Geology Reference
In-Depth Information
@V
@r
r
C
1
r
@V
@™
™
C
1
r sin ™
@V
@¥
¥
r
V
D
(4.73)
A formula for the Laplacian can be found
taking into account that
r
2
V
Drr
V
. We obtain
the following expression:
r
2
@V
@r
1
r
2
@
@r
1
r
2
sin ™
@
@™
2
V
D
r
C
sin ™
@V
@™
@
2
V
@¥
2
1
r
2
sin
2
™
C
(4.74)
Therefore, the spherical form of Laplace's
equation can be written as follows:
r
2
@V
@r
sin
@V
@™
@
@r
1
sin ™
@
@™
C
Fig. 4.21
Base
versors
for
the
transformation
from
Cartesian to spherical derivatives
@
2
V
@¥
2
D
0
1
sin
2
™
C
(4.75)
To transform Laplace's equation to spherical
coordinates, it is useful to introduce three or-
thogonal versors,
We can solve this equation by separation of
variables. First, let us try the following separation
of the potential into a radial component,
R
,anda
component depending only from colatitude and
longitude:
r, ™,and¥, which are directed
respectively toward increasing distance from the
origin, increasing colatitude (that is, southward),
and increasing longitude (i.e., eastward) at point
P
(
x
,
y
,
z
)(Fig.
4.21
). Our objective is to convert
the Cartesian derivatives of Laplace's equation
to spherical derivatives, which measure variations
of potential with respect to small increments of
r
,
™,and¥.
It is easy to show that the Cartesian com-
ponents of the three versors are given by the
following transformations:
8
<
V.r;™;¥/
D
R.r/Y .™;¥/
(4.76)
Substituting into (
4.75
)gives:
r
2
@R
@r
sin ™
@Y
@™
Y
@
@r
R
sin ™
@
@™
C
@
2
Y
@¥
2
D
0
R
sin
2
™
C
r
D
sin ™ cos ¥i
C
sin ™ sin ¥j
C
cos ™k
™
D
cos ™ cos
¥
i
C
cos ™ sin
¥
j
sin ™k
¥
D
sin ¥i
C
cos ¥j
:
Hence, dividing both sides by
YR
:
(4.72)
r
2
dR
dr
sin ™
@Y
@™
1
R
d
dr
1
Y sin ™
@
@™
D
2
,itis
necessary to find first an expression for the gra-
dient of the potential
V
in spherical coordinates.
The spatial derivatives @
V
/@
x
, @
V
/@
y
,and@
V
/@
z
can be transformed using simple chain rules and
expression for the gradient is:
To transform the Laplacian operator
r
@
2
Y
@¥
2
1
Y sin
2
™
(4.77)
where we have substituted partial derivatives at
the left-hand side by ordinary ones, because
R
