Keith Stroyan

This is a complete free text on the foundations of calculus using infinitesimals.  It is written at an upper undergraduate or beginning graduate level.

Table of Contents

A Brief Three Part Introduction to Calculus using Infinitesimals

1. Intuitive Proofs

  1. -Extreme Value Theorem

  2. -Infinitesimal Microscope

  3. -Fundamental Theorem

  4. -Continuity of f’[x]

  5. -Trig, Polar Coordinates

  6. -Holditch’s Formula

  7. -Leibniz’ Curvature

  8. -History of Calculus

  1. 2.Keisler’s Foundations

  2. -Small, Medium, & Large

  3. -Function Extension

  4. -Rigorous Extreme Values

  5. -Microscopic Tangency

  6. -Rigorous Integration

  7. -Inverse Function Theorem

  8. -Second Differences

  9. -Higher Order Smoothness

3. Local Linearity

  1. -Explicit Tangents

  2. -Implicit Tangents

  3. -Implicit Functions

  4. -Parametric Tangents

  5. -Coordinate Systems

  6. -Vector Fields

  7. -Stokes’ Theorem

  8. -Local ODE Stability

Mathematical Background: Foundations of Infinitesimal Calculus


H Jerome Keisler

Keisler writes:

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 limits.

The First Edition of this book was published in 1976, and a revised Second Edition was published in 1986, both by Prindle, Weber & Schmidt. The book is now out of print and the copyright has been returned to me as the author. I have decided (as of September 2002) to make the book available for free in electronic form at this site. These PDF files were made from the printed Second Edition. (2012 Dover Edition)


H.Jerome Keisler

This monograph is a companion to the online edition of Keisler’s textbook ``Elementary Calculus: An Approach Using Infinitesimals''. It can be used as a quick introduction to the infinitesimal approach to calculus for mathematicians, as background material for instructors, or as a text for an undergraduate seminar.