In particular, the numberings of Parts, equations, etc. Otherwise, the present edition is a re-typing of the old one, with only minor corrections, where necessary. A small Appendix D on this notion is therefore added. Most of the basic notions of synthetic differential geometry were already in the 1981 book the main exception being the general notion of “strong infinitesimal linearity” or “microlinearity”, which came into being just too late to be included. For the same reason, I have refrained from attempting an account of all the developments that have taken place since the First Edition only very minimal and incomplete pointers to the newer literature (1981– 2006) have been included as “Notes 2006” at the end of each of the Parts of the book. I do indeed intend to write a new book, but prefer it to be a sequel to the old one, rather than a rewriting of it. I realized that a rewriting would quickly lead to an almost new book. It is a compromise between a mere photographic reproduction of the First Edition, and a complete rewriting of it. Therefore I decided to bring out this Second Edition. I felt that there was still a need for the book, even though other accounts of the subject have in the meantime come into existence. The First Edition (1981) of “Synthetic Differential Geometry” has been out of print since the early 1990s. Models III.1 Models for Axioms 1, 2, and 3 III.2 Models for -stable geometric theories III.3 Axiomatic theory of well-adapted models (1) III.4 Axiomatic theory of well-adapted models (2) III.5 The algebraic theory of smooth functions III.6 Germ-determined T∞ -algebras III.7 The open cover topology III.8 Construction of well-adapted models III.9 W-determined algebras, and manifolds with boundary III.10 A field property of R and the synthetic role of germ algebras III.11 Order and integration in the Cahiers topos Appendices Bibliography Indexĩ6 97 98 102 107 112 114 120 122 126 129 129 136 141 146 152 162 168 173 179 190 196 204 220 227 Stokes’ Theorem Weil algebras Formal manifolds Differential forms in terms of simplices Open covers Differential forms as quantities Pure geometryġ 2 6 9 12 15 18 23 28 32 36 40 43 48 52 58 61 68 75 82 87 90 vĬontents Categorical logic II.1 Generalized elements II.2 Satisfaction (1) II.3 Extensions and descriptions II.4 Semantics of function objects II.5 Axiom 1 revisited II.6 Comma categories II.7 Dense class of generators II.8 Satisfaction (2) II.9 Geometric theories Application to proof of Jacobi identity The comprehensive axiom Order and integration Forms and currents Currents defined using integration.
#Differential geometry book that uses infinitesimals series#
Taylor series Some important infinitesimal objects Tangent vectors and the tangent bundle Vector fields and infinitesimal transformations Lie bracket – commutator of infinitesimal transformations Directional derivatives Functional analysis. Synthetic theory Basic structure on the geometric line Differential calculus Higher Taylor formulae (one variable) Partial derivatives Higher Taylor formulae in several variables. Preface to the Second Edition (2006) Preface to the First Edition (1981) I Synthetic Differential Geometry Second Edition 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. This is a second edition of Kock's classical text from 1981. 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.
For the first half of the book 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. The use of nilpotent elements allows one to replace the limit processes of calculus by purely algebraic calculations and notions. 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.