Consider f(x, y) = (y^2 - x^2)/2 and g(u, v) = uv. (a) Draw contour lines...

 

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Consider f(x, y) = (y^2 - x^2)/2 and g(u, v) = uv. (a) Draw contour lines for f (x, y) and g(u, v) in a single xy-plane and single uv- plane respectively. These should correspond to f = 0, 1,-1, 2, -2, 3, -3 and g = 0, 1, -1, 2, -2, 3, -3, and any axes-intercepts should be clearly marked. (b) At what points and for which c, do the lines v = u and v=-u cross the contour g = c? At what distance from the origin do these intersections occur? (c) At what points and for which c, do the lines x = 0 and y = 0 cross the contour f = c? At what distance from the origin do these intersections occur? (d) The above work suggests a geometric transformation will transform the graph off to the graph of g. You could use a 3D grapher, such as that available from the MATH1023 Canvas site, to convince yourself further. Say the transformation is u = T1(x, y) and v= T2(x, y). Find functions T1 and T2 that honor this transformation where preserving symmetry requires the distances from the origin to be equal. In other words, require √x2 + y2 = √u2 + v2, or simply x2 +y2 = u2 +v2. Consider f(x, y) = (y^2 - x^2)/2 and g(u, v) = uv. (a) Draw contour lines for f (x, y) and g(u, v) in a single xy-plane and single uv- plane respectively. These should correspond to f = 0, 1,-1, 2, -2, 3, -3 and g = 0, 1, -1, 2, -2, 3, -3, and any axes-intercepts should be clearly marked. (b) At what points and for which c, do the lines v = u and v=-u cross the contour g = c? At what distance from the origin do these intersections occur? (c) At what points and for which c, do the lines x = 0 and y = 0 cross the contour f = c? At what distance from the origin do these intersections occur? (d) The above work suggests a geometric transformation will transform the graph off to the graph of g. You could use a 3D grapher, such as that available from the MATH1023 Canvas site, to convince yourself further. Say the transformation is u = T1(x, y) and v= T2(x, y). Find functions T1 and T2 that honor this transformation where preserving symmetry requires the distances from the origin to be equal. In other words, require √x2 + y2 = √u2 + v2, or simply x2 +y2 = u2 +v2.

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