Hi, everyone!

I'm trying to prove a result in appendix C.2 of L. Evans, "Partial Differential Equations". Here it is listed as the Gauss-Green theorem and the statement is as follows:

Assume that $\displaystyle U$ is a bounded, open subset of $\displaystyle \mathbb R^n$ and $\displaystyle \partial U$ is $\displaystyle C^1$. Let $\displaystyle \nu = (\nu^1,\dots,\nu^n)$ denote the unit outward normal vector.

(i) Let $\displaystyle u : U\rightarrow \mathbb R$ such that $\displaystyle u$ is $\displaystyle C^1$. Then, $\displaystyle \int_U u_{x_i} dx = \int_{\partial U} u\nu^i dS$ for $\displaystyle i=1,\dots,n$.

(ii) For a vector field $\displaystyle \mathbf u \in C^1(U; \mathbb R^n)$ we have, $\displaystyle \int_U \text{div} \mathbf u \; dx = \int_{\partial U} \mathbf u \cdot \nu \; dS$.

The book claims that the second result is known as the divergence theorem* and follows from applying the first to each component of $\displaystyle \mathbf u = (u^1,\dots,u^n)$. That is, it should be enough to prove (i).

Anyway, I want to prove this for the case where $\displaystyle U$ is a simple "box", i.e. $\displaystyle U=[a_1,b_1]\times\dots\times[a_n,b_n]$. I really have no idea where to start. Could anyone point me in the right direction, give some reference or a sketch of a proof?

* Question: However, my lecturer called this same statement for Stokes theorem. I thought Stokes theorem had something to do with the integral of the curl in the interior? Is this a special case of Stokes theorem, or did he mix them up?