Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of the sphere x2 + y2 + z2 = 9. (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk x2 + y2 ≤ 9, oriented downward, and S2 = S1 ∪ S.)

Close off the hemisphere by attaching to it the disk of radius 3 centered at the origin in the plane . By the divergence theorem, we have

where is the interior of the joined surfaces .

Compute the divergence of :

Compute the integral of the divergence over . Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

So the volume integral is

From this we need to subtract the contribution of

that is, the integral of over the disk, oriented downward. Since in , we have

Parameterize by

where and . Take the normal vector to be

Then taking the dot product of with the normal vector gives

So the contribution of integrating over is

and the value of the integral we want is

(integral of divergence of F) - (integral over D) = integral over S

==>  486π/5 - (-81π/4) = 2349π/20