Basic Definition.

Let x,y,z be a system of Cartesian coordinatesMathworldPlanetmath on 3-dimensional Euclidean spaceMathworldPlanetmath, and let 𝐒,𝐣,𝐀 be the corresponding basis of unit vectorsMathworldPlanetmath. The divergenceMathworldPlanetmath of a continuously differentiable vector field


is defined to be the function


Another common notation for the divergence is βˆ‡β‹…π… (see gradientMathworldPlanetmath), a convenient mnemonic.

Physical interpretation.

In physical , the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivativePlanetmathPlanetmath of the net flow of the vector field across the surface of a small sphere relative to the surface areaMathworldPlanetmath of the sphere. To wit,


where S denotes the sphere of radius r about a point pβˆˆβ„3, and the integral is a surface integral taken with respect to 𝐍, the normal to that sphere.

The non-infinitesimal interpretationMathworldPlanetmathPlanetmath of divergence is given by Gauss’s Theorem. This theorem is a conservation law, stating that the volume total of all sinks and sources, i.e. the volume integral of the divergence, is equal to the net flow across the volume’s boundary. In symbols,


where VβŠ‚β„3 is a compactPlanetmathPlanetmath region with a smooth boundary, and S=βˆ‚β‘V is that boundary oriented by outward-pointing normals. We note that Gauss’s theorem follows from the more general Stokes’ Theorem, which itself generalizes the fundamental theorem of calculusMathworldPlanetmathPlanetmath.

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible – in this case, no flow can occur across any surface.

General definition.

The notion of divergence has meaning in the more general setting of Riemannian geometry. To that end, let 𝐕 be a vector field on a Riemannian manifoldMathworldPlanetmath. The covariant derivativeMathworldPlanetmath of 𝐕 is a type (1,1) tensor field. We define the divergence of 𝐕 to be the trace of that field. In terms of coordinatesMathworldPlanetmathPlanetmath (see tensor and Einstein summation convention), we have

Title divergence
Canonical name Divergence
Date of creation 2013-03-22 12:55:08
Last modified on 2013-03-22 12:55:08
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 11
Author rmilson (146)
Entry type Definition
Classification msc 26B12
Related topic SourcesAndSinksOfVectorField
Defines incompressible
Defines divergence theoremMathworldPlanetmath
Defines Gauss’s theorem