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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)
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