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

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

    (ii) For a vector field \mathbf u \in C^1(U; \mathbb R^n) we have, \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 \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 U is a simple "box", i.e. 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 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 \mathbb R^3 (which I suppose is reasonable in most applications, e.g. in physics). That is, the region U is always assumed to be a subset of \mathbb R^3, rather than the more general case where U is a subset of \mathbb R^n as Evans states.

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