Parabolic coordinates Let T be the transformation x = u2 - ν2, y = 2uν.a. Show that the lines u = a in the uν-plane map to parabolas inthe xy-plane that open in the negative x-direction with verticeson the positive x-axis.b. Show that the lines ν = b in the uv-plane map to parabolas inthe xy-plane that open in the positive x-direction with vertices on the negative x-axis.c. Evaluate J(u, ν).d. Use a change of variables to find the area of the regionbounded by x = 4 - y2/16 and x = y2/4 - 1.e. Use a change of variables to find the area of the curvedrectangle 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 = 2uν,y = u2 - ν2 on horizontal and vertical lines in the uν-plane.
Parabolic coordinates Let T be the transformation x = u2 - ν2, y = 2uν.
a. Show that the lines u = a in the uν-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 ν = 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, ν).
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 = 2uν,
y = u2 - ν2 on horizontal and vertical lines in the uν-plane.
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