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Assignment 8

# Assignment 8 - Math 237 Assignment 8 Due Friday Nov 28th 1...

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Math 237 Assignment 8 Due: Friday, Nov 28th 1. Consider the maps F : R 2 R 2 defined by ( u, v ) = F ( x, y ) = ( e x + y , e x - y ). a) Show that F has an inverse map by finding F - 1 explicitly. b) Find the derivative matrices DF ( x, y ) and DF - 1 ( u, v ) and verify that DF ( x, y ) DF - 1 ( u, v ) = I . c) Verify that the Jacobians satisfy ( x,y ) ( u,v ) = h ( u,v ) ( x,y ) i - 1 . 2. Find the Jacobian for the following mappings. a) ( x, y ) = T ( r, θ ) = ( r cos θ, r sin θ ). b) ( x, y, z ) = T ( r, θ, z ) = ( r cos θ, r sin θ, z ). c) ( x, y, z ) = T ( ρ, θ, φ ) = ( ρ cos θ sin φ, ρ sin θ sin φ, ρ cos φ ). d) ( u, v ) = T ( x, y ) = ( xy 2 , x 2 + y 2 ). 3. Evaluate the following integrals. a) RR D xe y dA where D is the region bounded by y = x , y = 0 and x = 1. b) RR D xy 2 dA where D is the region bounded by x + y = - 1, x + y 2
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