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 is a scalar field called "scalar potential", and A is a vector field, called a vector potential.
Statement of the theorem
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:
and is the nabla operator with respect to , not .
If and is therefore unbounded, and vanishes faster than as , then one has
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
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:
Fields with prescribed divergence and curl
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.
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. 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.)
The Hodge decomposition is closely related to the Helmholtz decomposition, generalizing from vector fields on R3 to differential forms on a Riemannian manifoldM. Most formulations of the Hodge decomposition require M to be compact. 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.
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 spaceH1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.
For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds:
where φ ∈ H1(Ω), v ∈ (H1(Ω))d.
Longitudinal and transverse fields
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. 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.
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 (L, M, N).
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.
^"Helmholtz' Theorem" (PDF). University of Vermont. Archived from the original (PDF) on 2012-08-13. Retrieved 2011-03-11.
^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.
^Stewart, A. M.; Longitudinal and transverse components of a vector field, Sri Lankan Journal of Physics 12, 33–42 (2011)
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 formulation
Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V. (1998). "Vector potentials in three dimensional non-smooth domains". Mathematical Methods in the Applied Sciences. 21 (9): 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.