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Infinitesimal Calculus (Dover Books on Mathematics) Dover Ed Edition

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James M. Henle (Author)
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James M. Henle (Author),
Eugene M. Kleinberg (Author)
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Eugene M. Kleinberg (Author)
4.8 out of 5 stars 20 customer reviews
ISBN-13: 080-0759428861
ISBN-10: 0486428869
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 144 pages
  • Publisher: Dover Publications; Dover Ed edition (July 22, 2003)
  • Language: English
  • ISBN-10: 0486428869
  • ISBN-13: 978-0486428864
  • Product Dimensions: 9.3 x 6 x 0.3 inches
  • Shipping Weight: 8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (20 customer reviews)
  • Amazon Best Sellers Rank: #238,128 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

83 of 88 people found the following review helpful
Very good for what it does, but doesn't do enough
By Bruce R. Gilson on August 22, 2003
Format: Paperback Verified Purchase
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.Read more ›
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52 of 53 people found the following review helpful
Keisler's book
By anonymous on November 9, 2004
Format: Paperback
Just a quick footnote to Gilson's excellent review. Keisler's out-of-print book is available for free online at:

[...]
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41 of 44 people found the following review helpful
The hidden wonders of Calculus revealed at last!!!
By A Customer on February 6, 2003
Format: Hardcover
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.Read more ›
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21 of 23 people found the following review helpful
Great second book on calculus
By Ken Braithwaite on February 28, 2006
Format: Paperback
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.
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14 of 17 people found the following review helpful
Fascinating Topic. Excellent Text Layout. Good Problems.
By Michael Wischmeyer on October 8, 2007
Format: Paperback
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.Read more ›
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Most Recent Customer Reviews

Five Stars

Very clear, should be the first book of anyone who want to know nonstandard analysis.

Published 1 month ago by changyau
Very Readable

Once you have taken an introductory analysis course, you should read this book. It contains exactly what every math major should know about non-standard analysis. Read more

Published 3 months ago by Rick Ferraro
Buy the printed edition, NOT the Kindle edition.

Outstanding book. I have nothing to add to previous reviews on content. But, DO NOT BUY THE KINDLE VERSION.

I have both the paperback and Kindle editions. Read more

Published 10 months ago by John Graber
But what book couldn't "do more"?

This is an amazingly good book, one which takes you by the hand and introduces you to the great mysteries of the hyperreals and granulars, as well as Nonstandard Analysis in... Read more

Published 22 months ago by William D. Fusfield
A very clear and illustrative exposition of a very important topic.

This is a very clear and easy to understand exposition of the hyperreals and infinitesimal calculus developed by Robinson. It contains good examples and graphical illustrations.

Published 23 months ago by numericalsciences
Superb

Reasonably clear for a higher math book, and as a bonus an amazingly entertaining review of a great deal of mathematics in a concise space, while introducing the main subject... Read more

Published on November 5, 2013 by Daniel L Pratt
amazing...

An excellent introductory treatise on the subject. Glad to see the early notions of calculus finally put on a firm theoretical footing - and then described to the student in an... Read more

Published on March 10, 2013 by Eric P. Patty
too little and too much

Since Keisler's wonderful Elementary Calculus: An Infinitesimal Approach is now available for free online, I see little reason to recommend Henle to anyone. Read more

Published on October 22, 2012 by Benjamin Crowell
A mathematical pearl for self studying, or just for relaxing and...

I've since the starting in school 58 ago been using calculation and mathematic and always liked mathematic and historic more than anything else. Read more

Published on July 2, 2012 by Gert Bo Thorgersen
The best introduction to nonstandard analysis

Since Robinson's pioneering work on nonstandard analysis in the early 1960s, many books have written on the subject including Robinson's own text Non-standard Analysis. Read more

Published on May 25, 2012 by ScienceThinker

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