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Let T(u, v) = (x(u, v), y(u, v)) be an invertible transformation from the uv-plane to the ry-plane. Let T1 denote its inverse so that T-1 (x, y) = (u(x,y), v(x, y)). a(x, y) (a) If a(u, v) a(u, v) a(x, y) is the Jacobian for T and is the Jacobian for T-1, use the multivariable chain rule to show that 8(x, y) 8(u, v) = 1. ο (u, υ) 8(α, )

Let T(u, v) = (x(u, v), y(u, v)) be an invertible transformation from the uv-plane to the ry-plane. Let T1 denote its inverse so that T-1 (x, y) = (u(x,y), v(x, y)). a(x, y) (a) If a(u, v) a(u, v) a(x, y) is the Jacobian for T and is the Jacobian for T-1, use the multivariable chain rule to show that 8(x, y) 8(u, v) = 1. ο (u, υ) 8(α, )

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you solve (a)?

Let T(u, v) = (x(u, v), y(u, v)) be an invertible transformation from the uv-plane to the ry-plane. Let T1 denote its inverse so that T-1 (x, y) = (u(x,y), v(x, y)). a(x, y) (a) If a(u, v) a(u, v) a(x, y) is the Jacobian for T and is the Jacobian for T-1, use the multivariable chain rule to show that a(x, y) a(u, v) = 1. a(u, v) a(x, y) 9 dA where (b) Use part (a) and the Change of Variables Theorem to evaluate {(x, y) : 1 < a² – y² < 4,0 < y<}. D= 2
Transcribed Image Text:Let T(u, v) = (x(u, v), y(u, v)) be an invertible transformation from the uv-plane to the ry-plane. Let T1 denote its inverse so that T-1 (x, y) = (u(x,y), v(x, y)). a(x, y) (a) If a(u, v) a(u, v) a(x, y) is the Jacobian for T and is the Jacobian for T-1, use the multivariable chain rule to show that a(x, y) a(u, v) = 1. a(u, v) a(x, y) 9 dA where (b) Use part (a) and the Change of Variables Theorem to evaluate {(x, y) : 1 < a² – y² < 4,0 < y<}. D= 2
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