The vectorfield $F$ is given by $F(x,y,z) = [x^2y+z^2, xcoz(z)-xy^2, x^3+3z]$.
Let $S$ be the cylindric surface defined by $x^2+4y^2=1$ where $0 \leq z \leq 8$.
Calculate $$\iint_S F \cdot \hat{N} dS$$ where the unitnormal $\hat{N}$ points away from the $z$ axis.
Once can easily find div $F = 3$. From there I simply used the divergence theorem and the fact that we're dealing with an ellipic cylinder to find the flux. We simply get $3*volume(T) = 3*\pi*1*\frac{1}{2}*8 = 12\pi$.
I thought I was done, but in the solution they find the flux across the surface like I did, and then they treat the surface where $z=0$ and where $z=8$. Doing this we get that the flux out of the surface at $z=0$ is equal to $0$ whereas the flux out of the surface at $z=8$ is equal to $12\pi$, and thus $$\iint_S F \cdot \hat{N} dS = 0$$
What they have found here is obviously the flux across the "cylindric" portion of the surface, excluding the ellipse at the bottom and the top. Shouldn't this be specified in the task itself? I thought since the inequalities are not strict, we are simply finding the flux across the entire surface, including at $z=0$ and $z=8$. Shouldn't the inequalities be strict (as in $0<z<8$ ), if we're only going to talk about the flux out of the "cylindric" part of the surface?
I hope the question is clear even tho my use of "cylindric part" is probably not the right terminology.
The question seems asking to find the flux across the lateral surface and the method used is the following