Geoscience Reference
In-Depth Information
1.3
Harmonic functions
Earlier we have defined the harmonic functions as solutions of Laplace's
equation
∆
V
=0
.
(1-21)
More precisely, a function is called
harmonic in a region v
of space if it
satisfies Laplace's equation at every point of
v
. If the region is the exterior
of a certain closed surface
S
, then it must in addition vanish like 1
/l
for
l
.Itcanbeshownthat
every harmonic function is analytic
(in the
region where it satisfies Laplace's equation); that is, it is continuous and
has continuous derivatives of any order and can be developed into a Taylor
series.
The simplest harmonic function is the reciprocal distance
→∞
1
l
1
=
(
x
(1-22)
ξ
)
2
+(
y
η
)
2
+(
z
ζ
)
2
−
−
−
between two points
P
(
ξ, η, ζ
)and
P
(
x, y, z
). It is the potential of a point
mass
m
=1
/G
, situated at the point
P
(
ξ, η, ζ
); compare (1-5) and (1-6).
It is easy to show that 1
/l
is harmonic. We form the following partial
derivatives with respect to
x, y, z
in the fashion of (1-8):
1
l
=
−
1
l
=
−
1
l
=
−
∂
∂x
x
−
ξ
∂
∂y
y
−
η
∂
∂z
z
−
ζ
,
,
;
l
3
l
3
l
3
∂x
2
1
=
−l
2
+3(
x − ξ
)
2
l
3
∂y
2
1
=
−l
2
+3(
y − η
)
2
l
3
∂
2
∂
2
,
,
(1-23)
l
l
∂z
2
1
=
−
∂
2
l
2
+3(
z
ζ
)
2
−
.
l
3
l
Adding the last three equations and recalling the definition of ∆, we find
∆
1
l
= 0 ;
(1-24)
that is, 1
/l
is harmonic.
The point
P
(
ξ, η, ζ
), where
l
is zero and 1
/l
is infinite, is the only point
to which we cannot apply the above derivation; 1
/l
is not harmonic at this
singular point.
As a matter of fact, the slightly more general potential (1-6) of an ar-
bitrary point mass
m
is also harmonic except at
P
(
ξ, η, ζ
), because (1-24)
remains unchanged if both sides are multiplied by
Gm
.
