In physics and mathematics, in the area of vector calculus, Helmholtz's theorem,[1][2] also known as the fundamental theorem of vector calculus,[3][4][5][6][7][8][9] states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz.[10]

As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form , where Φ is a scalar field, called scalar potential, and A is a vector field, called a vector potential.

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Statement of the theoremEdit

Let   be a vector field on a bounded domain  , which is twice continuously differentiable, and let   be the surface that encloses the domain  . Then   can be decomposed into a curl-free component and a divergence-free component:[11][12]

 

where

 

and   is the nabla operator with respect to  , not   .

If   and is therefore unbounded, and   vanishes faster than   as  , then one has[13]

 

DerivationEdit

Suppose we have a vector function   of which we know the curl,  , and the divergence,  , in the domain and the fields on the boundary. Writing the function using delta function in the form

 

where  is the Laplace operator, we have

 

where we have used the definition of the vector Laplacian:

 

differentiation/integration with respect to  by  and in the last line, linearity of function arguments:

 

Then using the vectorial identities

 

we get

 

Thanks to the divergence theorem the equation can be rewritten as

 

with outward surface normal  .

Defining

 
 

we finally obtain

 

  is the Green's function for the Laplacian, and in a more general setting it should be replaced by the appropriate Green's function - for example, in two dimensions it should be replaced by  . For higher dimensional generalization, see the discussion of Hodge decomposition below.

Another derivation from the Fourier transformEdit

Note that in the theorem stated here, we have imposed the condition that if   is not defined on a bounded domain, then   shall decay faster than  . Thus, the Fourier Transform of  , denoted as  , is guaranteed to exist. We apply the convention

 

The Fourier transform of a scalar field is a scalar field, and the Fourier transform of a vector field is a vector field of same dimension.

Now consider the following scalar and vector fields:

 

Hence

 

Fields with prescribed divergence and curlEdit

The term "Helmholtz theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R3 which are sufficiently smooth and which vanish faster than 1/r2 at infinity. Then there exists a vector field F such that

 

if additionally the vector field F vanishes as r → ∞, then F is unique.[13]

In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type.[13] The proof is by a construction generalizing the one given above: we set

 

where   represents the Newtonian potential operator. (When acting on a vector field, such as ∇ × F, it is defined to act on each component.)

Differential formsEdit

The Hodge decomposition is closely related to the Helmholtz decomposition, generalizing from vector fields on R3 to differential forms on a Riemannian manifold M. Most formulations of the Hodge decomposition require M to be compact.[14] Since this is not true of R3, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem.

Weak formulationEdit

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 square-integrable vector field u ∈ (L2(Ω))3 has an orthogonal decomposition:

 

where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and AH(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.

For a slightly smoother vector field uH(curl, Ω), a similar decomposition holds:

 

where φH1(Ω), v ∈ (H1(Ω))d.

Longitudinal and transverse fieldsEdit

A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component.[15] This terminology comes from the following construction: Compute the three-dimensional Fourier transform   of the vector field  . 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:

 
 
 

Since   and  ,

we can get

 
 

so this is indeed the Helmholtz decomposition.[16]

See alsoEdit

NotesEdit

  1. ^ On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958.
  2. ^ Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357
  3. ^ An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8.
  4. ^ J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, page 237, link from Internet Archive
  5. ^ Electromagnetic theory, Volume 1. By Oliver Heaviside. "The Electrician" printing and publishing company, limited, 1893.
  6. ^ Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854.
  7. ^ An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson. John Wiley & Sons, 1881.
    See also: Method of Fluxions.
  8. ^ Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
    See also: Green's Theorem.
  9. ^ A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.
  10. ^ See:
    • H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik, 55: 25–55. On page 38, the components of the fluid's velocity (u, v, w) are expressed in terms of the gradient of a scalar potential P and the curl of a vector potential (LMN).
    • However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society, vol. 9, part I, pages 1–62; see pages 9–10.
  11. ^ Aris (1962), pp. 70–72
  12. ^ "Helmholtz' Theorem" (PDF). University of Vermont.
  13. ^ a b c David J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, 1999, p. 556.
  14. ^ Cantarella, Jason; DeTurck, Dennis; Gluck, Herman (2002). "Vector Calculus and the Topology of Domains in 3-Space". The American Mathematical Monthly. 109 (5): 409–442. doi:10.2307/2695643. JSTOR 2695643.
  15. ^ Stewart, A. M.; Longitudinal and transverse components of a vector field, Sri Lankan Journal of Physics 12, 33-42 (2011)
  16. ^ Online lecture notes by Robert Littlejohn

ReferencesEdit

General referencesEdit

  • 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
  • Rutherford Aris, Vectors, tensors, and the basic equations of fluid mechanics, Prentice-Hall (1962), OCLC 299650765, pp. 70–72

References for the weak formulationEdit

  • Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V. (1998). "Vector potentials in three dimensional non-smooth domains". Mathematical Methods in the Applied Sciences. 21: 823–864. Bibcode:1998MMAS...21..823A. doi:10.1002/(sici)1099-1476(199806)21:9<823::aid-mma976>3.0.co;2-b.
  • 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 linksEdit