Elementary Calculus: An Infinitesimal Approach

On-line Edition. Copyright 2000 by H. Jerome Keisler, revised March 2024.

This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via epsilon, delta definitions.

The First Edition of this book was published in 1976, and a revised Second Edition was published in 1986, both by Prindle, Weber & Schmidt. When the Second Edition became out of print, the copyright was returned to me as the author. In September 2002 I decided to make the book available for free in electronic form at this site. These PDF files were made from the printed Second Edition, and are continually being revised with minor corrections.

A Third Edition of this book was published by Dover Publications, Inc. in 2012, with the agreement that this online version will continue to be freely available. This gives you the choice of downloading this free version or purchasing the printed book.

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

A picture of the cover

The whole book in one large file (28 megabytes)

Single chapters in much smaller files:

Preface to First and Second Editions

Contents and Introduction

Chapter 1 Real and Hyperreal Numbers

Chapter 2 Differentiation

Chapter 3 Continuous Functions

Chapter 4 Integration

Chapter 5 Limits, Analytic Geometry, and Approximations

Chapter 6 Applications of the Integral

Chapter 7 Trigonometric Functions

Chapter 8 Exponential and Logarithmic Functions

Chapter 9 Infinite Series

Chapter 10 Vectors

Chapter 11 Partial Differentiation

Chapter 12 Multiple Integrals

Chapter 13 Vector Calculus

Chapter 14 Differential Equations

Appendix and Index