Geoscience Reference
In-Depth Information
l
0
R sin /2)
P
P'
(
R
Ã
Fig. 1.9. Spatial distance between two points on a sphere
l
0
=2
R
sin
ψ
2
,
(1-141)
and the function
M
takes the simple form
1
4
R
2
sin
3
2
=
2
R
l
0
M
(
R, ψ
)=
.
(1-142)
For
ψ
, and we cannot use the original formula
(1-135) at the surface of the sphere
r
=
R
. In the transformed equation
(1-139), however, we have
V
→
0wehave
M
(
R, ψ
)
→∞
−
V
P
→
0for
ψ
→
0, and the singularity of
M
for
ψ
0 will be neutralized (provided
V
is differentiable twice at
P
).
Thus, we obtain the
gradient formula
→
2
π
π
R
V
P
+
R
2
∂V
∂r
1
V
−
V
P
sin
ϑ
dϑ
dλ
.
−
=
(1-143)
l
0
2
π
λ
=0
ϑ
=0
This equation expresses
∂V/∂r
on the sphere
r
=
R
in terms of
V
on this
sphere; thus, we now have
V
=
V
(
R, ϑ
,λ
)
.
V
P
=
V
(
R, ϑ, λ
)
,
(1-144)
Solution in terms of spherical harmonics
We may express
V
P
as
R
r
n
+1
=
∞
V
P
Y
n
(
ϑ, λ
)
.
(1-145)
n
=0
Differentiation yields
∞
(
n
+1)
R
n
+1
r
n
+2
∂V
∂r
=
−
Y
n
(
ϑ, λ
)
.
(1-146)
n
=0
