Geoscience Reference
In-Depth Information
that is,
d
x
and
g
differ only by a proportionality factor. This is best expressed
in the form
dx
W
x
dy
W
y
dz
W
z
=
=
.
(2-43)
In the coordinate system of Fig. 2.4, the curvature of the projection of
the plumb line onto the
xz
-plane is given by
κ
1
=
d
2
x
dz
2
;
(2-44)
this is equation (2-26) applied to the present case. Using (2-43), we have
dx
dz
=
W
W
z
.
(2-45)
We differentiate with respect to
z
, considering that
y
=0:
d
2
x
dz
2
W
z
W
xz
+
W
xx
W
zz
+
W
zx
.
1
W
2
dx
dz
dx
dz
=
−
W
x
(2-46)
z
In our particular coordinate system, the gravity vector coincides with the
z
-axis, so that its
x
-and
y
-components are zero:
W
x
=
W
y
=0
.
(2-47)
Figure 2.4 shows that we also have
dx
dz
=0
.
(2-48)
Therefore,
d
2
x
dz
2
=
W
z
W
xz
W
2
=
W
xz
W
z
=
W
zx
W
z
.
(2-49)
z
Considering
W
z
=
−
g
, we finally obtain
∂g
∂x
κ
1
=
1
g
(2-50)
and, similarly,
κ
2
=
1
g
∂g
∂y
.
(2-51)
These are the curvatures of the projections of the plumb line onto the
xz
-and
yz
-plane, the
z
-axis being vertical, that is, coinciding with the gravity vector.
The total curvature
κ
of the plumb line is given, according to differential
geometry (essentially Pythagoras' theorem), by
κ
=
κ
1
+
κ
2
=
1
g
2
x
+
g
2
y
.
(2-52)
g
