Geology Reference
In-Depth Information
The
i
th contravariant component of the curl of a vector (1.79) becomes
√
g
ξ
ijk
∂v
k
1
V
)
i
(
∇×
=
∂
u
j
.
(1.82)
The
divergence
of an arbitrary vector
V
is written as the scalar product of the
vector di
ff
erential operator
∇
and
V
. The divergence theorem of Gauss (A.17)
gives
V
∇·
V
d
V=
V
·
ν
d
S
,
(1.83)
S
where
. Thus, the integral of the divergence
of
V
throughout the volume is equal to the integral of the outward normal com-
ponent of
V
over the surface. If we now let the surface shrink, in the limit we
find that
V
is a volume enclosed by a surface
S
1
V
∇·
V
=
lim
S→
0
V
·
ν
d
S
,
(1.84)
S
with
V
representing the infinitesimal volume. Now take the infinitesimal volume to
be the parallelepiped bounded by the surfaces
u
1
and
u
1
du
1
,
u
2
and
u
2
du
2
,
u
3
+
+
and
u
3
du
3
. First, consider the integral of the outward normal component of
V
over the surfaces
u
1
and
u
1
+
du
1
. It is closely
+
V
b
3
)
u
1
−
V
b
3
)
u
1
du
2
du
3
du
1
du
2
du
3
·
(
b
2
×
·
(
b
2
×
.
(1.85)
+
In the limit, as the bounding surface shrinks to zero, it becomes the partial derivat-
ive with respect to
u
1
, times
du
1
, times
du
2
du
3
,or
∂
u
1
V
b
3
)
du
1
du
2
du
3
∂
·
(
b
2
×
.
(1.86)
From (1.6) and (1.5),
=
√
g
b
1
V
b
1
b
3
)
b
1
b
2
×
b
3
=
=
b
1
·
(
b
2
×
,
(1.87)
on taking the square root of (1.67) to replace the scalar triple product. Hence, (1.86)
becomes
∂
u
1
V
b
1
√
g
du
1
du
2
du
3
∂
u
1
v
√
g
du
1
du
2
du
3
∂
∂
1
·
=
,
(1.88)
1
of
V
, defined by (1.15).
Adding in the contributions of the other two pairs of sides of the parallelepiped,
the total integral of the outward normal component of
V
over the surface is
∂
b
1
recognising
V
·
as the contravariant component v
∂
u
i
v
√
g
du
1
du
2
du
3
i
.
(1.89)
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