$F(x,y,z)=\frac{(x,y,z)}{(x^2+y^2+z^2)^{3/2}}$
$S: 4x^2+4y^2+z^2=5, z>= 1 $
Find the flux when $\hat{n} \cdot \hat{z} > 0$
Solution: I attempted a solution using the divergence theorem. I added the surface $S1: x^2+y^2 <= 1, z=1$ to make a closed surface $S + S1$ and we make specially note that the vector function $F$ is defined over that closed surface and therefore making the usage of the divergence theorem valid. I then took the divergence of $F$ and got: $divF= \frac{3x^2+3y^2+3z^2-3x-3y-3z}{(x^2+y^2+z^2)^{5/2}}$ Here i realized that the integral $$\iiint_V divF \,dV$$
where V is the enclosed volume by the surface $S + S1$, would be tricky to solve without some sort of change of variables. But without going further down the rabbit hole I would like to know if I am on the correct path with my solution? Is there a much easier way of finding the flux?