View source for Helmholtz decomposition

{{Short description|Certain vector fields are the sum of an irrotational and a solenoidal vector field}}
{{Calculus |expanded=vector}}

In [[physics]] and [[mathematics]], the '''Helmholtz decomposition theorem''' or the '''fundamental theorem of vector calculus'''<ref name="murray1898"/><ref name="gibbs1901"/><ref name="heaviside1893"/><ref name="woolhouse1854"/><ref name="johnson1881"/><ref name="shaw1922"/><ref name="edwards1922"/> states that certain differentiable [[vector field]]s can be resolved into the sum of an [[irrotational vector field|irrotational]] ([[Curl (mathematics)|curl]]-free) vector field and a [[solenoidal]] ([[divergence]]-free) vector field. In [[physics]], often only the decomposition of sufficiently [[smooth function|smooth]], rapidly decaying [[vector field]]s in three dimensions is discussed. It is named after [[Hermann von Helmholtz]].

== Definition ==

For a vector field <math>\mathbf{F} \in C^1(V, \mathbb{R}^n)</math> defined on a domain <math>V \subseteq \mathbb{R}^n</math>, a Helmholtz decomposition is a pair of vector fields <math>\mathbf{G} \in C^1(V, \mathbb{R}^n)</math> and <math>\mathbf{R} \in C^1(V, \mathbb{R}^n)</math> such that:
<math display="block">
\begin{align}
\mathbf{F}(\mathbf{r}) &= \mathbf{G}(\mathbf{r}) + \mathbf{R}(\mathbf{r}), \\
\mathbf{G}(\mathbf{r}) &= - \nabla \Phi(\mathbf{r}), \\
\nabla \cdot \mathbf{R}(\mathbf{r}) &= 0.
\end{align}
</math>
Here, <math>\Phi \in C^2(V, \mathbb{R})</math> is a [[scalar potential]], <math>\nabla \Phi</math> is its [[gradient]], and <math>\nabla \cdot \mathbf{R}</math> is the [[divergence]] of the vector field <math>\mathbf{R}</math>. The irrotational vector field <math>\mathbf{G}</math> is called a ''gradient field'' and <math>\mathbf{R}</math> is called a ''[[solenoidal]] field'' or ''rotation field''. This decomposition does not exist for all vector fields and is not [[uniqueness|unique]].<ref name="glotzl2023"/>

== History ==

The Helmholtz decomposition in three dimensions was first described in 1849<ref name="stokes1849" /> by [[George Gabriel Stokes]] for a theory of [[diffraction]]. [[Hermann von Helmholtz]] published his paper on some [[hydrodynamics|hydrodynamic]] basic equations in 1858,<ref name="helmholtz1858" /><ref name="kustepeli2016" /> which was part of his research on the [[Helmholtz's theorems]] describing the motion of fluid in the vicinity of vortex lines.<ref name="kustepeli2016"/> 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.<ref name="glotzl2023"/><ref name="trancong1993" /><ref name="petrascheck2017"/> For [[Riemannian manifold|Riemannian manifolds]], the Helmholtz-Hodge decomposition using [[differential geometry]] and [[tensor calculus]] was derived.<ref name="glotzl2023" /><ref name="kustepeli2016"/><ref name="sprossig2009"/><ref name="bhatia2013" />

The decomposition has become an important tool for many problems in [[theoretical physics]],<ref name="kustepeli2016" /><ref name="sprossig2009" /> but has also found applications in [[animation]], [[computer vision]] as well as [[robotics]].<ref name="bhatia2013" />

== 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 function]]s that are defined on a [[bounded domain]]. Then, a [[vector potential]] <math>A</math> can be defined, such that the rotation field is given by <math>\mathbf{R} = \nabla \times \mathbf{A}</math>, using the [[Curl (mathematics)|curl]] of a vector field.<ref name="petrascheck2015" />

Let <math>\mathbf{F}</math> be a vector field on a bounded domain <math>V\subseteq\mathbb{R}^3</math>, which is twice continuously differentiable inside <math>V</math>, and let <math>S</math> be the surface that encloses the domain <math>V</math> with outward surface normal <math> \mathbf{\hat{n}}' </math>. Then <math>\mathbf{F}</math> can be decomposed into a curl-free component and a divergence-free component as follows:<ref name="vermont" />

<math display="block">\mathbf{F}=-\nabla \Phi+\nabla\times\mathbf{A},</math>
where
<math display="block">
\begin{align}
\Phi(\mathbf{r}) & =\frac 1 {4\pi} \int_V \frac{\nabla'\cdot\mathbf{F} (\mathbf{r}')}{|\mathbf{r} -\mathbf{r}'|} \, \mathrm{d}V' -\frac 1 {4\pi} \oint_S \mathbf{\hat{n}}' \cdot \frac{\mathbf{F} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}S' \\[8pt]
\mathbf{A}(\mathbf{r}) & =\frac 1 {4\pi} \int_V \frac{\nabla' \times \mathbf{F}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}V' -\frac 1 {4\pi} \oint_S \mathbf{\hat{n}}'\times\frac{\mathbf{F} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}S'
\end{align}
</math>

and <math>\nabla'</math> is the [[nabla operator]] with respect to <math>\mathbf{r'}</math>, not <math> \mathbf{r} </math>.

If <math>V = \R^3</math> and is therefore unbounded, and <math>\mathbf{F}</math> vanishes faster than <math>1/r</math> as <math>r \to \infty</math>, then one has<ref name="griffiths1999"/>

<math display="block">\begin{align}
\Phi(\mathbf{r}) & =\frac{1}{4\pi}\int_{\R^3} \frac{\nabla' \cdot \mathbf{F} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}V' \\[8pt]
\mathbf{A} (\mathbf{r}) & =\frac{1}{4\pi}\int_{\R^3} \frac{\nabla'\times\mathbf{F} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}V'
\end{align}</math>
This holds in particular if <math>\mathbf F</math> is twice continuously differentiable in <math>\mathbb R^3</math> and of bounded support.

