Problem 4: let f = (2z + 2)k be the flow field. answer the following to verify the divergence theorem: a) use definition to find the upward flux (normal vector pointing upward in z direction) along the hemisphere given in problem 1(b). b) use definition to find the downward flux on the circle x 2 + y 2 ≤ 3 on the xy-plane. c) using the results from parts (a) and (b), what is the net flux over the closed northern hemisphere? d) use the divergence theorem to verify that this answer agrees. [hint: dv = rho 2 sin ϕ drho dθ dϕ.]

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 cap , 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: