Regular value

Regular values and regular points are objects from differential geometry . Regular dots are used, among other things, in the definition of a submersion , important properties of regular values ​​follow from the theorem of regular values or Sard's theorem .

definition

Assume and be smooth manifolds and a -time differentiable map. A point is called the regular value of if for each the differential is surjective . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle r}$ ${\displaystyle n\in N}$ ${\displaystyle f}$ ${\displaystyle m\in f^{-1}(n)}$ ${\displaystyle D_{m}f}$

Trivially, every point of that is not in the image of is also a regular value. ${\displaystyle N}$ ${\displaystyle f}$

A point for which is surjective is called a regular point . If the differential is not surjective, one speaks of a critical point , and in the case of the image point, of a critical value. ${\displaystyle m}$ ${\displaystyle D_{m}f}$ ${\displaystyle D_{m}f}$ ${\displaystyle f(m)}$

literature

• Konrad Königsberger : Analysis Volume 2. 3rd revised edition. Springer, Berlin et al. 2000, ISBN 3-540-66902-7 .
• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis and Applications (= Applied Mathematical Sciences 75). Springer, New York NY 1988, ISBN 0-387-96790-7 .