Chapter Seven
Harmonic Functions
7.1. The Laplace equation.
The Fourier law of heat conduction says that the rate at which
heat passes across a surface
S
is proportional to the flux, or surface integral, of the
temperature gradient on the surface:
k
S
T
dA
.
Here
k
is the constant of proportionality, generally called the
thermal conductivity
of the
substance (We assume uniform stuff. ). We further assume no heat sources or sinks, and we
assume steadystate conditions—the temperature does not depend on time. Now if we take
S
to be an arbitrary closed surface, then this rate of flow must be 0:
k
S
T
dA
0.
Otherwise there would be more heat entering the region
B
bounded by
S
than is coming
out, or viceversa. Now, apply the celebrated Divergence Theorem to conclude that
B
T
dV
0,
where
B
is the region bounded by the closed surface
S
. But since the region
B
is completely
arbitrary, this means that
T
2
T
x
2
2
T
y
2
2
T
z
2
0.
This is the worldfamous
Laplace Equation
.
Now consider a slab of heat conducting material,
7.1
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View Full Documentin which we assume there is no heat flow in the
z
direction. Equivalently, we could assume
we are looking at the crosssection of a long rod in which there is no longitudinal heat
flow. In other words, we are looking at a twodimensional problem—the temperature
depends only on
x
and
y
, and satisfies the twodimensional version of the Laplace equation:
2
T
x
2
2
T
y
2
0.
Suppose now, for instance, the temperature is specified on the boundary of our region
D
,
and we wish to find the temperature
T
x
,
y
in region. We are simply looking for a solution
of the Laplace equation that satisfies the specified boundary condition.
Let’s look at another physical problem which leads to Laplace’s equation. Gauss’s Law of
electrostatics tells us that the integral over a closed surface
S
of the electric field
E
is
proportional to the charge included in the region
B
enclosed by
S
. Thus in the absence of
any charge, we have
S
E
dA
0.
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 Fall '09
 Peter
 Laplace, Analytic function, Holomorphic function, simply connected region

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