A differential geometric approach to the Helmholtz equations and an application to a space with a nontrivial Riemannian metric

The necessary and sufficient conditions for variationality are obtained fro m the requirement that a differential two-form be closed. The classical Hel mholtz equations are shown to follow from these equations. An application o f these results to the case in which one of the functions in these equation s is taken to be a Riemannian metric on a curved space is presented.