# a8 - x,y = x 2 y 2,x 2 y 2 a Use the chain rule in matrix...

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Math 237 Assignment 8 Due: Friday, Nov 20th 1. Convert the following points from Cartesian coordinates to polar coordinates with 0 θ < 2 π . a) ( - 3 , 3) b) (1 , - 1 / 3). 2. Convert the following points from polar coordinates to Cartesian coordinates. a) (3 , - π/ 6). b) (1 ,π/ 5). 3. For each of the indicated regions in polar coordinates, sketch the region and ﬁnd the area. a) The region enclosed by r = 4 - 2 cos θ . b) Inside both r = cos θ and r = sin 2 θ . 4. Convert the following equations in Cartesian coordinates to cylindrical coordinates. a) z = x 2 + y 2 . b) z 2 = x 2 - y 2 . 5. Convert the following equations in Cartesian coordinates to spherical coordinates. a) z = x 2 + y 2 . b) z 2 = x 2 - y 2 . c) x 2 = y 2 . 6. Consider the following maps T : R 2 R 2 . Find the image under T of the rectangle D = { ( x,y ) R 2 | 0 x 2 , - 2 y 2 } . a) T ( x,y ) = ( x cos( πy/ 3) ,x sin( πy/ 3)). b) T ( x,y ) = ( x 2 - y 2 ,xy ).

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2 7. Consider the maps F : R 2 R 2 and G : R 2 R 2 deﬁned by F ( u,v ) = ( e uv ,e u - v ) , G
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Unformatted text preview: ( x,y ) = ( x 2 + y 2 ,x 2 y 2 ) . a) Use the chain rule in matrix form to ﬁnd the derivative matrix D ( F ◦ G ). (You may leave D ( F ◦ G ) in terms of u,v,x,y ). b) Calculate D ( G ◦ F )(1 , 1). c) Use the linear approximation of mappings to approximate the image of ( u,v ) = (1 . 01 , . 98) under G ◦ F . 8. Let ( p,q ) = F ( u,v ) = (-ln( u-v ) , 1 2 ( u + v ) ) and ( u,v ) = G ( x,y ) = ( y + 1 2 e-x ,y-1 2 e-x ) . a) Find and sketch the image T of the square with vertices (0 , 0), (1 , 0), (0 , 1), and (1 , 1) under G . b) Find and sketch the image of T under F . c) Use the linear approximation of mappings to approximate the image of ( x,y ) = (0 . 01 , . 02) under F ◦ G ....
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a8 - x,y = x 2 y 2,x 2 y 2 a Use the chain rule in matrix...

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