A Primer of Infinitesimal AnalysisCambridge University Press, 7. apr. 2008 - 124 strani One of the most remarkable recent occurrences in mathematics is the re-founding, on a rigorous basis, the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this new and updated edition, basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of 'zero-square', or 'nilpotent' infinitesimal - that is, a quantity so small that its square and all higher powers can be set, to zero. The systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the "infinitesimal" methods figuring in traditional applications of the calculus to physical problems - a number of which are discussed in this book. This edition also contains an expanded historical and philosophical introduction. |
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Appendix assertion assigns associated sheaf functor axioms of BSIA Bell called Chapter classical commutative composable composite concept considered construct models containing contravariant coordinate ring correspond covering system covers f define elements enlarge exponential object extension fact fails false following properties formulated free variables function Grothendieck topos identified identity image infer infinitely internal language interpretation intuitionistic logic J-covering sieve know locally true manifold map f map ƒ map g McLarty means microneighbourhood microobjects Models for smooth models of BSIA Moerdijk natural transformation neighbourhood object F objects A,B open covers opposite category pair of objects particular point presheaf F presheaf topos principles real line recall representative result ring homomorphism satisfy sends sense SetA sheaves shown to follow smooth infinitesimal analysis smooth line smooth map sort statement subcategory subobject Suitable Theorem theory truth unique well-adapted model Yoneda embedding