Helmholtz Decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, 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.

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 $-\nabla \phi + \nabla \times \mathbf$, where $\phi$ is a scalar field called "scalar potential", and is a vector field, called a vector potential.

Statement of the theorem

Let $\mathbf$ be a vector field on a bounded domain $V\subseteq\mathbb^3$, which is twice continuously differentiable inside $V$, and let $S$ be the surface that encloses the domain $V$. Then $\mathbf$ can be decomposed into a curl-free component and a divergence-free component:

$\mathbf=-\nabla \Phi+\nabla\times\mathbf,$
where

\begin
\Phi(\mathbf) & =\frac 1 \int_V \frac \, \mathrmV' -\frac 1 \oint_S \mathbf' \cdot \frac \, \mathrmS' \\ pt\mathbf(\mathbf) & =\frac 1 \int_V \frac \, \mathrmV' -\frac 1 \oint_S \mathbf'\times\frac \, \mathrmS'
\end

and $\nabla'$ is the nabla operator with respect to $\mathbf$, not $\mathbf$.

If $V = \R^3$ and is therefore unbounded, and $\mathbf$ vanishes at least as fast as $1/r$ as $r \to \infty$, then one has David J. Griffiths, ''Introduction to Electrodynamics'', Prentice-Hall, 1999, p. 556.

\begin
\Phi(\mathbf) & =\frac\int_ \frac \, \mathrmV' \\ pt\mathbf (\mathbf) & =\frac\int_ \frac \, \mathrmV'
\end
This holds in particular if $\mathbf F$ is twice continuously differentiable in $\mathbb R^3$ and of bounded support.

Derivation

Suppose we have a vector function $\mathbf(\mathbf)$ of which we know the curl, $\nabla\times\mathbf$, and the divergence, $\nabla\cdot\mathbf$, in the domain and the fields on the boundary. Writing the function using delta function in the form
$\delta^3(\mathbf-\mathbf')=-\frac 1 \nabla^2 \frac\, ,$
where $\nabla^2:=\nabla\cdot\nabla$ is the Laplace operator, we have

\begin
\mathbf(\mathbf) &= \int_V \mathbf\left(\mathbf'\right)\delta^3 (\mathbf-\mathbf') \mathrmV' \\
&=\int_V\mathbf(\mathbf')\left(-\frac\nabla^2\frac\right)\mathrmV' \\
&=-\frac\nabla^2 \int_V \frac\mathrmV' \\
&=-\frac\left nabla\left(\nabla\cdot\int_V\frac\mathrmV'\right)-\nabla\times\left(\nabla\times\int_V\frac\mathrmV'\right)\right\\
&= -\frac \left nabla\left(\int_V\mathbf(\mathbf')\cdot\nabla\frac\mathrmV'\right)+\nabla\times\left(\int_V\mathbf(\mathbf')\times\nabla\frac\mathrmV'\right)\right\\
&=-\frac\left \nabla\left(\int_V\mathbf(\mathbf')\cdot\nabla'\frac\mathrmV'\right)-\nabla\times\left(\int_V\mathbf (\mathbf')\times\nabla'\frac\mathrmV'\right)\right\end

where we have used the definition of the vector Laplacian:
$\nabla^\mathbf=\nabla (\nabla\cdot\mathbf)-\nabla\times (\nabla\times\mathbf) \ ,$

differentiation/integration with respect to $\mathbf r'$by $\nabla'/\mathrm dV',$ and in the last line, linearity of function arguments:
$\nabla\frac=-\nabla'\frac\ .$

Then using the vectorial identities

$\begin \mathbf\cdot\nabla\psi &=-\psi(\nabla\cdot\mathbf)+\nabla\cdot (\psi\mathbf) \\ \mathbf\times\nabla\psi &=\psi(\nabla\times\mathbf)-\nabla \times (\psi\mathbf) \end$

we get
\begin
\mathbf(\mathbf)=-\frac\bigg &-\nabla\left(-\int_\frac\mathrmV'+\int_\nabla'\cdot\frac\mathrmV'\right)
\\&
-\nabla\times\left(\int_\frac\mathrmV'
- \int_\nabla'\times\frac\mathrmV'\right)\bigg
\end

