Characterization of wave physics using the rigorous Helmholtz decomposition based on the surface integral equation

Helmholtz decomposition (HD) is a fundamental tool of vector calculus and plays an important role in electromagnetics. In this work, arbitrary vector field defined on the open or closed surface is decomposed into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field by using the surface integral equation method. Unlike the popular loop-tree decomposition that is only a quasi-HD suitable for the circuit physics in the low frequency regime, the HD developed in this paper is rigorous and can capture both circuit and wave physics from very low frequency to high frequency regimes. The work could provide insightful physical interpretations for complex electromagnetic phenomena.