PlanetMath: Helmholtz decomposition

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Helmholtz decomposition

(Definition)

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:

$\displaystyle \mathbf{F} = -\nabla \varphi + \nabla \times \mathbf{A}$

(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:

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

"Helmholtz decomposition" is owned by invisiblerhino.

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Other names:

fundamental theorem of vector calcululs

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Cross-references: curl, vanish at infinity, divergence, side, term, scalar, definitions, vector potential, scalar potential, solenoidal, irrotational, vector
There is 1 reference to this entry.

This is version 2 of Helmholtz decomposition, born on 2008-04-15, modified 2008-04-16.
Object id is 10506, canonical name is HelmholtzDecomposition.
Accessed 619 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

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