270 Chapter 9Functions of several variables
wheref 1 = 1 f(x, y)is a function of the cartesian coordinates of a point in a plane. The
Laplace equation occurs in many branches of the physical sciences, and is the
fundamental equation in ‘potential theory’, when a physical system is described in
terms of a potential function; for example, the gravitational and electrostatic potential
functions in a region free of matter satisfy the Laplace equation in three dimensions
(see Chapter 10). The equation in two dimensions is important in flow theories; for
example in the theory of fluid flow and of heat conduction.
5As in Example 9.17, the position of a point in a plane can be specified in terms
of the polar coordinates rand θ, wherex 1 = 1 r 1 cos 1 θandy 1 = 1 r 1 sin 1 θ. The functionf
can therefore be treated as a function of rand θ,f 1 = 1 f(r, θ), and equation (9.37) can
be transformed from an equation in cartesian coordinates to an equation in polar
coordinates by the method described in Example 9.17. Example 9.18 shows how this
is done, with result
The differential operator
(9.38)
(read as ‘del-squared’ or ‘nabla-squared’) is called the Laplacian operator(although
the symbol and the name are usually reserved for the three-dimensional form; see
Chapter 10). The Laplace equation is then
∇
2f 1 = 10 (9.39)
and a solution fof the equation as known as a harmonic function.
EXAMPLE 9.18The two-dimensional Laplacian in polar coordinates
By the first of equations (9.36) in Example 9.17,
∂
∂
=
∂
∂
−
∂
∂
f
x
f
rr
f
cos
sin
θ
θ
θ
=
∂
∂
∂
∂
∂
∂
2222211
r
rr
r θ∇=
∂
∂
∂
∂
22222xy
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
=
22222222211
0
f
x
f
y
f
r
r
f
r
r
f
θ
5Pierre Simon de Laplace (1749–1827). His Traité de mécanique céleste(Treatise on celestial mechanics) in 5
volumes (1799–1825) marked the culmination of the Newtonian view of gravitation. Legend has it that whilst at
the École Militaire, where he taught elementary mathematics to the cadets, he examined, and passed, Napoleon in