Mathematic Methods HW Solutions 52

Mathematic Methods HW Solutions 52 - Chapter 10 9.18 9.19...

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Unformatted text preview: Chapter 10 9.18 9.19 9.20 9.21 52 − r −1 e θ , r −1 e r , 3 2eφ , er cos θ − eθ sin θ, 3 r −1 , r −3 , 0 2r−1 , 6, 2r−4 , −k2 eikr cos θ 11.4 Vector 11.5 ds2 = du2 + h2 dv 2 , hu = 1, hv = u(2v − v 2 )−1/2 , v dA = u(2v − v 2 )−1/2 du dv, ds = eu du + hv ev dv , eu = i(1 − v ) + j(2v − v 2 )1/2 , ev = −i(2v − v 2 )1/2 + j(1 − v ) au = eu = au , av = hv ev , av = ev /hv uv 2 ˙ ∂V =− = Fu v (2 − v ) ∂u uv 2 (v − 1) ˙ ∂V uv + 2uv ¨ ˙˙ = Fv + = −u−1 [v (2 − v )]1/2 1/2 ∂v [v (2 − v )] [v (2 − v )]3/2 11.6 m u − ¨ m 11.7 U = eu ∂U /∂u + ev u−1 v (2 − v ) ∂U /∂v · V = u−1 ∂ (uVu )/∂u + u−1 v (2 − v ) ∂Vv /∂v ∂U 1 1∂ ∂ ∂U 2 u + 2 v (2 − v ) U= v (2 − v ) u ∂u ∂u u ∂V ∂V 11.8 u−1 , u−1 k, 0 ...
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