divergence
Basic Definition.
Let be a system of Cartesian coordinates on -dimensional Euclidean space, and let be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field
is defined to be the function
Another common notation for the divergence is (see gradient), 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 derivative of the net flow of the vector field across the surface of a small sphere relative to the surface area of the sphere. To wit,
where denotes the sphere of radius about a point , and the integral is a surface integral taken with respect to , the normal to that sphere.
The non-infinitesimal interpretation of divergence is given by Gaussβs 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 volumeβs boundary. In symbols,
where is a compact region with a smooth boundary, and is that boundary oriented by outward-pointing normals. We note that Gaussβs theorem follows from the more general Stokesβ Theorem, which itself generalizes the fundamental theorem of calculus.
In light of the physical interpretation, a vector field with constant zero divergence is called incompressible β 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 be a vector field on a Riemannian manifold. The covariant derivative of is a type tensor field. We define the divergence of to be the trace of that field. In terms of coordinates (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 theorem |
Defines | Gaussβs theorem |