Sard's theorem: Difference between revisions

Revision as of 22:15, 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.

This also shows that a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension

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	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.		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.

			This also shows that [[manifold is not union of images of manifolds of smaller dimension\|a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension]]