=== Derivation ===

{{math proof| proof =
Suppose we have a vector function <math>\mathbf{F}(\mathbf{r})</math> of which we know the curl, <math>\nabla\times\mathbf{F}</math>, and the divergence, <math>\nabla\cdot\mathbf{F}</math>, in the domain and the fields on the boundary. Writing the function using the [[delta function]] in the form
<math display="block">\delta^3(\mathbf{r}-\mathbf{r}')=-\frac 1 {4\pi} \nabla^2 \frac{1}{|\mathbf{r}-\mathbf{r}'|}\, ,</math>
where <math>\nabla^2</math> is the [[Laplacian]] operator, we have

<math display="block">\begin{align}
\mathbf{F}(\mathbf{r}) &= \int_V \mathbf{F}\left(\mathbf{r}'\right)\delta^3 (\mathbf{r}-\mathbf{r}') \mathrm{d}V' \\
&=\int_V\mathbf{F}(\mathbf{r}')\left(-\frac{1}{4\pi}\nabla^2\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\right)\mathrm{d}V' \end{align}</math>

Now, changing the meaning of <math>\nabla^2</math> to the [[vector Laplacian]] operator (we have the right to do so because this laplacian is with respect to <math>\mathbf{r}</math> therefore it sees the vector field <math>\mathbf{F}(\mathbf{r'})</math> as a constant), we can move <math>\mathbf{F}(\mathbf{r'})</math> to the right of the<math>\nabla^2</math>operator.

<math display="block">\begin{align}\mathbf{F}(\mathbf{r})&=\int_V-\frac{1}{4\pi}\nabla^2\frac{\mathbf{F}(\mathbf{r}')}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' \\
&=-\frac{1}{4\pi}\nabla^2 \int_V \frac{\mathbf{F}(\mathbf{r}')}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' \\
&=-\frac{1}{4\pi}\left[\nabla\left(\nabla\cdot\int_V\frac{\mathbf{F}(\mathbf{r}')}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)-\nabla\times\left(\nabla\times\int_V\frac{\mathbf{F}(\mathbf{r}')}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right] \\
&= -\frac{1}{4\pi} \left[\nabla\left(\int_V\mathbf{F}(\mathbf{r}')\cdot\nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)+\nabla\times\left(\int_V\mathbf{F}(\mathbf{r}')\times\nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right] \\
&=-\frac{1}{4\pi}\left[-\nabla\left(\int_V\mathbf{F}(\mathbf{r}')\cdot\nabla'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)-\nabla\times\left(\int_V\mathbf{F} (\mathbf{r}')\times\nabla'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right]
\end{align}</math>

where we have used the vector Laplacian identity:
<math display="block">\nabla^{2}\mathbf{a}=\nabla (\nabla\cdot\mathbf{a})-\nabla\times (\nabla\times\mathbf{a}) \ ,</math>

differentiation/integration with respect to <math>\mathbf r'</math>by <math>\nabla'/\mathrm dV',</math> and in the last line, linearity of function arguments:
<math display="block"> \nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}=-\nabla'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\ .</math>

Then using the vectorial identities

<math display="block">\begin{align}
\mathbf{a}\cdot\nabla\psi &=-\psi(\nabla\cdot\mathbf{a})+\nabla\cdot (\psi\mathbf{a}) \\
\mathbf{a}\times\nabla\psi &=\psi(\nabla\times\mathbf{a})-\nabla \times (\psi\mathbf{a})
\end{align}</math>

we get
<math display="block">\begin{align}
\mathbf{F}(\mathbf{r})=-\frac{1}{4\pi}\bigg[
&-\nabla\left(-\int_{V}\frac{\nabla'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'+\int_{V}\nabla'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)
\\&
-\nabla\times\left(\int_{V}\frac{\nabla'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'
- \int_{V}\nabla'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\bigg].
\end{align}</math>

Thanks to the [[divergence theorem]] the equation can be rewritten as

<math display="block">\begin{align}
\mathbf{F} (\mathbf{r}) 
&=
-\frac{1}{4\pi}
\bigg[
    -\nabla\left(
        -\int_{V}
        \frac{
            \nabla'\cdot\mathbf{F}\left(\mathbf{r}'\right)
        }{
            \left|\mathbf{r}-\mathbf{r}'\right|
        } \mathrm{d}V'
        +
        \oint_{S}\mathbf{\hat{n}}'\cdot
        \frac{
            \mathbf{F}\left(\mathbf{r}'\right)
        }{
            \left|\mathbf{r}-\mathbf{r}'\right|
        }\mathrm{d}S'
    \right)
\\
&\qquad\qquad
-\nabla\times\left(\int_{V}\frac{\nabla'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'
-\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'\right)
\bigg]
\\
&= 
-\nabla\left[
  \frac{1}{4\pi}\int_{V}
  \frac{
    \nabla'\cdot\mathbf{F}\left(\mathbf{r}'\right)
  }{\left|
    \mathbf{r}-\mathbf{r}'
  \right|}
  \mathrm{d}V' 
  - 
  \frac{1}{4\pi} 
  \oint_{S}\mathbf{\hat{n}}' \cdot
  \frac{
    \mathbf{F}\left(\mathbf{r}'\right)
  }{
     \left|
       \mathbf{r}-\mathbf{r}'
     \right|
   }
 \mathrm{d}S'
\right]
\\
&\quad
+
\nabla\times
\left[
  \frac{1}{4\pi}\int_{V}
  \frac{
    \nabla '\times\mathbf{F}\left(\mathbf{r}'\right)
  }{
     \left|
       \mathbf{r}-\mathbf{r}'
     \right|
  }
  \mathrm{d}V' 
  -
  \frac{1}{4\pi}\oint_{S}
  \mathbf{\hat{n}}'
  \times
  \frac{
    \mathbf{F}\left(\mathbf{r}'\right)
  }{
    \left|
      \mathbf{r}-\mathbf{r}'
    \right|
   }
  \mathrm{d}S'
\right]
\end{align}</math>

with outward surface normal <math> \mathbf{\hat{n}}' </math>.

