is defined to be the function
Another common notation for the divergence is (see gradient), a convenient mnemonic.
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,
The non-infinitesimal interpretation 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 is a compact region with a smooth boundary, and 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 calculus.
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.
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
|Date of creation||2013-03-22 12:55:08|
|Last modified on||2013-03-22 12:55:08|
|Last modified by||rmilson (146)|