Infinitesimal calculus MOC

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

div⁡⃗𝐅=⃗∇⋅⃗𝐅=tr⁡𝐷⃗𝐅

where ⃗∇ is the gradient operator (sometimes called grad). Interpreting ⃗𝐅 as a velocity field of a fluid, the divergence represents the rate at which an infinitesimal volume changes with time.

Properties

• If div⁡⃗𝐅 =⃗𝟎 everywhere, then ⃗𝐅 is an Incompressible vector field, meaning it has a vector potential.
• Conversely, div⁡curl⁡⃗𝐅 =0 given continuous second order derivatives exist for ⃗𝐅.

Practice problems

See Practice problems.

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