Defining

<math display="block">\Phi(\mathbf{r})\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'</math>
<math display="block">\mathbf{A}(\mathbf{r})\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'</math>

we finally obtain
<math display="block">\mathbf{F}=-\nabla\Phi+\nabla\times\mathbf{A}.</math>
}}

===Solution space===

If <math>(\Phi_1, {\mathbf A_1})</math> is a Helmholtz decomposition of <math>\mathbf F</math>, then 
<math>(\Phi_2, {\mathbf A_2})</math> is another decomposition if, and only if,
:<math>\Phi_1-\Phi_2 = \lambda \quad </math> and <math>\quad \mathbf{A}_1 - \mathbf{A}_2 = {\mathbf A}_\lambda + \nabla \varphi,</math>
:where
:* <math> \lambda</math> is a [[harmonic function|harmonic scalar field]],
:* <math> {\mathbf A}_\lambda </math> is a vector field which fulfills <math>\nabla\times {\mathbf A}_\lambda = \nabla \lambda,</math>
:* <math> \varphi </math> is a scalar field.

Proof: 
Set <math>\lambda = \Phi_2 -  \Phi_1</math> and <math>{\mathbf B = A_2 - A_1}</math>. According to the definition
of the Helmholtz decomposition, the condition is equivalent to 
:<math> -\nabla \lambda + \nabla \times \mathbf B = 0 </math>. 

Taking the divergence of each member of this equation yields 
<math>\nabla^2 \lambda = 0</math>, hence <math>\lambda</math> is harmonic.

Conversely, given any harmonic function <math>\lambda</math>,
<math>\nabla \lambda </math> is solenoidal since
:<math>\nabla\cdot (\nabla \lambda) = \nabla^2 \lambda = 0.</math>

Thus, according to the above section, there exists a vector field <math>{\mathbf A}_\lambda</math> such that
<math>\nabla \lambda = \nabla\times {\mathbf A}_\lambda</math>.

If <math>{\mathbf A'}_\lambda</math> is another such vector field,	
then <math>\mathbf C = {\mathbf A}_\lambda -  {\mathbf A'}_\lambda</math> 
fulfills <math>\nabla \times {\mathbf C} = 0</math>, hence <math>C = \nabla \varphi</math>
for some scalar field <math>\varphi</math>.

=== Fields with prescribed divergence and curl ===
The term "Helmholtz theorem" can also refer to the following. Let {{math|'''C'''}} be a [[solenoidal vector field]] and ''d'' a scalar field on {{math|'''R'''<sup>3</sup>}} which are sufficiently smooth and which vanish faster than {{math|1/''r''<sup>2</sup>}} at infinity. Then there exists a vector field {{math|'''F'''}} such that

<math display="block">\nabla \cdot \mathbf{F} = d \quad \text{ and } \quad \nabla \times \mathbf{F} = \mathbf{C};</math>

if additionally the vector field {{math|'''F'''}} vanishes as {{math|''r'' → ∞}}, then {{math|'''F'''}} is unique.<ref name="griffiths1999"/>

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.<ref name="griffiths1999"/> The proof is by a construction generalizing the one given above: we set

<math display="block">\mathbf{F} = \nabla(\mathcal{G} (d)) - \nabla \times (\mathcal{G}(\mathbf{C})),</math>

where <math>\mathcal{G}</math> represents the [[Newtonian potential]] operator. (When acting on a vector field, such as {{math|∇ × '''F'''}}, 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 {{math|Ω}} is a bounded, simply-connected, [[Lipschitz domain]]. Every [[square-integrable]] vector field {{math|'''u''' ∈ (''L''<sup>2</sup>(Ω))<sup>3</sup>}} has an [[orthogonality|orthogonal]] decomposition:<ref name="amrouche1998" /><ref name="dautray1990" /><ref name="girault1986" />

<math display="block">\mathbf{u}=\nabla\varphi+\nabla \times \mathbf{A}</math>

where {{mvar|φ}} is in the [[Sobolev space]] {{math|''H''<sup>1</sup>(Ω)}} of square-integrable functions on {{math|Ω}} whose partial derivatives defined in the [[distribution (mathematics)|distribution]] sense are square integrable, and {{math|'''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 {{math|'''u''' ∈ ''H''(curl, Ω)}}, a similar decomposition holds:

<math display="block">\mathbf{u}=\nabla\varphi+\mathbf{v}</math>

where {{math|''φ'' ∈ ''H''<sup>1</sup>(Ω), '''v''' ∈ (''H''<sup>1</sup>(Ω))<sup>''d''</sup>}}.

