# 纪念微积分名著上线两周年

2020-03-24 09:10阅读：

纪念微积分名著上线两周年

毫无疑问，国内没有一本微积分教科书（电子版）可以上线，提供在线学习，

实在可悲也。

两年前，陈启清工程师把数学世界名著J.Keisler教师的“Elementary
Calculus ”上传到“无穷小微积分”基础数学网站，至今已经两年了。

当今，提倡在选学习，这是一件好事。

请见本文附件。

袁萌 陈启清
3月23日

附件：

PREFACE TO THE FIRST EDITION

The calculus was originally developed using the intuitive
concept of an infinitesimal, or an infinitely small number. But for
the past on

e hundred years infinitesimals have been banished from the calculus
course for reasons of mathematical rigor. Students have had to
learn the subject without the original intuition. This calculus
book is based on the work of Abraham Robinson, who in 1960 found a
way to make infinitesimals rigorous. While the traditional course
begins with the difficult limit concept, this course begins with
the more easily understood infinitesimals. It is aimed at the
average beginning calculus student and covers the usual three or
four semester sequence. The infinitesimal approach has three
important advantages for the student. First, it is closer to the
intuition which originally led. to the calculus. Second, the
central concepts of derivative and integral become easier for the
student to understand and use. Third, it teaches both the
infinitesimal and traditional approaches, giving the student an
extra tool which may become increasingly important in the future.
Before describing this book, I would like to put Robinson's work in
historical perspective. In the 1670's, Leibniz and Newton developed
the calculus based on the intuitive notion of infinitesimals.
Infinitesimals were used for another two hundred years, until the
first rigorous treatment of the calculus was perfected by
Weierstrass in the 1870's. The standard calculus course of today is
still based on the 'a, 6 definition' of limit given by Weierstrass.
In 1960 Robinson solved a three hundred year old problem by giving
a precise treatment of the calculus using infinitesimals.
Robinson's achievement will probably rank as one of the major
mathematical advances of the twentieth century. Recently,
infinitesimals have had exciting applications outside mathematics,
notably in the fields of economics and physics. Since it is quite
natural to use infinitesimals in modelling physical and social
processes, such applications seem certain to grow in variety and
importance. This is a unique opportunity to find new uses for
mathematics, but at present few people are prepared by training to
take advantage of this opportunity. Because the approach to
calculus is new, some instructors may need additional background
material. An instructor's volume, 'Foundations of
Infinitesimal

PREFACE TO THE FIRST EDITION v

Calculus,' gives the necessary background and develops the
theory in detail. The instructor's volume is keyed to this book but
is self-contained and is intended for the general mathematical
public. This book contains all the ordinary calculus topics,
including the traditional hmit definition, plus one exua tool-the
infinitesimals. Thus the student will be prepared for more advanced
courses as they are now taught. In Chapters 1 through 4 the basic
concepts of derivative, continuity, and integral are developed
quickly using infinitesimals. The traditional limit concept is put
off until Chapter 5, where it is motivated by approximation
problems. The later chapters develop transcendental functions,
series, vectors, partial derivatives, and multiple .integrals. The
theory differs from the traditional course, but the notation and
methods for solving practical problems are the same. There is a
variety of applications to both natural and social sciences. I have
included the following innovation for instructors who wish to
introduce the transcendental functions early. At the end of Chapter
2 on derivatives, there is a section beginning an alternate track
on transcendental functions, and each of Chapters 3 through 6 have
alternate track problem sets on transcendental functions. This
alternate track can be used to provide greater variety in the early
problems, or can be skipped in order to reach the integral as soon
as possible. In Chapters 7 and 8 the transcendental functions are
developed anew at a more leisurely pace. The book is written for
average students. The problems preceded by a square box go somewhat
beyond the examples worked out in the text and are intended for the
more adventuresome. I was originally led to write this book when it
became clear that Robinson's infinitesimal calculus col}ld be made
available to college freshmen. The theory is simply presented; for
example, Robinson's work used mathematical logic, but this book
does not. I first used an early draft of this book in a
one-semester course at the University of Wisconsin in 1969. In 1971
a two-semester experimental version was published. It has been used
at several colleges and at Nicolet High School near Milwaukee, and
was tested at five schools in a controlled experiment by Sister
Kathleen Sullivan in 1972-1974. The results (in her 1974 Ph.D.
thesis at the University of Wisconsin) show the viability of the
infinitesimal approach and will be summarized in an article in the
American Mathematical Monthly. I am indebted to many colleagues and
students who have given me encouragement and advice, and have
carefully read and used various stages of the manuscript. Special
thanks are due to Jon Barwise, University of Wisconsin; G. R.
Blakley, Texas A & M University; Kenneth A. Bowen, Syracuse
University; William P. Francis, Michigan Technological University;
A. W. M. Glass, Bowling Green University; Peter Loeb, University of
Illinois at Urbana; Eugene Madison and Keith Stroyan, University of
Iowa; Mark Nadel, Notre Dame University; Sister Kathleen Sullivan,
Barat College; an