Gauss Green theorem

Theorem 1 (Gauss-Green)

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

$${\int }_{\mathrm{\Omega }}\mathrm{div}f(x)𝑑x={\int }_{\partial \mathrm{\Omega }}⟨f(x),{\nu }_{\mathrm{\Omega }}(x)⟩𝑑\sigma (x).$$

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 $\partial \mathrm{\Omega }$. The scalar product in the second integral is sometimes written as $f_n(x)$ and represents the normal component of $f$ with respect to $\partial \mathrm{\Omega }$; hence the whole integral represents the flux of the vector field $f$ through $\partial \mathrm{\Omega }$;

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 $E\mathrm{\subset }{\mathrm{R}}^n$ be any measurable set. Then

$${\int }_E\mathrm{div}f(x)𝑑x={\int }_{{\partial }^{*}E}⟨{\nu }_E(x),f(x)⟩𝑑{\mathscr{H}}^{n-1}(x)$$

holds for every continuously differentiable function $f\mathrm{:}{\mathrm{R}}^n\mathrm{\to }{\mathrm{R}}^n$ with compact support (i.e. $f\mathrm{\in }{\mathrm{C}}_c^{\mathrm{1}}\mathit{}\mathrm{(}{\mathrm{R}}^n\mathrm{,}{\mathrm{R}}^n\mathrm{)}$) 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 $x\in ℱE$;

• ${\mathscr{H}}^{n-1}$ 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