Page 857 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 837

a
u v = (−a cosh(u)sin(v)i + sinh(u)cos(v)j),
h 1
u z = k
1
∂ ∂ ∂ F 3
∇· F(u,v, z) = (F 1 h 1 ) + (F 2 h 1 ) +
h 2 1 ∂u ∂v ∂z

1 ∂ F 3 ∂ F 2
∇× F(u,v, z) = − u u
h 1 ∂v ∂z

∂ F 1 1 ∂ F 3
+ − u v
∂z h 1 ∂u
1 ∂ ∂
+ (F 2 h 1 − (F 1 h 1 ) u z
h 2 1 ∂u ∂v
2
2
2
1 ∂ f ∂ f ∂ f
2
∇ f (u,v, z) = + +
h 2 1 ∂u 2 ∂v 2 ∂z 2
4.
√
2
2
h u = h v = u + v ,h z = 1
1 ∂ f 1 ∂ f ∂ f
∇ f (u,v, z) = u u + u v + u z
h u ∂u h v ∂v ∂z
1 ∂ ∂
∇· F(u,v, z) = (h u F 1 ) + (h v F 2 )
h 2 u ∂u ∂v
1 ∂ 2
+ h F 3
u
2
h ∂z
u
1 ∂ ∂
∇× F(u,v, z) = (F 3 ) − (h u F 2 ) u 1
h u ∂u ∂z
1 ∂ ∂
+ (h u F 1 ) − (F 3 ) u 2
h 2 v ∂z ∂u
1 ∂ ∂
+ (h v F 2 ) − (h u F 1 ) u 3
h 2 ∂u ∂v
v
2
2
1 ∂ f ∂ f ∂ ∂ f
2 2
∇ f = + + h u
h 2 u ∂u 2 ∂v 2 ∂z ∂z
CHAPTER THIRTEEN FOURIER SERIES
Section 13.1 Why Fourier Series?
3. If p(x) has degree k, then differentiating p(x), k + 1 times yields the zero function, while N n=1 b n sin(nx) can be
differentiated any number of times, and none of these derivatives is identically zero on [0,π].
Section 13.2 The Fourier Series of a Function
1. 4; the series (consisting of one term) converges to 4 on [−3,3].
3.
∞
1 2 (−1) n
sinh(π) + sinh(π) cos(nπx),
2
π π n + 1
n=1
converging to cosh(πx) for −1 ≤ x ≤ 1.

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October 14, 2010 17:50 THM/NEIL Page-837 27410_25_Ans_p801-866