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 or Helmholtz representation. It is named after Hermann von Helmholtz.
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 formula_1, where formula_2 is a scalar field called "scalar potential", and is a vector field, called a vector potential.
Statement of the theorem.
Let formula_3 be a vector field on a bounded domain formula_4, which is twice continuously differentiable, and let formula_5 be the surface that encloses the domain formula_6. Then formula_3 can be decomposed into a curl-free component and a divergence-free component:
where
and formula_10 is the nabla operator with respect to formula_11, not formula_12 .
If formula_13 and is therefore unbounded, and formula_3 vanishes faster than formula_15 as formula_16, then one has
Derivation.
Suppose we have a vector function formula_18 of which we know the curl, formula_19, and the divergence, formula_20, in the domain and the fields on the boundary. Writing the function using delta function in the form
where formula_22 is the Laplace operator, we have
where we have used the definition of the vector Laplacian:
differentiation/integration with respect to formula_25by formula_26 and in the last line, linearity of function arguments:
Then using the vectorial identities
we get
Thanks to the divergence theorem the equation can be rewritten as
with outward surface normal formula_31.
Defining
we finally obtain
formula_35 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 formula_36. 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 formula_3 is not defined on a bounded domain, then formula_3 shall decay faster than formula_15. Thus, the Fourier Transform of formula_3, denoted as formula_41, is guaranteed to exist. We apply the convention
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:
Hence
Fields with prescribed divergence and curl.
The term "Helmholtz theorem" can also refer to the following. Let be a solenoidal vector field and "d" a scalar field on which are sufficiently smooth and which vanish faster than at infinity. Then there exists a vector field such that
if additionally the vector field vanishes as , then 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
where formula_47 represents the Newtonian potential operator. (When acting on a vector field, such as , 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. 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 is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field has an orthogonal decomposition:
where is in the Sobolev space of square-integrable functions on whose partial derivatives defined in the distribution sense are square integrable, and , the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.
For a slightly smoother vector field , a similar decomposition holds:
where .
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 formula_50 of the vector field formula_3. 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
Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:
Since formula_58 and formula_59,
we can get
so this is indeed the Helmholtz decomposition.