On the Helmholtz Theorem and Its Generalization for Multi-Layers

Abstract

The decomposition of a vector field to its curl-free and divergence-free components in terms of a scalar and a vector potential function, which is also considered as the fundamental theorem of vector analysis, is known as the Helmholtz theorem or decomposition. In the literature, it is mentioned that the theorem was previously presented by Stokes, but it is also mentioned that Stokes did not introduce any scalar and vector potentials in his expressions, which causes a contradiction. Therefore, in this article, Stokess and Helmholtzs representations are examined in detail to reveal and emphasize their differences, similarities and important points. The Helmholtz theorem is obtained for all kinds of spaces by using the theory of distributions in a comprehensive and rigorous manner with detailed explanations, which also leads to a new surface version of the Helmholtz theorem or a new surface decomposition, resulting in the canonical form; hence, it is different than the one suggested previously in terms of two scalar functions. The generalized form of the Helmholtz theorem is also presented by employing the same approach when there is a multi-layer on the surface of discontinuity, which also corresponds to the extension of the theorem to fields with singularities of higher order.