In various branches of science and engineering, the quantities φand ψhave many
different physical interpretations. For example, in fluid dynamics, the velocity field
is derivable from a velocity potential φ, and the field lines are called streamlines. In
the study of heat flow, the heat flow vector is derivable from a potential φwhich rep-
resents temperature and the equipotential curves φ=Constant are called isothermal
curves (curves of constant temperature.) The field lines associated with this vector
field are termed heat flow lines. In the study of electric and magnetic fields the
potential functions from which these fields are derivable are termed, respectively,
the electric and magnetic field potentials. The field lines associated with these po-
tentials are called lines of electric and lines of magnetic force. Usually the harmonic
functions φand ψare expressed as the real and imaginary parts of a function of a
complex variable.
Laplace’s Equation
For F =F(x, y, z),a vector field which is both irrotational and solenoidal, then F
satisfies
curl F = 0 and div F= 0. (9 .74)
It has been shown that for these circumstances F is derivable from a scalar potential
function Φ.In particular, F = grad Φ = ∇Φ.Hence, Φmust be a solution of Laplace’s
equation ∇^2 Φ = 0 .That is,
div F = div (gradΦ) = ∇·(∇Φ) = ∇^2 Φ = 0.In expanded form the Laplace equation is expressed
∇^2 Φ =(
∂^2
∂x^2+ ∂2
∂y^2+ ∂2
∂z^2)
Φ = 0or ∇^2 Φ = ∂
(^2) Φ
∂x^2
+∂
(^2) Φ
∂y^2
+∂
(^2) Φ
∂z^2
= 0
This partial differential equation has many physical applications associated with it
and arises in many areas of science, physics and engineering. The Laplace equation
can be expressed in different forms depending upon the coordinate system in which
it is represented.
Three-dimensional Representations
In a rectangular right-handed (x, y, z)system of coordinates, the Laplace equation
is expressed as
∇^2 U=∂^2 U
∂x^2 +∂^2 U
∂y^2 +∂^2 U
∂z^2 = 0, U =U(x, y, z) (9 .75)