Thanks to the divergence theorem the equation can be rewritten as

$\begin \mathbf (\mathbf) &= -\frac \bigg[ -\nabla\left( -\int_ \frac \mathrmV' + \oint_\mathbf'\cdot \frac\mathrmS' \right) \\ &\qquad\qquad -\nabla\times\left(\int_\frac\mathrmV' -\oint_\mathbf'\times\frac\mathrmS'\right) \bigg] \\ &= -\nabla\left[ \frac\int_ \frac \mathrmV' - \frac \oint_\mathbf' \cdot \frac \mathrmS' \right] \\ &\quad + \nabla\times \left[ \frac\int_ \frac \mathrmV' - \frac\oint_ \mathbf' \times \frac \mathrmS' \right] \end$

with outward surface normal $\mathbf'$.

Defining

$\Phi(\mathbf)\equiv\frac\int_\frac\mathrmV'-\frac\oint_\mathbf'\cdot\frac\mathrmS'$
$\mathbf(\mathbf)\equiv\frac\int_\frac\mathrmV'-\frac\oint_\mathbf'\times\frac\mathrmS'$

we finally obtain
$\mathbf=-\nabla\Phi+\nabla\times\mathbf.$

Generalization to higher dimensions

In a $d$-dimensional vector space with $d\neq 3$, $-\frac$ should be replaced by the appropriate Green's function#Green's functions for the Laplacian, Green's function for the Laplacian, defined by
$\nabla^2 G(\mathbf,\mathbf') = \frac\fracG(\mathbf,\mathbf') = \delta^d(\mathbf-\mathbf')$
where Einstein summation convention is used for the index $\mu$. For example, $G(\mathbf,\mathbf')=\frac\ln\left, \mathbf-\mathbf'\$ in 2D.

Following the same steps as above, we can write
$F_\mu(\mathbf) = \int_V F_\mu(\mathbf') \frac\fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf' = \delta_\delta_\int_V F_\nu(\mathbf') \frac\fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf'$
where $\delta_$ 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 $\varepsilon$,
$\varepsilon_\varepsilon_ = (d-2)!(\delta_\delta_ - \delta_\delta_)$
which is valid in $d\ge 2$ dimensions, where $\alpha$ is a $(d-2)$-component multi-index. This gives
$F_\mu(\mathbf) = \delta_\delta_\int_V F_\nu(\mathbf') \frac\fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf' + \frac\varepsilon_\varepsilon_ \int_V F_\nu(\mathbf') \frac\fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf'$

We can therefore write
$F_\mu(\mathbf) = -\frac \Phi(\mathbf) + \varepsilon_\frac A_(\mathbf)$
where
$\begin \Phi(\mathbf) &= -\int_V F_\nu(\mathbf') \fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf'\\ A_ &= \frac\varepsilon_ \int_V F_\nu(\mathbf') \fracG(\mathbf,\mathbf') \,\mathrm^d \mathbf' \end$
Note that the vector potential is replaced by a rank-$(d-2)$ tensor in $d$ dimensions.

For a further generalization to manifolds, see the discussion of Hodge decomposition below.

Another derivation from the Fourier transform

Note that in the theorem stated here, we have imposed the condition that if $\mathbf$ is not defined on a bounded domain, then $\mathbf$ shall decay faster than $1/r$. Thus, the Fourier Transform of $\mathbf$, denoted as $\mathbf$, is guaranteed to exist. We apply the convention
$\mathbf(\mathbf) = \iiint \mathbf(\mathbf) e^ dV_k$

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:
\begin
G_\Phi(\mathbf) &= i \frac \\
\mathbf_\mathbf(\mathbf) &= i \frac \\ pt\Phi(\mathbf) &= \iiint G_\Phi(\mathbf) e^ dV_k \\
\mathbf(\mathbf) &= \iiint \mathbf_\mathbf(\mathbf) e^ dV_k
\end

