Geoscience Reference
In-Depth Information
Not only the potential of a point mass but also any other gravitational
potential is harmonic outside the attracting masses. Consider the potential
(1-12) of an extended body. Interchanging the order of differentiation and
integration, we find from (1-12)
∆
V
=
G
∆
v
l
dv
=
G
v
∆
1
l
dv
= 0 ;
(1-25)
that is, the potential of a solid body is also harmonic at any point
P
(
x, y, z
)
outside the attracting masses.
If
P
lies inside the attracting body, the above derivation breaks down,
since 1
/l
becomes infinite for that mass element
dm
(
ξ, η, ζ
) which coincides
with
P
(
x, y, z
), and (1-24) does not apply. This is the reason why the po-
tential of a solid body is not harmonic in its interior but instead satisfies
Poisson's differential equation (1-17).
1.4
Laplace's equation in spherical coordinates
The most important harmonic functions are the
spherical harmonics
. To find
them, we introduce spherical coordinates:
r
(radius vector; note that this is
a standard notation, although it does not represent a vector in the con-
temporary sense),
ϑ
(polar distance),
λ
(geocentric longitude), see Fig. 1.3.
Spherical coordinates are related to rectangular coordinates
x, y, z
by the
z
P
#
r
z
#
¸
y
x
y
x
Fig. 1.3. Spherical and rectangular coordinates