=== Derivation from the Fourier transform ===
Note that in the theorem stated here, we have imposed the condition that if <math>\mathbf{F}</math> is not defined on a bounded domain, then <math>\mathbf{F}</math> shall decay faster than <math>1/r</math>. Thus, the [[Fourier transform]] of <math>\mathbf{F}</math>, denoted as <math>\mathbf{G}</math>, is guaranteed to exist. We apply the convention
<math display="block">\mathbf{F}(\mathbf{r}) = \iiint \mathbf{G}(\mathbf{k}) e^{i\mathbf{k} \cdot \mathbf{r}} dV_k </math>

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:
<math display="block">\begin{align}
G_\Phi(\mathbf{k}) &= i \frac{\mathbf{k} \cdot \mathbf{G}(\mathbf{k})}{\|\mathbf{k}\|^2} \\
\mathbf{G}_\mathbf{A}(\mathbf{k}) &= i \frac{\mathbf{k} \times \mathbf{G}(\mathbf{k})}{\|\mathbf{k}\|^2}  \\ [8pt]
\Phi(\mathbf{r}) &= \iiint G_\Phi(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}} dV_k \\
\mathbf{A}(\mathbf{r}) &= \iiint \mathbf{G}_\mathbf{A}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}} dV_k
\end{align} </math>

Hence

<math display="block">\begin{align}
\mathbf{G}(\mathbf{k}) &= - i \mathbf{k} G_\Phi(\mathbf{k}) + i \mathbf{k} \times \mathbf{G}_\mathbf{A}(\mathbf{k}) \\ [6pt]
\mathbf{F}(\mathbf{r}) &= -\iiint i \mathbf{k} G_\Phi(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}} dV_k + \iiint i \mathbf{k} \times \mathbf{G}_\mathbf{A}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}} dV_k \\
&= - \nabla \Phi(\mathbf{r}) +  \nabla \times \mathbf{A}(\mathbf{r})
\end{align}</math>

=== 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'''.<ref name="stewart2011"/> This terminology comes from the following construction: Compute the three-dimensional [[Fourier transform]] <math>\hat\mathbf{F}</math> of the vector field <math>\mathbf{F}</math>. 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

<math display="block">\hat\mathbf{F} (\mathbf{k}) = \hat\mathbf{F}_t (\mathbf{k}) + \hat\mathbf{F}_l (\mathbf{k})</math>
<math display="block">\mathbf{k} \cdot \hat\mathbf{F}_t(\mathbf{k}) = 0.</math>
<math display="block">\mathbf{k} \times \hat\mathbf{F}_l(\mathbf{k}) = \mathbf{0}.</math>

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

<math display="block">\mathbf{F}(\mathbf{r}) = \mathbf{F}_t(\mathbf{r})+\mathbf{F}_l(\mathbf{r})</math>
<math display="block">\nabla \cdot \mathbf{F}_t (\mathbf{r}) = 0</math>
<math display="block">\nabla \times \mathbf{F}_l (\mathbf{r}) = \mathbf{0}</math>

Since <math>\nabla\times(\nabla\Phi)=0</math> and <math>\nabla\cdot(\nabla\times\mathbf{A})=0</math>,

we can get

<math display="block">\mathbf{F}_t=\nabla\times\mathbf{A}=\frac{1}{4\pi}\nabla\times\int_V\frac{\nabla'\times\mathbf{F}}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'</math>
<math display="block">\mathbf{F}_l=-\nabla\Phi=-\frac{1}{4\pi}\nabla\int_V\frac{\nabla'\cdot\mathbf{F}}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'</math>

so this is indeed the Helmholtz decomposition.<ref name="littlejohn"/>

== Generalization to higher dimensions ==

=== Matrix approach ===

The generalization to <math>d</math> dimensions cannot be done with a vector potential, since the rotation operator and the [[cross product]] are defined (as vectors) only in three dimensions. 

Let <math>\mathbf{F}</math> be a vector field on a bounded domain <math>V\subseteq\mathbb{R}^d</math> which decays faster than <math>|\mathbf{r}|^{-\delta}</math> for <math>|\mathbf{r}| \to \infty</math> and <math>\delta > 2</math>.

The scalar potential is defined similar to the three dimensional case as:
<math display="block">\Phi(\mathbf{r}) = - \int_{\mathbb{R}^d} \operatorname{div}(\mathbf{F}(\mathbf{r}')) K(\mathbf{r}, \mathbf{r}') \mathrm{d}V' = - \int_{\mathbb{R}^d} \sum_i \frac{\partial F_i}{\partial r_i}(\mathbf{r}') K(\mathbf{r}, \mathbf{r}') \mathrm{d}V',</math>
where as the integration kernel <math>K(\mathbf{r}, \mathbf{r}')</math> is again the [[fundamental solution]] of [[Laplace's equation]], but in d-dimensional space:
<math display="block">K(\mathbf{r}, \mathbf{r}') = \begin{cases}  \frac{1}{2\pi} \log{ | \mathbf{r}-\mathbf{r}' | } &  d=2,  \\    \frac{1}{d(2-d)V_d} | \mathbf{r}-\mathbf{r}' | ^{2-d} &  \text{otherwise}, \end{cases}</math>
with <math>V_d = \pi^\frac{d}{2} / \Gamma\big(\tfrac{d}{2}+1\big)</math> the volume of the d-dimensional [[unit ball]]s and <math>\Gamma(\mathbf{r})</math> the [[gamma function]].

For <math>d = 3</math>, <math>V_d</math> is just equal to <math>\frac{4 \pi}{3}</math>, yielding the same prefactor as above.
The rotational potential is an [[antisymmetric matrix]] with the elements:
<math display="block">A_{ij}(\mathbf{r}) = \int_{\mathbb{R}^d} \left( \frac{\partial F_i}{\partial x_j}(\mathbf{r}') - \frac{\partial F_j}{\partial x_i}(\mathbf{r}') \right) K(\mathbf{r}, \mathbf{r}') \mathrm{d}V'. </math>
Above the diagonal are <math>\textstyle\binom{d}{2}</math> 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 <math>\mathbf{A} = [A_1, A_2, A_3] = [A_{23}, A_{31}, A_{12}]</math>.
However, such a matrix potential can be written as a vector only in the three-dimensional case, because <math>\textstyle\binom{d}{2} = d</math> is valid only for <math>d = 3</math>.

