Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$.
Show that $$\iint_S \frac{\partial{f}}{\partial{n}} \,dS = 0$$
My main issue with this question is that I have no idea how to deal with $\frac{\partial{f}}{\partial{n}}$, once I understand what this is referring to I'm pretty sure I'll be able to do it. All help appreciated!
$\frac{\partial f}{\partial n}$ is shorthand for $ (\nabla f) \cdot n $, i.e. the directional derivative in the direction $n$. Really, what this question wants you to do is use the divergence theorem, $$ \int_V \nabla \cdot F \, dV = \int_S F \cdot n \, dS, $$ where here you have $F=\nabla f$, and the left-hand side is the integral of the Laplacian.