Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$? ($ k > -2$).
Using the divergence theorem, I got that the flux is:
$\frac{3\pi}{k}(1-(-1)^k)$ and directly calculating the flux using (x,y,z) as the normal unit vector I get $\frac{3\pi}{k+2}(1-(-1)^k)$. The latter makes more sense in the context of the question, but I'm not sure why they're different.
The calculations were done as follows: $divF = x^{k-1}+y^{k-1}+z^{k-1}$, and then from symmetry, we can look at the integral of $z^{k-1}$ on $S^2$ and multiply by 3. The integral was calculated using spherical coordinates. The other case was similar, but instead of the integral of $z^{k-1}$ we had the integral of $z^{k+1}$.
I'm doing something wrong, but I'm not sure what, and any help would be welcome.