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Synthetic Differential Geometry

Synthetic Differential Geometry

2nd Edition


Part of London Mathematical Society Lecture Note Series

  • Date Published: June 2006
  • availability: Available
  • format: Paperback
  • isbn: 9780521687386

£ 64.99

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About the Authors
  • 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.

    • Straightforward easy to read style with many exercises
    • No knowledge of differential geometry is presupposed
    • A much quoted classic now in 2nd edition, the two layers of the book (1981 and 2006) are clearly distinguished
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    Product details

    • Edition: 2nd Edition
    • Date Published: June 2006
    • format: Paperback
    • isbn: 9780521687386
    • length: 246 pages
    • dimensions: 229 x 152 x 14 mm
    • weight: 0.37kg
    • contains: 4 b/w illus. 142 exercises
    • availability: Available
  • Table of Contents

    Preface to the second edition (2005)
    Preface to the first edition (1981)
    Part I. The Synthetic Ttheory:
    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 epsilon-stable geometric theories
    3. Well-adapted models (1)
    4. Well-adapted models (2)
    5. The algebraic theory of smooth functions
    6. Germ-determined T-infinity-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

  • Author

    Anders Kock, Aarhus Universitet, Denmark
    Anders Kock is an Associate Professor of Mathematics at the University of Aarhus, Denmark.

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