I am given that my surface S is given by;
$$ S = (\boldsymbol x : |\boldsymbol x|=1) $$ $$\boldsymbol F(\boldsymbol x)=\boldsymbol x/r^3 $$ Compute $$\int_S \boldsymbol F\cdot d\boldsymbol S $$
My Approach,
I use spherical polars and get that $$d\boldsymbol S = sin(\theta) \boldsymbol e_r d\theta d\phi $$ and thus I calculate $$\int_S \boldsymbol F\cdot d\boldsymbol S = \int_{\phi =0}^{2\pi}\int_{\theta=0}^\pi sin(\theta)d\theta d\phi=4\pi$$
However Gauss' Thm gives $$\int_S \boldsymbol F\cdot d\boldsymbol S = \int_V \nabla \cdot \boldsymbol FdV $$ but $$\nabla \cdot \boldsymbol F = 0$$ Which contradicts my above result? Im wondering if I have made a mistake in the computation of my first integral or maybe Divergence Thm doesnt hold because F is undefined at the origin?
EDIT: I think I realise now that the integral over the volume of the divergence is not 0 and instead given by,
$$ \int_V \nabla \cdot \boldsymbol FdV = \int_{\phi =0}^{2\pi}\int_{\theta=0}^\pi \int_0^1 \delta(r)sin(\theta)drd\theta d\phi =4\pi $$ And i Think this is correct and thus divergence thm still holds?