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 
.
a. Let 
denote the hemispherical <u>c</u>ap 
, parameterized by
with 
and 
. Take the normal vector to to be
Then the upward flux of 
through is
b. Let 
be the disk that closes off the hemisphere , parameterized by
with 
and 
. Take the normal to to be
Then the downward flux of 
through is
c. The net flux is then 
.
d. By the divergence theorem, the flux of across the closed hemisphere 
with boundary 
is equal to the integral of 
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 . Just to confirm, we could compute the integral in spherical coordinates:
