- Helmholtz decomposition
In
mathematics , in the area ofvector calculus , Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decayingvector field can be resolved into irrotational (curl-free) andsolenoidal (divergence-free) component vector fields.This implies that any vector field can be considered to be generated by a pair of potentials: a
scalar potential and avector potential .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 ofgradient and curl as follows:: where represents theNewtonian potential operator.If , we say is solenoidal or
divergence -free and thus the Helmholtz decomposition of collapses to: In this case, is known as the "vector potential" for .Likewise, if then is said to be curl-free or irrotational and thus the Helmholtz decomposition of collapses then to: In this case, is known as the "scalar potential" for .
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::.
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 is a bounded, simply-connected,
Lipschitz domain . Every vector field has an orthogonal decomposition::where and . For a slightly smoother vector field , a similar decomposition holds::where and .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 . 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::Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:::so 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. 95-101References 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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