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
8月31日
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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