Helmholtz decomposition

In vector analysis, the Helmholtz decomposition of a vector field on $\scriptstyle \mathbb {R} ^{3}$ is the writing of the vector field as a sum of two vector fields, one a divergence-free field and one a curl-free field. The decomposition is called after the German physiologist and physicist Hermann von Helmholtz (1821 – 1894).

Mathematical formulation

The Helmholtz decomposition may be formulated as follows. Any vector field F(r) that is sufficiently often differentiable and vanishes at infinity can be written as,

$\mathbf {F} ={\boldsymbol {\nabla }}\times \mathbf {A} -{\boldsymbol {\nabla }}\Phi$

with

${\begin{aligned}\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '\\\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int {\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '\\\end{aligned}}$

The primed nabla operator ∇' acts on primed coordinates and the unprimed ∇ acts on unprimed coordinates.

Note that

${\boldsymbol {\nabla }}\cdot ({\boldsymbol {\nabla }}\times \mathbf {V} )=0\quad {\hbox{and}}\quad {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\Psi )=0$

holds for any vector field V(r) and any scalar function Ψ(r). Hence it follows that the first term of F is divergence-free and the second curl-free.

As a corollary follows that the specification of both the divergence and the curl of a vector field at all points of space gives the field uniquely.

Decomposition in transverse and longitudinal components

A vector field F(r) with $\scriptstyle \mathbf {r} \in \mathbb {R} ^{3}$ can be decomposed in a transverse $\scriptstyle \mathbf {F} _{\perp }(\mathbf {r} )$ and longitudinal component $\scriptstyle \mathbf {F} _{\parallel }(\mathbf {r} )$:

$\mathbf {F} (\mathbf {r} )=\mathbf {F} _{\perp }(\mathbf {r} )+\mathbf {F} _{\parallel }(\mathbf {r} ),$

where

${\boldsymbol {\nabla }}\cdot \mathbf {F} _{\perp }(\mathbf {r} )=0,\qquad {\boldsymbol {\nabla }}\times \mathbf {F} _{\parallel }(\mathbf {r} )=\mathbf {0} .$

Thus, the arbitrary field F(r) can be decomposed in a part that is divergence-free, the transverse component, and a part that is curl-free, the longitudinal component.

Proof of decomposition

The decomposition is formulated in r-space. By a Fourier transformation the decomposition may be formulated in k-space. This is advantageous because differentiations in r-space become multiplications in k-space. We will show that divergence in r-space becomes an inner product in k-space and a curl becomes a cross product. Thus, we define the mutually inverse Fourier transforms,

${\begin{aligned}{\tilde {\mathbf {F} }}(\mathbf {k} )&={\frac {1}{(2\pi )^{3/2}}}\int e^{-i\mathbf {k} \cdot \mathbf {r} }\,\mathbf {F} (\mathbf {r} )\,d^{3}\mathbf {r} \\\mathbf {F} (\mathbf {r} )&={\frac {1}{(2\pi )^{3/2}}}\int e^{i\mathbf {k} \cdot \mathbf {r} }\,{\tilde {\mathbf {F} }}(\mathbf {k} )\,d^{3}\mathbf {r} \\\end{aligned}}$

An arbitrary vector field in k-space can be decomposed in components parallel and perpendicular to k,

${\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} )\equiv {\hat {\mathbf {k} }}{\big (}{\hat {\mathbf {k} }}\cdot {\tilde {\mathbf {F} }}(\mathbf {k} ){\big )},\qquad {\hbox{with}}\qquad {\hat {\mathbf {k} }}\equiv {\frac {\mathbf {k} }{|\mathbf {k} |}},$

${\tilde {\mathbf {F} }}_{\perp }(\mathbf {k} )\equiv {\tilde {\mathbf {F} }}(\mathbf {k} )-{\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} ),$

so that

${\tilde {\mathbf {F} }}(\mathbf {k} )={\tilde {\mathbf {F} }}_{\perp }(\mathbf {k} )+{\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} ).$

Clearly,

$\mathbf {k} \cdot {\tilde {\mathbf {F} }}_{\perp }(\mathbf {k} )=0\qquad {\hbox{and}}\qquad \mathbf {k} \times {\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} )=0.$

