Critical Points

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.