As in the three-dimensional case, the gradient field is defined as
<math display="block">
\mathbf{G}(\mathbf{r}) = - \nabla \Phi(\mathbf{r}).
</math>
The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix:
<math display="block">\mathbf{R}(\mathbf{r}) = \left[ \sum\nolimits_k \partial_{r_k} A_{ik}(\mathbf{r}); {1 \leq i \leq d} \right].</math>
In three-dimensional space, this is equivalent to the rotation of the vector potential.<ref name="glotzl2023" /><ref name="glotzl2020" />

=== Tensor approach ===

In a <math>d</math>-dimensional vector space with <math>d\neq 3</math>, <math display="inline">-\frac{1}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}</math> can be replaced by the appropriate [[Green's function#Green's functions for the Laplacian|Green's function for the Laplacian]], defined by
<math display="block">
\nabla^2 G(\mathbf{r},\mathbf{r}') = \frac{\partial}{\partial r_\mu}\frac{\partial}{\partial r_\mu}G(\mathbf{r},\mathbf{r}') = \delta^d(\mathbf{r}-\mathbf{r}')
</math>
where [[Einstein notation|Einstein summation convention]] is used for the index <math>\mu</math>. For example, <math display="inline">G(\mathbf{r},\mathbf{r}')=\frac{1}{2\pi}\ln\left|\mathbf{r}-\mathbf{r}'\right|</math> in 2D.

Following the same steps as above, we can write
<math display="block">
F_\mu(\mathbf{r}) = \int_V F_\mu(\mathbf{r}') \frac{\partial}{\partial r_\mu}\frac{\partial}{\partial r_\mu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
 = \delta_{\mu\nu}\delta_{\rho\sigma}\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
</math>
where <math>\delta_{\mu\nu}</math> 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]] <math>\varepsilon</math>,
<math display="block">
\varepsilon_{\alpha\mu\rho}\varepsilon_{\alpha\nu\sigma} = (d-2)!(\delta_{\mu\nu}\delta_{\rho\sigma} - \delta_{\mu\sigma}\delta_{\nu\rho})
</math>
which is valid in <math>d\ge 2</math> dimensions, where <math>\alpha</math> is a <math>(d-2)</math>-component [[Multi-index notation|multi-index]]. This gives
<math display="block">
F_\mu(\mathbf{r}) = \delta_{\mu\sigma}\delta_{\nu\rho}\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
+ \frac{1}{(d-2)!}\varepsilon_{\alpha\mu\rho}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
</math>

We can therefore write
<math display="block">
F_\mu(\mathbf{r}) = -\frac{\partial}{\partial r_\mu} \Phi(\mathbf{r}) + \varepsilon_{\mu\rho\alpha}\frac{\partial}{\partial r_\rho} A_{\alpha}(\mathbf{r})
</math>
where
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= -\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\nu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'\\
A_{\alpha} &= \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
\end{aligned}
</math>
Note that the vector potential is replaced by a rank-<math>(d-2)</math> tensor in <math>d</math> dimensions.

Because <math>G(\mathbf{r},\mathbf{r}')</math> is a function of only <math>\mathbf{r}-\mathbf{r}'</math>, one can replace <math>\frac{\partial}{\partial r_\mu}\rightarrow - \frac{\partial}{\partial r'_\mu}</math>, giving
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r'_\nu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'\\
A_{\alpha} &= -\frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\sigma'}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
\end{aligned}
</math>
[[Integration_by_parts#Higher_dimensions|Integration by parts]] can then be used to give
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= -\int_V G(\mathbf{r},\mathbf{r}')\frac{\partial}{\partial r'_\nu}F_\nu(\mathbf{r}') \,\mathrm{d}^d \mathbf{r}' + \oint_{S} G(\mathbf{r},\mathbf{r}') F_\nu(\mathbf{r}') \hat{n}'_\nu \,\mathrm{d}^{d-1} \mathbf{r}'\\
A_{\alpha} &= \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V G(\mathbf{r},\mathbf{r}') \frac{\partial}{\partial r_\sigma'}F_\nu(\mathbf{r}')  \,\mathrm{d}^d \mathbf{r}'- \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \oint_{S} G(\mathbf{r},\mathbf{r}') F_\nu(\mathbf{r}')  \hat{n}'_\sigma \,\mathrm{d}^{d-1} \mathbf{r}'
\end{aligned}
</math>
where <math>S=\partial V</math> is the boundary of <math>V</math>. These expressions are analogous to those given above for [[#Three-dimensional_space|three-dimensional space]].

For a further generalization to manifolds, see the discussion of [[Hodge decomposition]] [[#Differential forms|below]].

