Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem,[1][2] also known as the fundamental theorem of vector calculus,[3][4][5][6][7][8][9] 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.[10]

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 ${\displaystyle -\operatorname {grad} \Phi +\operatorname {curl} \mathbf {A} }$, where Φ is a scalar field, called scalar potential, and A is a vector field, called a vector potential.

Statement of the theorem

Let ${\displaystyle \mathbf {F} }$  be a vector field on a bounded domain ${\displaystyle V\subseteq \mathbb {R} ^{3}}$ , which is twice continuously differentiable, and let ${\displaystyle S}$  be the surface that encloses the domain ${\displaystyle V}$ . Then ${\displaystyle \mathbf {F} }$  can be decomposed into a curl-free component and a divergence-free component:[11][12]

${\displaystyle \mathbf {F} =-\nabla \Phi +\nabla \times \mathbf {A} ,}$

where

{\displaystyle {\begin{aligned}\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{aligned}}}

and ${\displaystyle \nabla '}$  is the nabla operator with respect to ${\displaystyle \mathbf {r'} }$ , not ${\displaystyle \mathbf {r} }$  .

If ${\displaystyle V=\mathbb {R} ^{3}}$  and is therefore unbounded, and ${\displaystyle \mathbf {F} }$  vanishes faster than ${\displaystyle 1/r}$  as ${\displaystyle r\to +\infty }$ , then the second component of both scalar and vector potential are zero. That is,[13]

{\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\text{all space}}{\frac {\nabla '\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\\[8pt]\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\text{all space}}{\frac {\nabla '\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\end{aligned}}}

Derivation

Suppose we have a vector function ${\displaystyle \mathbf {F} (\mathbf {r} )}$  of which we know the curl, ${\displaystyle \nabla \times \mathbf {F} }$ , and the divergence, ${\displaystyle \nabla \cdot \mathbf {F} }$ , in the domain and the fields on the boundary. Writing the function using delta function in the form

${\displaystyle \delta ^{3}(\mathbf {r} -\mathbf {r} ')=-{\frac {1}{4\pi }}\nabla ^{2}{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\ ,}$

where ${\displaystyle \nabla ^{2}:=\nabla \cdot \nabla }$ is the Laplace operator, we have

{\displaystyle {\begin{aligned}\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'\\&=-{\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{aligned}}}

where we have used the definition of the vector Laplacian:

${\displaystyle \nabla ^{2}\mathbf {a} =\nabla (\nabla \cdot \mathbf {a} )-\nabla \times (\nabla \times \mathbf {a} )\ ,}$

derivation/integration with respect to ${\displaystyle \mathbf {r} '}$ by ${\displaystyle \nabla '/\mathrm {d} V',}$ and in the last line, linearity of function arguments:

${\displaystyle \nabla {\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}=-\nabla '{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\ .}$

Then using the vectorial identities

{\displaystyle {\begin{aligned}\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{aligned}}}

we get

{\displaystyle {\begin{aligned}\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{aligned}}}

Thanks to the divergence theorem the equation can be rewritten as

{\displaystyle {\begin{aligned}\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{aligned}}}

with outward surface normal ${\displaystyle \mathbf {\hat {n}} '}$ .

Defining

${\displaystyle \Phi (\mathbf {r} )\equiv {\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}$
${\displaystyle \mathbf {A} (\mathbf {r} )\equiv {\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}$

we finally obtain

${\displaystyle \mathbf {F} =-\nabla \Phi +\nabla \times \mathbf {A} .}$

${\displaystyle -{\frac {1}{4\pi \left|\mathbf {r} -\mathbf {r} '\right|}}}$  is the Green's function for the Laplacian, and in a more general setting it should be replaced by the appropriate Green's function - for example, in two dimensions it should be replaced by ${\displaystyle {\frac {1}{2\pi }}\ln \left|\mathbf {r} -\mathbf {r} '\right|}$ . For higher dimensional generalization, 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 ${\displaystyle \mathbf {F} }$  is not defined on a bounded domain, then ${\displaystyle \mathbf {F} }$  shall decay faster than ${\displaystyle 1/r}$ . Thus, the Fourier Transform of ${\displaystyle \mathbf {F} }$ , denoted as ${\displaystyle \mathbf {G} }$ , is guaranteed to exist. We apply the convention

${\displaystyle \mathbf {F} (\mathbf {r} )=\iiint \mathbf {G} (\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}}$

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:

{\displaystyle {\begin{aligned}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{aligned}}}

Hence

{\displaystyle {\begin{aligned}\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{aligned}}}

