Helmholtz decomposition

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 phi 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} = - abla,mathcal{G} ( abla cdot mathbf{F}) + abla imes mathcal{G}( abla imes mathbf{F})where mathcal{G} represents the Newtonian potential operator.

If ablacdotmathbf{F}=0, we say mathbf{F} is solenoidal or divergence-free and thus the Helmholtz decomposition of mathbf{F} collapses to: mathbf{F} = abla imes mathcal{G}( abla imes mathbf{F}) = abla imes mathbf{A}In this case, mathbf{A} is known as the "vector potential" for mathbf{F}.

Likewise, if abla imesmathbf{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} = - abla,mathcal{G} ( abla cdot mathbf{F}) = - abla phi.In this case, phi 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} = - abla phi + abla imes mathbf{A} .

Applicability to differential forms

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

Weaker formulation

The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Omega is a bounded, simply-connected, Lipschitz domain. Every vector field mathbf{u}in(L^2(Omega))^3 has an orthogonal decomposition::mathbf{u}= ablaphi+mathrm{curl},mathbf{A}where phiin H^1(Omega) and mathbf{A}in H(mathrm{curl},Omega). For a slightly smoother vector field mathbf{u}in H(mathrm{curl},Omega), a similar decomposition holds::mathbf{u}= ablaphi+mathbf{v}where phiin H^1(Omega) and mathbf{v}in(H^1(Omega))^d.

Longitudinal and transverse fields

A terminology often used in physics is the curl-free component of a vector field is called the longitudinal component and the divergence-free component is called the transverse component. [ [http://arxiv.org/abs/0801.0335 [0801.0335 Longitudinal and transverse components of a vector field ] ] This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field F, which we call ilde{mathbf{F. Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have: ilde{mathbf{F(mathbf{k}) = ilde{mathbf{F_l(mathbf{k}) + ilde{mathbf{F_t(mathbf{k}):mathbf{k} cdot ilde{mathbf{F_t(mathbf{k}) = mathbf{k} imes ilde{mathbf{F_l(mathbf{k}) = 0Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive::mathbf{F} = mathbf{F}_t+mathbf{F}_l: abla cdot mathbf{F}_t = abla imes mathbf{F}_l = 0so this is indeed the Helmholtz decomposition. [ [http://bohr.physics.berkeley.edu/classes/221/0708/notes/hamclassemf.pdf Online lecture notes by Robert Littlejohn] ]

References

General references

* George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92-93
* George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists International Edition, 6th edition, Academic Press: San Diego (2005) pp. 95-101

References for the weak formulation

* 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.

External links

* [http://mathworld.wolfram.com/HelmholtzsTheorem.html Helmholtz theorem] on MathWorld


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