== Differential forms ==
The [[Hodge decomposition#Hodge decomposition|Hodge decomposition]] is closely related to the Helmholtz decomposition,<ref name="warner1983"/> generalizing from vector fields on '''R'''<sup>3</sup> to [[differential forms]] on a [[Riemannian manifold]] ''M''.  Most formulations of the Hodge decomposition require ''M'' to be [[compact space|compact]].<ref name="cantarella2002" /> Since this is not true of '''R'''<sup>3</sup>, 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 <math>|\mathbf{r}|^{-\delta}</math> with <math>\delta > 1</math> at infinity.<ref name="petrascheck2015" /><ref name="petrascheck2017"/><ref name="gregory1996" /> However, [[Otto Blumenthal]] showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than <math>|\mathbf{r}|^{-\delta}</math> with <math>\delta > 0</math>, which is substantially less strict.
To achieve this, the kernel <math>K(\mathbf{r}, \mathbf{r}')</math> in the convolution integrals has to be replaced by <math>K'(\mathbf{r}, \mathbf{r}') = K(\mathbf{r}, \mathbf{r}') - K(0, \mathbf{r}')</math>.<ref name="blumenthal1905" />
With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.<ref name="trancong1993" /><ref name="petrascheck2017"/><ref name="glotzl2020" /><ref name="gurtin1962"/>

For all [[Analytic function|analytic]] vector fields that need not go to zero even at infinity, methods based on [[Integration by parts|partial integration]] and the [[Cauchy formula for repeated integration]]<ref name="cauchy1823" /> 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 function]]s.<ref name="glotzl2023" />

== Uniqueness of the solution ==

In general, the Helmholtz decomposition is not uniquely defined.
A [[harmonic function]] <math>H(\mathbf{r})</math> is a function that satisfies <math>\Delta H(\mathbf{r}) = 0</math>.
By adding <math>H(\mathbf{r})</math> to the scalar potential <math>\Phi(\mathbf{r})</math>, a different Helmholtz decomposition can be obtained:

<math display="block">\begin{align}
\mathbf{G}'(\mathbf{r}) &= \nabla (\Phi(\mathbf{r}) + H(\mathbf{r})) = \mathbf{G}(\mathbf{r}) + \nabla H(\mathbf{r}),\\
\mathbf{R}'(\mathbf{r}) &= \mathbf{R}(\mathbf{r}) - \nabla H(\mathbf{r}).
\end{align}</math>

For vector fields <math>\mathbf{F}</math>, decaying at infinity, it is a plausible choice that scalar and rotation potentials also decay at infinity. 
Because <math>H(\mathbf{r}) = 0</math> is the only harmonic function with this property, which follows from [[Liouville's theorem (complex analysis)|Liouville's theorem]], this guarantees the uniqueness of the gradient and rotation fields.<ref name="axler1992" />

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.<ref name="petrascheck2015" />

=== 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 flow|Stokes equation]] is obtained. This depends only on the velocity of the particles in the flow, but no longer on the static pressure, allowing the equation to be reduced to one unknown. However, both equations, the Stokes and linearized equations, are equivalent. The operator <math>P\Delta</math> is called the [[Stokes operator]].<ref name="chorin1990" />

=== Dynamical systems theory ===

In the theory of [[dynamical system]]s, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute [[Lyapunov function]]s in some cases.<ref name="suda2019" /><ref name="suda2020" /><ref name="zhou2012" />

For some dynamical systems such as the [[Lorenz system]] ([[Edward N. Lorenz]], 1963<ref name="lorenz1963" />), a simplified model for [[atmosphere|atmospheric]] [[convection]], a [[closed-form expression]] of the Helmholtz decomposition can be obtained:
<math display="block">\dot \mathbf{r} = \mathbf{F}(\mathbf{r}) = \big[a (r_2-r_1), r_1 (b-r_3)-r_2, r_1 r_2-c r_3 \big].</math>
The Helmholtz decomposition of <math>\mathbf{F}(\mathbf{r})</math>, with the scalar potential <math>\Phi(\mathbf{r}) = \tfrac{a}{2} r_1^2 + \tfrac{1}{2} r_2^2 + \tfrac{c}{2} r_3^2</math> is given as:

<math display="block">\mathbf{G}(\mathbf{r}) = \big[-a r_1, -r_2, -c r_3 \big],</math>
<math display="block">\mathbf{R}(\mathbf{r}) = \big[+ a r_2, b r_1 - r_1 r_3, r_1 r_2 \big].</math>

The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable [[fixed point (mathematics)|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]].<ref name="glotzl2023" /><ref name="peitgen1992" />

=== 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).<ref name="manduca2021" /> 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.<ref name="bhatia2013" /><ref name="bhatia2014" />

== See also ==
{{Portal|Mathematics|Physics}}
* [[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 <math> \nabla \times \mathbf{A} </math>.
* [[Scalar–vector–tensor decomposition]]
* [[Hodge theory]] generalizing Helmholtz decomposition
* [[Polar factorization theorem]]
* ''Helmholtz–Leray decomposition'' used for defining the [[Leray projection]]

== Notes ==
{{Reflist|30em|refs=

<ref name="amrouche1998">Cherif Amrouche, [[Christine Bernardi]], [[Monique Dauge]], [[Vivette Girault]]: ''Vector potentials in three dimensional non-smooth domains''. In: ''[[Mathematical Methods in the Applied Sciences]]'' 21(9), 1998, pp. 823–864, {{doi|10.1002/(sici)1099-1476(199806)21:9<823::aid-mma976>3.0.co;2-b}}, {{bibcode|1998MMAS...21..823A }}.</ref>

<ref name="axler1992">Sheldon Axler, Paul Bourdon, Wade Ramey: ''Bounded Harmonic Functions''. In: ''Harmonic Function Theory'' (= Graduate Texts in Mathematics 137). Springer, New York 1992, pp. 31–44, {{doi|10.1007/0-387-21527-1_2}}.</ref>

<ref name="bhatia2013">Harsh Bhatia, Gregory Norgard, Valerio Pascucci, Peer-Timo Bremer: ''The Helmholtz-Hodge Decomposition – A Survey''. In: ''[[Institute of Electrical and Electronics Engineers|IEEE]] Transactions on Visualization and Computer Graphics'' 19.8, 2013, pp. 1386–1404, {{doi|10.1109/tvcg.2012.316}}.</ref>

