Sard s lemma

Sard's lemma, also known as Sard's theorem or the Morse-Sard theorem, is a result of mathematical analysis characterising the image of the critical points of a smooth function F from one Euclidean space to another as having Lebesgue measure 0 (and so small, in a definite sense). More precisely, if

F: Rⁿ → R^m

is smooth, and C is the critical set of F (the set in Rⁿ of the points x at which the Jacobian matrix of F has rank < m), then

F(C)

has measure 0, for the usual measure on R^m. Here C can be the whole of Rⁿ, for example, when n < m; but in that case the image will be small in the sense of measure.

There are many variants on this lemma, which plays a basic role in singularity theory amongst other fields. The case m = 1 was proved by A. P. Morse in 1939, and the general case by Arthur Sard in 1942.

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