Sard’s theorem
Theorem.
Let $U$ be an open set in ${R}^{m}$, $V$ be an open set in ${R}^{n}$ and $f\mathrm{:}U\mathrm{\to}V$ be a ${C}^{k}$ function^{}. If $k\mathrm{\ge}m\mathrm{/}n$, then the set of critical values of $f$ has $n$-dimensional measure zero^{}.
References
- 1 S.G. Krantz, H. R. Parks, The Implicit Function Theorem^{}: History, Theory, and Applications, Birkhäuser, Boston, c. 2002
- 2 H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969
Title | Sard’s theorem |
---|---|
Canonical name | SardsTheorem |
Date of creation | 2013-03-22 16:12:33 |
Last modified on | 2013-03-22 16:12:33 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 7 |
Author | Mathprof (13753) |
Entry type | Theorem |
Classification | msc 57R35 |