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 <em>F</em>) - (integral over <em>D</em>) = integral over <em>S</em>
==> 486π/5 - (-81π/4) = 2349π/20