Geology Reference
In-Depth Information
This is to be divided by the infinitesimal enclosed area,
S
, of the parallelogram,
(
b
2
b
3
)
du
2
du
3
S
=
×
b
3
)
·
(
b
2
×
,
(1.76)
to give the right side of (1.72). The unit outward normal vector
ν
is in the direction
of the reciprocal base vector
b
1
. Hence,
b
1
√
b
1
b
1
ν
=
b
1
=
V
b
3
)
,
(1.77)
√
(
b
2
×
b
3
)
·
(
b
2
×
·
using the first expression in (1.6
)
for the reciprocal base vector
b
1
. Replacing
V
with its expression in (1.69) as
√
g, and collecting terms, the component of the curl
in the direction of
b
1
is given by
∂
∂
u
2
(
V
b
2
)
1
√
g
−
∂
b
1
(
∇×
V
)
·
=
·
b
3
)
∂
u
3
(
V
·
∂v
3
∂
u
2
−
∂
u
3
1
√
g
∂v
2
=
,
(1.78)
where we have used the definition of the covariant components of
V
givenbythe
second of expressions (1.15). The remaining two components of the curl are found
by simply permuting indices. Expanding
∇×
V
in the unitary base system, as in
(1.17), it then becomes
∂v
3
∂
u
2
−
∂
u
3
b
1
∂v
1
∂
u
3
−
∂
u
1
b
2
∂v
2
∂
u
1
−
∂
u
2
b
3
1
√
g
∂v
2
∂v
3
∂v
1
∇×
V
=
+
+
b
1
b
2
b
3
1
√
g
∂/∂
u
1
∂/∂
u
2
∂/∂
u
3
=
.
(1.79)
v
1
v
2
v
3
The expressions for the cross product (1.70) and the curl (1.79) can be condensed
by introduction of the
permutation symbol
ξ
ijk
. This is defined as
⎩
1,
i
,
j
,
k
cyclic,
−
ξ
ijk
=
1,
i
,
j
,
k
anticyclic,
0, otherwise.
(1.80)
Cyclic permutations of the subscripts are (1,2,3), (2,3,1), (3,1,2), while anticyclic
permutations are (1,3,2), (3,2,1), (2,1,3). In a right-handed Cartesian system of
co-ordinates, we will later find that the permutation symbol is a third-order tensor,
called the
alternating tensor
. Using the permutation symbol, the
i
th covariant com-
ponent of the cross product (1.70) becomes
W
)
i
=
√
gξ
ijk
v
j
k
(
V
×
w
.
(1.81)
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