---
tag: MAX2ZYG6H9
kind: concept
name: Morse index
msc: [57R70, "58E05", "58E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_theory
---

## Summary

The Morse index of a [[critical point|0YTCMFB4HC]] is the number of independent directions in which the function decreases to second order.
It is the largest dimension of a subspace on which the [[Hessian|0693Y2METT]] is negative definite.
In Morse theory, passing a [[nondegenerate critical point|MGPPCDFQDG]] of index $\lambda$ attaches a cell of dimension $\lambda$ to the sublevel set.

## Explanations

### L1 human

At the bottom of a bowl no direction goes down, so the index is $0$.
At a mountain pass one direction goes down on each side, so the index is $1$.
At the top of a hill in a surface every direction goes down, so the index is $2$.
For example, $x^2 - y^2 - z^2$ on $\R^3$ has a critical point of index $2$ at the origin, since it decreases along the $y$ and $z$ axes.