Laplace's and Poisson's Equations 157rise meaning of this formula and demonstrating that (3.2.1), (3.2.4), and
(3.2.11) all follow as particular cases requires a higher level of abstraction
and is done in differential geometry, where (3.2.12) is known as Stokes's
Theorem. The details are beyond the scope of this book and can be found
in most books on differential geometry, for example, in Section 8.2 of the
book Differentiable Manifolds: A First Course by L. Conlon, 1993.
The following exercise provides yet another application of Stokes's The-
orem. One of the results, namely, (3.2.14), will be used later in the analysis
of the magnetic dipole.
EXERCISE 3.2.9. c Let S be piece-wise smooth orientable surface and its
boundary OS, a simple closed piece-wise smooth curve. Assume that the
orientations of S and dS agree. Denote by n the field of unit normal
vectors on S, and by u, the field of unit tangent vectors on dS. (a) Show
that, for every continuously differentiable scalar function f,
//grad/ x nda = - I fuds. (3.2.13)
s dsHint: the left- and right-hand sides of (3.2.13) are vectors; denote them by v and
w, respectively. The claim is that v • b = w -b for every constant vector b; in the
cartesian coordinates, it is enough to verify this equality for only three vectors,
i,j,K instead of b. Note that, by Stokes's Theorem, w = f/(curl(/6)) -nda.
' s
By (3.1.30), curl(/6) = grad/ x b. Then use the properties of the scalar triple
product, (b) Taking f = TQ • r in (3.2.13), where TQ is a constant vector
and r is the vector field defined on page HO, and using (3.1.34), show that
(p(ro • r)uds =—ro x nda. (3.2.14)
as s3.2.4 Laplace's and Poisson's Equations
This section is a very brief introduction to potential theory and harmonic
functions. Gauss's Theorem will be the main tool in the investigation.
A scalar function / is called harmonic in a domain G if / is twice con-
tinuously differentiable in G and V^2 / = 0. Recall that V^2 / = div(grad/),
so that, in cartesian coordinates, V^2 / = fxx + fyy in M^2 and V^2 / =
fxx + fyy + fzz in K^3. Similar expression exist in every M", n > 3. While
we restrict our discussion to M^3 , all the results of this section are valid in
every Mn, n > 2.