) because the limit of any convergent sequence (in the SUBSPACE sense) in the set stays in the set. 10.5.3. a) Let I and J be connected in R. Then I and J are intervals by Theorem 10.56. Hence I ∩ J is empty or an interval, hence connected by definition or Theorem 10.56. Let A = {(x, y) : y = x2 } and B = {(x, y) : y = 1}. Then A and B are connected in R2 but A ∩ B = {(−1, 1), (1, 1)} T is not connected. b) If E = α∈A Eα is empty or contains a single point, then E is connected by definition. If E contains two points, say a, b, then a, b ∈ Eα for every α ∈ A. But Eα is an interval, hence (a, b) ⊂ Eα for all α ∈ A, i.e., (a, b) ⊂ E. Hence E is an interval, so connected by Theorem 10.56. 10.5.4. a) If E is connected in R then E is an interval, hence E o is either empty or an interval, hence connected by definition or Theorem 10.56. b) The set E = B1 (0, 0) ∪ B1 (3, 0) ∪ {(x, 0) : 1 ≤ x ≤ 2} is connected in R2 , but E o = B1 (0, 0) ∪ B1 (3, 0) is not. 10.5.5. Suppose A is not connected. Then there is a pair of open sets U, V which separates A. We claim that E ∩ U 6= ∅. If E ∩ U = ∅ then since A ∩ U 6= ∅, there exists a point x ∈ U ∩ (A \ E). But E ⊆ A ⊆ E implies A \ E ⊆ E \ E = ∂E. Thus x ∈ ∂E ∩ U . Since U is open it follows that E ∩ U 6= ∅, a contradiction. This verifies the claim. Similarly, E ∩ V 6= ∅. Thus the pair U, V separates E, which contradicts the fact that E is connected. 10.5.6. For each x ∈ X, f is constant on Bx . Since X is compact and is covered by {Bx }x∈X , there exist x1 , . . . , xN such that N [ X= Bxj . j=1

Let x ∈ X. Then x ∈ Bxj for some 1 ≤ j ≤ N , so f (x) = f (xj ). It follows that f (x) ∈ {f (x1 ), . . . , f (xN )} for all x ∈ X. In particular, f (X) ⊂ {f (x1 ), . . . , f (xN )}, so f (X) is finite, say X = {y1 , . . . , yM }. But a finite set is nonempty connected if and only if it is a single point. Indeed, if M > 1, then set r = min{ρ(yj , yk ) : j, k ∈ [1, M ]} and notice that the Br (yj )’s are open, nonempty, and disjoint, hence separate f (X). Hence, N = 1, i.e., f (x) = f (x1 ) for all x ∈ X. 10.5.7. Suppose H is compact. Let E := {Uα }α∈A be a relatively open covering of H. Choose Vα , open in Rn , such that Uα = H ∩ Vα . Then {Vα }α∈A is an open covering of H. Since H is compact, there exists a finite subset A0 of A such that {Vα }α∈A0 covers H. In particular, {Uα }α∈A0 is a finite subcovering of E which covers H. Conversely, suppose every relatively open covering of H has a finite subcover. If {Vα }α∈A is an open covering of H then {H ∩ Vα }α∈A is a relatively open covering of H. Therefore, there exists a finite subset A0 of A such that {H ∩ Vα }α∈A0 covers H. In particular, {Vα }α∈A0 covers H and H is compact. 10.5.8. a) By Remark 10.11, ∅ and X are clopen. b) Suppose E is clopen and ∅ ⊂ E ⊂ X. Then U = E and V = X \ E are nonempty open sets, U ∩ V = ∅, and X = U ∪ V . Therefore, X is not connected. Conversely, if X is not connected then there exist nonempty open sets U and V such that U ∩ V = ∅ and X = U ∪ V . Thus E := U = X \ V is clopen and ∅ ⊂ E ⊂ X. In particular, X contains more than two clopen sets. 10.5.9. Let E be a nonempty, proper subset of X. By Theorem 10.34, E has no boundary if and only if E \ E o = ∂E = ∅, i.e., if and only if E = E o . Thus E has no boundary if and only if E is clopen. This happens, by Exercise 10.5.8, if and only if X is not connected. 10.5.10. a) Suppose E is polygonally connected but some pair of open sets U, V separates E. Let x1 ∈ E ∩ U , x2 ∈ E ∩ V . Since E is polygonally connected, there is a continuous function f : [0, 1] → E with f (0) = x1 and f (1) = x2 . By Theorems 10.56 and 10.62, f ([0, 1]) is connected. But since f ([0, 1]) ⊆ E, U, V separates f ([0, 1]), a contradiction. b) Let x ∈ U . Since E is open, choose r > 0 such that Br (x) ⊂ E. Let y ∈ Br (x) and let P be a polygonal path from x0 to x which lies in E. Then the path P ∪ L(x; y) goes from x0 to y and lies in E, i.e., y ∈ U . Therefore, Br (x) ⊆ U and U is open. 95

