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Sard's theorem (Theorem)
Theorem 1   Let $ U$ be an open set in $ R^m$, $ V$ be an open set in $ R^n$ and $ f: U \to V$ be a $ C^k$ function. If $ k \geq m/n$, then the set of critical values of $ f$ has $ n$-dimensional measure zero.

Bibliography

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



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Cross-references: measure zero, critical values, function, open set

This is version 4 of Sard's theorem, born on 2006-08-31, modified 2006-10-06.
Object id is 8305, canonical name is SardsTheorem2.
Accessed 896 times total.

Classification:
AMS MSC: 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings)
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