Geoscience Reference
In-Depth Information
equations
x
=
r
sin
ϑ
cos
λ,
y
=
r
sin
ϑ
sin
λ,
z
=
r
cos
ϑ
;
(1-26)
or inversely by
r
=
x
2
+
y
2
+
z
2
,
ϑ
=tan
−
1
x
2
+
y
2
z
,
(1-27)
λ
=tan
−
1
y
x
.
To get Laplace's equation in spherical coordinates, we first determine the
element of arc (element of distance)
ds
in these coordinates. For this purpose
we form
dx
=
∂x
∂r
dr
+
∂x
∂ϑ
dϑ
+
∂x
∂λ
dλ ,
dy
=
∂y
∂r
dr
+
∂y
∂ϑ
dϑ
+
∂y
(1-28)
∂λ
dλ ,
dz
=
∂z
∂r
dr
+
∂z
∂ϑ
dϑ
+
∂z
∂λ
dλ .
By differentiating (1-26) and substituting it into the elementary formula
ds
2
=
dx
2
+
dy
2
+
dz
2
,
(1-29)
we obtain
ds
2
=
dr
2
+
r
2
dϑ
2
+
r
2
sin
2
ϑdλ
2
.
(1-30)
We might have found this well-known formula more simply by geometrical
considerations, but the approach used here is more general and can also be
applied to ellipsoidal (harmonic) coordinates.
In (1-30) there are no terms with
dr dϑ
,
dr dλ
,and
dϑ dλ
. This expresses
the evident fact that spherical coordinates are orthogonal: the spheres
r
=
constant, the cones
ϑ
= constant, and the planes
λ
= constant intersect each
other orthogonally.
The general form of the element of arc in arbitrary orthogonal coordinates
q
1
,q
2
,q
3
is
ds
2
=
h
1
dq
1
+
h
2
dq
2
+
h
3
dq
3
.
(1-31)
