Mathematics of Physics and Engineering

158 The Three Theorems

The equation

V^2 / - g, (3.2.15)

where g is a known continuous function, is called Poisson's equation,
after the French mathematician SIMEON DENIS POISSON (1781-1840). The
particular case of the Poisson equation,

V^2 / = 0, (3.2.16)

is important enough to have its own name, and is called Laplace's
equation, after the French mathematician PIERRE-SIMON LAPLACE
(1749-1827). Potential theory is the branch of mathematics studying
these two equations; the large number of problems in both mathematics
and physics that are reduced to either (3.2.15) or (3.2.16) justifies the allo-
cation of a whole branch of mathematics to the study of just two equations.
This section provides the most basic introduction to potential theory. While
the results we discuss are true in both two and three dimensions, our pre-
sentation will be in R^3 , both for the sake of concreteness, and because in K^2
complex numbers provide a much more efficient method to study Laplace's
and Poisson's equations; see Theorem 4.2.6 on page 205 below.

Let / be a continuously differentiable function in a domain G in E^3 ,

and S, a closed orientable piece-wise smooth surface. Denote by ris =

ns{P) the outside unit normal vector to S at the point P. The normal

derivative of / at P is, by definition,

^•(P) = V/(P) • ns(P); where V/ = grad/. (3.2.17)

an

EXERCISE S^.IO.*^ Assume that f is a harmonic function in a domain G

and S, a closed orientable piece-wise smooth surface in G. Assume that the

domain enclosed by S is a subset of G. (a) Show that

-i-da = 0. (3.2.18)

an

s

Hint: Apply Gauss's Theorem to F = V/. (b) Show that (3.2.18) can fail

if the domain enclosed by S contains points that are not in G. Hint: take

/ = r_1 and G, the unit ball around the origin with the center removed.

The following result is the analog of the integration-by-parts formula

from the one-variable calculus.

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