nLab critical point

Definition

For $f : X \to Y$ a smooth function, a critical point is a point $x \in X$ at which the derivative $d f : T X \to T Y$ has rank strictly less than the dimension of $Y$.

The collection of all critical points is also called the critical locus of $f$.

The $f$-image of a critical point is known as a critical value. A point in $Y$ that is not a critical value is known as a regular value.

A formalization of this in cohesive geometry is at cohesive (infinity,1)-topos – infinitesimal cohesion – critical locus.

Applications

• Critical loci are used to study topology in terms of Morse theory.

• Critical loci of functionals on jet bundles are studied in variational calculus. See also at shell.

Related concepts

• Morse–Sard theorem

• extremum

• locus

• derived critical locus

Rereferences

• René Thom, p. 18 of: Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)