Inmathematics,Sard's theorem, also known asSard's lemma or theMorse–Sard theorem, is a result inmathematical analysis that asserts that the set ofcritical values (that is, theimage of the set ofcritical points) of asmooth functionf from oneEuclidean space ormanifold to another is anull set, i.e., it hasLebesgue measure 0. This makes the set of critical values "small" in the sense of ageneric property. The theorem is named forAnthony Morse andArthur Sard.

More explicitly,^[1] let

${\displaystyle f:{\mathbb{R}}^n\to {\mathbb{R}}^m}$

be${\displaystyle C^k}$, (that is,${\displaystyle k}$ timescontinuously differentiable), where${\displaystyle k\ge max\{n-m+1,1\}}$. Let${\displaystyle X\subset {\mathbb{R}}^n}$ denote thecritical set of${\displaystyle f,}$ which is the set of points${\displaystyle x\in {\mathbb{R}}^n}$ at which theJacobian matrix of${\displaystyle f}$ hasrank. Then theimage${\displaystyle f(X)}$ has Lebesgue measure 0 in${\displaystyle {\mathbb{R}}^m}$.

Intuitively speaking, this means that although${\displaystyle X}$ may be large, its image must be small in the sense of Lebesgue measure: while${\displaystyle f}$ may have many criticalpoints in the domain${\displaystyle {\mathbb{R}}^n}$, it must have few criticalvalues in the image${\displaystyle {\mathbb{R}}^m}$.

More generally, the result also holds for mappings betweendifferentiable manifolds${\displaystyle M}$ and${\displaystyle N}$ of dimensions${\displaystyle m}$ and${\displaystyle n}$, respectively. The critical set${\displaystyle X}$ of a${\displaystyle C^k}$ function

${\displaystyle f:N\to M}$

consists of those points at which thedifferential

${\displaystyle df:TN\to TM}$

has rank less than${\displaystyle m}$ as a linear transformation. If${\displaystyle k\ge max\{n-m+1,1\}}$, then Sard's theorem asserts that the image of${\displaystyle X}$ has measure zero as a subset of${\displaystyle M}$. This formulation of the result follows from the version for Euclidean spaces by taking acountable 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 underdiffeomorphism.

There are many variants of this lemma, which plays a basic role insingularity theory among other fields. The case${\displaystyle m=1}$ was proven byAnthony P. Morse in 1939,^[2] and the general case byArthur Sard in 1942.^[1]

A version for infinite-dimensionalBanach manifolds was proven byStephen Smale.^[3]

The statement is quite powerful, and the proof involves analysis. Intopology it is often quoted — as in theBrouwer fixed-point theorem and some applications inMorse theory — in order to prove the weaker corollary that “a non-constant smooth map hasat least one regular value”.

In 1965 Sard further generalized his theorem to state that if${\displaystyle f:N\to M}$ is${\displaystyle C^{\mathrm{\infty }}}$ and if${\displaystyle A_r\subseteq N}$ is the set of points${\displaystyle x\in N}$ such that${\displaystyle d{f}_x}$ has rank less or equal than${\displaystyle r}$, then theHausdorff dimension of${\displaystyle f(A_r)}$ is at most${\displaystyle r}$.^[4][5]

• Generic property

• Hirsch, Morris W. (1976),Differential Topology, New York: Springer, pp. 67–84,ISBN 0-387-90148-5.
• Sternberg, Shlomo (1964),Lectures on Differential Geometry, Englewood Cliffs, NJ:Prentice-Hall,MR 0193578,Zbl 0129.13102.

• Lemmas in mathematical analysis
• Smooth functions
• Multivariable calculus
• Singularity theory
• Theorems in mathematical analysis
• Theorems in differential geometry
• Theorems in measure theory

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