Elementary Calculus: An Infinitesimal Approach

This book is concerned with the infinitesimal approach originally set forth by Newton and Leibnitz, using Non-standard analysis. The author has moved the theoretical material from Chapter One to an Appendix in this edition. A new chapter on differential equations has been added and the transcendental functions have been fully integrated into the first section. This book should be of interest to first and second year undergraduate mathematics students.
 

About the Authors
    H. Jerome Keisler is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.

    His Ph.D. advisor was Alfred Tarski at Berkeley; his dissertation is Ultraproducts and Elementary Classes (1961).

Title Elementary Calculus: An Infinitesimal Approach
 

Author(s) H. Jerome Keisler
 

Publisher: Prindle Weber & Schmidt; 2 Sub edition (December 1985), Revised in February, 2012
 

Hardcover 913 pages
 

Language: English
 

ISBN-10: 0871509113
 

ISBN-13: 978-0871509116

eBook: http://www.math.wisc.edu/~keisler/calc.html

Mathematical Analysis II

 This final text in the Zakon Series on Mathematics Analysis follows the release of the author's Basic Concepts of Mathematics and Mathematical Analysis I and completes the material on Real Analysis that is the foundation for later courses in functional analysis, harmonic analysis, probability theory, etc.

The first chapter extends calculus to n-dimensional Euclidean space and, more generally, Banach spaces, covering the inverse function theorem, the implicit function theorem, Taylor expansions, etc. Some basic theorems in functional analysis, including the open mapping theorem and the Banach-Steinhaus uniform boundedness principle, are also proved.

The text then moves to measure theory, with a complete discussion of outer measures, Lebesgue measure, Lebesgue-Stieltjes measures, and differentiation of set functions. The discussion of measurable functions and integration in the following chapter follows an innovative approach, carefully choosing one of the equivalent definitions of measurable functions that allows the most intuitive development of the material. Fubini's theorem, the Radon-Nikodym theorem, and the basic convergence theorems (Fatou's lemma, the monotone convergence theorem, dominated convergence theorem) are covered.

Finally, a chapter relates antidifferentiation to Lebesgue theory, Cauchy integrals, and convergence of parametrized integrals.

Nearly 500 exercises allow students to develop their skills in the area. 

Title Mathematical Analysis II
Author(s) Elias Zakon
Publisher: Trillia Group, 2009.
Hardcover 424 pages
Language: English
ISBN-10: 1-931705-03-8 

eBook: http://www.trillia.com/zakon-analysisII.html

Elementary Calculus: An Infinitesimal Approach

 This book is concerned with the infinitesimal approach originally set forth by Newton and Leibnitz, using Non-standard analysis. The author has moved the theoretical material from Chapter One to an Appendix in this edition. A new chapter on differential equations has been added and the transcendental functions have been fully integrated into the first section. This book should be of interest to first and second year undergraduate mathematics students.

Title Elementary Calculus: An Infinitesimal Approach
Author(s) H. Jerome Keisler
Publisher: Prindle Weber & Schmidt; 2 Sub edition (December 1985), Revised in February, 2012
Hardcover 913 pages
Language: English
ISBN-10: 0871509113
ISBN-13: 978-0871509116
eBook: http://www.vias.org/calculus/