Sard's theorem
Template:Short description In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard.
StatementEdit
More explicitly,<ref name="Sard1942">Template:Citation</ref> let
- <math>f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m</math>
be <math>C^k</math>, (that is, <math>k</math> times continuously differentiable), where <math>k\geq \max\{n-m+1, 1\}</math>. Let <math>X \subset \mathbb R^n</math> denote the critical set of <math>f,</math> which is the set of points <math>x\in \mathbb{R}^n</math> at which the Jacobian matrix of <math>f</math> has rank <math><m</math>. Then the image <math>f(X)</math> has Lebesgue measure 0 in <math>\mathbb{R}^m</math>.
Intuitively speaking, this means that although <math>X</math> may be large, its image must be small in the sense of Lebesgue measure: while <math>f</math> may have many critical points in the domain <math>\mathbb{R}^n</math>, it must have few critical values in the image <math>\mathbb{R}^m</math>.
More generally, the result also holds for mappings between differentiable manifolds <math>M</math> and <math>N</math> of dimensions <math>m</math> and <math>n</math>, respectively. The critical set <math>X</math> of a <math>C^k</math> function
- <math>f:N\rightarrow M</math>
consists of those points at which the differential
- <math>df:TN\rightarrow TM</math>
has rank less than <math>m</math> as a linear transformation. If <math>k\geq \max\{n-m+1,1\}</math>, then Sard's theorem asserts that the image of <math>X</math> has measure zero as a subset of <math>M</math>. This formulation of the result follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism.
VariantsEdit
There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case <math>m=1</math> was proven by Anthony P. Morse in 1939,<ref>Template:Citation</ref> and the general case by Arthur Sard in 1942.<ref name="Sard1942" />
A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.<ref>Template:Citation</ref>
The statement is quite powerful, and the proof involves analysis. In topology it is often quoted — as in the Brouwer fixed-point theorem and some applications in Morse theory — in order to prove the weaker corollary that “a non-constant smooth map has at least one regular value”.
In 1965 Sard further generalized his theorem to state that if <math>f:N\rightarrow M</math> is <math>C^\infty</math> and if <math>A_r\subseteq N</math> is the set of points <math>x\in N</math> such that <math>df_x</math> has rank less or equal than <math>r</math>, then the Hausdorff dimension of <math>f(A_r)</math> is at most <math>r</math>.<ref>Template:Citation and also Template:Citation</ref><ref>Template:Citation</ref>