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A7 - F ◦ G and use the chain rule in matrix form to ﬁnd...

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Math 237 Assignment 7 Due: Friday, Nov 14th 1. Consider the following maps T : R 2 R 2 . Find the image under T of the rectangle D = { ( x, y ) R 2 | 1 x 3 , 2 y 3 } . a) T ( x, y ) = ( x + y, x - y ). b) T ( x, y ) = ( x cos( πy/ 3) , x sin( πy/ 3)). c) T ( x, y ) = ( xy, x 2 - y 2 ) 2. Invent a transformation that transforms a) The ellipse x 2 + 6 xy + 10 y 2 = 2 onto the unit circle. b) The disc x 2 + y 2 1 onto the square 0 u 2, 0 v 1. 3. 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 ( x, y ) = ( x 2 + y 2 , x 2 y 2 ) . a) Calculate the composite map
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Unformatted text preview: F ◦ G and use the chain rule in matrix form to ﬁnd the derivative matrix D ( F ◦ G ). b) Calculate D ( G ◦ F )(1 , 1). 4. Let F : R 2 → R 2 be deﬁned by F ( x,y ) = ± p x 2 + y 2 , x √ x 2 + y 2 ² . Use the linear approximation of mappings to ﬁnd the approximate image of the point (3 . 1 , 3 . 9) under F ....
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