Chapter 13.7, Problem 57E

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 J(u, v).

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.