Parabolic coordinates Let T be the transformation x = u2 -

Chapter 13, Problem 57

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Parabolic coordinates Let T be the transformation x = u2 - v2, y = 2uv. a. Show that the lines u = a in the uv-plane map to parabolas in the xy-plane that open in the negative x-direction with vertices on the positive x-axis. b. Show that the lines v = b in the uv-plane map to parabolas in the xy-plane that open in the positive x-direction with vertices on the negative x-axis. c. Evaluate J1u, v2. d. Use a change of variables to find the area of the region bounded by x = 4 - y2>16 and x = y2>4 - 1. e. Use a change of variables to find the area of the curved rectangle above the x-axis bounded by x = 4 - y2>16, x = 9 - y2>36, x = y2>4 - 1, and x = y2>64 - 16. f. Describe the effect of the transformation x = 2uv, y = u2 - v2 on horizontal and vertical lines in the uv-plane.