Hence

\begin
\mathbf(\mathbf) &= - i \mathbf G_\Phi(\mathbf) + i \mathbf \times \mathbf_\mathbf(\mathbf) \\ pt\mathbf(\mathbf) &= -\iiint i \mathbf G_\Phi(\mathbf) e^ dV_k + \iiint i \mathbf \times \mathbf_\mathbf(\mathbf) e^ dV_k \\
&= - \nabla \Phi(\mathbf) + \nabla \times \mathbf(\mathbf)
\end

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

$\nabla \cdot \mathbf = d \quad \text \quad \nabla \times \mathbf = \mathbf;$

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

$\mathbf = - \nabla(\mathcal (d)) + \nabla \times (\mathcal(\mathbf)),$

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

Solution space

For two Helmholtz decompositions $(\Phi_1, )$ $(\Phi_2, )$ of $\mathbf F$, there holds
:$\Phi_1-\Phi_2 = \lambda,\quad =_\lambda + \nabla \varphi,$
:where
:* $\lambda$ is an harmonic scalar field,
:* $_\lambda$ is a vector field determined by $\lambda$,
:* $\varphi$ is any scalar field.

Proof:
Setting $\lambda = \Phi_2 - \Phi_1$ and $$, one has, according to the
definition of the Helmholtz decomposition,
:$-\nabla \lambda + \nabla \times \mathbf B = 0$.
Taking the divergence of each member of this equation yields
$\nabla^2 \lambda = 0$, hence $\lambda$ is harmonic.

Conversely, given any harmonic function $\lambda$,
$\nabla \lambda$ is solenoidal since
:$\nabla\cdot (\nabla \lambda) = \nabla^2 \lambda = 0.$
Thus, according to the above section, there exists a vector field $_\lambda$ such that
$\nabla \lambda = \nabla\times _\lambda$.
If $_\lambda$ is another such vector field,
then $\mathbf C = _\lambda - _\lambda$
fulfills $\nabla \times = 0$, hence $C = \nabla \varphi$
for some scalar field $\varphi$ (and conversely).

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.

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. Every square-integrable vector field has an orthogonal decomposition:

$\mathbf=\nabla\varphi+\nabla \times \mathbf$

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:

$\mathbf=\nabla\varphi+\mathbf$

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 $\hat\mathbf$ of the vector field $\mathbf$. 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

$\hat\mathbf (\mathbf) = \hat\mathbf_t (\mathbf) + \hat\mathbf_l (\mathbf)$
$\mathbf \cdot \hat\mathbf_t(\mathbf) = 0.$
$\mathbf \times \hat\mathbf_l(\mathbf) = \mathbf.$

Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:

$\mathbf(\mathbf) = \mathbf_t(\mathbf)+\mathbf_l(\mathbf)$
$\nabla \cdot \mathbf_t (\mathbf) = 0$
$\nabla \times \mathbf_l (\mathbf) = \mathbf$

Since $\nabla\times(\nabla\Phi)=0$ and $\nabla\cdot(\nabla\times\mathbf)=0$,

we can get

$\mathbf_t=\nabla\times\mathbf=\frac\nabla\times\int_V\frac\mathrmV'$
$\mathbf_l=-\nabla\Phi=-\frac\nabla\int_V\frac\mathrmV'$

so this is indeed the Helmholtz decomposition.Online lecture notes by Robert Littlejohn
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See also

* Clebsch representation for a related decomposition of vector fields
* Darwin Lagrangian for an application
* Poloidal–toroidal decomposition for a further decomposition of the divergence-free component $\nabla \times \mathbf$.
* Scalar–vector–tensor decomposition
* Hodge theory generalizing Helmholtz decomposition
* Polar factorization theorem.

Notes

References

General 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

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

{{DEFAULTSORT:Helmholtz Decomposition
Vector calculus
Theorems in analysis
Analytic geometry
Hermann von Helmholtz