The following separation of variables theorem is a key step in the proof of the Morse Lemma. In this exposition the term smooth stands for infinitely many times differentiable.
The Morse Lemma is the following theorem, which follows from the above reduction with separation of variables.
Proof of the separation of variables theorem.
By assumption, the matrix of second derivatives at
is non zero. So, there exist an invertible matrix such
that the entry of the matrix
be the linear system of coordinates on
obtained by changing the standard coordinates with . Now we have
that at the second partial derivative with respect to of
. The restriction of
the function to the intersection of with the -coordinate
axis has at a non-degenerate
local maximum or minimum, according to the sign of
at . In particular, the restiction
to the interval
on is strictly
convex or concave, if we choose to be sufficiently small.
Having chosen , we may choose , such that for every
of to the interval
is defined and
convex or concave with a non-degenerate critical point in the interior.
The position of this critical point is at
is implicitely given by
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