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Lecture 28
Let
U, V
be
connected open sets
of
R
n
, and
let
f
:
U
→
V
be a
diffeomorphism.
Then
+1
if
f
is
orient. preserving,
deg(
f
) =
(5.124)
if
f
is
orient. reversing.
−
1
We
showed that
given any
ω
∈
Ω
n
c
(
V
),
f
∗
ω
=
ω.
(5.125)
±
V
U
Let
ω
=
φ
(
x
)
dx
1
∧ · · · ∧
dx
n
,
where
φ
∈ C
0
∞
(
V
). Then
∂f
i
f
∗
ω
=
φ
(
f
(
x
)) det
∂x
j
(
x
)
dx
1
∧ · · · ∧
dx
n
,
(5.126)
so,
∂f
i
φ
(
f
(
x
))
det
dx
=
φ
(
x
)
dx.
(5.127)
∂x
j
±
V
U
Notice
that
∂f
i
f
is
orientation preserving
⇐
det
∂x
j
(
x
)
>
0
,
(5.128)
⇒
∂f
i
f
is
orientation reversing
⇐
det
∂x
j
(
x
)
<
0
.
(5.129)
⇒
So, in
general,
∂f
i
dx.
(5.130)
φ
(
f
(
x
)) det
(
x
)
∂x
j
U
As usual,
we
assumed that
f
∈ C
∞
.
Remark.
The
above
is
true
for
φ
∈ C
0
1
, a
compactly
supported
continuous
function.
The
proof
of
this
is
in section 5 of the Supplementary
Notes. The theorem
is
true
even
if only
f
∈ C
1
(the
notes
prove it for
f
∈ C
2
).
Today we
show
how
to compute the degree in
general.
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 Fall '04
 unknown
 Topology, Sets, Metric space, Open set, General topology, regular value, σpi

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