Divergence

The divergence of a vector field, also called the flux density, is a scalar measure of a vector field's tendency to move away (i.e. to diverge) from a given point. It is given by

d i v ⃗ 𝐅 = ⃗ \nabla \cdot ⃗ 𝐅 = t r 𝐷 ⃗ 𝐅

where $⃗ \nabla$ is the gradient operator (sometimes called $g r a d$ ). Interpreting $⃗ 𝐅$ as a velocity field of a fluid, the divergence represents the rate at which an infinitesimal volume changes with time.

Properties

If $d i v ⃗ 𝐅 = ⃗ 𝟎$ everywhere, then $⃗ 𝐅$ is an Incompressible vector field, meaning it has a vector potential.
Conversely, $d i v c u r l ⃗ 𝐅 = 0$ given continuous second order derivatives exist for $⃗ 𝐅$ .

Practice problems

See Practice problems.

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