Generalized Helmholtz theorem for multi-forms

Abstract

The classical theory of Helmholtz is extended to the case of one- and two-forms in the space of two, three and four dimensions. We show that the Helmholtz decomposition into the scalar, vector and tensor potential is invariant with respect to a transformation which is a generalization of the classical Abelian gauge transformation. Applications of the theorem to the ideal compressible fluids, classical electrodynamics and continuum dislocation theory indicate that there exists a possibility of participation of the constitutive equations in the decomposition of multi-forms.