# Helmholtz decomposition

The Helmholtz theorem states that any vector $\mathbf{F}$ may be decomposed into an irrotational (curl-free) and a solenoidal (divergence-free) part under certain conditions (given below). More precisely, it may be written in the form:

 $\mathbf{F}=-\nabla\varphi+\nabla\times\mathbf{A}$ (1)

where $\varphi$ is a scalar potential and $\mathbf{A}$ is a vector potential. By the definitions of scalar and vector potentials it follows that the first term on the right-hand side is irrotational and the second is solenoidal. The general conditions for this to be true are:

1. 1.

The divergence of $\mathbf{F}$ must vanish at infinity.

2. 2.

The curl of $\mathbf{F}$ must also vanish at infinity.

Title Helmholtz decomposition HelmholtzDecomposition 2013-03-22 17:59:40 2013-03-22 17:59:40 invisiblerhino (19637) invisiblerhino (19637) 5 invisiblerhino (19637) Definition msc 26B12 fundamental theorem of vector calcululs