Calc 3 - outward flux question

[chongj12]

The region in question is bounded by:
the cylinder (x^2)+(y^2)=(R^2)
the parabola x = y-((y^2)/R)
the planes z = H, y = 0, and z = 0

and the velocity field is:

F = yz(i)+xz(j)+xy(k)

and we need to calculate the outward flux of the field of the region at z = H (the top of the region).

Ive tried doing this 2 ways:
Doing the double integral of div(F)
Doing the double integral of F (dot) k dA
the solution to this problem uses the 2nd of the two; my question is this:
why are the 2 methods in disagreement with each other? (when i take div(F) i get zero, so flux is zero)

 

[nickjer]

The divergence theorem applies to a closed volume. You are calculated the flux through a piece of the surface area and not the whole surface area of the whole volume. So you can't use the divergence theorem.