The Potential

Equation CHAPTER

4

4.1 Potential Equation

The equation for the steady-state temperature distribution in two dimensions
(see Chapter 5) is

∂^2 u

∂x^2

+∂

(^2) u
∂y^2

= 0.

The same equation describes the equilibrium (time-independent) displace-
ments of a two-dimensional membrane, and so is an important common part
of both the heat and wave equations in two dimensions. Many other physical
phenomena — gravitational and electrostatic potentials, certain fluid flows —
and an important class of functions are described by this equation, thus mak-
ing it one of the most important of mathematics, physics, and engineering.
The analogous equation in three dimensions is

∂^2 u

∂x^2 +

∂^2 u

∂y^2 +

∂^2 u

∂z^2 =^0.

Either equation may be written∇^2 u=0 and is commonly called thepotential
equationorLaplace’s equation.
The solutions of the potential equation (calledharmonic functions)have
many interesting properties. An important one, which can be understood in-
tuitively, is the maximum principle: If∇^2 u=0inaregion,thenucannot
have a relative maximum or minimum inside the region unlessuis constant.
(Thus, if∂u/∂xand∂u/∂yarebothzeroatsomepoint,itisasaddlepoint.)
Ifuis thought of as the steady-state temperature distribution in a metal plate,

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