divergence

Basic Definition.

Let $x,y,z$ be a system of Cartesian coordinates on $3$-dimensionalEuclidean space, and let $𝐢,𝐣,𝐤$ be thecorresponding basis of unit vectors. The divergence of a continuouslydifferentiable vector field

$$𝐅=F^1𝐢+F^2𝐣+F^3𝐤,$$

is defined to be the function

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

Another common notation for the divergenceis $\nabla \cdot 𝐅$ (see gradient), a convenient mnemonic.

Physical interpretation.

In physical , the divergence of a vector fieldis the extent to which the vector field flow behaves like a source ora sink at a given point. Indeed, an alternative, but logicallyequivalent definition, gives the divergence as the derivative of thenet flow of the vector field across the surface of a small sphererelative to the surface area of the sphere. To wit,

$$(\mathrm{div}𝐅)(p)=\underset{r\to 0}{lim}{\int }_S(𝐅\cdot 𝐍)𝑑S/\left(4\pi r^2\right),$$

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

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

$${\int }_V\mathrm{div}𝐅dV={\int }_S(𝐅\cdot 𝐍)𝑑S,$$

where$V\subset {ℝ}^3$ 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 constantzero 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 ofRiemannian geometry. To that end, let $𝐕$ be a vector field on aRiemannian manifold. The covariant derivative of $𝐕$ is a type$(1,1)$ tensor field. We define the divergence of $𝐕$ to be thetrace of that field. In terms of coordinates (see tensor and Einsteinsummation convention), we have

$$\mathrm{div}𝐕=V^i{}_{;i}.$$