Advanced Engineering Mathematics, 7th Edition

20.6 Applications

INTRODUCTION

In Sections 20.2, 20.3, and 20.5 we demonstrated how Laplace’s partial differential equation can be solved with conformal mapping methods, and we interpreted a solution u = u(x, y) of the Dirichlet problem as either the steady-state temperature at the point (x, y) or the equilibrium displacement of a membrane at the point (x, y). Laplace’s equation is a fundamental partial differential equation that arises in a variety of contexts. In this section we will establish a general relationship between vector fields and analytic functions and use our conformal mapping techniques to solve problems involving electrostatic force fields and two-dimensional fluid flows.

Vector Fields

A vector field F(x, y) = P(x, y)i +