Elementary Calculus: An Infinitesimal Approach

Elementary Calculus: An Infinitesimal Approach By H. Jerome Keisler This mathematics textbook presents calculus through the lens of hyperreal numbers and infinitesimals, offering an alternative to traditional calculus instruction. Published by Dover and available freely online, it provides students with a foundation in calculus using Abraham Robinson's hyperreal number system. The text introduces innovative teaching tools, including the concept of an infinite-magnification microscope and infinite-resolution telescope, to help visualize complex mathematical concepts. Through these pedagogical devices, Keisler explains fundamental calculus concepts like derivatives, integrals, and continuity using infinitesimal numbers rather than conventional ε–δ definitions. This book represents a significant departure from standard calculus texts, offering a perspective that aligns more closely with historical approaches to calculus while incorporating modern mathematical rigor. The work's emphasis on infinitesimals provides students with an intuitive path to understanding calculus concepts before transitioning to traditional methods.

Readers value this textbook's clear explanations of calculus concepts using infinitesimals rather than limits. Many note it makes calculus more intuitive and accessible for beginners. Likes: - Rigorous but understandable approach - Well-structured progression of topics - Helpful examples and exercises - Free PDF availability - Practical applications included Dislikes: - Some readers found the infinitesimal approach confusing if they later took standard calculus courses - Limited availability of physical copies - Few solution guides or supplementary materials - Some sections could use more practice problems Ratings: Goodreads: 4.3/5 (12 ratings) Amazon: 4.5/5 (11 ratings) Notable review: "This book made calculus click for me in a way that epsilon-delta proofs never did. The infinitesimal approach feels more natural." - Goodreads user Another reader noted: "Good for self-study but may not align with standard college courses that use the limit-based approach."

Calculus Made Easy by Silvanus P. Thompson This text presents calculus through simple explanations and practical examples using infinitesimals, sharing Keisler's focus on intuitive understanding over formal proofs.

Non-standard Analysis by Abraham Robinson The foundational text that developed the hyperreal number system forms the mathematical basis for Keisler's approach to calculus.

A Primer of Infinitesimal Analysis by John L. Bell This book develops calculus using nilpotent infinitesimals within a synthetic differential geometry framework, providing an alternative infinitesimal approach to Keisler's methods.

Infinitesimal Calculus by James M. Henle and Eugene M. Kleinberg The text builds calculus from infinitesimals using a construction similar to Keisler's, while incorporating more advanced topics in analysis.

Analysis by Its History by E. Hairer and G. Wanner This work presents calculus through its historical development, including the original infinitesimal methods that align with Keisler's perspective.

🔍 Keisler's textbook was one of the first modern calculus books to successfully implement Abraham Robinson's 1960s Non-standard Analysis approach, making complex concepts more approachable for beginners. 🎓 The book was initially banned from the University of California, Berkeley's mathematics department in the 1970s due to controversy over its non-traditional approach, but later gained acceptance in many institutions. 📚 The metaphor of an "infinite-magnification microscope" used throughout the book was inspired by Leibniz's original vision of calculus, which also employed the concept of infinitesimals. 🌐 Though originally published in 1976, the book gained renewed attention when it became freely available online in 2012, making it one of the earliest comprehensive calculus textbooks to embrace open access. 🧮 The hyperreal number system used in the book actually includes numbers that are infinitely large and infinitely small, extending the real number line in ways that match how early calculus pioneers thought about these concepts.