Sard's theorem

In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the of the set of critical points) 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. More explicitly, let be , (that is, times continuously differentiable), where . Let denote the critical set of which is the set of points at which the Jacobian matrix of has rank . Then the has Lebesgue measure 0 in . Intuitively speaking, this means that although may be large, its image must be small in the sense of Lebesgue measure: while may have many critical points in the domain , it must have few critical values in the image . More generally, the result also holds for mappings between differentiable manifolds and of dimensions and , respectively. The critical set of a function consists of those points at which the differential has rank less than as a linear transformation. If , then Sard's theorem asserts that the image of has measure zero as a subset of . 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. There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case 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.

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Related concepts (4)

Critical point (mathematics)

Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Sard's theorem