# Critical Points

• James J. Callahan
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

At a regular point, the linear terms of a function determine its local behavior,and there is a local coordinate change that transforms the function into one of the new coordinates. At a critical point, the linear terms vanish, but there is still an analogous result for the quadratic terms, called Morse’s lemma.However, the quadratic terms may not determine the local behavior, but when they do (the critical point is then said to be nondegenerate),Morse’s lemma provides a local coordinate change that transforms the function into a sum of positive and negative squares of the new coordinates. In this chapter we analyze Morse’s lemma and use it to characterize critical points.

## Keywords

Quadratic Form Saddle Point Symmetric Matrix Hessian Matrix Local Behavior
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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