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The easiest way to prove this result is to modify the proof of Theorem 12.39. First notice that except for continuity of the bounding functions, Ω is a type I region with φ = 0 and ψ = f . The first issue to look at: Is Ω a Jordan region? As in the proof of Theorem 12.39, ∂Ω has a top T , a bottom B, and sides S, and both B and S are of volume zero. We cannot yet conclude that T is of volume zero because ψ = f is not continuous this time. However, since f is integrable, choose a grid G = {R1 , . . . , RN } on E such that U (f, G) − L(f, G) < ². Set Mj = sup f (Rj ) and mj = inf f (Rj ) and observe that if Hj := Rj × [mj , Mj ], then SN |Hj | = (Mj − mj )|Rj | and j=1 Hj contains T . Let H represent the grid generated by the Hj ’s, i.e., the partitions Pk (H), k = 1, 2, are the same as those of G, and the partition P3 (H) := {Mj , mj : j = 1, 2, . . . , N } arranged in increasing order. Then H is a grid on T , and X

V (T ; H) =

|Hj | =

Hj ∩T 6=∅

(Mj − mj )|Rj | = U (f, G) − L(f, G) < ².

Rj ∩E6=∅

We conclude by Theorem 12.4ii that ∂Ω is of volume zero, hence Ω is a Jordan region. Now repeat the proof of the second half of Theorem 12.39. Since continuity of φ and ψ were not used there, we conclude that ZZ Z ZZ ZZZ f (x,y)

Vol (Ω) =

dV =

dz d(x, y) =

Ω

f dV.

Rd 12.3.9. a) These inequalities can be verified by repeating the proof of Lemma 12.30 with (X) c f (x, y) dy in Rd place of c f (x, y) dy. b) If f is integrable on R then the inequalities of part a) become equalities for both X = L and X = U . This proves part b) c) Since Z 1 Z 1 (L) f (x, y) dy = x(1 − 0) = x and (U ) f (x, y) dy = 1(1 − 0) = 1, 0

we have

Z (L)

f (x, y) dy

1 6= 1 = 2

dx =

µ Z (U )

f (x, y) dy

dx.

Hence by part b), f cannot be integrable on [0, 1] × [0, 1]. Rβ 12.3.10. Let ² > 0 and choose a < A < B < b such that |F (y) − α f (x, y) dx| < ² for all a < α < A and B < β < b. Then ¯Z Ã ! ¯ Z β ¯ d ¯ ¯ ¯ F (y) − f (x, y) dx dy ¯ < ²(d − c), ¯ ¯ c ¯ α i.e., that

