(937, #5) Find a parametric representation of the part of the sphere x2 + y2 + z2 = 4 that lies above the cone z = sqrt(x2 + y2)

(937, #5) Find a parametric representation of the part of the sphere x² + y² + z² = 4 that lies above the cone
z = sqrt(x² + y²).

Solution: The curves intersect when x² + y² + x² + y² = 4. That is, when = x² + y² = 2. Thus one parametric representation is

x = sqrt(2) r cos(t), y = sqrt(2) r sin(t) , z = sqrt(4 - 2 r²)

where r = 0..1 and t = 0 ..2*Pi.

plot3d([sqrt(2)*r*cos(t), sqrt(2)*r*sin(t), sqrt(4 - 2*r^2)], r = 0..1, t = 0..2*Pi,
grid = [10,25], color = blue, tickmarks = [2,3,2], labels = [`x`, `y`, `z`],
labelfont = [HELVETICA, BOLD, 10],
view = [-1.5..1.5, -1.5..1.5, 0..2]);