Transforming back, we get

$\mathbf {F} _{\perp }(\mathbf {r} )\equiv {\frac {1}{(2\pi )^{3/2}}}\int e^{i\mathbf {k} \cdot \mathbf {r} }\,{\tilde {\mathbf {F} }}_{\perp }(\mathbf {k} )\,d^{3}\mathbf {k} ,\qquad \mathbf {F} _{\parallel }(\mathbf {r} )\equiv {\frac {1}{(2\pi )^{3/2}}}\int e^{i\mathbf {k} \cdot \mathbf {r} }\,{\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} )\,d^{3}\mathbf {k} ,$

which satisfy the properties

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {F} _{\perp }(\mathbf {r} )&={\frac {1}{(2\pi )^{3/2}}}\int e^{i\mathbf {k} \cdot \mathbf {r} }\,\mathbf {k} \cdot {\tilde {\mathbf {F} }}_{\perp }(\mathbf {k} )\,d^{3}\mathbf {k} =0\\{\boldsymbol {\nabla }}\times \mathbf {F} _{\parallel }(\mathbf {r} )&={\frac {1}{(2\pi )^{3/2}}}\int e^{i\mathbf {k} \cdot \mathbf {r} }\,\mathbf {k} \times {\tilde {\mathbf {F} }}_{\parallel }(\mathbf {k} )\,d^{3}\mathbf {k} =0.\end{aligned}}$

Hence we have found the required decomposition.

Integral expressions for the transverse and longitudinal components

We assume that the curl and the divergence of the vector field F(r)

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )={\boldsymbol {\nabla }}\times \mathbf {F} _{\perp }(\mathbf {r} )\quad {\hbox{and}}\quad {\boldsymbol {\nabla }}\cdot \mathbf {F} (\mathbf {r} )={\boldsymbol {\nabla }}\cdot \mathbf {F} _{\parallel }(\mathbf {r} )$

are given. Then

${\begin{aligned}\mathbf {F} _{\perp }(\mathbf {r} )&={\frac {1}{4\pi }}{\boldsymbol {\nabla }}\times \int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '\qquad \qquad \qquad (1)\\\mathbf {F} _{\parallel }(\mathbf {r} )&=-{\frac {1}{4\pi }}{\boldsymbol {\nabla }}\int {\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} _{\parallel }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '\qquad \qquad \qquad \quad (2)\\\end{aligned}}$

are, respectively, the perpendicular (transverse, divergence-free) and parallel (longitudinal, curl-free) components. The operator ∇ acts on unprimed coordinates and ∇' acts on primed coordinates. Note that the two components of F(r) are indeed uniquely determined once the curl and the divergence of F(r) are known.

Proof of integral expressions

We will confirm the integral forms, equations (1) and (2), of the components. We will show that they lead to identities.

For the perpendicular (transverse) component we note that for any vector A,

${\boldsymbol {\nabla }}\times {\big (}{\boldsymbol {\nabla }}\times \mathbf {A} {\big )}={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$

and insert this in

${\begin{aligned}{\boldsymbol {\nabla }}\times \mathbf {F} _{\perp }(\mathbf {r} )&={\frac {1}{4\pi }}{\boldsymbol {\nabla }}\times {\Big (}{\boldsymbol {\nabla }}\times \int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '{\Big )}\\&=-{\frac {1}{4\pi }}\nabla ^{2}\int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '+{\frac {1}{4\pi }}{\boldsymbol {\nabla }}{\Big (}{\boldsymbol {\nabla }}\cdot \int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '{\Big )}.\end{aligned}}$

Below we will show that second term vanishes. Use for the first term the following equation for the Dirac delta function,

$\nabla ^{2}{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}=-4\pi \delta (\mathbf {r} -\mathbf {r} ')$

Hence the first term becomes (note that the unprimed nabla may be moved under the integral)

${\begin{aligned}&-{\frac {1}{4\pi }}\int {\Big (}{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} '){\Big )}\nabla ^{2}{\Big (}{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}{\Big )}d^{3}\mathbf {r} '=\int {\Big (}{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} '){\Big )}\delta (\mathbf {r} -\mathbf {r} ')d^{3}\mathbf {r} '\\&={\boldsymbol {\nabla }}\times \mathbf {F} _{\perp }(\mathbf {r} ),\end{aligned}}$

so that we indeed end up with an identity.

Before turning to the parallel (longitudinal) term we prove that the second term vanishes. To that end we introduce a shorthand notation

${\boldsymbol {\nabla }}\cdot \int {\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '\equiv \sum _{\alpha =x,y,z}\nabla _{\alpha }\int {\frac {G_{\alpha }(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '.$

Move the divergence under the integral and use

$\nabla _{\alpha }{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}=-\nabla _{\alpha }'{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}.$

By partial integration and using that the integrand vanishes for the integral limits, we can let −∇'_α act on G_α(r' ) (this trick is known as the turnover rule for the anti Hermitian operator ∇'_α). Then from

${\boldsymbol {\nabla }}'\cdot \mathbf {G} (\mathbf {r} ')\equiv {\boldsymbol {\nabla }}'\cdot {\big (}{\boldsymbol {\nabla }}'\times \mathbf {F} _{\perp }(\mathbf {r} '){\big )}=0,$

(because the divergence of the curl of any vector is zero) follows the vanishing of the second term.

(To be continued)