Helmholtz decomposition

In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational vector field and a solenoidal vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz.

Definition

For a vector field defined on a domain, a Helmholtz decomposition is a pair of vector fields and such that:
Here, is a scalar potential, is its gradient, and is the divergence of the vector field. The irrotational vector field is called a gradient field and is called a solenoidal field or rotation field. This decomposition does not exist for all vector fields and is not unique.

History

The Helmholtz decomposition in three dimensions was first described in 1849 by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. Their derivation required the vector fields to decay sufficiently fast at infinity. Later, this condition could be relaxed, and the Helmholtz decomposition could be extended to higher dimensions. For Riemannian manifolds, the Helmholtz-Hodge decomposition using differential geometry and tensor calculus was derived.
The decomposition has become an important tool for many problems in theoretical physics, but has also found applications in animation, computer vision as well as robotics.

Three-dimensional space

Many physics textbooks restrict the Helmholtz decomposition to the [|three-dimensional space] and limit its application to vector fields that decay sufficiently fast at infinity or to bump functions that are defined on a bounded domain. Then, a vector potential can be defined, such that the rotation field is given by, using the curl of a vector field.
Let be a vector field on a bounded domain, which is twice continuously differentiable inside, and let be the surface that encloses the domain with outward surface normal. Then can be decomposed into a curl-free component and a divergence-free component as follows:
where
and is the nabla operator with respect to, not.
If and is therefore unbounded, and vanishes faster than as, then one has
This holds in particular if is twice continuously differentiable in and of bounded support.

Derivation

Solution space

If is a Helmholtz decomposition of, then
is another decomposition if, and only if,
Proof:
Set and. According to the definition
of the Helmholtz decomposition, the condition is equivalent to
Taking the divergence of each member of this equation yields
, hence is harmonic.
Conversely, given any harmonic function,
is solenoidal since
Thus, according to the above section, there exists a vector field such that
If is another such vector field,
then
fulfills, hence
for some scalar field.

Fields with prescribed divergence and curl

The term "Helmholtz theorem" can also refer to the following. Let be a solenoidal vector field and d a scalar field on which are sufficiently smooth and which vanish faster than at infinity. Then there exists a vector field such that
if additionally the vector field vanishes as, then 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.

Weak formulation

The Helmholtz decomposition can be generalized by reducing the regularity assumptions. Suppose is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field has an orthogonal decomposition:
where is in the Sobolev space of square-integrable functions on whose partial derivatives defined in the distribution sense are square integrable, and, the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.
For a slightly smoother vector field, a similar decomposition holds:
where.

Derivation from the Fourier transform

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

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.

Generalization to arbitrary dimensions

Informally speaking, in, the Helmholtz decomposition can be expressed by where is any scalar function that solves the Poisson equation, where is the divergence of the vector field in, and is divergence free:. Thus the existence of Helmholtz decomposition a consequence of the existence of the solution of the Poisson equation.

Matrix approach

The generalization to dimensions cannot be done with a vector potential, since the rotation operator and the cross product are defined only in three dimensions.
Let be a vector field in which decays faster than for and.
The scalar potential is defined similar to the three dimensional case as:
where as the integration kernel is again the fundamental solution of Laplace's equation, but in d-dimensional space:
with the volume of the d-dimensional unit balls and the gamma function.
For, is just equal to, yielding the same prefactor as above.
The rotational potential is an antisymmetric matrix with the elements:
Above the diagonal are entries which occur again mirrored at the diagonal, but with a negative sign.
In the three-dimensional case, the matrix elements just correspond to the components of the vector potential.
However, such a matrix potential can be written as a vector only in the three-dimensional case, because is valid only for.
As in the three-dimensional case, the gradient field is defined as
The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix:
In three-dimensional space, this is equivalent to the rotation of the vector potential.

Tensor approach

In a -dimensional vector space with, can be replaced by the appropriate Green's function for the Laplacian, defined by
where Einstein summation convention is used for the index. For example, in 2D.
Following the same steps as above, we can write
where is the Kronecker delta. In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol,
which is valid in dimensions, where is a -component multi-index. This gives
We can therefore write
where
Note that the vector potential is replaced by a rank- tensor in dimensions.
Because is a function of only, one can replace, giving
Integration by parts can then be used to give
where is the boundary of. These expressions are analogous to those given above for three-dimensional space.
For a further generalization to manifolds, see the discussion of Hodge decomposition [|below].

Differential forms

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. 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.

Extensions to fields not decaying at infinity

Most textbooks only deal with vector fields decaying faster than with at infinity. However, Otto Blumenthal showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than with, which is substantially less strict.
To achieve this, the kernel in the convolution integrals has to be replaced by.
With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.
For all analytic vector fields that need not go to zero even at infinity, methods based on partial integration and the Cauchy formula for repeated integration can be used to compute closed-form solutions of the rotation and scalar potentials, as in the case of multivariate polynomial, sine, cosine, and exponential functions.