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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 set of critical values (that is, the image 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. == Statement == 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 image 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 second countable 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sard's theorem」の詳細全文を読む スポンサード リンク
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