Infinitesimal Calculus

by Eugene M. Kleinberg and James M. Henle

Book Overview

Introducing calculus at the basic level, this text covers hyperreal numbers and hyperreal line, continuous functions, integral and differential calculus, fundamental theorem, infinite sequences and series, infinite polynomials, more. 1979 edition.

• Format: Paperback
• Language: English
• ISBN: 0486428869
• ISBN13: 9780486428864
• Release Date: July 2003
• Publisher: Dover Publications
• Length: 144 Pages
• Weight: 0.50 lbs.
• Dimensions: 0.3" x 6.0" x 9.3"

Customer Reviews

A 24 carat gem in a tiny package.

I got this book from the library expecting to get a brief review of the Calculus. After reading the text I ordered it right away. This is an exceptionally beautiful book that gives one a rigorous understanding of real analysis. During my read I missed fully understanding portions of the text so I had to read some parts more than once. Hence it took me about a month to finish the 144 page book amongst other readings. If you are looking for a great Calc book with a goodly amount of rigor and you have the time and inclination to keep at it this book will not let you down. What amazes me is how much of import and beauty is cramped within the 144 pages of this text. In comparing this book to many of the huge over-priced, glitzy Calculus textbooks; so prevalent today, this book gives one a ton of return for the money and the effort expended in understanding its contents. Dover Books has done us a great service once again by making such a valuable and excellent work so affordable and in a great presentation format with regards to font size, diagrams etc. What a tremendous bargain. Definitely a 'must have'.

Simply the best!

One of the best math books I've ever come across, and one of the few for infinitesimals. It has an excellent format and very nice extra info to the sides it almost reads like a novel, get it and then download keisler's infinitesimal book free!

Fascinating Topic. Excellent Text Layout. Good Problems.

The short text, Infinitesimal Calculus (1979) by James M. Henle and Eugene M. Kleinberg, is a fascinating and enjoyable introduction to nonstandard analysis. The two authors use a precise and rigorous definition of the intuitively attractive concept of the infinitesimal as the basis for a new and exciting look at calculus. I have encountered few mathematics book that I have enjoyed as much. The authors indicate that the only prerequisite assumed for their book is a good foundation in high school mathematics. Be that as it may, a reader will find a year or two of standard calculus (and even a class in real analysis) to be helpful. Note that Henle and Kleinberg focus on the proofs of calculus theorems, not the techniques used in solving problems. In the introductory chapter Henle and Kleinberg provide a concise overview of infinitesimals, wetting the reader's appetite for this easier approach to proving calculus theorems. But we quickly discover that we need to know a little bit about mathematical logic, language, and structure in order to develop hyperreal numbers and the hyperreal line. Only then can we begin using infinitesimals in our proofs. The discussion of continuous functions is consequently deferred to chapter 5. Thereafter, this little text moves along in a familiar pattern: Continuous Functions are followed by Integral Calculus (chapter 6), Differential Calculus (7), The Fundamental Theorem (8), Infinite Sequences and Series (9), and finally Infinite Polynomials (10). Chapter 11 is The Topology of the Real Line, essentially the classical theorems of real analysis. Lastly, chapter 12, Standard Calculus and Sequences of Functions, examines the relationship between the nonstandard definitions and the standard ones, as well proving some deeper theorems of analysis with the help of hyperhyperreal numbers (no, this is not a typo). The page layout is quite attractive. The basic text occupies the rightmost two-thirds of each page; the leftmost third is used for clarification of details, example problems, and historical notes. Exercises are found throughout the chapters. Few problems are difficult, but using infinitesimals does require some practice. On occasion the authors slip a little humor into the problems. The authors confidently expected the infinitesimal approach (developed by Abraham Robinson in the 1960s) to replace the more traditional (and cumbersome) epsilon-delta methodology as a foundation for calculus. Nonetheless, with the clear exception of H. Jerome Keisler's text, Elementary Calculus: An Approach Using Infinitesimals (first edition 1976; 2nd edition 1986), nearly all texts still use the epsilon-delta approach. An earlier reviewer pointed out that Keisler kindly posted his text online. I recently checked and it is still available.

Keisler's book

Just a quick footnote to Gilson's excellent review. Keisler's out-of-print book is available for free online at: http://www.math.wisc.edu/~keisler/calc.html

The hidden wonders of Calculus revealed at last!!!

If one is to buy into Plato's theory of perfect forms, then I must say that this comes infinitesimally close to being a "perfect introductory Calculus book". I couldn't help but get the impression that this was a book that was crafted to be enjoyed. Even without looking at the content, its physical properties are admirable. It's much smaller than those over-size Calculus textbooks you're used to lugging around in school, yet the print is large enough that it's easilly readable. The organization is quite impressive. The book allows you to delve into the complexities of hyperreals from the get-go, or skip the technicalities and still understand enough of the concepts to apply to the rest of the book. But the most remarkable trait of this book is that it is actually entertaining!!! Not because it consists of a lot of lame jokes that detract from the book's mathematical content as other "friendly Calculus" books sometimes do, but because the authors actually appear to be competent writers as well as mathematicians! Background is intermixed with theory, and in the midst of it, you'll find lots of interesting little anecdotes interwoven in the sidebars that enlighten your perspective of mathematical concepts and the personalities of the matematicians who discovered them. Content-wise, the book is completely rigourous, concise, and very consistent. It's such a tiny book that I was sure that it must have skipped something important, but comparing it to the much longer long-winded Spivak book, I couldn't find anything missing...except epsilons and deltas. That of course is the main goal of the book, to take the traditional introductory material of a first-year Calculus class and apply the techniques of Nonstandard Analysis, which were discovered in the last few decades. The result is that the authors have created a very concrete and rigorous treatment of Calculus that has all of the traditional uglyness removed from it. The authors even provide one epsilon-delta proof in the beginning, just to show how much more cumbersome it is compared to their elegant hyperreal system. The system itself is very abstract, but the authors take us to the point where we can see that abstraction and intuition do converge! Amazing. My only warning about this book is that it may not help you very much with your current curriculum, simply because the approaches it uses are so different than norm. Most of the topics in this book are not even covered during undergraduate studies, much less a first year class. If this book was actually used as a textbook for a real math course, I'd be the first to enroll!