Helmholtz decomposition

Every sufficiently nice vector field splits into a curl-free part (a gradient) plus a divergence-free part (a curl).

The Helmholtz decomposition writes any sufficiently smooth, sufficiently decaying vector field $\mathbf{F}$ as $\mathbf{F}=-\nabla\phi + \nabla\times\mathbf{A}$ -- a curl-free (irrotational) part built from a scalar potential, plus a divergence-free (solenoidal) part built from a vector potential.

Examples

An electric field (curl-free, from a scalar potential) and a magnetic field (divergence-free, from a vector potential) are the two halves of this decomposition in electromagnetism.