# a8 - a) The region enclosed by r = cos 2 . b) Inside r = 3...

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Math 237 Assignment 8 Due: Friday, Mar 18th * 1. For each of the indicated regions in polar coordinates, sketch the region and ﬁnd the area. a) The region enclosed by r = sin 3 θ . b) Inside both r = 2 + 2 cos θ and r = 2 - 2 cos θ . * 2. Convert the following equations in Cartesian coordinates to cylindrical coordinates. a) z = x 2 + y 2 . b) 1 = x 2 - y 2 . * 3. Convert the following equations in Cartesian coordinates to spherical coordinates. a) x = y . b) ( x 2 + y 2 + z 2 ) 2 = z . * 4. Consider ( u,v ) = F ( x,y ) = ( y + e x ,e x - y ). a) Find and sketch the image of the square with vertices (0 , 0), (0 , 1), (1 , 1), and (1 , 0) under F . b) Use the linear approximation for mappings to approximate the image of ( x,y ) = (0 . 01 , 0 . 02) under F . * 5. Let ( u,v ) = F ( x,y ) = ( x ln( y - x 4 ) , (2 + y x ) 3 / 2 ) . Suppose that G ( u,v ) has continuous partial derivatives with G (0 , 8) = (1 , - 1) and DG (0 , 8) = ± - 2 1 - 4 3 ² . Use the linear approximation to approximate ( G F )(0 . 9 , 2 . 1). 6. For each of the indicated regions in polar coordinates, sketch the region and ﬁnd the area.
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Unformatted text preview: a) The region enclosed by r = cos 2 . b) Inside r = 3 cos and outside r = 1 + cos . 7. Let T ( x,y ) = ( x + 2 y, 3 x-y ) and let D = { ( x,y ) R 2 | -1 x 2 , y 2 } . Find the image of D under T . 8. Let T ( x,y ) = ( x 2 + y 2 ,x 2-y 2 ) and let D = { ( x,y ) R 2 | 1 x 2 , y 2 } . Find the image of D under T . 9. Let ( p,q ) = F ( u,v ) = ( v cos( uv + 2) ,v u 2 + 5 ) and ( u,v ) = G ( x,y ) = ( xy-xy 2 ,xe xy-2 ). a) Use the chain rule in matrix form to nd D ( F G )(1 , 2). b) Use the linear approximation for mappings to approximate the image of ( x,y ) = (1 . 1 , 1 . 9) under F G . NOTE: Only * questions will be graded...
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## This note was uploaded on 12/05/2011 for the course MATH 237 taught by Professor Wolczuk during the Winter '08 term at Waterloo.

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