Helmholtz Theorem for Differential Forms in 3-D Euclidean Space

Abstract

There are significant differences between Helmholtz and Hodge's decomposition theorems, but both share a common flavor. This paper is a first step to bring them together.We here produce Helmholtz theorems for differential 1−forms and 2−forms in 3-D Euclidean space, E 3 . We emphasize their common structure in order to facilitate the understanding of another paper, soon to be made public, where a Helmholtz theorem for arbitrary differential forms in arbitrary Euclidean space is presented and which allows one to connect (actually to derive from it) an improvement of Hodge's decomposition theorem.