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.