Editing Helmholtz decomposition (section)

== Differential forms ==
The [[Hodge decomposition#Hodge decomposition|Hodge decomposition]] is closely related to the Helmholtz decomposition,<ref name="warner1983"/> generalizing from vector fields on '''R'''<sup>3</sup> to [[differential forms]] on a [[Riemannian manifold]] ''M''.  Most formulations of the Hodge decomposition require ''M'' to be [[compact space|compact]].<ref name="cantarella2002" /> Since this is not true of '''R'''<sup>3</sup>, 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.