Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition. It is named for Hermann von Helmholtz.

This implies that any such vector field F can be considered to be generated by a pair of potentials: a scalar potential φ and a vector potential A.

Statement of the theorem

Let F be a vector field on R³, which is twice continuously differentiable and which, together with its divergence and curl, vanishes faster than 1/r² at infinity. Then F is a sum of a gradient and a curl as follows:

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

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

If F has zero divergence, ∇·F = 0, then F is called solenoidal or divergence-free, and the Helmholtz decomposition of F collapses to

$\mathbf {F} =\nabla \times {\mathcal {G}}(\nabla \times \mathbf {F} )=\nabla \times \mathbf {A} .$

In this case, A is known as a vector potential for F. This particular choice of vector potential is divergence-free, which in physics is referred to as the Coulomb gauge condition.

Likewise, if F has zero curl, ∇×F = 0, then F is called irrotational or curl-free, and the Helmholtz decomposition of F collapses to

$\mathbf {F} =-\nabla \,{\mathcal {G}}(\nabla \cdot \mathbf {F} )=-\nabla \varphi .$

In this case, φ is known as a scalar potential for F.

In general F is the sum of these two terms,

$\mathbf {F} =-\nabla \varphi +\nabla \times \mathbf {A}$

where the negative gradient of the scalar potential is the irrotational component, and the curl of the vector potential is the solenoidal component.

Fields with prescribed divergence and curl

The term "Helmholtz Theorem" can also refer to the following. Let C be a 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

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

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

In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type. The proof is by a construction generalizing the one given above: we set

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

Differential forms

The Hodge decomposition is closely related to the Helmholtz decomposition, generalizing from vector fields on R³ to differential forms on a Riemannian manifold M. Most formulations of the Hodge decomposition require M to be compact. Since this is not true of R³, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem.

Weak formulation

The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L²(Ω))³ has an orthogonal decomposition:

$\mathbf {u} =\nabla \varphi +\mathrm {curl} \,\mathbf {A}$

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 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 u ∈ H(curl,Ω), a similar decomposition holds:

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

where φ ∈ H¹(Ω) and 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. This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field F, which we call ${\tilde {\mathbf {F} }}$. 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

${\tilde {\mathbf {F} }}(\mathbf {k} )={\tilde {\mathbf {F} }}_{l}(\mathbf {k} )+{\tilde {\mathbf {F} }}_{t}(\mathbf {k} )$

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

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

$\mathbf {F} =\mathbf {F} _{t}+\mathbf {F} _{l}$

$\nabla \cdot \mathbf {F} _{t}=\nabla \times \mathbf {F} _{l}=0$

so this is indeed the Helmholtz decomposition.

Integral formulas

If the vector field is smooth and is rapidly decreasing along with its first and second derivatives, then the vector fields F_l and F_t appearing in the Helmholtz decomposition

$\mathbf {F} \left(\mathbf {x} \right)=\mathbf {F} _{l}\left(\mathbf {x} \right)+\mathbf {F} _{t}\left(\mathbf {x} \right)$

can be expressed as the integrals

${\begin{aligned}\mathbf {F} _{l}(\mathbf {x} )&=-{1 \over 4\pi }{\nabla }\left({\nabla }\cdot \int {\frac {\mathbf {F} (\mathbf {x} ')\,d^{3}x'}{\mid \mathbf {x} -\mathbf {x} ^{\prime }\mid }}\right)\\&=-{1 \over 4\pi }\nabla \int {\frac {\nabla ^{\prime }\cdot \mathbf {F} (\mathbf {x} ^{\prime })}{\mid \mathbf {x} -\mathbf {x} ^{\prime }\mid }}d^{3}x^{\prime }\end{aligned}}$

and

$\mathbf {F} _{t}\left(\mathbf {x} \right)={1 \over 4\pi }{\nabla }\times {\nabla }\times \int {\mathbf {F} \left(\mathbf {x} ^{\prime }\right)\,d^{3}x^{\prime } \over \mid \mathbf {x} -\mathbf {x} ^{\prime }\mid }.$

The curl of the first vector field is zero, since it is a gradient, and the divergence of the second vector field is zero, because it is a curl.

The integral decomposition is a consequence of the expression of the Newtonian potential of F. This is the rapidly decreasing vector field W such that

$\nabla ^{2}\mathbf {W} =-\mathbf {F} .$

The potential can be expressed as the integral

$\mathbf {W} (\mathbf {x} )={\frac {1}{4\pi }}\int {\frac {\mathbf {F} (\mathbf {x} ')\,d^{3}x'}{|\mathbf {x} -\mathbf {x} '|}}.$

The vector identity

$-\nabla ^{2}\mathbf {W} =-\nabla \left(\nabla \cdot \mathbf {W} \right)+\nabla \times \nabla \times \mathbf {W}$

is then precisely the Helmholtz decomposition, for the left-hand side is F, and the right-hand side is F_l + F_t.

See also

• Darwin Lagrangian for an application

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

References for the weak formulation

• C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. "Vector potentials in three dimensional non-smooth domains." Mathematical Methods in the Applied Sciences, 21, 823–864, 1998.
• R. Dautray and J.-L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990.
• V. Girault and P.A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.

External links

• Helmholtz theorem on MathWorld