Helmholtz decomposition

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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 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.

DefinitionEdit

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 unique.<ref name="glotzl2023"/>

HistoryEdit

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 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 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 spaceEdit

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 <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 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.

DerivationEdit

Template:Math proof'\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 spaceEdit

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 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 curlEdit

The term "Helmholtz theorem" can also refer to the following. Let Template:Math be a solenoidal vector field and d a scalar field on Template:Math which are sufficiently smooth and which vanish faster than Template:Math at infinity. Then there exists a vector field Template:Math 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 Template:Math vanishes as Template:Math, then Template:Math 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 Template:Math, it is defined to act on each component.)

Weak formulationEdit

The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Template:Math is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field Template:Math has an orthogonal decomposition:<ref name="amrouche1998" /><ref name="dautray1990" /><ref name="girault1986" />

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

where Template:Mvar is in the Sobolev space Template:Math of square-integrable functions on Template:Math whose partial derivatives defined in the distribution sense are square integrable, and Template:Math, the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.

For a slightly smoother vector field Template:Math, a similar decomposition holds:

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

where Template:Math.

Derivation from the Fourier transformEdit

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 fieldsEdit

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 dimensionsEdit

Matrix approachEdit

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 balls 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 approachEdit

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 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 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. 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 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.

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

Differential formsEdit

The Hodge decomposition is closely related to the Helmholtz decomposition,<ref name="warner1983"/> 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.<ref name="cantarella2002" /> 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 infinityEdit

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 vector fields that need not go to zero even at infinity, methods based on 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 functions.<ref name="glotzl2023" />

Uniqueness of the solutionEdit

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, 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.

ApplicationsEdit

ElectrodynamicsEdit

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 dynamicsEdit

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 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 theoryEdit

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

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 roboticsEdit

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" />

NotesEdit

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ReferencesEdit

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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), Template:Oclc, pp. 70–72

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