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Lecture 38
We begin with a review from last time.
Let
X
be
an
oriented
manifold,
and
let
D
⊆
X
be
a
smooth
domain.
Then
Bd (
D
) =
Y
is an oriented (
n
−
1)dimensional manifold.
We deFned integration over
D
as follows.
±or
ω
∈
Ω
n
c
(
X
) we want to make sense
of the integral
ω.
(6.161)
D
We look at some special cases:
Case 1:
Let
p
∈
Int
D
, and let
φ
:
U
→
V
be an oriented parameterization of
X
at
p
, where
V
⊆
Int
D
. ±or
ω
∈
Ω
c
n
(
X
), we deFne
ω
=
ω
=
φ
∗
ω
=
φ
∗
ω.
(6.162)
D
V
U
R
n
This is just our old deFnition for
ω.
(6.163)
V
Case 2:
Let
p
∈
Bd (
D
), and let
φ
:
U
→
V
be an oriented parameterization of
D
at
p
.
That is,
φ
maps
U
∩
H
n
onto
V
∩
D
. ±or
ω
∈
Ω
n
c
(
V
), we deFne
ω
=
φ
∗
ω.
(6.164)
D
H
n
We showed last time that this deFnition does not depend on the choice of parameter
ization.
General case: ±or each
p
∈
Int
D
, let
φ
:
U
p
→
V
p
be an oriented parameterization
of
X
at
p
with
V
p
⊆
Int
D
.
±or each
p
∈
Bd (
D
), let
φ
:
U
V
p
be and oriented
p
→
parameterization of
D
at
p
.
Let
U
=
U
p
,
(6.165)
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 Fall '04
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