Given that you mention the divergence theorem, and that part (b) is asking you to find the downward flux through the disk  , I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere
, I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere  .
.
a. Let  denote the hemispherical <u>c</u>ap
 denote the hemispherical <u>c</u>ap  , parameterized by
, parameterized by

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

Then the upward flux of  through
 through  is
 is



b. Let  be the disk that closes off the hemisphere
 be the disk that closes off the hemisphere  , parameterized by
, parameterized by

with  and
 and  . Take the normal to
. Take the normal to  to be
 to be

Then the downward flux of  through
 through  is
 is


c. The net flux is then  .
.
d. By the divergence theorem, the flux of  across the closed hemisphere
 across the closed hemisphere  with boundary
 with boundary  is equal to the integral of
 is equal to the integral of  over its interior:
 over its interior:

We have

so the volume integral is

which is 2 times the volume of the hemisphere  , so that the net flux is
, so that the net flux is  . Just to confirm, we could compute the integral in spherical coordinates:
. Just to confirm, we could compute the integral in spherical coordinates:
