## Smooth Manifolds and ObservablesSmooth Manifolds and Observables is about the differential calculus, smooth manifolds, and commutative algebra. While these theories arose at different times and under completely different circumstances, this book demonstrates how they constitute a unified whole. The motivation behind this synthesis is the mathematical formalization of the process of observation in classical physics. The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. This unique textbook contains a large number of exercises and is intended for advanced undergraduates, graduate students, and research mathematicians and physicists. |

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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