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Synthetic Differential Geometry 2nd Edition by Anders Kock ISBN 0521687381 9780521687386

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Synthetic Differential Geometry 2nd Edition Anders Kock Digital Instant Download

Author(s): Anders Kock
ISBN(s): 9780521687386, 0521687381
Edition: 2
File Details: PDF, 1.11 MB
Year: 2006
Language: english
SKU: EB-2032134 Category: Tag:

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ISBN 10: 0521687381 
ISBN 13: 9780521687386
Author: Anders Kock

Synthetic Differential Geometry is a method of reasoning in differential geometry and differential calculus, based on the assumption of sufficiently many nilpotent elements on the number line, in particular numbers d such that d2=0. The use of nilpotent elements allows one to replace the limit processes of calculus by purely algebraic calculations and notions. For the first half of the book, first published in 2006, familiarity with differential calculus and abstract algebra is presupposed during the development of results in calculus and differential geometry on a purely axiomatic/synthetic basis. In the second half basic notions of category theory are presumed in the construction of suitable Cartesian closed categories and the interpretation of logical formulae within them. This is a second edition of Kock’s classical text from 1981. Many notes have been included, with comments on developments in the field from the intermediate years, and almost 100 new bibliographic entries have been added.

Synthetic Differential Geometry 2nd Table of contents:

Part I: The Synthetic Theory

  1. Basic structure on the geometric line

  2. Differential calculus

  3. Taylor formulae – one variable

  4. Partial derivatives

  5. Taylor formulae – several variables

  6. Some important infinitesimal objects

  7. Tangent vectors and the tangent bundle

  8. Vector fields

  9. Lie bracket

  10. Directional derivatives

  11. Functional analysis – Jacobi identity

  12. The comprehensive axiom

  13. Order and integration

  14. Forms and currents

  15. Currents – Stokes’ theorem

  16. Weil algebras

  17. Formal manifolds

  18. Differential forms in terms of simplices

  19. Open covers

  20. Differential forms as quantities

  21. Pure geometry

Part II: Categorical Logic

  1. Generalized elements

  2. Satisfaction (1)

  3. Extensions and descriptions

  4. Semantics of function objects

  5. Axiom 1 revisited

  6. Comma categories

  7. Dense class of generators

  8. Satisfaction (2)

  9. Geometric theories

Part III: Models

  1. Models for axioms 1, 2, and 3

  2. Models for ε-stable geometric theories

  3. Well-adapted models (1)

  4. Well-adapted models (2)

  5. The algebraic theory of smooth functions

  6. Germ-determined T∞-algebras

  7. The open cover topology

  8. Construction of well-adapted models

  9. Manifolds with boundary

  10. Field property – germ algebras

  11. Order and integration in Cahiers topos

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Tags: Anders Kock, Synthetic Differential