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...

differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which...

Rokhlin's theorem (geometric topology) S–cobordism theorem (differential topology) Sard's theorem (differential geometry) Scott core theorem (3-manifolds)...

Zbl 0101.16103. Smale, S. (1965). "An infinite dimensional version of Sard's theorem". Amer. J. Math. 87 (4): 861–866. doi:10.2307/2373250. JSTOR 2373250...

theorem. Alternatively, the requirement that det(Dφ) ≠ 0 can be eliminated by applying Sard's theorem. For Lebesgue measurable functions, the theorem...

a single point of Sn. In the smooth case, it follows directly from Sard's Theorem. Therefore the homotopy group is the trivial group. When i = n, every...

requires more sophisticated results from mathematical analysis such as Sard's theorem and the coarea formula. In even greater generality, using the Lebesgue...

given above is the more commonly used one, e.g., in the formulation of Sard's theorem. Given a submersion f : M → N {\displaystyle f\colon M\to N} between...

complement of the set of critical values is called a regular value. Sard's theorem states that the set of critical values of a smooth map has measure zero...

embedding theorem Critical value Sard's theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes' theorem De Rham...

nullcline or some other curve describing terminal conditions. Using Sard's theorem, whose hypothesis is a special case of the transversality of maps, it...

a closed set of Lebesgue measure zero", as in Sard's theorem. There are many such genericity theorems. One example is the following: Genericity—For any...

then a generic point of N is not a critical value of f." (This is by Sard's theorem.) There are many different notions of "generic" (what is meant by "almost...

volume of the unit ball in R n . {\displaystyle \mathbb {R} ^{n}.} Sard's theorem Smooth coarea formula Federer, Herbert (1969), Geometric measure theory...

the field that has connections, for example, to rectifiability and Sard's theorem. Definition: Given f : R n → R m {\displaystyle f\colon \mathbb {R}...

topology and in spline interpolation. His fame stems primarily from Sard's theorem, which says that the set of critical values of a differential function...

Theory of Sets. He is also known for his work on the Morse–Sard theorem and the Federer–Morse theorem. Anthony Morse should not be confused with Marston Morse...

and diffeomorphism of manifolds are distinct properties. Although Sard's Theorem does not hold in general, every continuous map f : X → R n {\displaystyle...

differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions:...

(completing in a suitable Sobolev norm, applying the implicit function theorem and Sard's theorem for Banach manifolds, and using elliptic regularity to recover...

3297)  famous for Morse code, nor Anthony Morse, famous for the Morse–Sard theorem. Morse, Harold Marston (1924). "A fundamental class of geodesics on any...

has also been developed separately within differential topology using Sard's theorem. See for example: Guillemin & Pollack 1974, pp. 94−116 Shastri 2011...

together with Kronrod, he rediscovered Sard's lemma, unknown in USSR at the time. Later, he worked on uniqueness theorems for elliptic and parabolic differential...

saddle pointPages displaying wikidata descriptions as a fallback Sard's lemma – Theorem in mathematical analysisPages displaying short descriptions of redirect...

Berkeley; known for the Morse–Kelley set theory, Morse–Sard theorem and the Federer–Morse theorem John Mylopoulos (Sc.B. 1966) – Professor Emeritus of Computer...

the two Fundamental Theorems. Another method of proof of existence, global analysis, uses Sard's lemma and the Baire category theorem; this method was pioneered...

lines or circles are null sets in R 2 . {\displaystyle \mathbb {R} ^{2}.} Sard's lemma: the set of critical values of a smooth function has measure zero...

Fekete's lemma Fundamental lemma of the calculus of variations Hopf lemma Sard's lemma (singularity theory) Stechkin's lemma (functional and numerical analysis)...

boundary value problem, and then applies mean curvature flow and the Sard–Smale Theorem on regular values of Fredholm operators to prove a contradiction for...

set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory and real analytic geometry. Examples:...