Critical point (mathematics)

A continuously differentiable map between two differentiable manifolds has at one point a critical or stationary point if there is not surjective differential. Otherwise, it is a regular point. Is there one or several critical points in the inverse image of a point is called critical or stationary him value, otherwise: regular value.

Definition

It should be an open set and a continuously differentiable function. Value is a critical or stationary point if and only if it is not injective, that is, if applies, the total differential respectively. Is a critical or steady-state value, if there is a critical point.

Examples

  • In particular, the definition includes the one-dimensional special case. Is a continuously differentiable function, clearly then a critical point of when the derivative of at the point disappears, so true. For example, the polynomial given as accurately applies if is. So are and the critical points of.
  • A continuously differentiable real-valued mapping in real variable has exactly one critical point, then at the point when at this point its gradient is equal to the zero vector, so if there all partial derivatives vanish:

Properties

The set of critical points of a function can be great, for example, each point in the archetype of a constant map is critical. By definition, every point is critical if, even in the case of immersion.

The set of Sard says, however, that the set of critical values ​​of a sufficiently differentiable mapping has degree zero; so there are " very few " critical values ​​. At these locations, 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 can be determined using the Hessian matrix, whether it is a degenerate critical point. This is exactly the case when the Hessian matrix is singular, that is not invertible is. With features without degenerate critical points dealt the Morse theory.

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

  • Analysis
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