PlanetMath: Gauss Green theorem

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Gauss Green theorem

(Theorem)

Theorem 1 (Gauss-Green) Let $\Omega\subset \mathbf R^n$ be a bounded open set with boundary, let $\nu_\Omega\colon \partial \Omega\to \mathbf R^n$ be the exterior unit normal vector to $\Omega$ in the point 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

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

Some remarks on notation. The operator $\mathrm{div} f$ is the divergence of the vector field , 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 and represents the normal component of with respect to $\partial \Omega$ ; hence the whole integral represents the flux of the vector field 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 functions.

Theorem 2 (generalized Gauss-Green) Let $E\subset \mathbf R^n$ be any measurable set. Then

$\displaystyle \int_E \mathrm{div} f(x)\, dx = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$

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 which is a subset of the topological boundary $\partial E$ ;
$\nu_E(x)$ is the exterior normal vector to , which is defined when $x\in\mathcal F E$ ;
$\mathcal H^{n-1}$ is the -dimensional Hausdorff measure.