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Helmholtz decomposition
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(Definition)
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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: \begin{equation} \mathbf{F} = -\nabla \varphi + \nabla \times \mathbf{A} \end{equation}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.
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"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, theorem
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 1987 times total.
Classification:
| AMS MSC: | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) |
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Pending Errata and Addenda
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