## Smooth Manifolds and ObservablesThe author isvery pleased that hisbook, ?rst publishedinRussian in2000 by MCCME Publishers, is now appearing in English under the auspices of such a truly classical publishing house as Springer-Verlag. In this edition several pertinent remarks by the referees (to whom the author expresses his gratitude) were taken into account, and new exercises were added (mostly) to the ?rst half of the book, thus achieving a better balancewiththesecond half.Besides, sometyposandminorerrors,noticed in the Russian edition, were corrected. Weare extremely grateful to all our readers whoassisted usinthistiresomebughunt.Weareespeciallygrateful toA.DeParis,I.S.Krasil’schik,andA.M.Verbovetski,whodemonstrated their acute eyesight, truly of degli Lincei standards. The English translation was carried out by A.B. Sossinsky (Chapters 1–8), I.S. Krasil’schik(Chapter 9), and S.V. Duzhin (Chapters 10–11)and reduced to a common denominator by the ?rst of them; A.M. Astashov prepared new versions of the ?gures; all the T X-nical work was done by E M.M. Vinogradov. In the process of preparing this edition, the author was supported bythe Istituto Nazionaledi Fisica Nucleare and the Istituto Italianoper gliStudi Filoso?ci. It is only thanks to these institutions, and to the e?cient help of Springer-Verlag, that the process successfully came to its end in such a short period of time. Jet Nestruev Moscow–Salerno April 2002 Preface The limits of my language are the limits of my world. |

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### Contents

Introduction | 1 |

Cutoff and Other Special Smooth Functions on R | 13 |

Algebras and Points | 21 |

Smooth Manifolds Algebraic Definition | 37 |

Charts and Atlases | 53 |

Smooth Maps | 65 |

Equivalence of Coordinate and Algebraic Definitions | 77 |

Spectra and Ghosts | 85 |

The Differential Calculus as a Part of Commutative Algebra | 95 |

Smooth Bundles | 143 |

Vector Bundles and Projective Modules | 163 |

Observability Principle Set Theory and the Foundations of Mathematics | 209 |

### Common terms and phrases

A-module algebra of smooth algebraic definition arbitrary atlas atlases bijective Boolean algebra boundary called chart coincides commutative algebra compact configuration space consider construction coordinate Corollary corresponding cotangent defined denoted described Diff diffeomorphism differential calculus differential operator direct sum dual space elements equations equivalence Example Exercise fact fiber finite formula function f functor geometric R-algebra given hence homomorphism identified isomorphism jets Jl(M Lemma linear mathematics matrix maximal ideals measuring devices Mobius band module of sections morphism multiplication natural neighborhood notion observability open set operator of order point z C M polynomial projective modules proof Proposition prove quotient R-algebra homomorphism R-linear R-points reader scalar smooth algebra smooth envelope smooth functions smooth manifold smooth map spectrum structure subalgebra subbundle submanifold subset Suppose surjective tangent bundle tangent space tangent vector theorem topology total space vanishes vector bundle vector field vector space zero