Foundations of Infinitesimal Calculus by K.D. Stroyan

About this book :-
"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 Detail :-
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 Author :-
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.

Book 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

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