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# a9 - F-1 explicitly b Find the derivative matrices DF x,y...

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Math 237 Assignment 9 Due: Friday, Nov 26th 1. Invent an invertible transformation that transforms the ellipse 3 x 2 + 6 xy + 4 y 2 = 4 onto the unit circle and determine the inverse map. 2. Invent an invertible transformation f : R 3 R 3 that maps the ellipsoid x 2 +8 y 2 +6 z 2 + 4 xy - 2 xz + 4 yz = 9 onto the unit sphere. 3. Consider the maps F : R 2 R 2 defined by ( u, v ) = F ( x, y ) = ( y + xy, y - xy ). a) Show that F has an inverse map by finding
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Unformatted text preview: 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 . 4. Find the Jacobian of ( u,v ) = T ( x,y ) = ( x 2 + y 2 ,x 2-y 2 )....
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