Customer Reviews

Calculus: The Elements

5 star:		(6)
4 star:		(0)
3 star:		(0)
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1 star:		(0)

 

 

I am at this time familiar with four other books which use infinitesimals in whole or in part to explain the calculus, an approach which was ruled out in the 19th century but has more recently been shown (by Abraham Robinson) to be mathematically justified, and an approach that is closer to the way most scientists USE calculus than the rigorously (epsilon/delta) defined limits of traditional calculus texts. One of those four, by H. J. Keisler (who edited Robinson's papers) is now out of print, though I have been informed that it is available by download on the Internet (of course, printing a book of over 1000 pages will add to your costs!). The other three have been earlier reviewed by me. Henle and Kleinberg's book, "Infinitesimal Calculus," is a slim book, which explains the concepts well, but does not go much into the applications or techniques. I gave it 4 stars, because what it does, it does well, but it doesn't do everything one might want in a calculus text. Bell's book, "A Primer of Infinitesimal Analysis," is harder going but covers more in the way of applications. Neither of the two provides a lot of practice in techniques. Sparks' book, "Calculus Without Limits: Almost," is a very intuitive approach, geared very much to the prospective working scientist, but short on the kind of proofs that mathematicians would insist on. Of the three, I think Sparks' book was, up to now, the best for teaching that I've seen. But after looking at Comenetz' book, I prefer it, provided that you are teaching relatively good students.

If your students just barely got through algebra, they would find this book a little rough going. Sparks' more intuitive approach might be better. But for a prospective physicist or engineer who is not too weak in math to see (and try his hand at) some proofs, I would recommend this book. One of the strong points is that Comenetz never strays far from the idea that calculus is not just slopes and areas, but can be applied to a lot of types of problems (though almost exclusively taking physics-related ones). Many people think of a derivative as a slope or an integral as an area, and fail to see that anything that can be thought of as a rate of change or the accumulation of small bits is an example of the type of things calculus handles. Comenetz tries to dispel this attitude.

As I said, this is a harder book than Sparks' book (though easier, I think, than Bell's!) But if you want to understand things on a more fundamental basis, I think this book is the best of the lot. The other three books which I reviewed all got four stars from me; this gets five. So this should clue you into the fact that I think this the best one of the bunch.

It's no secret to anybody that knows me that I am Calculus illiterate. I've taken three courses, failing once and just passing the other two times. I entered into the Master's program in Hydrology and soon began realizing that I would actually need to understand the principles of Calculus in order to successfully complete my coursework and thesis research.

I picked up a copy of Michael Comenetz' book "Calculus: The Elements" and began reading it, hoping that by some cosmic chance I might understand something that he had written. I'd done this with several other Calculus texts, only to find that I was still clueless.

Dr. Comenetz' book was the answer to my problems. He lays out fundamental concepts in easy to understand format. For the first time in my entire life, I understand the basics of Calculus and it's all thanks to this book!

I highly recommend this book if you're a "math illiterate" like myself but need to understand the complex subject of Calculus. Other texts that I've encountered simply have not been sufficient. THIS particular text has changed my entire mindset on the subject. Before the book, I thought "There's no reason to ever use this subject, and I'll never understand it anyway." After reading it, I'm understanding Calculus and actually considering taking an advanced course in the subject.

Thanks Dr. Comenetz!

This book presents all of the key topical areas to provide a basis for understanding calculus. It begins with the most basic or fundamental building blocks (this is the hardpart for most of us)and leads to a complete development of the topic. The fundamentals are presented and all you have to do is put forth the time and effort to stay with the material. You will be rewarded. Those who are also interested in proofs will be provided all of the details. The liberal use of technical examples help the reader grasp the fundamentals.

This is by far the book to have to grasp and retain the fundamentals of calculus. It starts a topic by building from the most basic(the part that is the hardest for most) and expands it a conclusion. The text uses numerous technical applications to help in conveying the concepts. The reader must of course dedicate the time and effort to fully understand the concepts. Proofs are used throughout and provide added value to those so inclined to have a rigorous presentation.

In high school I kept trying to puzzle out _why_ analysis works the way it does, trying, in other words, to translate math into the sort of verbal propositions I was already familiar with. It wasn't until college that I realized the best way to "understand" calculus and differential equations is simply to work through lots of problem sets. (That's the way we "understand" most things -- and it's one reason kids cry so much.) Like any calc text, Comenetz's is full of problem sets. What sets his approach apart from six other textbooks I've used is that his argument is as elegantly spun as his mathematics, and his problem sets include the kind of challenges that leverage logic to bring big picture focus.

People learn in different ways. I believe working though problem sets will always be the foundation of learning. But for those who, like myself, feel compelled to make verbal sense of the world of math (as physicists do), this textbook provides a rare opportunity to powerfully integrate two kinds of learning.

It's no doubt that Stewart's book is the most popuplar textbook on Calculus. It's comprehensive and standard. However, it's a pain to read through every page and do all the exercises. In that regard, I've found Michael Comenetz "Calculus: the Elements" most suitable for students without a solid background who intend to major in physics, math, chemistry, and engineering. Comenetz' book is not only comprehensive but very concisely written. Problems are well chosen--unlike Stewart's that has repetitive/similar problems all over the textbook. Yet, my advice would be "keep Stewart's as a reference while learn from Comenetz's" This, based on my own experience, is the most effective to achieve high scores in tests and excellent grades.