In his lectures on physical potential theory, Lejeune Dirichlet developed a concept later named “Dirichlet's Principle” by Riemann. This concept relates the boundary of a field (its external appearance) and the dynamics of its internal nature. Combined with the surfaces we've covered here, and the topology on the next page, Dirichlet's principle will allow Riemann to treat complex functions from a purely physical, rather than mathematical standpoint. The subject of Dirichlet's principle will be covered by videos. Clicking the labels above the videos will take you to pages including comments, questions, and transcripts.
This is an introductory video on metaphor and Dirichlet's principle, which develops the concept of potential and the relation of the boundary to the interior:
The concept of potential was first developed by Leibniz, who called it “dead force” or vis mortua, in contrast to vis viva, the “living force” of motion. Learn more about that here:
We'll return to Dirichlet's principle after working through connectivity on the next page.