<ref name="bhatia2014">Hersh Bhatia, Valerio Pascucci, Peer-Timo Bremer: ''The Natural Helmholtz-Hodge Decomposition for Open-Boundary Flow Analysis''. In: ''[[Institute of Electrical and Electronics Engineers|IEEE]] Transactions on Visualization and Computer Graphics'' 20.11, Nov. 2014, pp. 1566–1578, Nov. 2014, {{doi|10.1109/TVCG.2014.2312012}}.</ref>

<!-- <ref name="bladel1958">Jean Bladel: ''On Helmholtz's Theorem in Finite Regions''. Midwestern Universities Research Association, 1958.</ref> -->

<ref name="blumenthal1905">[[Otto Blumenthal]]: ''Über die Zerlegung unendlicher Vektorfelder''. In: ''[[Mathematische Annalen]]'' 61.2, 1905, pp. 235–250, {{doi|10.1007/BF01457564}}.</ref>

<ref name="cantarella2002">{{cite journal| jstor=2695643| title=Vector Calculus and the Topology of Domains in 3-Space| first1=Jason |last1=Cantarella |first2=Dennis |last2=DeTurck | first3=Herman|last3=Gluck|journal=The American Mathematical Monthly|volume=109|issue=5|year=2002 |pages=409–442 | doi=10.2307/2695643 }}</ref>

<ref name="cauchy1823">{{cite book|author-link =Augustin-Louis Cauchy|first=Augustin-Louis|last = Cauchy|chapter=Trente-Cinquième Leçon|title=Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal|publisher= Imprimerie Royale|location= Paris|date= 1823|pages= 133–140 |url=https://gallica.bnf.fr/ark:/12148/bpt6k62404287/f146.item |language = fr}}</ref>

<ref name="chorin1990">Alexandre J. Chorin, Jerrold E. Marsden: ''A Mathematical Introduction to Fluid Mechanics'' (= Texts in Applied Mathematics 4). Springer US, New York 1990, {{doi|10.1007/978-1-4684-0364-0}}.</ref>

<ref name="dautray1990">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.</ref>

<ref name="edwards1922">Joseph Edwards: ''A Treatise on the Integral Calculus''. Volume 2. Chelsea Publishing Company, 1922.</ref> 

<ref name="gibbs1901">[[J. W. Gibbs]], [[Edwin Bidwell Wilson]]: ''[https://archive.org/stream/117714283#page/236/mode/2up Vector Analysis]''. 1901, p. 237, link from [[Internet Archive]].</ref>

<ref name="girault1986">[[Vivette Girault|V. Girault]], P.A. Raviart: ''Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms.'' Springer Series in Computational Mathematics. Springer-Verlag, 1986.</ref>

<ref name="glotzl2020">Erhard Glötzl, Oliver Richters: ''Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates''. 2020, {{arXiv|2012.13157}}.</ref>

<ref name="glotzl2023">Erhard Glötzl, Oliver Richters: ''Helmholtz decomposition and potential functions for n-dimensional analytic vector fields''. In: ''[[Journal of Mathematical Analysis and Applications]]'' 525(2), 127138, 2023, {{doi|10.1016/j.jmaa.2023.127138}}, {{arXiv|2102.09556v3}}. ''Mathematica'' worksheet at {{doi|10.5281/zenodo.7512798}}.</ref>

<ref name="gregory1996">R. Douglas Gregory: ''Helmholtz's Theorem when the domain is Infinite and when the field has singular points''. In: ''[[The Quarterly Journal of Mechanics and Applied Mathematics]]'' 49.3, 1996, pp. 439–450, {{doi|10.1093/qjmam/49.3.439}}.</ref>

<ref name="griffiths1999">[[David J. Griffiths]]: ''Introduction to Electrodynamics''. Prentice-Hall, 1999, p. 556.</ref>

<ref name="gurtin1962">Morton E. Gurtin: ''On Helmholtz’s theorem and the completeness of the Papkovich-Neuber stress functions for infinite domains''. In: ''[[Archive for Rational Mechanics and Analysis]]'' 9.1, 1962, pp. 225–233, {{doi|10.1007/BF00253346}}.</ref>

<ref name="heaviside1893">[[Oliver Heaviside]]: ''Electromagnetic theory''. Volume 1, "The Electrician" printing and publishing company, limited, 1893.</ref>

<ref name="helmholtz1858">[[Hermann von Helmholtz]]: ''Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen''. In: ''[[Journal für die reine und angewandte Mathematik]]'' 55, 1858, pp. 25–55, {{doi|10.1515/crll.1858.55.25}} ([http://resolver.sub.uni-goettingen.de/purl?GDZPPN002150212 sub.uni-goettingen.de],  [https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002150212 digizeitschriften.de]). On page 38, the components of the fluid's velocity (''u'',&nbsp;''v'',&nbsp;''w'') are expressed in terms of the gradient of a scalar potential P and the curl of a vector potential&nbsp;(''L'',&nbsp;''M'',&nbsp;''N'').</ref>

<ref name="johnson1881">[[William Woolsey Johnson]]: ''An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions''. John Wiley & Sons, 1881.<br />See also: [[Method of Fluxions]].</ref>

<!-- <ref name="koenigsberger1906">[[Leo Koenigsberger]]: ''Hermann von Helmholtz''. Clarendon Press, 1906, p. 357.</ref> -->

<ref name="kustepeli2016">Alp Kustepeli: ''On the Helmholtz Theorem and Its Generalization for Multi-Layers''. In: ''[[Electromagnetics]]'' 36.3, 2016, pp. 135–148, {{doi|10.1080/02726343.2016.1149755}}.</ref>

