Chapter 14, Problem 55RE

Flux across hemispheres and paraboloids Let S be the hemisphere x² + y² + z² = a², for z ≥ 0, and let T be the paraboloid z = a – (x² + y²)/a, for z ≥ 2= 0, where a > 0. Assume the surfaces have outward normal vectors.

a.    Verify that S and T have the same base (x² + y² ≤ a²) and the same high point (0, 0, a).

b.    Which surface has the greater area?

c.    Show that the flux of the radial field F = 〈x, y, z〉 across S is 2πa³.

d.    Show that the flux of the radial field F = 〈x, y, z〉 across T is 3πa³/2.