Hodge Theory via Orientable Closed Surfaces

Hodge theory is a powerful framework for modeling and analyzing vector fields, as found in fluid and electro- dynamics. However, its full power and modern formulation, especially on manifolds, remain largely inaccessible to non-specialists due to the substantial prerequisite in modern mathematics. This article aims to fill this gap by providing a comprehensive, self-contained introduction to Hodge theory, specifically tailored for an audience versed in traditional vector/tensor calculus and seeking to understand the modern formulation of this theory. We choose orientable closed surfaces as our pedagogical setting, due to their conceptual simplicity, mathematical tractability, and physical relevance (as boundaries of three-dimensional regions). Crucially, this setting is perfect for elucidating the complete structure of Hodge theory, particularly its topological and geometric aspects---elements frequently absent or obscured in treatments rooted in classical physics. We accomplish this through a parallel development of the modern exterior calculus and a dedicated 2D vector calculus on surfaces, followed by a series of specifically designed analytical and numerical examples. Furthermore, recognizing the fragmented historical development of the subject---a primary source of the conceptual gap for modern readers---we include a concise historical synopsis to bridge this divide.