Foundations of Infinitesimal Calculus by Howard Jerome Keisler
Foundations of Infinitesimal Calculus - Table of Contents
1. Real and Hyperreal Numbers
2. Differentiation
3. Continuous Functions
4. Integration
5. Limits, Analytic Geometry, and Approximations
6. Applications of the Integral
7. Trigonometric Functions
8. Exponential and Logarithmic Functions
9. Infinite Series
10. Vectors
11. Partial Differentiation
12. Multiple Integrals
13. Vector Calculus
14. Differential Equations
What You Will Learn in Foundations of Infinitesimal Calculus
"Foundations of Infinitesimal Calculus" by Howard Jerome Keisler is a clear and rigorous guide to understanding "infinitesimal calculus" using the "hyperreal numbers". Unlike traditional calculus texts, it presents a logically sound approach to derivatives, integrals, and continuity with infinitesimals, providing an intuitive yet precise framework for advanced learners.
The book explores core concepts of calculus through the lens of "nonstandard analysis", offering an alternative to the epsilon-delta approach. Keisler develops the foundations step by step, explaining how infinitely small and large quantities can be used to rigorously define "derivatives" and "integrals". This approach makes the reasoning behind calculus more accessible and restores the intuitive methods originally used by pioneers like Leibniz and Newton, now formalized with modern mathematics.
Ideal for instructors, advanced undergraduates, or anyone studying "mathematical analysis", this text bridges the gap between elementary calculus and higher-level theory. It provides exercises and explanations that reinforce understanding and demonstrate the power of infinitesimal methods. "Foundations of Infinitesimal Calculus" equips readers with both a conceptual and technical grasp of calculus, preparing them to explore further topics in "nonstandard analysis" and advanced mathematics.
Book Details & Specifications
Title:
Foundations of Infinitesimal Calculus by Howard Jerome Keisler
Publisher:
University of Iowa
Year:
2022
Pages:
213
Type:
PDF
Language:
English
ISBN-10 #:
0871502151
ISBN-13 #:
978-0871502155
License:
CC BY-NC-SA 3.0
Amazon:
Amazon
About the Author: Howard Jerome Keisler
The author
Howard Jerome Keisler
is an American mathematician and professor emeritus at the "University of Wisconsin–Madison", known for his pioneering work in "Nonstandard Analysis" and mathematical logic. He studied under Alfred Tarski and has contributed significantly to rigorous approaches in modern calculus and model theory. His book "Foundations of Infinitesimal Calculus" introduces students to a "Foundational Theory" of calculus using "Infinitesimal Calculus", offering an alternative to traditional epsilon-delta methods.
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