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Unformatted text preview: Math 237 Assignment 8 Due: Friday, Nov 28th 1. Consider the maps 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) 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 ) = ∂ (u,v )
∂ (x,y ) −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 , x2 + y 2 ).
3. Evaluate the following integrals.
xey dA where D is the region bounded by y = x, y = 0 and x = 1. a)
D xy 2 dA where D is the region bounded by x + y = −1, x + y 2 = 1. b)
D
2
0 c) √
0 √ d)
R 4−x2 (4 − y 2 )3/2 dy dx. 1
x2 + y 2 dA where R is the region outside x2 + y 2 = 4 and inside x2 + y 2 = 4x. y 2 dA, where Dxy is bounded by the ellipse x2 + 6xy + 10y 2 = 2. e)
Dxy 4. Let Dxy be the region in the xy plane enclosed by the lines y = 2 − x, y = 4 − x,
y = x and y = 0. Let (x, y ) = F (u, v ) = (u + uv, u − uv ).
a) Show that F has an inverse map F −1 deﬁned on Dxy .
b) Find the image Duv of Dxy under F −1 .
x−y e x+y
dx dy .
x+y c) Use the mapping F to evalute
D ...
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This note was uploaded on 01/04/2012 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.
 Spring '08
 WOLCZUK
 Derivative, Matrices

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