How to use the theorem of Gauß 
Let $f:\mathbb{R}^3$/{$0$}$\rightarrow \mathbb{R}$ be a function with $f(x)=\frac{a}{|x|}$ and $a>0$. Show that $f$ is harmonic, and that for every compact 3-dimensional, delimited undermanifold $M$ in $\mathbb{R}^3$ with a smooth limit and $0$ as an inner point, we have:
  $$\int_{\partial M}<grad f,v>d \mu_{\partial M}=-4a\Pi$$
  I already showed, that f is harmonic. We newly introduced this topic, so I'm not sure how to start with this problem. I think I have to use the theorem of Gauß, but could someone explain how to use it? How do you get that exact value with no specific given manifold?

Thanks in advance.
 A: Let $M$ be a smooth $3$-dimensional manifold whose outer boundary (surface) is given by $S_1$ and inner boundary (surface) is a sphere $S_2$ of radius $r>0$ centered at the origin. So the boundary $\partial M$ of $M$ is $S_1\dot\cup S_2$, where $\dot\cup$ is a notation for disjoint union. 
Then 
$$
\begin{align*}
0 &= \int_M 0 \: d\mu_M \\ 
&= \int_M \Delta f \: d\mu_M \\ 
&= \int_M \text{div}(\text{grad}f) d\mu_M \\ 
&= \int_{\partial M} \langle \text{grad}f, v\rangle d\mu_{\partial M} \\ 
&= \int_{S_1} \langle \text{grad}f, v\rangle d\mu_{S_1} -\int_{S_2} \langle \text{grad}f, v\rangle d\mu_{S_2}, \mbox{ where }v \mbox{ is unit normal}. \\ 
\end{align*}
$$
This implies 
$$
\begin{align*}
\int_{S_1} \langle \text{grad}f, v\rangle d\mu_{S_1} 
&= \int_{S_2} \langle \text{grad}f, v\rangle d\mu_{S_2} \\ 
&= \int_0^{2\pi} \int_0^{\pi} 
\left\langle 
-\frac{a}{r^2}
\left(
\sin \phi\cos \theta,
\sin \phi\sin \theta,
\cos \phi 
\right), \right. \\
&\left. \hspace{4mm} \underbrace{(\sin \phi\cos \theta ,\sin \phi\sin \theta ,\cos \phi )}_{v}
\right\rangle  
r^2 \sin \phi \: d\phi d\theta \\ 
&= \int_{0}^{2\pi} \int_{0}^{\pi} -\dfrac{a}{r^2} r^2 \sin \phi \: d\phi d\theta \\ 
&=  \int_{0}^{2\pi} \int_{0}^{\pi} -a \sin \phi \: d\phi d\theta \\ 
&=2\pi a \cos\phi\Big|_{0}^{\pi} \\ 
&= 2\pi a (-1-1) \\ 
&= -4\pi a. \\ 
\end{align*}
$$
Now let $r\rightarrow 0$. Then we conclude that 
$$
\int_{\partial M} \langle \text{grad}f, v\rangle d\mu_{\partial M} = -4\pi a. 
$$
