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Math Help - Reference for a derivation of the multidimensional integration by parts formula

  1. #1
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    Reference for a derivation of the multidimensional integration by parts formula

    The general integration by parts formula in multiple dimensions is given by

    \int_{\Omega} \nabla u \cdot \mathbf{v} = \int_{\partial\Omega} (u \mathbf{v})\cdot n-\int_\Omega u \nabla\cdot\mathbf{v}

    Is there a good reference that details the derivation of this formula? More specifically, I'm looking for a derivation of the above result that does not use differential forms.

    Thanks.
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  2. #2
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    I found a reference that develops the formula in a general setting:

    Integration on infinite-dimensional ... - Google Books

    This comes from the book, "Integration on Infinite-Dimensional Surfaces and Its Applications," by Uglanov.

    Integration on Infinite-Dimensional Surfaces and Its Applications

    I've not had an opportunity to obtain the book, but from a quick glance, this appears to be the sort of derivation that I'm looking for.

    For the time being, I'll leave the thread as unresolved in case someone has a different reference they've found useful.
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  3. #3
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    Re: Reference for a derivation of the multidimensional integration by parts formula

    It turns out that Daniel Stroock has a full derivation of the divergence theorem in two of his books, "A Concise Introduction to the Theory of Integration" and "Essentials of Integration Theory for Analysis". The result in the first book is slightly more general that the second (few derivatives assumed on F), but the proof in the second is simpler. Specifically, he states


    Theorem (Divergence Theorem). Again let G be a smooth region in \mathbb{R}^N and U an open neighborhood of \bar{G}. If F:U\rightarrow\mathbb{R}^N is continuously differentiable and either G is bounded or F\equiv 0 off a compact subset of U, then

    \int\limits_G div F(x) dx = \int\limits_{\partial G} (F(x),n(x))_{\mathbb{R}^N} \lambda_{\partial G}(dx)


    In order to get integration by parts, let F=fg for f:\mathbb{R}^N\rightarrow\mathbb{R}^N and g:\mathbb{R}^N\rightarrow\mathbb{R} and use the chain rule.
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