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Unformatted text preview: Math 237 Tutorial 9 Problems 1: Consider the map F : R2 → R2 deﬁned by (u, v ) = F (x, y ) = (ex+y , ex−y ). a) Show that F has an inverse map by ﬁnding F −1 explicitly. b) Verify that the Jacobians satisfy
∂ (x,y ) ∂ (u,v ) ∂ (u,v ) ∂ (x,y ) −1 = . 2: Invent an invertible transformation G that transforms the region bounded by x = y 2 , x = y 2 + 1, y = 0, y = 1 onto a square Duv in the uv -plane. Find the inverse mapping G−1 and verify that the Jacobian of G−1 is non zero on Duv . 3: Let (u, v, w) = F (x, y, z ) = (x3 + yz, −x + yz, xz 2 ). a) Find the Jacobian of F . b) Using your result in a), what can you say about the existence of F −1 at (1, 0, 1)? c) Approximate the area of the image of a small rectangle prism with area ∆x∆y ∆z containing the point (1, 0, 1) under F . ...
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This note was uploaded on 01/27/2011 for the course MATH 237 taught by Professor Wolczuk during the Fall '08 term at Waterloo.
- Fall '08