Let $B_r=\{x\in \mathbb R^3 : |x|\le r\}$ and let $dS_r$ denote the area element on $\partial B_r$. Set $$E(x)=C\int_{\partial B_R}\nabla_x |x-y|^{-1} dS_y$$
Show that for $|x|< R$, $E$ is zero and for $r<R$, the flux $\int_{\partial B_r} E(x)\cdot \nu \ dS_x$ is zero.
First of all, what does $\nabla_x$ stand for? The $\nabla$ without subscripts usually stands for the gradient, but I'm not sure about $\nabla_x$.
For the flux integral, is it better to use the divergence theorem or use the definition? In the former case, how do I find the divergence (i.e., how to differentiate a line integral)?
Here $E(x)$ is the electric field of a surface charge on a sphere. The second result is Gauss's law applied to $B_r$. The first result then follows by symmetry.