# tut8 - F At what points a does the mapping F have an...

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MATH 237 Tutorial 8 Wednesday, July 5th, 2006 1. Let D xy be the region in the xy –plane which is enclosed by the lines y = 2 - x, y = 4 - x, x = 0 and y = 0. Find the image, D uv in the uv –plane, of D xy under the mapping ( u, v ) = F ( x, y ) = ± x + y 2 , x - y x + y ² and sketch both regions. 2. Consider the mappings F : R 2 R 2 and G : R 2 R 2 de±ned by F ( u, v ) = (e u + v , e u - v ) , G ( x, y ) = ( xy, x 2 - y 2 ). a) Calculate the composite map F G and its derivative matrix D ( F G ). b) Use the chain rule in matrix form to verify your result in (a). 3. Consider the mapping F : R 2 R 2 given by ( u, v ) = F ( x, y ) = ( y + e x , y - e x ). a) Find and sketch the image under F of the square with vertices (0 , 0) , (1 , 0) , (0 , 1) and (1 , 1). b) Find F - 1 explicitly. c) Find the derivative matrices D F ( x, y ) and D F - 1 ( u, v ) and verify that D F - 1 ( u, v ) D F ( x, y ) = I where I is the identity matrix. 4. Consider the mapping F : R 2 R 2 de±ned by ( u, v ) = F ( x, y ) = ( x - y, xy ). a) Calculate the derivative matrix of
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Unformatted text preview: F . At what points a does the mapping F have an inverse mapping in some neighbourhood of a ? b) Consider the region in the ±rst quadrant bounded by the lines x-y = 0 , x-y = 2 and the curves xy = 1 , xy = 4. Give a clearly labelled sketch of D xy in the xy – plane and of its image in the uv –plane. 5. a) Consider the transformation u = x-y, v = 1 x + y . How is the area of a small bounded region in the xy –plane, A xy , containing the point (0 , 1) related to the area of its image in the uv –plane, A uv ? b) De±ne D xy = { ( x, y ) ∈ R 2 : 1 ≤ x + y ≤ 2 , y ≥ , x ≥ y } . Sketch D xy and its image under the transformation in (a). In both sketches provide the equations for the boundaries....
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