Critical point (math)

A continuously differentiable mapping between two differentiable manifolds has a critical or stationary point at one point if the differential is not surjective there . In the one-dimensional case, this means that their derivative is 0 there. Otherwise it is a regular point . If there are one or more critical points in the archetype of a point, it is called the critical or stationary value , otherwise: regular value .

definition

Let it be an open set and a continuously differentiable function. ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\rightarrow \mathbb {R} ^{m}}$

A value is called the critical or stationary point of if it is not surjective, that is, if it holds, where denotes the total differential . ${\displaystyle x_{0}\in U}$ ${\displaystyle f}$ ${\displaystyle \operatorname {D} f(x_{0})}$ ${\displaystyle \operatorname {rang} \left(\operatorname {D} f\left([x_{0}]\right)\right)<m}$ ${\displaystyle \operatorname {D} f}$

One is critical or stationary value when there is a critical point with there. ${\displaystyle y\in \mathbb {R} ^{m}}$ ${\displaystyle x_{0}\in U}$ ${\displaystyle f(x_{0})=y}$

Examples

• The definition contains in particular the one-dimensional special case. If there is a continuously differentiable function, then there is a critical point of if and only if the derivative of vanishes at the point , i.e. it holds. For example , if the polynomial function is given, then exactly if is. So and are the critical points of . ${\displaystyle f\colon U\subseteq \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f'(x)=0}$ ${\displaystyle \textstyle f(x)=6x-{\frac {1}{2}}x^{2}-{\frac {1}{3}}x^{3}}$ ${\displaystyle f'(x)=6-x-x^{2}=0}$ ${\displaystyle x\in \{-3,2\}}$ ${\displaystyle x=2}$ ${\displaystyle x=-3}$ ${\displaystyle p}$
• A continuously differentiable real-valued mapping in real variables has a critical point at the point if and only if its gradient is equal to the zero vector at this point , i.e. if all partial derivatives disappear there: ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle x=(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}}$

${\displaystyle {\frac {\partial f}{\partial x_{1}}}(x)=\dotsb ={\frac {\partial f}{\partial x_{n}}}(x)=0}$.

properties

The set of critical points of a function can be large, for example every point in the archetype of a constant mapping is critical. According to the definition, every point is also critical if it holds, even in the case of immersion . ${\displaystyle n<m}$

The set of Sard , however, states that the set of critical values of a sufficiently differentiable mapping measure has zero; so there are “very few” critical values. At these points the principle of regular value fails: The archetype of a critical value is generally not a manifold .

degeneration

In the case of a real-valued function , the Hessian matrix can be used to determine whether it is a degenerate critical point. This is exactly the case if the Hessian matrix is ​​singular, i.e. not invertible . The Morse theory deals with functions without degenerate critical points .

If there is no degeneracy, it can also be determined with real-valued functions whether it is a local minimum , a local maximum or a saddle point of the function.

Individual evidence

1. ^ John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 , p. 113
2. ^ John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 , p. 132