ABSTRACT
This chapter outlines the applications in the two—dimensional setting of Laplace's equation, discusses the invariance of Laplace's equation under translations and rotations of coordinates, and introduces the two basic boundary—value problems for Laplace's equation, namely the Dirichlet and Neumann problems. It expresses Laplace's equation in terms of polar coordinates, in order to obtain the mean—value theorem for harmonic functions and solve the Dirichlet problem for the annular region between concentric circles. The chapter utilizes the close relationship between Laplace's equation and complex variable theory to solve problems of two—dimensional ideal fluid flow, steady—state temperatures and electrostatics. It includes a few brief biographical sketches of the key individuals who contributed to the early development of potential theory. Uniqueness of solutions of the Dirichlet problem for such regions follows at once from the Maximum/Minimum Principle, since the maximum and minimum of the difference of two solutions occurs on the boundary, where the difference is zero.
