Geology Reference
In-Depth Information
v
1 b 1 + v
2 b 2 + v
3 b 3
w
1 b 1 + w
2 b 2 + w
3 b 3
V × W =
×
v
2 ( b 2 ×
v
3 ( b 3 ×
2
3
3
3
1
1
=
w
v
w
b 3 )
+
w
v
w
b 1 )
v
1 ( b 1 ×
1
2
2
+
w
v
w
b 2 )
V v
2 b 1
v
3 b 2
v
1 b 3
2
3
3
3
1
1
1
2
2
=
w
v
w
+
w
v
w
+
w
v
w
b 1
b 2
b 3
= g
1
2
3
v
v
v
,
(1.70)
1
2
3
w
w
w
making use of the definitions (1.6) of the reciprocal base vectors.
The curl of an arbitrary vector V is written as the cross product of the vector
di
ff
erential operator
and V . The integral theorem of Stokes (A.25) gives,
(
∇×
V )
·
ν d
S=
V
·
d s ,
(1.71)
S
C
where the surface
, with d s an infinitesimal tan-
gential vector displacement along the bounding curve, in a direction such that the
surface is on the left. ν is the unit outward normal to
S
is bounded by a closed curve
C
. If the bounding curve is
allowed to shrink to a point, in the limit the component of the curl in the direction
of ν is given by Stokes' theorem as
S
1
S
(
∇×
V )
·
ν
=
lim
C→ 0
V
·
d s ,
(1.72)
C
with S the infinitesimal enclosed area. If we take the bounding curve to be a
parallelogram in the u 1
surface formed by sides b 2 du 2 and b 3 du 3
in the u 2 and
u 3 directions, the sides parallel to the u 3 direction contribute
V
b 3 du 3 u 2
V
b 3 du 3 u 2 ,
·
·
(1.73)
+ du 2
while the sides parallel to the u 2 direction contribute
V
b 2 du 2 u 3
V
b 2 du 2 u 3 ,
·
+
·
(1.74)
+ du 3
to the integral on the right side of (1.72). In the limit as the bounding curve shrinks
to a point, the first contribution is replaced by the partial derivative with respect to
u 2 , times du 2 , times du 3 , and the second is replaced by the negative of the partial
derivative with respect to u 3 , times du 3 , times du 2 . Thus, the integral on the right
side of (1.72) becomes
u 2 ( V
b 2 ) du 2 du 3
·
b 3 )
u 3 ( V
·
.
(1.75)
Search WWH ::




Custom Search