Gauss Green theoremGaussGreenTheorem2005-02-11 08:30:292005-02-18 05:44:01Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{theorem}{Theorem}\begin{theorem}[Gauss-Green]
Let $\Omega\subset \R^n$ be a bounded open set with $C^1$ boundary, let $\nu_\Omega\colon \partial \Omega\to \R^n$ be the exterior unit normal vector to $\Omega$ in the point $x$ and let $f\colon \overline{\Omega}\to \R^n$ be a vector function in $C^0(\overline\Omega,\R^n)\cap C^1(\Omega,\R^n)$. Then
\[
\int_\Omega \mathrm{div} f(x)\, dx
=\int_{\partial \Omega} \langle f(x),\nu_\Omega(x)\rangle \, d\sigma(x).
\]
\end{theorem}
Some remarks on notation.
The operator $\mathrm{div} f$ is the divergence of the vector field $f$, which is sometimes written as $\nabla \cdot f$.
In the right-hand side we have a surface integral, $d\sigma$ is the corresponding area measure on $\partial \Omega$.
The scalar product in the second integral is sometimes written as $f_n(x)$
and represents the \emph{normal component} of $f$ with respect to $\partial \Omega$; hence the whole integral represents the \emph{flux} of the vector field $f$ through $\partial \Omega$;
This theorem can be easily extended to \emph{piecewise} regular domains.
However the more general statement of this Theorem involves the theory of \emph{perimeters} and $BV$ functions.
\begin{theorem}[generalized Gauss-Green]
Let $E\subset \R^n$ be any measurable set.
Then
\[
\int_E \mathrm{div} f(x)\, dx
= \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)
\]
holds for every continuously differentiable function $f\colon \R^n\to\R^n$ with compact support (i.e.\ $f\in\mathcal C^1_c(\R^n,\R^n)$) where
\begin{itemize}
\item
$\partial^* E$ is the \emph{essential boundary} of $E$ which is a subset of the topological boundary $\partial E$;
\item $\nu_E(x)$ is the exterior normal vector to $E$, which is defined when $x\in\mathcal F E$;
\item $\mathcal H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.
\end{itemize}
\end{theorem}