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Lecture 34
6.6
Orientation of Manifolds
Let
X
be
an
n
dimensional
manifold
in
R
N
. Assume that
X
is
a
closed
subset of
R
N
.
Let
f
:
X
→
R
be
a
C
∞
map.
Definition 6.21.
We
remind you
that
the support
of
f
is
defined
to
be
supp
f
=
{
x
∈
X
:
f
(
x
) = 0
}
.
(6.69)
Since
X
is
closed,
we
don’t
have to
worry
about whether
we are taking
the closure
in
X
or
in
R
n
.
Note
that
f
∈
C
∞
0
(
X
)
⇐⇒
supp
f
is
compact.
(6.70)
Let
ω
∈
Ω
k
(
X
).
Then
supp
ω
=
{
p
∈
X
:
ω
p
�
=
0
}
.
(6.71)
We
use
the
notation
ω
∈
Ω
k
supp
ω
is
compact.
(6.72)
c
(
X
)
⇐⇒
We
will
be
using partitions
of
unity, so
we remind
you
of the definition:
Definition 6.22.
A
collection of
functions
{
ρ
i
∈
C
0
∞
(
X
) :
i
= 1
,
2
,
3
, . . .
}
is
a
partition
of unity
if
1. 0
≤
ρ
i
,
2. For
every compact
set
A
⊆
X
, there exists
N >
0
such
that
supp
ρ
i
∩
A
=
φ
for
all
i > N
,
3.
ρ
i
=
1.
Suppose
the
collection of
sets
U
=
{
U
α
:
α
∈
I
}
is
a
covering
of
X
by open
subsets
U
α
of
X
.
Definition 6.23.
The
partition of unity
ρ
i
, i
= 1
,
2
,
3
, . . . ,
is
subordinate to
U
if for
every
i
, there
exists
α
∈
I
such that supp
ρ
i
⊆
U
α
.
Claim.
Given a
collection of sets
U
=
{
U
α
:
α
∈
I
}
,
there exists a partition of
unity
subordinate
to
U
.
1
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˜
˜
Proof.
For
each
α
∈
I
,
let
U
α
be an
open
set in
R
N
such
that
U
α
=
U
α
∩
X
.
We
˜
define
the
collection of
sets
U
=
{
˜
Let
U
α
:
α
∈
I
}
.
±
˜
˜
U
=
U
α
.
(6.73)
From
our
study of
Euclidean space, we know
that there exists
a
partition
of unity
˜
0
(
˜
U
)
,
i
= 1
,
2
,
3
, . . .
,
subordinate to
U
˜
. Let
ι
X
:
X
U
˜
be the inclusion
map.
ρ
i
∈
C
∞
→
Then
ρ
i
=
ρ
˜
i
◦
ι
X
=
ι
∗
ρ
˜
i
,
(6.74)
X
which
you should check.
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 Fall '04
 unknown
 Topology, Open set, Volume form

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