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 (curl-free) vector field and a solenoidal (divergence-free) 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

is a scalar potential is its gradient is the divergence is called a gradient field is called a solenoidal field or rotation field. This decomposition may be calculated for vector fields that satisfy certain regularity or decay conditions.

A decomposition exists for all smooth or sufficiently rapidly decaying vector fields in classical terms. If arbitrary growth at infinity is allowed, a decomposition still exists for any regular vector field in the distributional sense, though it is no longer unique due to the presence of non-trivial harmonic fields. Existence in this generalized framework follows from the surjectivity of the Laplace operator on the space of distributions. The surjectivity of this class of elliptic operators was proved by Lars Hörmander in his work on linear partial differential operators.

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

In physics, the term Helmholtz theorem is sometimes used to describe a special case of Helmholt's decomposition theorem for 3 dimensions. Then, a vector potential, using the curl of a vector field.

can be decomposed into a curl-free component and a divergence-free component as follows:

is the nabla operator.

, then one has

and of bounded support.

Derivation

Solution space

is another decomposition if, and only if,

where

• is a harmonic scalar field,

• is a scalar field.

. According to the definition of the Helmholtz decomposition, the condition is equivalent to

1. .

is harmonic.

is solenoidal since

.

.

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

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

Weak formulation

The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). 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

. Thus, the Fourier transform

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.

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. 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:

,

we can get

so this is indeed the Helmholtz decomposition.

Generalization to arbitrary dimensions

.

Matrix approach

dimensions cannot be done with a vector potential, since the rotation operator and the cross product are defined (as vectors) only in three dimensions.

.

is again the fundamental solution of Laplace's equation the volume of the d-dimensional unit balls the gamma function.

, yielding the same prefactor as above. The rotational potential is an antisymmetric matrix.

In three-dimensional space, this is equivalent to the rotation of the vector potential.

Tensor approach

can be replaced by the appropriate Green's function for the Laplacian where Einstein summation convention in 2D.

is the Kronecker delta (and the summation convention is again used). In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol-component multi-index

dimensions.

Integration by parts. 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

at infinity. However, Otto Blumenthal. 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.

Uniqueness of the solution

In general, the Helmholtz decomposition is not uniquely defined. A harmonic function, a different Helmholtz decomposition can be obtained:

is the only harmonic function with this property, which follows from Liouville's theorem, this guarantees the uniqueness of the gradient and rotation fields.

This uniqueness does not apply to the potentials: In the three-dimensional case, the scalar and vector potential jointly have four components, whereas the vector field has only three. The vector field is invariant to gauge transformations and the choice of appropriate potentials known as gauge fixing is the subject of gauge theory. Important examples from physics are the Lorenz gauge condition and the Coulomb gauge. An alternative is to use the poloidal–toroidal decomposition.

Applications

Electrodynamics

The Helmholtz theorem is of particular interest in electrodynamics, since it can be used to write Maxwell's equations in the potential image and solve them more easily. The Helmholtz decomposition can be used to prove that, given electric current density and charge density, the electric field and the magnetic flux density can be determined. They are unique if the densities vanish at infinity and one assumes the same for the potentials.

Fluid dynamics

In fluid dynamics, the Helmholtz projection plays an important role, especially for the solvability theory of the Navier-Stokes equations. If the Helmholtz projection is applied to the linearized incompressible Navier-Stokes equations, the Stokes equation is called the Stokes operator.

Dynamical systems theory

In the theory of dynamical systems, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute Lyapunov functions in some cases.

For some dynamical systems such as the Lorenz system (Edward N. Lorenz, 1963), a simplified model for atmospheric convection, a closed-form expression is given as:

The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable fix point for some parameter range. For other parameters, the rotation field ensures that a strange attractor is created, causing the model to exhibit a butterfly effect.

Medical Imaging

In magnetic resonance elastography, a variant of MR imaging where mechanical waves are used to probe the viscoelasticity of organs, the Helmholtz decomposition is sometimes used to separate the measured displacement fields into its shear component (divergence-free) and its compression component (curl-free). In this way, the complex shear modulus can be calculated without contributions from compression waves.

Computer animation and robotics

The Helmholtz decomposition is also used in the field of computer engineering. This includes robotics, image reconstruction but also computer animation, where the decomposition is used for realistic visualization of fluids or vector fields.

See also

• Clebsch representation for a related decomposition of vector fields

• Darwin Lagrangian for an application

• Poloidal–toroidal decomposition.

• Scalar–vector–tensor decomposition

• Hodge theory generalizing Helmholtz decomposition

• Polar factorization theorem

• Helmholtz–Leray decomposition used for defining the Leray projection

Notes

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

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