400px|thumb|right|The x-coordinates of the red circles are [[stationary points; the blue squares are inflection points.]]
In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a . Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not holomorphic). Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined).
This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points. In particular, if is a plane curve, defined by an implicit equation the critical points of the projection onto the parallel to the are the points where the tangent to are parallel to the that is the points where In other words, the critical points are those where the implicit function theorem does not apply.
Critical point of a single variable function
A critical point of a function of a single real variable, , is a value in the domain of where is not differentiable or its derivative is 0 (i.e. The image of a critical point under is a called a critical value. A point in the complement of the set of critical values is called a regular value. Sard's theorem states that the set of critical values of a smooth map has measure zero.
Some authors give a slightly different definition: a critical point of is a point of where the rank of the Jacobian matrix of is less than . With this convention, all points are critical when .
These definitions extend to differential maps between differentiable manifolds in the following way. Let <math>f:V \to W</math> be a differential map between two manifolds and of respective dimensions and . In the neighborhood of a point of and of , charts are diffeomorphisms <math>\varphi : V \to \R^m</math> and <math>\psi : W \to \R^n.</math> The point is critical for if <math>\varphi(p)</math> is critical for <math>\psi \circ f \circ \varphi^{-1}.</math> This definition does not depend on the choice of the charts because the transitions maps being diffeomorphisms, their Jacobian matrices are invertible and multiplying by them does not modify the rank of the Jacobian matrix of <math>\psi \circ f \circ \varphi^{-1}.</math> If is a Hilbert manifold (not necessarily finite dimensional) and is a real-valued function then we say that is a critical point of if is not a submersion at .
Application to topology
Critical points are fundamental for studying the topology of manifolds and real algebraic varieties. In particular, they are the basic tool for Morse theory and catastrophe theory.
The link between critical points and topology already appears at a lower level of abstraction. For example, let <math>V</math> be a sub-manifold of <math>\mathbb R^n,</math> and be a point outside <math>V.</math> The square of the distance to of a point of <math>V</math> is a differential map such that each connected component of <math>V</math> contains at least a critical point, where the distance is minimal. It follows that the number of connected components of <math>V</math> is bounded above by the number of critical points.
In the case of real algebraic varieties, this observation associated with Bézout's theorem allows us to bound the number of connected components by a function of the degrees of the polynomials that define the variety.
See also
- Singular point of a curve
- Singularity theory
- Nullcline