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 Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... Chemistry In chemistry, a critical point is the conditions ( temperature, pressure) at which the liquid state of the matter ceases to exist. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

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 In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

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. For non-mathematical singularity theories, see singularity. ...