Gauss Green theorem

Theorem 1 (Gauss-Green)

Let ΩRn be a bounded open set with C1 boundary, let νΩ:ΩRn be the exterior unit normal vector to Ω in the point x and let f:Ω¯Rn be a vector function in C0(Ω¯,Rn)C1(Ω,Rn). Then


Some remarks on notation. The operator divf is the divergenceMathworldPlanetmath of the vector fieldMathworldPlanetmath f, which is sometimes written as f. In the right-hand side we have a surface integral, dσ is the corresponding area measure on Ω. The scalar productMathworldPlanetmath in the second integral is sometimes written as fn(x) and represents the normal componentPlanetmathPlanetmathPlanetmath of f with respect to Ω; hence the whole integral represents the flux of the vector field f through Ω;

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 ERn be any measurable setMathworldPlanetmath. Then


holds for every continuously differentiable function f:RnRn with compact support (i.e. fCc1(Rn,Rn)) where

  • *E is the essential boundary of E which is a subset of the topological boundary E;

  • νE(x) is the exterior normal vector to E, which is defined when xE;

  • n-1 is the (n-1)-dimensional Hausdorff measureMathworldPlanetmath.

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