<ref name="littlejohn">Robert Littlejohn: [http://bohr.physics.berkeley.edu/classes/221/1112/notes/hamclassemf.pdf ''The Classical Electromagnetic Field Hamiltonian'']. Online lecture notes, berkeley.edu.</ref>

<ref name="lorenz1963">[[Edward N. Lorenz]]: ''Deterministic Nonperiodic Flow''. In: ''[[Journal of the Atmospheric Sciences]]'' 20.2, 1963, pp. 130–141, {{doi|10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2}}.</ref>

<ref name="murray1898">[[Daniel Murray (mathematician)|Daniel Alexander Murray]]: ''An Elementary Course in the Integral Calculus''. American Book Company, 1898. p. 8.</ref>

<ref name="peitgen1992">Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe: ''Strange Attractors: The Locus of Chaos''. In: ''Chaos and Fractals''. Springer, New York, pp. 655–768. {{doi|10.1007/978-1-4757-4740-9_13}}.</ref>

<ref name="petrascheck2015">Dietmar Petrascheck: ''The Helmholtz decomposition revisited''. In: ''[[European Journal of Physics]]'' 37.1, 2015, Artikel 015201, {{doi|10.1088/0143-0807/37/1/015201}}.</ref>

<ref name="petrascheck2017">D. Petrascheck, R. Folk: ''Helmholtz decomposition theorem and Blumenthal’s extension by regularization''. In: ''Condensed Matter Physics'' 20(1), 13002, 2017, {{doi|10.5488/CMP.20.13002}}.</ref>

<ref name="shaw1922">James Byrnie Shaw: ''Vector Calculus: With Applications to Physics''. D. Van Nostrand, 1922, p. 205.<br />See also: [[Green's theorem]].</ref>

<ref name="sprossig2009">Wolfgang Sprössig: ''On Helmholtz decompositions and their generalizations – An overview''. In: ''[[Mathematical Methods in the Applied Sciences]]'' 33.4, 2009, pp. 374–383, {{doi|10.1002/mma.1212}}.</ref>

<ref name="stewart2011">A. M. Stewart: ''Longitudinal and transverse components of a vector field''. In: ''Sri Lankan Journal of Physics'' 12, pp. 33–42, 2011, {{doi|10.4038/sljp.v12i0.3504}} {{arxiv|0801.0335}}</ref>

<ref name="stokes1849">[[George Gabriel Stokes]]: ''On the Dynamical Theory of Diffraction''. In: ''Transactions of the [[Cambridge Philosophical Society]]'' 9, 1849, pp. 1–62. {{doi|10.1017/cbo9780511702259.015}}, see pp. 9–10.</ref>

<ref name="suda2019">Tomoharu Suda: ''Construction of Lyapunov functions using Helmholtz–Hodge decomposition''. In: ''Discrete & Continuous Dynamical Systems – A'' 39.5, 2019, pp. 2437–2454, {{doi|10.3934/dcds.2019103}}.</ref>

<ref name="suda2020">Tomoharu Suda: ''Application of Helmholtz–Hodge decomposition to the study of certain vector fields''. In: ''[[Journal of Physics]] A: Mathematical and Theoretical'' 53.37, 2020, pp. 375703. {{doi|10.1088/1751-8121/aba657}}.</ref>

<ref name="trancong1993">Ton Tran-Cong: ''On Helmholtz’s Decomposition Theorem and Poissons’s Equation with an Infinite Domain''. In: ''[[Quarterly of Applied Mathematics]]'' 51.1, 1993, pp. 23–35, {{JSTOR|43637902}}.</ref>

<ref name="vermont">{{cite web |url=http://www.cems.uvm.edu/~oughstun/LectureNotes141/Topic_03_(Helmholtz'%20Theorem).pdf |title=Helmholtz' Theorem |publisher=University of Vermont| access-date=2011-03-11 | archive-url=https://web.archive.org/web/20120813005804/http://www.cems.uvm.edu/~oughstun/LectureNotes141/Topic_03_(Helmholtz'%20Theorem).pdf| archive-date=2012-08-13| url-status=dead}}</ref>

<ref name="warner1983">Frank W. Warner: ''The Hodge Theorem''. In: ''Foundations of Differentiable Manifolds and Lie Groups''. (= Graduate Texts in Mathematics 94). Springer, New York 1983, {{doi|10.1007/978-1-4757-1799-0_6}}.</ref>

<ref name="woolhouse1854">[[Wesley Stoker Barker Woolhouse]]: ''Elements of the differential calculus''. Weale, 1854.</ref>

<ref name="zhou2012">Joseph Xu Zhou, M. D. S. Aliyu, Erik Aurell, Sui Huang: ''Quasi-potential landscape in complex multi-stable systems''. In: ''[[Journal of the Royal Society Interface]]'' 9.77, 2012, pp. 3539–3553, {{doi|10.1098/rsif.2012.0434}}.</ref>

<ref name="manduca2021">Armando Manduca: ''MR elastography: Principles, guidelines, and terminology''. In: ''[[Magnetic Resonance in Medicine]]'', 2021, {{doi|10.1002/mrm.28627}}{{PMID|33296103}}.</ref>

}}

==References==

{{refbegin}}
* [[George B. Arfken]] and Hans J. Weber, ''Mathematical Methods for Physicists'', 4th edition, Academic Press: San Diego (1995) pp.&nbsp;92–93
* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists – International Edition'', 6th edition, Academic Press: San Diego (2005) pp.&nbsp;95–101
* [[Rutherford Aris]], ''Vectors, tensors, and the basic equations of fluid mechanics'', Prentice-Hall (1962), {{oclc|299650765}}, pp.&nbsp;70–72 
{{refend}}

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