RdRβ c

f (x, y) dx dy converges to Z

Rd c

F (y) dy as α → a+ and β → b−. Hence it follows from Fubini’s Theorem Z

f (x, y) dx dy = c

lim

f (x, y) dx dy

α→a+,β→b−

α d

βZ

lim

f (x, y) dy dx =

α→a+,β→b−

f (x, y) dy dx. a

12.4 Change of Variables. Z

π/2

12.4.1. a) 0

sin(r2 )r dr dθ =

π (1 − cos 4). 4

b) By Fubini’s Theorem, and the substitution u = 2y − y 2 , du = (2 − 2y) dy, we have Z 0

p 3

(2y − y 2 )2 dy dx =

Z p 3 1 1 2/3 3 u du = . (2y − y 2 )2 dx dy = 2 0 10 125

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Chapter 13 13.1 Curves. 13.1.1. (ψ, I) runs clockwise. Its speed is kψ 0 (t)k = k(a cos t, −a sin t)k = a. (σ, J) runs counterclockwise. Its speed is k(−2a sin 2t, 2a cos 2t)k = 2a. 13.1.2. Since φ0 (t) = a 6= 0, C is smooth. Since ta + x0 = ua + x0 implies t = u, C is simple. Since φ(0) = x0 and φ(1) = x0 + a, C contains x0 and x0 + a. Finally, if θ is the angle between φ(t1 ) − φ(0) and φ(t2 ) − φ(0), then by (3) in 8.1 we have t1 t2 t1 a · (t2 a) = = ±1. cos θ = |t1 | |t2 | kak2 |t1 | |t2 | Hence θ = 0 or π. 13.1.3. The trace of φ(θ) := (f (θ) cos θ, f (θ) sin θ) on I := [0, 2π] coincides with the graph r = f (θ). Since φ0 (θ) = (f 0 (θ) cos θ − f (θ) sin θ, f 0 (θ) sin θ + f (θ) cos θ) we have kφ0 (θ)k2 = |f 0 (θ)|2 + |f (θ)|2 6= 0. Therefore, (φ, I) is a smooth curve. 13.1.4. Let C = (φ, (0, 1]) where φ(t) = (t, sin(1/t)) and set tk = 2/((2k + 1)π) for k ∈ N. Since sin(1/tk ) = (−1)k , it is clear that kφ(tk ) − φ(tk+1 )k ≥ 2 for each k ∈ N. Hence by definition, kCk ≥

k X

kφ(tj ) − φ(tj+1 )k ≥ 2k

j=1

for each k ∈ N, i.e., kCk = ∞. 13.1.5. a) This curve evidently lies on the cone x2 + y 2 = z 2 . It spirals around this cone from φ(0) = (0, 1, 1) to φ(2π) = (0, e2π , e2π ). Since φ0 (t) = (et (sin t + cos t), et (cos t − sin t), et ), the arc length is given by Z

2π

kφ0 (t)k dt =

L(C) =

√ Z 3

2π

et dt =

√ 2π 3(e − 1).

√ b) This curve forms a “script vee” from (−1, 1) through (0, 0) to (1, 1). Since x = y 3/2 implies x0 = 3 y/2, we have by the explicit form (see the formula which follows (3)) that Z L(C) = 2

1 + 9y/4 dy = 0

√ p 2( 133 − 1) . 4 + 9y dy = 27

2 √ c) Since φ(t) =√t (1, 1, 1), this curve is the straight line from (0, 0, 0) to (4, 4, 4). Hence its arc length is 2 2 2 4 + 4 + 4 = 4 3. d) By definition, φ0 (t) = (−3 cos2 t sin t, 3 sin2 t cos t), hence

p kφ0 (t)k = 3 cos4 t sin2 t + sin4 t cos2 t = 3| sin t cos t|. Therefore,

π/2

L(C) = 4 0

¯π/2 3| sin t cos t| dt = 6 sin2 t ¯0 = 6.

13.1.6. a) If φ(t) = (3 cos t, 3 sin t) and I = [0, π/2], then kφ0 (t)k = k(−3 sin t, 3 cos t)k = 3, hence Z

π/2

xy 2 ds = C

3 cos t · 9 sin2 t · 3 dt = 27.

b) If φ(t) = (a cos t, b sin t) and I = [0, π/2], then kφ0 (t)k = k(−a sin t, b cos t)k =

p a2 + (b2 − a2 ) cos2 t,

135

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