Helmholtz decomposition
The Helmholtz theorem states that any vector $\mathbf{F}$ may be decomposed into an irrotational (curlfree) and a solenoidal (divergencefree) part under certain conditions (given below). More precisely, it may be written in the form:
$$\mathbf{F}=\nabla \phi +\nabla \times \mathbf{A}$$  (1) 
where $\phi $ 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 righthand side is irrotational and the second is solenoidal. The general conditions for this to be true are:

1.
The divergence^{} of $\mathbf{F}$ must vanish at infinity.

2.
The curl of $\mathbf{F}$ must also vanish at infinity.
Title  Helmholtz decomposition 

Canonical name  HelmholtzDecomposition 
Date of creation  20130322 17:59:40 
Last modified on  20130322 17:59:40 
Owner  invisiblerhino (19637) 
Last modified by  invisiblerhino (19637) 
Numerical id  5 
Author  invisiblerhino (19637) 
Entry type  Definition 
Classification  msc 26B12 
Synonym  fundamental theorem of vector calcululs 