# Elementary Calculus: An Infinitesimal Approach 3rd Edition

Book Preface

Elementary Calculus: An Infinitesimal Approach, Third Edition

This book contains all the usual topics in an elementary calculus course, plus one extra concept that makes the basic ideas more easily understood by beginning students. This concept is the infinitesimal, or infinitely small number. The calculus was originally developed in the 1670’s by Leibniz and Newton using infinitesimals in an informal way, and infinitesimals were given a rigorous foundation by Abraham Robinson in 1960.

The work is designed as the main textbook for a three or four semester calculus course. It has been used at both the college and high school levels. Because of its unique approach with infinitesimals, it can also serve as a supplementary book for courses based on other textbooks, and as an interesting refresher for those who have had calculus courses in the past. In addition, it can be a starting point for mathematicians, scientists, and engineers who want to learn about the recent use of infinitesimals beyond calculus.

The First Edition of this book was published in 1976, and a considerably revised Second Edition was published in 1986, both by Prindle, Weber, and Schmidt. When the Second Edition went out of print in 1992, the copyright was returned to me as the author. Since 2002, the Second Edition with minor revisions has been available free in digital form under a Creative Commons License. It can be downloaded at http://www..math.wisc.edu/~keisler/calc. It is listed by the California Digital Textbook Initiative as meeting all content standards for calculus.

Many people find it easier and more convenient to learn mathematics from a printed book. With this Third Edition, Dover Publications has given people the option of purchasing a high-quality affordable printed version of the free digital book.

I have written a higher-level companion to this book, Foundations of Infinitesimal Calculus (2007). This companion is intended for instructors and advanced students. It is a self-contained treatment of the mathematical background for this book, and can be a bridge to more advanced topics. It is freely available online at http://www.Math.wisc.edu/~keisler/foundations.

Since 1960, modern infinitesimals have been applied to many areas of mathematics, as well as to the physical sciences and economics. For the most part, these applications have been reported in professional journal articles and in proceedings of conferences, but I will mention two books on the subject. Nonstandard Methods in Stochastic Analysis and Mathematical Physics, by Sergio Albeverio, et. al., is currently available from Dover Publications. The forthcoming book Infinitesimal Methods in Mathematical Economics by Robert M. Anderson is available online at http://www.econ.berkeley.edu/~anderson/Book.pdf.

H. Jerome Keisler

June, 2011

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