An Introduction to Hodge Theory on Compact Orientable Surfaces with Boundary

This note is a sequel to a previous exposition of Hodge theory, aiming to provide a non-mathematician- friendly introduction to Hodge theory on surfaces with boundary. The central ingredients include five versions of the Helmholtz-Hodge decompositions. It also discusses the finite-dimensional spaces of harmonic vector fields with boundary conditions, whose dimensionalities equal the first Betti number of the underlying surface. Then, the Helmholtz-Hodge decompositions are used to solve differential equations on surfaces, especially the Cauchy-Stokes system. Finally, illustrative examples in physcis are presented.