# divergence

## Basic Definition.

Let $x,y,z$ be a system of Cartesian coordinates on $3$-dimensional Euclidean space, and let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field

 $\mathbf{F}=F^{1}\mathbf{i}+F^{2}\mathbf{j}+F^{3}\mathbf{k},$

is defined to be the function

 $\operatorname{div}\mathbf{F}=\frac{\partial F^{1}}{\partial x}+\frac{\partial F% ^{2}}{\partial y}+\frac{\partial F^{3}}{\partial z}.$

Another common notation for the divergence is $\nabla\cdot\mathbf{F}$ (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,

 $(\operatorname{div}\mathbf{F})(p)=\lim_{r\rightarrow 0}\int_{S}\!\!(\mathbf{F}% \cdot\mathbf{N})dS\;/\left(4\pi r^{2}\right),$

where $S$ denotes the sphere of radius $r$ about a point $p\in\mathbb{R}^{3}$, and the integral is a surface integral taken with respect to $\mathbf{N}$, the normal to that sphere.

The non-infinitesimal interpretation of divergence is given by Gaussβ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 volumeβs boundary. In symbols,

 $\int_{V}\operatorname{div}\mathbf{F}\,dV=\int_{S}(\mathbf{F}\cdot\mathbf{N})\,dS,$

where $V\subset\mathbb{R}^{3}$ is a compact region with a smooth boundary, and $S=\partial V$ is that boundary oriented by outward-pointing normals. We note that Gaussβs 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 $\mathbf{V}$ be a vector field on a Riemannian manifold. The covariant derivative of $\mathbf{V}$ is a type $(1,1)$ tensor field. We define the divergence of $\mathbf{V}$ to be the trace of that field. In terms of coordinates (see tensor and Einstein summation convention), we have

 $\operatorname{div}\mathbf{V}=V^{i}{}_{;i}\ .$
Title divergence Divergence 2013-03-22 12:55:08 2013-03-22 12:55:08 rmilson (146) rmilson (146) 11 rmilson (146) Definition msc 26B12 SourcesAndSinksOfVectorField incompressible divergence theorem Gaussβs theorem