Geoscience Reference
In-Depth Information
only to the extensive literature, e.g., the topic by Moritz (1995), the inter-
net page www.inas.tugraz.at/forschung/InverseProblems/AngerMoritz.html
or Anger and Moritz (2003).
1.14
The radial derivative of a harmonic function
For later application to problems related with the vertical gradient of gravity,
we will now derive an integral formula for the derivative along the radius
vector
r
of an arbitrary harmonic function which we denote by
V
. Such a
harmonic function satisfies Poisson's integral (1-123):
2
π
π
V
(
r, ϑ, λ
)=
R
(
r
2
− R
2
)
4
π
V
(
R, ϑ
,λ
)
l
3
sin
ϑ
dϑ
dλ
.
(1-134)
λ
=0
ϑ
=0
Forming the radial derivative
∂V/∂r
,wenotethat
V
(
R, ϑ
,λ
)doesnot
depend on
r
. Thus, we need only to differentiate (
r
2
R
2
)
/l
3
, obtaining
−
2
π
π
∂V
(
r, ϑ, λ
)
∂r
R
4
π
M
(
r, ψ
)
V
(
R, ϑ
,λ
)sin
ϑ
dϑ
dλ
,
=
(1-135)
λ
=0
ϑ
=0
where
r
2
=
− R
2
l
3
∂
∂r
1
l
5
(5
R
2
r
r
3
Rr
2
cos
ψ
3
R
3
cos
ψ
)
.
(1-136)
M
(
r, ψ
)
≡
−
−
−
Applying (1-135) to the special harmonic function
V
1
(
r, ϑ, λ
)=
R/r
,where
∂V
1
∂r
R
r
2
and
V
1
(
R, ϑ
,λ
)=
R
=
−
R
=1
,
(1-137)
we obtain
2
π
π
R
r
2
R
4
π
M
(
r, ψ
)sin
ϑ
dϑ
dλ
.
−
=
(1-138)
λ
=0
ϑ
=0
Multiplying both sides of this equation by
V
(
r, ϑ, λ
) and subtracting it from
(1-135) gives
2
π
π
∂V
∂r
+
R
R
4
π
V
P
)sin
ϑ
dϑ
dλ
,
r
2
V
P
=
M
(
r, ψ
)(
V
−
(1-139)
λ
=0
ϑ
=0
where
V
=
V
(
R, ϑ
,λ
)
.
V
P
=
V
(
r, ϑ, λ
)
,
(1-140)
In order to find the radial derivative at the surface of the sphere of radius
R
,wemustset
r
=
R
.Then
l
becomes (Fig. 1.9)
