PlanetMath: divergence

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divergence

(Definition)

Basic Definition.

Let

be a system of Cartesian coordinates on

-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

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

is defined to be the function

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

Another common notation for the divergence is $\nabla \cdot\mathbf{F}$ (see gradient), a convenient mnemonic.

In physical terms, 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,

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

where

denotes the sphere of radius

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 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,

$\displaystyle \int_V \operatorname{div}\mathbf{F}\, dV = \int_S (\mathbf{F}\cdot \mathbf{N})\, dS,$

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 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 net flow can occur across any closed 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 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