Difference between revisions of "Level set"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX done) |
||
| Line 1: | Line 1: | ||
| − | ''of a function | + | ''of a function $f$ on $\mathbf{R}^n$'' |
| − | The set of points in | + | The set of points in $\mathbf{R}^n$ on which $f= \text{const}$. If the function $f$ is given on a square $Q$ of the plane $\mathbf{R}^2$ and has partial derivatives there which also satisfy a [[Lipschitz condition]], then for almost-all $c$ in the interval $\min f \le c \le \max f$ the level set |
| + | $$ | ||
| + | M_c = \{ x \in Q \ :\ f(x) = c \} | ||
| + | $$ | ||
| + | consists of a finite number of regular curves (on them, $\mathrm{grad}\,f \ne 0$). Cf. [[Sard theorem]]. | ||
| − | + | {{TEX|done}} | |
| − | |||
| − | |||
Latest revision as of 19:57, 3 September 2017
of a function $f$ on $\mathbf{R}^n$
The set of points in $\mathbf{R}^n$ on which $f= \text{const}$. If the function $f$ is given on a square $Q$ of the plane $\mathbf{R}^2$ and has partial derivatives there which also satisfy a Lipschitz condition, then for almost-all $c$ in the interval $\min f \le c \le \max f$ the level set $$ M_c = \{ x \in Q \ :\ f(x) = c \} $$ consists of a finite number of regular curves (on them, $\mathrm{grad}\,f \ne 0$). Cf. Sard theorem.
How to Cite This Entry:
Level set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Level_set&oldid=41809
Level set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Level_set&oldid=41809
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article