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4.3. STOKES’S THEOREM
99
4.3
Stokes’s Theorem
4.3.1
Orientation of the Boundary
•
Let
M
be an
n
dimensional orientable manifold with boundary
∂
M
, which
is an (
n

1)dimensional manifold without boundary.
•
Let
M
be oriented.
•
Then an orientation on
M
naturally induces an orientation on
∂
M
.
•
Let
p
∈
∂
M
and
{
e
2
, . . . ,
e
n
}
be a basis in
T
p
∂
M
.
•
Let
N
∈
T
p
M
be a tangent vector at
p
that is transverse to
∂
M
and points
out of
M
.
•
Then
{
N
,
e
2
, . . . ,
e
n
}
forms a basis in
T
p
M
.
•
Then, by deﬁnition, the basis
{
e
2
, . . . ,
e
n
}
has the same orientation as the
basis
{
N
,
e
2
, . . . ,
e
n
}
. That is,
{
e
2
, . . . ,
e
n
}
is positively oriented in
∂
M
if
{
N
,
e
2
, . . . ,
e
n
}
is positively oriented in
M
.
4.3.2
Stokes’ Theorem
•
Theorem 4.3.1
Let M be an ndimensional manifold and V be a p
dimensional compact oriented submanifold with boundary
∂
V in M. Let
ω
∈
Λ
p

1
M be a smooth
(
p

1)
form in M. Then
Z
V
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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