math237_a8

math237_a8 - SOLUTIONS FOR ASSIGNMENT 8 Problem Set 7 A1(ii...

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Unformatted text preview: SOLUTIONS FOR ASSIGNMENT 8 Problem Set 7: A1 (ii): Find the image of the square D = { ( x, y ) ∈ R 2 | 1 ≤ x ≤ 2 , 2 ≤ y ≤ 3 } under the transformation T : R 2 → R 2 , T ( x, y ) = ( xy, x 2 − y 2 ) . Since the boundary of D is made of straight lines of equations x = k and y = l , for k = 1 , k = 2 and l = 2 , l = 3 , we study the images of x = k and y = l under the map T . If we denote by: u = xy v = x 2 − y 2 , then, for x = k : u = ky, v = k 2 − y 2 , and if we eliminate y we get: v = k 2 − u 2 k 2 . Thus the images of the straight lines x = k under T are parabolas of equations: v = k 2 − u 2 k 2 . SImilarly, for y = l : u = xl, v = x 2 − l 2 , and if we eliminate x we get: v = u 2 l 2 − l 2 , i.e. the images if the straight lines y = l under T are parabolas of equations: v = u 2 l 2 − l 2 . In conclusion the straight lines that make the boundaries of D transform into parabolas under the map T as it is seen in fig. (1). Problem Set 7: A3: Consider the map F : R 2 → R 2 defined by F ( x, y ) = parenleftBigg radicalBig x 2 + y 2 , x √ x 2 + y 2 parenrightBigg . Use the linear approximation in the matrix form to find the approximate image of the point (3 . 1 , 3 . 9) under F...
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math237_a8 - SOLUTIONS FOR ASSIGNMENT 8 Problem Set 7 A1(ii...

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