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Author : John Lane Bell
Publisher : Cambridge University Press
Page : 140 pages
File Size : 40,9 Mb
Release : 1998-07-28
Category : Mathematics
ISBN : 0521624010
This is the first elementary book to employ the concept of infinitesimals.
Author : John L. Bell
Publisher : Cambridge University Press
Page : 7 pages
File Size : 49,9 Mb
Release : 2008-04-07
Category : Mathematics
ISBN : 9780521887182
A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.
Author : Anonim
Publisher : Unknown
Page : 124 pages
File Size : 47,5 Mb
Release : 2008
Category : Calculus
ISBN : 0511370962
Author : E.I. Gordon,A.G. Kusraev,Semën Samsonovich Kutateladze
Publisher : Springer Science & Business Media
Page : 435 pages
File Size : 48,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401700634
Infinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics. The book sets forth the basics of the theory, as well as the most recent applications in, for example, functional analysis, optimization, and harmonic analysis. The concentric style of exposition enables this work to serve as an elementary introduction to one of the most promising mathematical technologies, while revealing up-to-date methods of monadology and hyperapproximation. This is a companion volume to the earlier works on nonstandard methods of analysis by A.G. Kusraev and S.S. Kutateladze (1999), ISBN 0-7923-5921-6 and Nonstandard Analysis and Vector Lattices edited by S.S. Kutateladze (2000), ISBN 0-7923-6619-0
Author : James M. Henle,Eugene M. Kleinberg
Publisher : Courier Corporation
Page : 144 pages
File Size : 55,9 Mb
Release : 2014-01-15
Category : Mathematics
ISBN : 9780486151014
Introducing calculus at the basic level, this text covers hyperreal numbers and hyperreal line, continuous functions, integral and differential calculus, fundamental theorem, infinite sequences and series, infinite polynomials, more. 1979 edition.
Author : Oswald Veblen & N. J. Lennes
Publisher : Unknown
Page : 128 pages
File Size : 50,6 Mb
Release : 2024-05-21
Category : Electronic
ISBN : OCLC:760704450
Author : N J Lennes,Oswald Veblen
Publisher : Unknown
Page : 236 pages
File Size : 49,8 Mb
Release : 2020-01-11
Category : Electronic
ISBN : 1659154324
A course dealing with the fundamental theorems of infinitesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician. It appears in the curriculum of nearly every university and is taken by students as "Advanced Calculus" in their last collegiate year, or as part of "Theory of Functions" in the first year of graduate work. This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources. The book may also be used as a basis for a rather short theoretical course on real functions, such as is now given from time to time in some of our universities.The general aim has been to obtain rigor of logic with a minimum of elaborate machinery. It is hoped that the systematic use of the Heine-Borel theorem has helped materially toward this end since by means of this theorem it is possible to avoid almost entirely the sequential division or "pinching" process so common in discussions of this kind. The definition of a limit by means of the notion "value approached" has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of limits...
Author : H. Jerome Keisler
Publisher : Unknown
Page : 968 pages
File Size : 53,5 Mb
Release : 1976
Category : Mathematics
ISBN : STANFORD:36105007502482
Author : John L. Bell
Publisher : Springer Nature
Page : 313 pages
File Size : 49,6 Mb
Release : 2019-09-09
Category : Mathematics
ISBN : 9783030187071
This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
Author : F. William Lawvere,Stephen H. Schanuel
Publisher : Cambridge University Press
Page : 409 pages
File Size : 41,6 Mb
Release : 2009-07-30
Category : Mathematics
ISBN : 9780521894852
This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists.
Author : Amir Alexander
Publisher : Simon and Schuster
Page : 368 pages
File Size : 53,5 Mb
Release : 2014-07-03
Category : Science
ISBN : 9781780745336
On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.
Author : Nader Vakil
Publisher : Cambridge University Press
Page : 587 pages
File Size : 45,5 Mb
Release : 2011-02-17
Category : Mathematics
ISBN : 9781107002029
A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.
Author : Oswald Veblen
Publisher : Legare Street Press
Page : 0 pages
File Size : 45,5 Mb
Release : 2022-10-27
Category : Electronic
ISBN : 1016649142
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Author : Moshe Machover
Publisher : Cambridge University Press
Page : 304 pages
File Size : 44,6 Mb
Release : 1996-05-23
Category : Mathematics
ISBN : 0521479983
This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations. A rigorous axiomatic presentation of Zermelo-Fraenkel set theory is given, demonstrating how the basic concepts of mathematics have apparently been reduced to set theory. This is followed by a presentation of propositional and first-order logic. Concepts and results of recursion theory are explained in intuitive terms, and the author proves and explains the limitative results of Skolem, Tarski, Church and Gödel (the celebrated incompleteness theorems). For students of mathematics or philosophy this book provides an excellent introduction to logic and set theory.
Author : Oswald Veblen
Publisher : Unknown
Page : 190 pages
File Size : 45,6 Mb
Release : 2019-09-05
Category : Electronic
ISBN : 169116500X
A course dealing with the fundamental theorems of infinitesimal calculus in a rigorousmanner is now recognized as an essential part of the training of a mathematician. Itappears in the curriculum of nearly every university, and is taken by students as "Advanced Calculus" in their last collegiate year, or as part of "Theory of Functions" in the first year of graduate work. This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources. The book may also be used as a basis for a rather short theoretical course on real functions, such as is now given from time to time in some of our universities.The general aim has been to obtain rigor of logic with a minimum of elaborate machinery.It is hoped that the systematic use of the Heine-Borel theorem has helped materially toward this end, since by means of this theorem it is possible to avoid almost entirely the sequential division or "pinching" process so common in discussions of this kind. The definition of a limit by means of the notion "value approached" has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of limits, and in general has largely decreased the number of "'s and _'s. The theory of limits is developed for multiple-valued functions, which gives certain advantages in the treatmentof the definite integral.