Sard's theorem

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 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.

Contents 1 Statement 2 Variants 3 See also 4 References 5 Further reading Statement More explicitly,[1] let {displaystyle fcolon mathbb {R} ^{n}rightarrow mathbb {R} ^{m}} be {displaystyle C^{k}} , (that is, {displaystyle k} times continuously differentiable), where {displaystyle kgeq max{n-m+1,1}} . Let {displaystyle Xsubset mathbb {R} ^{n}} denote the critical set of {displaystyle f,} which is the set of points {displaystyle xin mathbb {R} ^{n}} at which the Jacobian matrix of {displaystyle f} has rank {displaystyle

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