Topology from the Differentiable ViewpointPrinceton University Press, 14. 12. 1997 - Počet stran: 64 This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. |
Obsah
Smooth manifolds and smooth maps | 1 |
The theorem of Sard and Brown | 10 |
Proof of Sards theorem | 19 |
Oriented manifolds | 26 |
Vector fields and the Euler number | 32 |
16 | 54 |
20 | 60 |
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1-manifold boundary points boundaryless manifold Brouwer C₁ clearly codimension compact manifold completes the proof corresponds critical points critical values defined DEFINITION deg(f degree of f denote derivative df diffeomor diffeomorphism differential topology Euler number example f is smooth f¹(y f¹(z finite fixed point follows framed manifold framed submanifold ƒ and g homotopic to g homotopy classes Hopf hyperplane identity map isolated zero Let f linear mapping M C R manifold ƒ˜¹(y manifold of dimension manifold with boundary map f Math measure zero mod 2 degree nondegenerate nonsingular one-one open set open subset parametrization by arc-length Pontryagin manifold positively oriented basis PROBLEM proof of Theorem regular value Sard's theorem set of critical smooth homotopy smooth manifold smooth map smoothly homotopic smoothly isotopic stereographic projection subspace sufficiently small tangent space unit ball unit sphere V₁ value of f vector field vector space
Oblíbené pasáže
Strana 59 - Debreu, G., Theory of Value, New York, Wiley, 1959. [6] Dhrymes, PJ, On a Class of Utility and Production Functions Yielding Everywhere Differentiable Demand Functions, Review of Economic Studies, 34(1967), 399-408.