Proof of divergence theorem

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I am trying to proof the classical Gauss's divergence theorem given by,

For a bounded domain $D\in \mathbb{R}^N$ with a smooth boundary $\partial D$. For the function $F=(F_1,F_2,...,F_N)$ it is given by

$\int_{D}div(F)dx=\int_{\partial D}F\cdot v dS_x$, where $S$ is the surface of $D$ and $v$ denotes the surface normal vector.

I have proven this for $N=1$ by

$\int_{a}^bf'(x)dx=1\cdot f(b)+(-1)\cdot f(a)$ for any $(a,b)\in \mathbb{R}$, i.e. $F(x)=f(x)$ and $div(F)=f'(x)$.

But for general $N\in \mathbb{N}$, any hints?

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