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

where  is the interior of the joined surfaces
 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:
. 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
 over the disk, oriented downward. Since  in
 in  , we have
, we have

Parameterize  by
 by

where  and
 and  . Take the normal vector to be
. Take the normal vector to be

Then taking the dot product of  with the normal vector gives
 with the normal vector gives

So the contribution of integrating  over
 over  is
 is

and the value of the integral we want is
(integral of divergence of <em>F</em>) - (integral over <em>D</em>) = integral over <em>S</em>
==>  486π/5 - (-81π/4) = 2349π/20