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微积分手机版配套参考书:无穷小微积分基础

(2018-08-31 08:16:09)

微积分手机版配套参考书:无穷小微积分基础

新学年就要开始了,一些担任00后一年级新生微积分必修课的数学教员敢于“吃螃蟹”,使用电子投影仪讲授微积分手机版教材。

我们与你们始终站在一起。未了给你们加油,我们把J.Keisler撰写的微积分手机版配套参考书(无穷小微积分基础,2007年版本)投放给你们,以便讲课时更有“底气”,不转向。

    预祝你们成功!

 袁萌   陈启清   831  

附:无穷小微积分基础的作者前言

PREFACE

In 1960 Abraham Robinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based on innitesimals. Robinson’s achievement was one of the major mathematical advances of the twentieth century. This is an exposition of Robinson’s innitesimal calculus at the advanced undergraduate level. It is entirely self-contained but is keyed to the 2000 digital edition of my rst year college text Elementary Calculus: An Innitesimal Approach [Keisler 2000] and the second printed edition [Keisler 1986]. Elementary Calculus: An Innitesimal Approach is available free online at   www.math.wisc.edu/Keisler/calc. This monograph can be used as a quick introduction to the subject for mathematicians, as background material for instructors using the book Elementary Calculus, or as a text for an undergraduate seminar(讨论班). This is a major revision of the rst edition of Foundations of Innitesimal Calculus [Keisler 1976], which was published as a companion to the rst (1976) edition of Elementary Calculus, and has been out of print for over twenty years. A companion to the second (1986) edition of Elementary Calculus was never written. The biggest changes are: (1) A new chapter on dierential equations, keyed to the corresponding new chapter in Elementary Calculus. (2) The axioms for the hyperreal number system are changed to match those in the later editions of Elementary Calculus. (3) An account of the discovery of Kanovei and Shelah [KS 2004] that the hyperreal number system, like the real number system, can be built as an explicitly denable mathematical structure. Earlier constructions of the hyperreal number system depended on an arbitrarily chosen parameter such as an ultralter. The basic concepts of the calculus were originally developed in the seventeenth and eighteenth centuries using the intuitive notion of an innitesimal, culminating in the work of Gottfried Leibniz (1646-1716) and Isaac Newton (1643-1727). When the calculus was put on a rigorous basis in the nineteenth century, innitesimals were rejected in favor of the ε,δ approach, because mathematicians had not yet discovered a correct treatment of innitesimals. Since then generations of students have been taught that innitesimals do not exist and should be avoided.

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