16.8
Stokes' Theorem
Whereas
Green's Theorem
relates a double integral over a
plane region
D
to a line integral
around its
plane boundary curve
,
Stokes' Theorem
relates a surface integral over a
surface
S
to a line integral around the
boundary curve
of
S
(which is a space curve).
Let
C
be the boundary curve of an
oriented surface
S
with the unit normal vector
n
. The
positive
orientation
of
C
is defined as the
counterclockwise direction
centered at
n
. That is, the
positive
orientation
of
C
is the direction of the four remaining fingers when the thumb is in the direction of
n
in the
right hand rule
.
Stokes' Theorem
Let
S
be an oriented piecewise-smooth surface that is bounded by a simple, closed,
piecewise-smooth boundary curve
C
with
positive orientation
. Let
F
be a vector field whose components
have continuous partial derivatives on an open region in
R
3
that contains
S
. Then
C
F
·
d
r
=
S
curl
F
·
d
S
.
Since
C
F
·
d
r
=
C
F
·
T
ds
and
S
curl
F
·
d
S
=
S
curl
F
·
n
dS,
Stokes' Theorem says that the line integral around the boundary curve of
S
of the tangential com-
ponent of
F
is equal to the surface integral over
S
of the normal component of the curl of
F
.
The positively oriented boundary curve of the oriented surface
S
is often written as
∂S
, so Stokes'
Theorem can be expressed as
S
curl
F
·
d
S
=
∂S
F
·
d
r
as well.
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