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

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

Synonym:

fundamental theorem of vector calcululs

Type of Math Object:

Definition

Major Section:

Reference

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