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| '''Sard's theorem''', also known as '''Sard's lemma''' or the '''Morse–Sard theorem''', is a result in [[mathematical analysis]] which asserts that the [[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 has [[Lebesgue measure]] 0 – they form a [[null set]]. In particular, for [[real-valued function]]s, the set of the critical values, which belong to any bounded interval, is finite. This makes the set of critical values "small" in the sense of a [[generic property]]. It is named for [[Anthony Morse]] and [[Arthur Sard]].
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| == Statement ==
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| More explicitly ({{harvtxt|Sternberg|1964|loc=Theorem II.3.1}}; {{harvtxt|Sard|1942}}), let
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| :<math>f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m</math>
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| 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</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>.
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| 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>.
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| More generally, the result also holds for mappings between [[second-countable space|second countable]] [[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
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| :<math>f:N\rightarrow M</math>
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| consists of those points at which the [[pushforward (differential)|differential]]
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| :<math>df:TN\rightarrow TM</math>
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| 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]].
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| == Variants ==
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| 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 {{harv|Morse|1939}}, and the general case by [[Arthur Sard]] in 1942 {{harv|Sard|1942}}.
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| A version for infinite-dimensional [[Banach manifold]]s was proven by [[Stephen Smale]] {{harv|Smale|1965}}.
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| The statement is quite powerful, and the proof is involved analysis. In topology it is often quoted — as in the [[Brouwer fixed point theorem]] and some applications in [[Morse theory]] — in order to use the weaker corollary that “a non-constant smooth map has ''a'' regular value”, and sometimes “...hence also a regular point”.
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| In 1965 Sard further generalized his theorem to state that if <math>f:M\rightarrow N</math> is <math>C^k</math> for <math>k\geq \max\{n-m+1, 1\}</math> and if <math>A_r\subseteq M</math> is the set of points <math>x\in M</math> such that <math>df_x</math> has rank less than or equal to <math>r</math>, then <math>f(A_r)</math> has [[Hausdorff dimension]] at most <math>r</math>.
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| ==See also==
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| *[[Jacobian criterion]] / [[generic smoothness]], similar statements in [[algebraic geometry]]
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| ==References==
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| * {{citation
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| | first= Shlomo
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| | last=Sternberg
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| | authorlink=Shlomo Sternberg
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| | title=Lectures on differential geometry
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| | publisher=[[Prentice-Hall]]
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| | place=Englewood Cliffs, NJ
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| | year=1964
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| | pages = xv+390
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| | mr = 0193578
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| | zbl = 0129.13102
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| }}.
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| * {{citation
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| | first= Anthony P.
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| | last=Morse
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| | author-link = Anthony Morse
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| | title=The behaviour of a function on its critical set
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| | journal=[[Annals of Mathematics]]
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| | volume=40
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| | issue=1
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| |date=January 1939
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| | pages=62–70
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| | jstor=1968544
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| | doi=10.2307/1968544
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| | mr=1503449
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| | zbl=
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| }}.
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| * {{citation
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| | first=Arthur
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| | last=Sard
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| | author-link=Arthur Sard
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| | title=The measure of the critical values of differentiable maps
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| | url=http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07811-6/home.html
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| | journal=[[Bulletin of the American Mathematical Society]]
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| | volume=48
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| | year=1942
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| | issue=12
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| | pages=883–890
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| | mr= 0007523
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| | zbl= 0063.06720
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| | doi=10.1090/S0002-9904-1942-07811-6
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| }}.
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| * {{citation
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| | first=Arthur
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| | last=Sard
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| | author-link =
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| | title=Hausdorff Measure of Critical Images on Banach Manifolds
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| | journal=[[American Journal of Mathematics]]
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| | volume=87
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| | year=1965
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| | pages=158–174
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| | doi=10.2307/2373229
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| | issue=1
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| | jstor=2373229
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| | mr=0173748
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| | zbl=0137.42501
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| }} and also {{Citation
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| | first=Arthur
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| | last=Sard
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| | author-link =
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| | title = Errata to ''Hausdorff measures of critical images on Banach manifolds''
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| | journal=[[American Journal of Mathematics]]
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| | volume=87
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| | year=1965
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| | pages=158–174
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| | issue=3
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| | jstor = 2373074
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| | doi = 10.2307/2373229
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| | mr = 0180649
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| | zbl = 0137.42501
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| }}.
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| * {{citation
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| | first=Stephen
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| | last=Smale
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| | authorlink=Stephen Smale
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| | title=An Infinite Dimensional Version of Sard's Theorem
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| | journal=[[American Journal of Mathematics]]
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| | volume=87
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| | year=1965
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| | pages=861–866
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| | jstor= 2373250
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| | doi=10.2307/2373250
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| | issue=4
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| | mr=0185604
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| | zbl=0143.35301
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| }}.
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| [[Category:Lemmas]]
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| [[Category:Smooth functions]]
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| [[Category:Multivariable calculus]]
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| [[Category:Singularity theory]]
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| [[Category:Theorems in analysis]]
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| [[Category:Theorems in differential geometry]]
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| [[Category:Theorems in measure theory]]
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