Helmholtz decomposition

The Helmholtz theorem states that any vector 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:

𝐅=-φ+×𝐀 (1)

where φ is a scalar potential and 𝐀 is a vector potentialMathworldPlanetmath. 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 divergenceMathworldPlanetmath of 𝐅 must vanish at infinity.

  2. 2.

    The curl of 𝐅 must also vanish at infinity.

Title Helmholtz decomposition
Canonical name HelmholtzDecomposition
Date of creation 2013-03-22 17:59:40
Last modified on 2013-03-22 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