Helmholtz decomposition

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In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields.

This implies that any vector field $\mathbf{F}$ can be considered to be generated by a pair of potentials: a scalar potential $φ$ and a vector potential $\mathbf{A}$ .

The resulting Helmholtz decomposition of a vector field, which is twice continuously differentiable and with rapid enough decay at infinity, splits the vector field into a sum of gradient and curl as follows:

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) + \nabla \times \mathcal{G}(\nabla \times \mathbf{F})$

where $\mathcal{G}$ represents the Newtonian potential.

If $\nabla\cdot\mathbf{F}=0$ , we say $\mathbf{F}$ is solenoidal or divergence-free and thus the Helmholtz decomposition of $\mathbf{F}$ collapses to

$\mathbf{F} = \nabla \times \mathcal{G}(\nabla \times \mathbf{F}) = \nabla \times \mathbf{A}$

In this case, $\mathbf{A}$ is known as the vector potential for $\mathbf{F}$ .

Likewise, if $\nabla\times\mathbf{F}=\mathbf{0}$ then $\mathbf{F}$ is said to be curl-free or irrotational and thus the Helmholtz decomposition of $\mathbf{F}$ collapses then to

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) = - \nabla \phi.$

In this case, $φ$ is known as the scalar potential for $\mathbf{F}$ .

In general the negative gradient of the scalar potential is equated with the irrotational component, and the curl of the vector potential is equated with the solenoidal component:

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

1 Applicability to differential forms
2 Weaker formulation
3 References
4 External links

[edit] Applicability to differential forms

The Hodge decomposition generalizes the Helmholtz decomposition from vector fields to differential forms.

[edit] Weaker formulation

The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose $Ω$ is a bounded, simply-connected, Lipschitz domain. Every vector field $\mathbf{u}\in(L^2(\Omega))^3$ has an orthogonal decomposition

$\mathbf{u}=\nabla\phi+\mathrm{curl}\,\psi$

where $\phi\in H^1(\Omega)$ and $\psi\in H(\mathrm{curl},\Omega)$ .

C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. "Vector potentials in three dimensional non-smooth domains." Mathematical Methods in the Applied Sciences, 21, 823–864, 1998.
R. Dautray and J.-L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990.
V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.