Calculate the flux of the vector field $\displaystyle \textbf{A}(\rho, \varphi, z) = \left(z \frac{\rho^2-1}{\rho}, 0, 0\right)$ out through the surface of $\displaystyle x^2 +y^2 +(z-2)^2 \leq 4$ (cylinder coordinates).

I've almost solved this problem using the divergence theorem, but I need to calculate the flux through the above sphere when the radius is $\displaystyle \epsilon$ (because of the singularity along the z-axis).

The problem is that I don't know how to get the normal vector. I know it's $\displaystyle |r'_\varphi \times r'_z|$, but I'm not quite sure how to get $\displaystyle r'_\varphi$ and $\displaystyle r'_z$.

Does anyone have a good online resource for some extra reading by the way?