文档介绍：Chapter Seven
Harmonic Functions
. 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:
XX 4
k T 6 dA.
S
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
steady-state 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:
XX 4
k T 6 dA = 0.
S
Otherwise there would be more heat entering the region B bounded by S than ing out, or
vice-versa. Now, apply the celebrated Divergence Theorem to conclude that
XXX 4 4
Ý 6 TÞdV = 0,
B
where B is the region bounded by the closed surface S. But since the region B pletely
arbitrary, this means that
/ / /
4 6 4 = 2T + 2T + 2T =
T / / / 0.
x2 y2 z2
This is the world-famous Laplace Equation.
Now consider a slab of heat conducting material,
in which we assume there is no heat flow in the z-direction. Equivalently, we could assume we are
looking at the cross-section of a long rod in which there is no longitudinal heat flow. In other
words, we are looking at a two-dimensional problem—the temperature depends only