Sard's lemma
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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 → Rm
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 Rm. 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. A version for infinite-dimensional Banach manifolds was proved by Stephen Smale.
