Sard's theorem: Difference between revisions

Revision as of 00:40, 13 January 2008

Statement

Suppose $M$ and $N$ are differential manifolds and $f : M \to N$ is a smooth map between them. Then, the set of regular values of $f$ is a subset of measure zero in $N$ .

Applications

Suppose $M$ and $N$ are differential manifolds, and the dimension of $M$ is strictly less than the dimension of $N$ . Then, if $f : M \to N$ is a smooth map, the image $f (M)$ has measure zero as a subset of $N$ . In particular, $f$ cannot be surjective.

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 Suppose <math>M</math> and <math>N</math> are [[differential manifold]]s and <math>f:M \to N</math> is a [[smooth map]] between them. Then, the set of [[regular value]]s of <math>f</math> is a [[measure zero subset of a differential manifold|subset of measure zero]] in <math>N</math>.
+==Applications==
+Suppose <math>M</math> and <math>N</math> are differential manifolds, and the dimension of <math>M</math> is strictly less than the dimension of <math>N</math>. Then, if <math>f:M \to N</math> is a [[smooth map]], the image <math>f(M)</math> has measure zero as a subset of <math>N</math>. In particular, <math>f</math> cannot be surjective.