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Difference between revisions of "Level set"

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''of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582201.png" />''
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''of a function $f$ on $\mathbf{R}^n$''
  
The set of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582202.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582203.png" />. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582204.png" /> is given on a square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582205.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582206.png" /> and has partial derivatives there which also satisfy a [[Lipschitz condition|Lipschitz condition]], then for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582207.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582208.png" /> the level set
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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
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$$
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M_c = \{ x \in Q \ :\ f(x) = c \}
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$$
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consists of a finite number of regular curves (on them, $\mathrm{grad}\,f \ne 0$). Cf. [[Sard theorem]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l0582209.png" /></td> </tr></table>
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consists of a finite number of regular curves (on them, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058220/l05822010.png" />). Cf. [[Sard theorem|Sard theorem]].
 

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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article