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 $dX:TX\to TY$ vanishes.

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

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 spaces are studied in variational calculus.

Related concepts

• extremum

• derived critical locus