Editing Sard's theorem

{{Short description|Theorem in mathematical analysis}}
In [[mathematics]], '''Sard's theorem''', also known as '''Sard's lemma''' or the '''Morse&ndash;Sard theorem''', is a result in [[mathematical analysis]] that asserts that the set of [[critical value (critical point)|critical value]]s (that is, the [[image (mathematics)|image]] of the set of [[critical point (mathematics)|critical point]]s) 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]].

== 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]].

== Variants ==
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>{{citation  | first= Anthony P.  | last=Morse   | author-link = Anthony Morse   | title=The behaviour of a function on its critical set  | journal=[[Annals of Mathematics]]  | volume=40  | issue=1  |date=January 1939  | pages=62–70  | jstor=1968544  | doi=10.2307/1968544  | bibcode=1939AnMat..40...62M | mr=1503449  |postscript=.}}</ref> and the general case by [[Arthur Sard]] in 1942.<ref name="Sard1942" />

A version for infinite-dimensional [[Banach manifold]]s was proven by [[Stephen Smale]].<ref>{{citation  | first=Stephen  | last=Smale  | author-link=Stephen Smale  | title=An Infinite Dimensional Version of Sard's Theorem  | journal=[[American Journal of Mathematics]] | volume=87   | year=1965  | pages=861–866  | jstor= 2373250  | doi=10.2307/2373250  | issue=4  | mr=0185604   | zbl=0143.35301 |postscript=. }}</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>{{citation  | first=Arthur  | last=Sard  | title=Hausdorff Measure of Critical Images on Banach Manifolds  | journal=[[American Journal of Mathematics]]  | volume=87  | year=1965  | pages=158–174  | doi=10.2307/2373229  | issue=1  | jstor=2373229  | mr=0173748  | zbl=0137.42501 }} and also {{Citation  | first=Arthur  | last=Sard  | title = Errata to ''Hausdorff measures of critical images on Banach manifolds''  | journal=[[American Journal of Mathematics]]  | volume=87  | year=1965  | pages=158–174  | issue=3  | jstor = 2373074  | doi = 10.2307/2373229  | mr = 0180649  | zbl = 0137.42501 |postscript=. }}</ref><ref>{{citation |title=Show that <var>f(C)</var> has Hausdorff dimension at most zero |date=July 18, 2013 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/446049 }}</ref>

==See also==
* [[Generic property#Definitions: topology|Generic property]]

==References==
{{Reflist}}

==Further reading==

* {{citation |first=Morris W. |last=Hirsch |author-link=Morris Hirsch |title=Differential Topology |location=New York |publisher=Springer |year=1976 |isbn=0-387-90148-5 |pages=67–84 |postscript=. }}
* {{citation | first= Shlomo  | last=Sternberg  | author-link=Shlomo Sternberg  | title=Lectures on Differential Geometry  | publisher=[[Prentice-Hall]] | place=Englewood Cliffs, NJ | year=1964  | mr = 0193578  | zbl = 0129.13102 |postscript=. }}

{{Manifolds}}
{{Measure theory}}

[[Category:Lemmas in mathematical analysis]]
[[Category:Smooth functions]]
[[Category:Multivariable calculus]]
[[Category:Singularity theory]]
[[Category:Theorems in mathematical analysis]]
[[Category:Theorems in differential geometry]]
[[Category:Theorems in measure theory]]