(937, #1) Find a parametric representation for the upper half of the ellipsoid 3 x2 + 2 y2 + z2 = 1.

Solution: Take

x = (1/sqrt(3)) r cos(t), y = (1/sqrt(2)) r sin(t), z = sqrt(1-r2)

where r = ..1, t = 0..2 Pi. Then

3 x2 + 2 y2 + z2 = 3( 1/3 *r2 cos2(t)) + 2(1/2 * r2 sin2 (t)) + 1 -r2 =

r2(cos2(t) + sin2(t)) + 1 -r2 =1,

so the points (x,y,z) lie on the ellipsoid.

The Maple code for the surface is:

plot3d([(1/sqrt(3))*r* cos(t), (1/sqrt(2))*r *sin(t), sqrt(1-r^2)], r = 0..1, t = 0..2*Pi,
grid = [10,25], color = blue, tickmarks = [3,3,3]);