# Gauss Green theorem

###### Theorem 1 (Gauss-Green)

Let $\Omega\subset\mathbf{R}^{n}$ be a bounded open set with $C^{1}$ boundary, let $\nu_{\Omega}\colon\partial\Omega\to\mathbf{R}^{n}$ be the exterior unit normal vector to $\Omega$ in the point $x$ and let $f\colon\overline{\Omega}\to\mathbf{R}^{n}$ be a vector function in $C^{0}(\overline{\Omega},\mathbf{R}^{n})\cap C^{1}(\Omega,\mathbf{R}^{n})$. Then

 $\int_{\Omega}\mathrm{div}f(x)\,dx=\int_{\partial\Omega}\langle f(x),\nu_{% \Omega}(x)\rangle\,d\sigma(x).$

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 normal component of $f$ with respect to $\partial\Omega$; hence the whole integral represents the flux of the vector field $f$ through $\partial\Omega$;

This theorem can be easily extended to piecewise regular domains. However the more general statement of this Theorem involves the theory of perimeters and $BV$ functions.

###### Theorem 2 (generalized Gauss-Green)

Let $E\subset\mathbf{R}^{n}$ 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 $f\colon\mathbf{R}^{n}\to\mathbf{R}^{n}$ with compact support (i.e. $f\in\mathcal{C}^{1}_{c}(\mathbf{R}^{n},\mathbf{R}^{n})$) where

• $\partial^{*}E$ is the essential boundary of $E$ which is a subset of the topological boundary $\partial E$;

• $\nu_{E}(x)$ is the exterior normal vector to $E$, which is defined when $x\in\mathcal{F}E$;

• $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

 Title Gauss Green theorem Canonical name GaussGreenTheorem Date of creation 2013-03-22 15:01:51 Last modified on 2013-03-22 15:01:51 Owner paolini (1187) Last modified by paolini (1187) Numerical id 13 Author paolini (1187) Entry type Theorem Classification msc 26B20 Synonym divergence theorem Related topic GreensTheorem Related topic GeneralStokesTheorem Related topic IntegrationWithRespectToSurfaceArea Related topic ClassicalStokesTheorem Related topic FluxOfVectorField