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A72_soln

# A72_soln - Math 237 Assignment 7 Solutions 1 Consider the...

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Math 237 Assignment 7 Solutions 1. Consider the following maps T : R 2 R 2 . Find the image under T of the square D = { ( x, y ) R 2 | 1 x 3 , 0 y 2 } . a) T ( x, y ) = (2 x y, 2 x + y ). Solution: We have u = 2 x y and v = 2 x + y , so For x = 1, 0 y 2 we get u = 2 y so 0 y 2 0 u 2 and v = 2 + y = 2 + (2 u ) = 4 u . For x = 3, 0 y 2 we get u = 6 y so 0 y 2 4 u 6 and v = 6 + y = 6 + (6 u ) = 12 u . For y = 2, 1 x 3 we get u = 2 x 2 so 1 x 3 0 u 4 and v = 2 x + 2 = ( u + 2) + 2 = u + 4. For y = 0, 1 x 3 we get u = 2 x so 1 x 3 2 u 6 and v = 2 x = u . b) T ( x, y ) = (2 x y, x 2 2 x + 3 + y ). Solution: We have u = 2 x y and v = x 2 2 x + 3 + y , so For x = 1, 0 y 2 we get u = 2 y so 0 y 2 0 u 2 and v = 2 + y = 2 + (2 u ) = 4 u . For x = 3, 0 y 2 we get u = 6 y so 0 y 2 4 u 6 and v = 6 + y = 6 + (6 u ) = 12 u . For y = 2, 1 x 3 we get u = 2 x 2 = 2( x 1) so 1 x 3 0 u 4 and v = x 2 2 x + 3 + 2 = ( x 1) 2 + 4 = u 2 4 + 4. For y = 0, 1 x 3 we get u = 2 x so 1 x 3 2 u 6 and v = x 2 2 x + 3 = ( x 1) 2 + 2 = ( u 2 1 ) 2 + 2.

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A72_soln - Math 237 Assignment 7 Solutions 1 Consider the...

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