Editing Morse theory (section)

===Morse lemma===
Let <math>p</math> be a non-degenerate critical point of <math>f \colon M \to \reals.</math>  Then there exists a [[Chart (topology)|chart]] <math>\left(x_1, x_2, \ldots, x_n\right)</math> in a [[Neighborhood (topology)|neighborhood]] <math>U</math> of <math>p</math> such that <math>x_i(p) = 0</math> for all <math>i</math> and
<math display="block">f(x) = f(p) - x_1^2 - \cdots - x_{\gamma}^2 + x_{\gamma +1}^2 + \cdots + x_n^2</math>
throughout <math>U.</math> Here <math>\gamma</math> is equal to the index of <math>f</math> at <math>p</math>. As a corollary of the Morse lemma, one sees that non-degenerate critical points are [[Isolated point|isolated]].  (Regarding an extension to the complex domain see [[Method of steepest descent#Complex Morse lemma|Complex Morse Lemma]]. For a generalization, see [[Morse–Palais lemma]]).