Editing Sard's theorem (section)

== Statement ==
More explicitly,<ref name="Sard1942">{{citation | first=Arthur  | last=Sard  | author-link=Arthur Sard  | title=The measure of the critical values of differentiable maps  | url=http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07811-6/home.html  | journal=[[Bulletin of the American Mathematical Society]]  | volume=48   | year=1942   | issue=12   | pages=883–890   | mr= 0007523  | zbl= 0063.06720   | doi=10.1090/S0002-9904-1942-07811-6 |postscript=. | doi-access=free}}</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 point (mathematics)|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 of a matrix|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 manifold]]s <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 [[pushforward (differential)|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]].