Let $x_{1}, ..., x_{n}$ be $n$ points in $\mathbb{R}^3$, and let $\mathbb{v_{1}, ..., v_{n}}$ be their radius vectors. We define the function $$\phi(x, y, z)=\sum_{i=1}^{n} \frac{q_{i}}{4\pi||\mathbb{r}-\mathbb{v_{i}}||},$$ where $\mathbb{r}=(x, y, z)$ and $q_{1}, q_{2}, ... q_{n}$ are constants. I'm supposed to find $$\iint_{S} E \cdotp d\textbf{S},$$ where $S$ is an arbitrary bounded, closed surface which doesn't contain $x_{1}, ... x_{n}$, and $E=-\nabla \phi.$
I was able to find $E(x, y, z)=\frac{1}{4\pi}\sum_{i=1}^{n} \frac{q_{i}}{||\mathbb{r}-\mathbb{v_{i}}||^3}(x-v_{ix}, y-v_{iy}, z-v_{iz})$, where $v_{i}=(v_{ix}, v_{iy}, v_{iz})$, but I can't really figure out where to go from here. Is $E$ the curl of another vector field, so maybe I can use reverse Stokes' theorem? I can't really use the divergence theorem, because I don't know if $S$ is the boundary of any set $V$ which I could integrate through.