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where the g
i
are C
•
functions defined on
V
with g
i
(
0
) =∂f/∂x
i
(
0
).
Proof.
Consider the function t Æ f(tx
1
,tx
2
,...,tx
n
). By the Fundamental Theorem of
Calculus
df
dt
1
(
)
=
Ú
(
)
fx x
,
,...,
x
tx
,
tx
,...,
tx
dt
12
n
1 2
n
0
n
È
Í
˘
˙
∂
∂
f
x
1
Â
Ú
(
)
=
tx
,
tx
,...,
tx
x
dt
.
12
ni
0
i
i
=
1
We can therefore define the functions g
i
by
Ú
∂
f
x
1
(
)
=
(
)
gxx
,
,...,
x
tx
,
tx
,...,
tx
dt
i
12
n
1 2
n
∂
0
i
and the lemma is proved. (One needs C
•
here because if f is only C
r
, then the g
i
will
not be C
r
in general.)
The main result of this section is the next theorem.
4.6.3. Theorem.
(The Morse Lemma) Let f be a real-valued C
•
function defined in
a neighborhood of a point
p
in
R
n
. If
p
is a nondegenerate critical point for f, then
there is a local diffeomorphism j from a neighborhood
V
of
p
onto a neighborhood
U
of
0
in
R
n
so that
(
)
(
j
-
1
2
2
2
2
)
=
()
---+
f
o
y
,
y
,...,
y
f
p
y
...
y
y
++
...
y
12
n
n
k
k
+
1
on
U
. The integer k is well defined and called the
index
of
p
.
Proof.
Without loss of generality assume that
p
=
0
and f(
p
) =
0
. By Lemma 4.6.2
we can write f as
n
Â
(
)
=
(
)
fx x
,
,...,
x
xg x x
,
,...,
x
.
12
n
i
i
12
n
i
=
1
Since
p
is a critical point of f,
∂
∂
f
x
()
=
()
=
g
0
0
0.
i
i
This means that we can apply Lemma 4.6.2 to the g
i
to get
n
Â
(
)
=
(
)
gxx
,
,...,
x
xh xx
,
,...,
x
.
i
12
n
j
ij
12
n
j
=
1
