I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux of
$$ f(x,y,z) = (\sin(yz),y+\sqrt{x^2 + z^2}, 1-z) $$
through the surface
$$ \Omega = \{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = 1, 0 < z <1 \} $$
The divergence of f is $0$, so evaluating $\int_\Omega div(f)d\Omega $ doesn't give me the solution. I tried to calculate $ \oint_{E =\partial\Omega}f \cdot n $ d$E$, but I don't know exactly how to do this. Can anybody held me? What exactly is $d\Omega$ and how should I parameterize it? Thanks!!
The issue is that you cannot apply the Divergence Theorem until you close up the surface. Put the top and bottom faces on your cylinder, and then the net flux will be $0$. So now calculate (directly) the flux outwards across the top and bottom.