In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
mathematics, in the area of
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
, rapidly decaying
vector field in three dimensions can be resolved into the sum of an
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
(
curl-free) vector field and a
solenoidal
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
(
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...
.
As an irrotational vector field has a
scalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
and a solenoidal vector field has a
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
, 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 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 inside
, and let
be the surface that encloses the domain
. Then
can be decomposed into a curl-free component and a divergence-free component:
where
and
is the nabla operator with respect to
, not
.
If
and is therefore unbounded, and
vanishes at least as fast as
as
, then one has
[ David J. Griffiths, ''Introduction to Electrodynamics'', Prentice-Hall, 1999, p. 556.]
This holds in particular if
is twice continuously differentiable in
and of bounded support.
Derivation
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 mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
in the form
where
is the Laplace operator, we have
where we have used the definition of the
vector Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
:
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
Generalization to higher dimensions
In a
-dimensional vector space with
,
should be replaced by the appropriate
Green's function#Green's functions for the Laplacian, 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 mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
(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
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
,
which is valid in
dimensions, where
is a
-component
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
. This gives
We can therefore write
where
Note that the vector potential is replaced by a rank-
tensor in
dimensions.
For a further generalization to manifolds, see the discussion of
Hodge decomposition
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
below.
Another 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
Fields with prescribed divergence and curl
The term "Helmholtz theorem" can also refer to the following. Let be a
solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
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 In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...
operator. (When acting on a vector field, such as , it is defined to act on each component.)
Solution space
For two Helmholtz decompositions
of
, there holds
:
:where
:*
is an
harmonic scalar field,
:*
is a vector field determined by
,
:*
is any scalar field.
Proof:
Setting
and
, one has, according to the
definition of the Helmholtz decomposition,
:
.
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
(and conversely).
Differential forms
The
Hodge decomposition
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
is closely related to the Helmholtz decomposition, generalizing from vector fields on R
3 to
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
on a
Riemannian manifold ''M''. Most formulations of the Hodge decomposition require ''M'' to be
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
. Since this is not true of R
3, 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 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 In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. T ...
. Every
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
vector field has an
orthogonal decomposition:
where is in the
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
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 .
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.
Online lecture notes by Robert Littlejohn
/ref>
See also
* Clebsch representation In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field \boldsymbol(\boldsymbol) is:
\boldsymbol = \boldsymbol \varphi + \psi\, \boldsymbol \chi,
where the scalar fields \varphi(\boldsymbol), \psi(\bol ...
for a related decomposition of vector fields
* Darwin Lagrangian for an application
* Poloidal–toroidal decomposition for a further decomposition of the divergence-free component .
* Scalar–vector–tensor decomposition
* Hodge theory generalizing Helmholtz decomposition
* Polar factorization theorem
In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987), with antecedents of Knott-Smith (1984) and Rachev (1985), that generalizes many existing results among which are the polar ...
.
Notes
References
General references
* George B. Arfken
George Brown Arfken (November 20, 1922 – October 8, 2020) was an American theoretical physicist and the author of several mathematical physics texts. He was a physics professor at Miami University from 1952 to 1983 and the chair of the Miami Un ...
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
Rutherford "Gus" Aris (September 15, 1929 – November 2, 2005) was a chemical engineer, control theorist, applied mathematician, and a Regents Professor Emeritus of Chemical Engineering at the University of Minnesota (1958–2005).
Early ...
, ''Vectors, tensors, and the basic equations of fluid mechanics'', Prentice-Hall (1962), , pp. 70–72
References for the weak formulation
*
* 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
Helmholtz theorem
on MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
{{DEFAULTSORT:Helmholtz Decomposition
Vector calculus
Theorems in analysis
Analytic geometry
Hermann von Helmholtz