Geology Reference
In-Depth Information
From (6.24),
ν
1
1
2
2
R
z
=
−
−
μ
,
(6.29)
νμ
giving
1
dz
dR
=−
ν
−
μ
2
1
.
(6.30)
μ
ν
2
−
Similarly, di
ff
erentiating (6.21) with respect to
R
,forfixedμ,gives
dz
dR
=
2
μ
R
z
=
μ
ν
ν
2
−
1
2
.
(6.31)
1
−
μ
2
1
−
μ
Thus, at the points of intersection, the slopes of the co-ordinate surfaces are negat-
ive reciprocals and hence they are orthogonal.
Elimination of μ
2
from the pair of equations (6.24) yields
k
2
R
2
k
2
ν
z
2
k
2
=
1
4
z
2
2
ν
−
+
+
+
0,
(6.32)
2
yields
while elimination of ν
k
2
R
2
+
k
2
μ
z
2
k
2
=
1
4
+
z
2
2
−
+
μ
0.
(6.33)
Given
R
2
and
z
2
, these two equations may be regarded as quadratic equations in
ν
2
, respectively. Taking the azimuthal dependence to be proportional to
exp
im
φ, where
m
is the azimuthal number, in the auxiliary cylindrical co-ordinates
(
R
,φ,
z
), on multiplying through by
R
2
, the governing Laplace equation becomes
2
and μ
R
∂χ
∂
R
2
R
∂
∂
R
R
2
∂
χ
∂
z
2
−
m
2
+
χ
=
0.
(6.34)
To convert to prolate spheroidal co-ordinates we require
R
∂χ
R
∂χ
∂ν
∂ν
∂
R
+
R
∂χ
∂μ
∂μ
∂
R
,
∂
R
=
(6.35)
and
∂χ
∂
z
=
∂χ
∂ν
∂ν
∂
z
+
∂χ
∂μ
∂μ
∂
z
,
(6.36)
and then
R
∂χ
∂
R
R
∂χ
∂ν
∂ν
∂
R
+
R
∂χ
∂μ
∂ν
∂
R
∂
∂
R
∂
∂ν
∂ν
∂
R
∂
∂ν
∂μ
∂
R
=
R
∂χ
∂μ
∂μ
R
∂χ
∂ν
∂μ
∂
R
,
+
∂
∂μ
∂μ
∂
R
∂
R
+
∂
∂ν
∂
R
(6.37)
∂μ
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