Geoscience Reference
In-Depth Information
Comparing Fig. 2.4 with Fig. 2.3, we see that
z
now takes the place of
y
. Therefore, instead of (2-26) we have for the curvature of the intersection
of the level surface with the
xz
-plane:
d
2
z
dx
2
.
K
1
=
(2-29)
If we differentiate
W
(
x, y, z
)=
W
0
with respect to
x
, considering that
y
is zero and
z
is a function of
x
,weget
dz
dx
W
x
+
W
z
=0
,
dz
dx
2
(2-30)
d
2
z
dx
2
dz
dx
+
W
zz
W
xx
+2
W
xz
+
W
z
=0
,
where the subscripts denote partial differentiation:
∂
2
W
∂x ∂z
, ... .
W
x
=
∂W
∂x
,W
xz
=
(2-31)
Since the
x
-axis is tangent at
P
,weget
dz/dx
=0at
P
,sothat
d
2
z
dx
2
W
xx
W
z
=
−
.
(2-32)
Since the
z
-axis is vertical, we have, using (2-22),
W
z
=
∂W
∂z
=
∂W
∂H
=
−
g.
(2-33)
Therefore, Eq. (2-29) becomes
K
1
=
W
xx
g
.
(2-34)
The curvature of the intersection of the level surface with the
yz
-plane is
found by replacing
x
with
y
:
K
2
=
W
yy
g
.
(2-35)
The mean curvature
J
of a surface at a point
P
is defined as the arith-
metic mean of the curvatures of the curves in which two mutually perpen-
dicular planes through the surface normal intersect the surface (Fig. 2.5).
Hence, we find
1
2
(
K
1
+
K
2
)=
W
xx
+
W
yy
2
g
J
=
−
−
.
(2-36)
