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Thread: How to prove Gauss-Green Theorem?

  1. #1
    May 2011

    How to prove Gauss-Green Theorem?

    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?
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  2. #2
    May 2011

    Re: How to prove Gauss-Green Theorem?

    Let me add something:

    I have been watching an MIT OpenCourseWare lecture on the divergence theorem (MIT OpenCourseWare | Mathematics | 18.02 Multivariable Calculus, Fall 2007 | Video Lectures | Lecture 29: Divergence Theorem (cont.)) where Prof. Denis Auroux sketches a proof for a special case of the divergence theorem. That is, under the assumption that the region U is 'vertically simple' and the vector field has only one non-zero component (which makes alot of ugly terms dissappear during the integration).

    Now, assuming that $\displaystyle U$ is a box should be even easier than assuming it is 'vertically simple', so I suppose I could apply the same ideas as in his proof. However, everywhere I look, I keep seeing the divergence theorem formulated for vector fields in $\displaystyle \mathbb R^3$ (which I suppose is reasonable in most applications, e.g. in physics). That is, the region $\displaystyle U$ is always assumed to be a subset of $\displaystyle \mathbb R^3$, rather than the more general case where $\displaystyle U$ is a subset of $\displaystyle \mathbb R^n$ as Evans states.

    How should I proceed to prove the theorem for a general vector field $\displaystyle \mathbf u$ on the domain $\displaystyle U \subseteq \mathbb R^n$ ?
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