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# a9 - 6 Invent an invertible transformation f R 3 → R 3...

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Math 237 Assignment 9 Due: Friday, Mar 25th * 1. Invent an invertible transformation that transforms the ellipse x 2 + 4 xy + 5 y 2 = 5 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 + 2 y 2 + 2 z 2 + 2 xy + 2 xz + 2 yz = 1 onto the unit sphere. * 3. Consider the maps F : R 2 R 2 deﬁned by ( u,v ) = F ( x,y ) = ( y + e - x ,y - e - x ). 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 . * 4. Find the Jacobian of ( u,v ) = T ( x,y ) = ( x 2 + 2 xy 3 ,x 2 y ). 5. 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.
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Unformatted text preview: 6. 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. 7. 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 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 . 8. Find the Jacobian of ( u,v ) = T ( x,y ) = ( x 2 + y 2 ,x 2-y 2 )....
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