gives We can therefore write where Note that the vector potential is replaced by a rank- tensor in dimensions. For a further generalization to manifolds, 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 is not defined on a bounded domain, then shall decay faster than . Thus, the Fourier Transform of , denoted as , 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 represents the Newtonian potential operator. (When acting on a vector field, such as , it is defined to act on each component.) Solution space For two Helmholtz decompositions of , there holds where is an harmonic scalar field, is a vector field determined by , is any scalar field. Proof: Setting and , one has, according to the definition of the Helmholtz decomposition, . Taking the divergence of each member of this equation yields , hence is harmonic. Conversely, given any harmonic function , is solenoidal since Thus, according to the above section, there exists a vector field such that . If is another such vector field, then fulfills , hence for some scalar field (and conversely). 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