Foundations of Infinitesimal Calculus by K.D. Stroyan

Foundations of Infinitesimal Calculus - Table of Contents

PART 1: NUMBERS AND FUNCTIONS
1. Numbers
2. Functional Identities

PART 2: LIMITS
3. The Theory of Limits
4. Continuous Functions

PART 3: VARIABLE DIFFERENTIATION
5. The Theory of Derivatives
6. Pointwise Derivatives
7. The Mean Value Theorem
8. Higher Order Derivatives

PART 4: INTEGRATION
9. Basic Theory of the Definite Integral

PART 5: MULTIVARIABLE DIFFERENTIATION
10. Derivatives of Multivariable Functions

PART 6: DIFFERENTIAL EQUATIONS
11. Theory of Initial Value Problems

PART 7: INFINITE SERIES
12. The Theory of Power Series
13. The Theory of Fourier Series

What You Will Learn in Foundations of Infinitesimal Calculus

"Foundations of Infinitesimal Calculus" by "K. D. Stroyan" presents a rigorous approach to calculus based on "infinitesimals", offering an alternative to the traditional epsilon–delta framework. Using ideas from "nonstandard analysis", the book rebuilds core calculus concepts such as limits, continuity, derivatives, and integrals from first principles. This approach reflects the intuitive methods originally envisioned by early mathematicians while maintaining full mathematical rigor. A central strength of the book is its clear explanation of infinitesimal reasoning. Stroyan introduces infinitely small and infinitely large numbers in a systematic way, showing how classical results like the Mean Value Theorem and Taylor’s theorem follow naturally. The text also covers advanced topics such as higher-order derivatives, improper integrals, and elements of "multivariable calculus", all within the infinitesimal framework. Designed for advanced undergraduates, graduate students, and readers interested in the foundations of calculus, the book emphasizes conceptual clarity and logical structure. It is especially valuable for those seeking a deeper understanding of calculus through "nonstandard analysis", bridging intuition and rigor while offering fresh insight into classical results.

Book Details & Specifications

Title: Foundations of Infinitesimal Calculus by K.D. Stroyan
Publisher: University of Iowa
Year: 1997
Pages: 182
Type: PDF
Language: English
ISBN-10 #: 0126741506
ISBN-13 #: 978-0126741506
License: University Educational Resource
Amazon: Amazon

About the Author: Keith Duncan Stroyan

The author Keith Duncan Stroyan is an American mathematician best known for his work in "nonstandard analysis" and "infinitesimal calculus". He served as a professor at the University of Iowa, Iowa, United States and focused on building rigorous foundations for calculus using infinitesimals, blending intuition with formal mathematical structure. Stroyan is widely admired for his clear teaching style and contributions to "real analysis" and "mathematics education". His books and research help students understand deep concepts through precise reasoning, making him an influential figure in both "mathematical foundations" and advanced calculus instruction.

Free Calculus Books PDF | Download Textbooks & Solutions

Mathematics Book Categories