TITLE: Proof of the Gauss-Green Theorem
QUESTION [30 upvotes]: I can't seem to find any references that gives a proof of the Gauss-Green theorem:

Let $U\subset\mathbb{R}^{n}$ be an open, bounded set with $\partial U$ being $C^1$. Suppose $u\in C^{1}(\bar{U})$, then $$\int_{U}{\frac{\partial u}{\partial x_i}}dx=\int_{\partial U}u\nu^{i}dS\;\;\;\;(i=1,\ldots,n),$$
  where $\nu=(\nu^1,\ldots\nu^n)$ denotes the outward-pointing unit normal vector field to the region $U$.

Evans' PDE textbook presents the theorem (with no proof) in the appendix, and proceeds to use it to derive Green's forumlas and the formula for $n$-dimensional integration by parts. So I'd really like to have a proof of the theorem for future reference.

REPLY [15 votes]: There is a simple proof of Gauss-Green theorem if one begins with the assumption of Divergence theorem, which is familiar from vector calculus,
\begin{equation}
\int_{U}\mathrm{div}\,\mathbf{w}\,dx =
\int_{\partial U} \mathbf{w}\cdot\mathbf{\nu}\,dS,
\end{equation}
where $\mathbf{w}$ is any $C^\infty$ vector field on $U\in\Bbb{R}^n$ and $\mathbf{\nu}$
is the outward normal on $\partial U$.
Now, given the scalar function $u$ on the open set $U$, we can construct the vector field 
\begin{equation}
\mathbf{w}=(0,\ldots,0,u,0,\ldots,0),
\end{equation}
where $u$ is the $i$th component. Then, following the Divergence theorem, 
we have
\begin{equation}
\int_U \mathrm{div}\,\mathbf{w}\,dx=\int_U u_{x_i}\,dx
=\int_{\partial U}\mathbf{w}\cdot\mathbf{\nu}\,dS
=\int_{\partial U}u\nu^i\,dS.
\end{equation}
In Evans' book (Page 712), the Gauss-Green theorem is stated without proof and the Divergence theorem is shown as a consequence of it. This may be opposite to what most people are familiar with.