Elementary Calculus: An Infinitesimal Approach

by H. Jerome Keisler

This first-year calculus book is centered around the use of infinitesimals. It contains all the ordinary calculus topics, including the basic concepts of the derivative, continuity, and the integral, plus traditional limit concepts and approximation problems. Additional subjects include transcendental functions, series, vectors, partial derivatives, and multiple integrals. 2007 edition.

Genres: Mathematics, Calculus, Nonfiction, Science

1008 pages, Paperback

First published January 1, 1976

Reviews

[Janez Perme · Rating 5 out of 5]

It is a definitely a good book for a beginner in calculus. It shows certain elementary stuff, that one just needs to know, before diving deep into mathematical analysis. I wouldn't say that it is the only resource for calculus that one should own, but it definitely gets you going. If I would be a true beginner in math, I would start with this one. If you are planning to read this piece, by all means, do it slowly and unconditionally understand every step you make. Mind you: there are some proofs stated in the book, but you won't find proof for Stokes theorem, for instance. Also, differential equations and series are only glossed over. But as I said, enough to learn basics and get you ready for next steps in learning this wonderful science called math. Keisler is a good teacher and a good writer.

[Juan Pablo Spaventa · Rating 5 out of 5]

My goal is to finish this book by the time a vaccine is made for coronavirus. Let's go!

[Daniel Lyons · Rating 4 out of 5]

Tremendous! If you have struggled with Calculus because it seemed so unintuitive, try this book. The infinitesimal approach is much more intuitive and, as we now know, just as expressive and powerful as the limit approach. This is the first calculus book I've found that I've been able to self-study and make progress with.

[Robert Murphy · Rating 4 out of 5]

Keisler has taken the Abraham Robison approach to Analysis and made it an accessible college calculus textbook. This tome is good for two semesters of college (Calc I & II) or two years of high school (AB & BC Calculus).
However, it is a bit dated at times. The explanations are geared towards engineers from the 60's and the problems are extremely uneven in difficulty at times. Teachers, assign odds until you know what you're getting into (I have no answer key and one might not exist)!
The infinitesimal approach is so much more intutive and coincides with the way calculus was developed in history. Limits should wait until chapter 3, instead of wrecking students from the very beginning. I have experienced great success teaching from this book, and I highly recommend it, if you are able to supplement with your own material at times.

[Jandle Bearer · Rating 5 out of 5]

I never understood why infinitesimals were not considered rigorous, even rough handling of them leads to them to the same conclusions as limit calculus, it couldn't just be a coincidence that they work. Finding this book was vindicating in that respect.
This book provides a very clear and intuitive description of infinitesimal calculus, which falls within the domain of Non-standard analysis; which, by the way, has been shown to be logically equivalent to Real analysis. You will not learn about the construction of the hyperreal numbers, or Model theory, you will (like with limit calculus) learn the tools for using calculus, in this case transfer and standardisation. like limit calculus books you not learn higher analysis, using mathematical logic, constructing the number system and such, you will learn only the tools you need; but like the use of limits, infinitesimal calculus's tools make sense, in that you don't need higher analysis to justify to yourself using it or to make sense of it.
While this book is the go to resource for infinitesimal calculus if you are interested in gaining a basic understanding of its construction I would recommend 'Infinitesimal calculus' by Henle which provides an approachable construction of the hyppereal's and proof of the transfer theorem; there is also 'Foundations of infinitesimal calculus' by Keisler which covers Standardisation and slightly more rigorous proofs for calculus, both single and multivariate (including vector calculus).

[Nedam · Rating 1 out of 5]

"the student should be able to apply the calculus himself in new situations"

"A student can get a score of 100t/(t + 1) on his math exam"

Since I and the other 49.584% (according to the World Bank) female human beings have been summarily kicked out of this book as potential math students, I have no choice but to move on to another book that will allow for my existence.

I dedicate this review to Hypatia, (born c. 355 CE—murdered 415, Alexandria) - mathematician, astronomer, and philosopher.

[Muhammad al-Khwarizmi · Rating 3 out of 5]

The infinitesimal-based approach is different but, in my opinion, not superior. It's just another way of looking at things. This is a pretty solid text for those who want a more theoretical introduction to calculus. If you want a scientist's or engineer's perspective on the topic (i.e. applied calculus but still with a decent understanding of what you are doing), I recommend looking into a decent engineering mathematics book.