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