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1485
SECTION 14.8
.
.
Stokes’ Theorem
1199
14.8
STOKES’ THEOREM
Recall that, after introducing the curl in section 14.5, we observed that for a piecewise,
smooth, positively oriented, simple closed curve
C
in the
xy
plane enclosing the region
R
,
we could rewrite Green’s Theorem in the vector form
±
C
F
·
d
r
=
²²
R
(
∇×
F
)
·
k
dA
,
(8.1)
where
F
(
x
,
y
)isavector ﬁeld of the form
F
(
x
,
y
)
=±
M
(
x
,
y
)
,
N
(
x
,
y
)
,
0
²
.In this section,
we generalize this result to the case of a vector ﬁeld deﬁned on a surface in three dimensions.
Suppose that
S
is an oriented surface with unit normal vector
n
.If
S
is bounded by the simple
closed curve
C
,we determine the orientation of
C
using a righthand rule like the one used
to determine the direction of a cross product of two vectors. Align the thumb of your right
hand so that it points in the direction of one of the unit normals to
S
. Then if you curl your
ﬁngers, they will indicate the
positive orientation
on
C
,as indicated in Figure 14.52a. If
the orientation of
C
is opposite that indicated by the curling of the ﬁngers on your right
hand, as shown in Figure 14.52b, we say that
C
has
negative orientation.
The vector form
of Green’s Theorem in (8.1) generalizes as follows.
S
C
z
n
y
x
FIGURE 14.52a
Positive orientation
S
C
z
y
x
n
FIGURE 14.52b
Negative orientation
THEOREM 8.1
(Stokes’ Theorem)
Suppose that
S
is an oriented, piecewisesmooth surface with unit normal vector
n
,
bounded by the simple closed, piecewisesmooth boundary curve
∂
S
having positive
orientation. Let
F
(
x
,
y
,
z
)beavector ﬁeld whose components have continuous ﬁrst
partial derivatives in some open region containing
S
. Then,
²
∂
S
F
(
x
,
y
,
z
)
·
d
r
=
S
(
F
)
·
n
dS
.
(8.2)
Notice right away that the vector form of Green’s Theorem (8.1) is a special case
of (8.2), as follows. If
S
is simply a region in the
xy
plane, then a unit normal to the
surface at every point on
S
is the vector
n
=
k
. Further,
dS
=
dA
(i.e., the surface area
of the plane region is simply the area) and (8.2) simpliﬁes to (8.1). The proof of Stokes’
Theorem for the special case considered below hinges on Green’s Theorem and the chain
rule.
One important interpretation of Stokes’ Theorem arises in the case where
F
represents
a force ﬁeld. Note that in this case, the integral on the left side of (8.2) corresponds to the
work done by the force ﬁeld
F
as the point of application moves along the boundary of
S
.
Likewise, the right side of (8.2) represents the net ﬂux of the curl of
F
over the surface
S
.
A general proof of Stokes’ Theorem can be found in more advanced texts. We prove it here
only for a special case of the surface
S
.
PROOF(Special Case)
We consider here the special case where
S
is a surface of the form
S
={
(
x
,
y
,
z
)

z
=
f
(
x
,
y
)
,
for (
x
,
y
)
∈
R
}
,
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CHAPTER 14
.
.
Vector Calculus
1486
where
R
is a region in the
xy
plane with piecewisesmooth boundary
∂
R
, where
f
(
x
,
y
) has
continuous ﬁrst partial derivatives and for which
∂
R
is the projection of the boundary of the
surface
∂
S
onto the
xy
plane (see Figure 14.53).
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