Fields with prescribed divergence and curl

The term "Helmholtz theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R3 which are sufficiently smooth and which vanish faster than 1/r2 at infinity. Then there exists a vector field F such that

${\displaystyle \nabla \cdot \mathbf {F} =d\quad {\text{ and }}\quad \nabla \times \mathbf {F} =\mathbf {C} ;}$

if additionally the vector field F vanishes as r → ∞, then F is unique.[13]

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.[13] The proof is by a construction generalizing the one given above: we set

${\displaystyle \mathbf {F} =-\nabla ({\mathcal {G}}(d))+\nabla \times ({\mathcal {G}}(\mathbf {C} )),}$

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

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.[14] 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 Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L2(Ω))3 has an orthogonal decomposition:

${\displaystyle \mathbf {u} =\nabla \varphi +\nabla \times \mathbf {A} }$

where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and AH(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.

For a slightly smoother vector field uH(curl, Ω), a similar decomposition holds:

${\displaystyle \mathbf {u} =\nabla \varphi +\mathbf {v} }$

where φH1(Ω), v ∈ (H1(Ω))d.

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.[15] This terminology comes from the following construction: Compute the three-dimensional Fourier transform ${\displaystyle {\hat {\mathbf {F} }}}$  of the vector field ${\displaystyle \mathbf {F} }$ . 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

${\displaystyle {\hat {\mathbf {F} }}(\mathbf {k} )={\hat {\mathbf {F} }}_{t}(\mathbf {k} )+{\hat {\mathbf {F} }}_{l}(\mathbf {k} )}$
${\displaystyle \mathbf {k} \cdot {\hat {\mathbf {F} }}_{t}(\mathbf {k} )=0.}$
${\displaystyle \mathbf {k} \times {\hat {\mathbf {F} }}_{l}(\mathbf {k} )=\mathbf {0} .}$

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

${\displaystyle \mathbf {F} (\mathbf {r} )=\mathbf {F} _{t}(\mathbf {r} )+\mathbf {F} _{l}(\mathbf {r} )}$
${\displaystyle \nabla \cdot \mathbf {F} _{t}(\mathbf {r} )=0}$
${\displaystyle \nabla \times \mathbf {F} _{l}(\mathbf {r} )=\mathbf {0} }$

Since ${\displaystyle \nabla \times (\nabla \Phi )=0}$  and ${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$ ,

we can get

${\displaystyle \mathbf {F} _{t}=\nabla \times \mathbf {A} ={\frac {1}{4\pi }}\nabla \times \int _{V}{\frac {\nabla '\times \mathbf {F} }{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'}$
${\displaystyle \mathbf {F} _{l}=-\nabla \Phi =-{\frac {1}{4\pi }}\nabla \int _{V}{\frac {\nabla '\cdot \mathbf {F} }{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'}$

so this is indeed the Helmholtz decomposition.[16]

Notes

1. ^ On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958.
2. ^ Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357
3. ^ An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8.
4. ^ J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, page 237, link from Internet Archive
5. ^ Electromagnetic theory, Volume 1. By Oliver Heaviside. "The Electrician" printing and publishing company, limited, 1893.
6. ^ Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854.
7. ^ An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson. John Wiley & Sons, 1881.
8. ^ Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
9. ^ A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.
10. ^ See:
• H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik, 55: 25–55. On page 38, the components of the fluid's velocity (u, v, w) are expressed in terms of the gradient of a scalar potential P and the curl of a vector potential (LMN).
• However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society, vol. 9, part I, pages 1–62; see pages 9–10.
11. ^ Aris (1962), pp. 70–72
12. ^ "Helmholtz' Theorem" (PDF). University of Vermont.
13. ^ a b c David J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, 1999, p. 556.
14. ^ Cantarella, Jason; DeTurck, Dennis; Gluck, Herman (2002). "Vector Calculus and the Topology of Domains in 3-Space". The American Mathematical Monthly. 109 (5): 409–442. doi:10.2307/2695643. JSTOR 2695643.
15. ^ Stewart, A. M.; Longitudinal and transverse components of a vector field, Sri Lankan Journal of Physics 12, 33-42 (2011)
16. ^ Online lecture notes by Robert Littlejohn

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), OCLC 299650765, pp. 70–72

References for the weak formulation

• Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V. (1998). "Vector potentials in three dimensional non-smooth domains". Mathematical Methods in the Applied Sciences. 21: 823–864. Bibcode:1998MMAS...21..823A. doi:10.1002/(sici)1099-1476(199806)21:9<823::aid-mma976>3.0.co;2-b.
• 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.