Physical Applications of Harmonic Functions

An earnest effort has been made to present this tutorial paper with an emphasis on analytical, logical and intuitive thinking; geometric reasoning; and physical description in a way that is more congenial and receptive to the readers, especially engineering students, to fully comprehend the practical power of harmonic functions in solving physical problems in physics and engineering.Mathematically, all these physical problems can be formulated in terms of Laplace's equation.Furthermore, harmonic functions are solutions of Laplace's equation and have continuous second partial derivatives.This paper commences to give a brief introduction to harmonic functions and proceeds to delve into the general properties of harmonic functions, particularly important are mean value property and maximum modulus property.It then moves on to examine the connection between Laplace's equation and complex analytical functions and illustrate it by working out examples from electrostatics and fluid flow.These examples reveal the unifying power of mathematics: physical problems from electrostatics and fluid flow can be treated by the same mathematical methods.There exists an analogy between them: electrostatic equipotential lines and electrostatic lines of force correspond to the equipotential lines of the velocity potential and the streamlines of fluid flow respectively.As harmonic functions are the real and imaginary parts of complex analytical function, they remain harmonic under conformal mapping so that conformal mapping becomes a powerful tool in solving boundary value problems.Consequently, conformal mapping can be effectively and efficiently used to solve problems by mapping a given domain onto one for which the solution of the given problem is known or can be solved more easily.The solution thus obtained is then mapped back to the given domain.