(938, #27) Find the area of the surface with the parametric equations

x = u v, y = u + v, z = u - v, u2 + v2 <= 1.

Solution: Set r(u, v) = <u v, u + v, u - v>. Then

Consequently

The surface area is given by

The parametrization u = r cos(t), v = r sin(t) is used to graph this surface:

plot3d([ r^2*cos(t)*sin(t), r*(cos(t) + sin(t)),r*(cos(t) - sin(t))],
r = 0..1, t = 0..2*Pi, color = red, labels = [`x`, `y`, `z`],
labelfont = [HELVETICA, BOLD, 10], tickmarks = [2,2,2],
grid = [10, 25] );