Customer Reviews

Infinitesimal Calculus (Dover Books on Mathematics)

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The calculus was created, as many know, by Newton and Leibniz. Newton's concept of calculus was based on continuity, while Leibniz used a conceptual framework based on infinitesimals: numbers smaller than any real number, but less than zero. In the 19th century, a rigorous basis was established for Newton's conceptual framework, but it became an article of faith that infinitesimals could not be rigorously used as a basis for calculus. However, in the 20th century, a rigorous basis was established for an infinitesimal-based treatment of the calculus, as a result of Abraham Robinson's "nonstandard analysis." This involves expanding the real number system to a much larger number system, the "hyperreal number system."

In the physical sciences, it is common to use an intuitive treatment of calculus that includes infinitesimals; however, nearly all books on basic calculus avoid them and ignore Robinson's ideas. I only know of two exceptions: a book by H. J. Keisler (who edited Robinson's papers) and this one. Each has its advantages and disadvantages.

Keisler's book is unfortunately out of print and nearly unobtainable. It is a complete textbook of calculus, using the approach through nonstandard analysis. Its treatment of the hyperreal number system, however, I find hard to understand. By contrast, this book has a very much clearer treatment of the hyperreals; I think I finally understand how they are constructed after reading this book. But this book is _not_ a complete textbook of calculus. It covers the theory, and covers it extremely well, but does not even attempt to teach how to _use_ calculus. Therefore, it would not be appropriate as a sole textbook in a calculus class, for example.

I have read other work by Henle, and it is clear that his forte is explaining unusual number systems. He does a great job in this book at what he does. I just wish he had added more material on how to actually _use_ calculus. Unfortunately, the reader will have to augment this book by another, and since no other in-print book that I know of uses this nonstandard-analysis-based approach, there will be a disconnect if anyone tries to combine it with another 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

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!

This perhaps requires a little more mathematical maturity than one has in first year, but it is an exceptionally clear development of calculus with infinitesimals. The proofs and defintions are much simpler and more direct this way.

They also get into a bit of topology and compactness, which again is simpler than in traditional calculus.

Deals only with the theory of calculus, not with the applications.

You really need to learn epsilon-delta anyway, so this makes a great second book on calculus. Highly recommended.

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.

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'.

A true gem. Best intro to infinitesimals/non-standard analysis, accessible and "light" but nevertheless instructive and illuminating. Very well written, with clear and well-motivated arguments and a lot of historical and other comments running down a coloumn parallel to the main text.

This book is an extremely good & very readable introduction to HyperReal numbers & their applications to the infinitestimal calculus. The author clearly shows how to replace the somewhat clumsy epsilon-delta approach to calculus with "true" infinitestimals that are consistency defined HyperReal numbers. The book also clearly describes the logical underpinnings of the HyperReal construction.

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!

I have not read this book, as yet (though I have recently ordered it from Amazon), but I wanted to steer readers to some of my better finds on this subject. The most comprehensive math book for those wishing to go from say, basic fractions to college level trig and calculus, is a textbook called : "Technical Mathematics with Calculus" by Paul Calter (Prentice Hall). My copy is the second addition dated 1990. This is THE book for anyone who wishes to go from grade-school math up to college level. A big book at about 1000 pages, it's incredibly well done. I don't know if it's still in print, though.

One reader commented that the Keisler book is available free on-line. He is correct. In fact, I recently came across a site which is a treasure trove of numerous free e-books on quite a few subjects.
The site is called: (e-booksdirectory.com).

You should check it out--there's good stuff there. At this site I discovered two e-books which so impressed me at initial viewing, that I have ordered them in hardcopy from Amazon. They are "Calculus Without Limits" by John C. Sparks and "Calculus" by Benjamin Crowell. The first book seems to come at the subject in a novel and quirky style. Any deviation from a dry, formula approach is always welcome. The second book (which looks more accessible to beginners), reminds me of Silvanus Thompson's classic "Calculus Made Easy". The Keisler book is available (at the above site), as a free PDF download and reminds me somewhat of the well known Kline book, which I also have.
While ordering my two books from Amazon, I stumbled onto two more promising finds: "Infinitesimal Calculus" by James M. Henle and "The Calculus Lifesaver" by Adrian Banner. It was because of this discovery that I placed this comment here. I have not received the books yet, so I cannot comment on them, but they look like good bets. You may want to check them out, too.