The Geomagnetic Field - Quantitative Plate Tectonics - page 128

Therefore, taking the limit as R ! 0ofthis
expression, we obtain:
Z

Gauss' Theorem of the Arithmetic Mean
The value of a harmonic function at a point is
the average of the function over any spherical
neighbor of harmonicity about the point .

I 1

r

1
r

dS

1
r r

@§

@n § @

2 §dV D

@n

R

S. R /

Proof Let † be a sphere centered on the point P
2 R .If R is the radius of †, then the representa-
tion formula ( 4.70 ) assumes the form:

4 §.P/

The Green's third identity immediately fol-
lows from this identity.

I I

1
4 R

@§
@n dS C

1
4 R 2

§.P/ D §dS

The Green's third identity implies that the
potential at any point P has three components:

† †

According to the first Green's identity, the first
integral vanishes. Therefore:

1. The potential associated with the volume R ,
with density (1/4 r ) r

2 §;
2. A potential associated with the surface S ( R ),
with density (1/4 r )@§/@ n ;
3. Another potential associated with the surface
S ( R ), with density (1/4 )§@(1/ r )/@ n .

4 R 2 I
†

1

§.P/ D §dS Dh § i † (4.71)

This identity proves the theorem.

If § is harmonic, then the volume compo-
nent in ( 4.68 ) is zero, and the identity simplifies
to:

Gauss' theorem reveals two key intercon-
nected features of a harmonic function, namely,
the fact that its value at a point coincides with
the average over a spherical neighbor, and the
lack of maxima and minima within the region
of harmonicity. These properties were used in
Sect. 3.3 to find a numerical solution to Laplace's
equation. In the next section, we shall face
the problem to find an analytic solution to this
equation.

1
r

1
r

dS

I

1
4

@§

@n § @

§.P/ D

@n

S . R /

(4.70)

This important and surprising result is called
representation formula (e.g., Blakely 1996 ). It
says that the value of a harmonic function can
be calculated from the values it takes over the
boundary S ( R ) of the harmonicity region R and
from normal derivatives along the same bound-
ary. However, if we consider any subset E
R such that P 2 E , this is clearly a region of
harmonicity for §. Therefore, we can use the
boundary values over S ( E ) to determine § ( P )as
well. Therefore, if the Dirichlet boundary value
problem ensures the unicity of a harmonic func-
tion given the boundary values, the converse
is not true, in the sense that we cannot deter-
mine uniquely the boundary values starting from
a known value of § in R . We shall see that
Eq. ( 4.70 ) is an invaluable tool for the manipu-
lation of potential field data. An important corol-
lary of Green's third identity is the following
Gauss ' theorem of the arithmetic mean .

4.8 Spherical Harmonic

Expansion
of the Geomagnetic Field
and the IGRF

In Sect. 3.3 , we mentioned the fact that the
scalar magnetic potential V satisfies Laplace's
Eq. ( 3.29 ) in any region R where the current den-
sity is zero (including magnetization currents).
Now we want to find a solution to this equation
for the geomagnetic field. To this purpose, it
is convenient to use spherical coordinates ( r , ™,
¥) (see Sect. 2.3 ) instead of the usual Carte-
sian coordinates ( x , y , z ), the laws of transforma-
tion between the two systems being given by
Eqs. ( 2.27 ) and( 2.28 ) .