mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

that asserts that the set of

critical value Critical value or threshold value can refer to: * A quantitative threshold in medicine, chemistry and physics * Critical value (statistics), boundary of the acceptance region while testing a statistical hypothesis * Value of a function at a crit ...

s (that is, the

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

of the set of critical points) of a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

''f'' from one

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

to another is a

null set In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notio ...

, i.e., it has

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

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, let :

f\colon \mathbb^n \rightarrow \mathbb^m

C^k

, (that is,

k

times continuously differentiable), where

k\geq \max\

. Let

X \subset \mathbb R^n

denote the '' critical set'' of

f,

which is the set of points

x\in \mathbb^n

at which the

Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...

f

has rank

.  Then the

f(X)

has Lebesgue measure 0 in

\mathbb^m

. Intuitively speaking, this means that although

X

may be large, its image must be small in the sense of Lebesgue measure: while

f

may have many critical ''points'' in the domain

\mathbb^n

, it must have few critical ''values'' in the image

\mathbb^m

. More generally, the result also holds for mappings between

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...

M

and

N

of dimensions

m

and

n

, respectively. The critical set

X

of a

C^k

function :

f:N\rightarrow M

consists of those points at which the differential :

df:TN\rightarrow TM

has rank less than

m

as a linear transformation. If

k\geq \max\

, then Sard's theorem asserts that the image of

X

has measure zero as a subset of

M

. This formulation of the result follows from the version for Euclidean spaces by taking a

countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

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 In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

Variants

There are many variants of this lemma, which plays a basic role in

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

among other fields. The case

m=1

was proven by Anthony P. Morse in 1939, and the general case by Arthur Sard in 1942. A version for infinite-dimensional Banach manifolds was proven by Stephen Smale. The statement is quite powerful, and the proof involves analysis. In

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

it is often quoted — as in the

Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...

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

f:N\rightarrow M

C^\infty

and if

A_r\subseteq N

is the set of points

x\in N

such that

df_x

has rank less or equal than

r

, then the

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...

f(A_r)

is at most

r

Statement

Variants

See also

References

Further reading