Free Calculus: A Liberation from Concepts and Proofs (Hardcover)

~ (Author)
Key Phrases: abstract space, simplest differential equation, weak equation, Official Calculus, Strang's Calculus, Differential Equations of Second Order (more...)

Editorial Reviews

Product Description

Conventional calculus is too hard and too complex. Students are forced to learn too many theorems and proofs. In Free Calculus, the author suggests a direct approach to the two fundamental concepts of calculus -- differentiation and integration -- using two inequalities. Regular calculus is condensed into a single concise chapter. This makes the teaching of physics in step with the calculus teaching.

Contents: Calculus in Terms of Images; Official Calculus; Differential Equations of First Order; Differential Equations of Second Order; Free Calculus; Appendix: Calculus of Functional Analysis Becomes Elementary Algebra.

Product Details

• Hardcover: 108 pages
• Publisher: World Scientific Publishing Company (May 23, 2008)
• Language: English
• ISBN-10: 9812704582
• ISBN-13: 978-9812704580
• Product Dimensions: 9 x 6.2 x 0.6 inches
• Shipping Weight: 12 ounces (View shipping rates and policies)
• Average Customer Review: 2.0 out of 5 stars  See all reviews (4 customer reviews)
• Amazon.com Sales Rank: #856,784 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

2 of 2 people found the following review helpful:

1.0 out of 5 stars Spectacularly bad book, November 15, 2008

By	James M. Cargal (Montgomery, AL USA) - See all my reviews (REAL NAME)

This book is not merely a terrible translation. It is an awful book for anyone who does not know calculus and of limited value for anyone else. The author has a good idea or so on how to look at calculus but it is presented in such a concise and cryptic manner it is obtuse for experts. There are plenty of great books that try to motivate calculus for beginners. (My web site lists a few.) A good example is: The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides). This book however is not merely bad it is a candidate for one of the worst math books of the decade, if not for all time. Consider, for example the promise on the cover: "A liberation from concepts and proofs." Most books for beginners give a liberation from proofs. This is reasonable as calculus was developed nearly two-hundred years before rigorous proofs were discovered. But "freedom from concepts?" The whole point is concepts. If you are just getting into calculus this book would be disastrous for you. If on the other hand you are an expert, if you find anything of value here, I would love to hear about it.

2 of 2 people found the following review helpful:

1.0 out of 5 stars Very Disappointing, November 1, 2008

By	Mike Cargal (Columbus, GA) - See all my reviews (REAL NAME)

I read a review of this book as an explanation of Calculus "sticking to plain language". While that may be technically accurate, but it's not plain english. It reads like a very poor translation. Because of this, what *might* have been a nice succinct description turns into a real "head scratcher" of grammatical nightmare. Even working hard to get past the very poor english, I found that it hardly achieved its goal of clearly explaining calculus without assuming a lot of a priori knowledge.

2 of 3 people found the following review helpful:

3.0 out of 5 stars A brave attempt to simplify calculus., November 28, 2008

By	Michael (Cambridge, MA, USA) - See all my reviews

It is indeed a pity that there are so many grammar bloopers and other expositional faults in this book, especially because the mathematics is sound, and bears a great promise in simplifying calculus and therefore making it more accessible. I am partially to blame for these shortcomings, because I saw some of the manuscript and thought that the publisher would take care of the presentational details, well, unfortunately I was wrong...

Let me spell out the good idea that the author has, at least as I understand it and discussed it with him. The idea is that you don't need the full generality of the 19th century "rigorous" definitions and proofs to understand differentiation and integration, as you encounter them in introductory calculus. Why? Because the functions that you encounter there are much nicer than continuous (most of them are analytic, i.e., can be defined by converging power series), and can be handled with much less technical overhead. For example, as the author suggests, unfortunately not too eloquently (but, then again, true words aren't eloquent, as Lao Tzu pointed out in Tao Te Ching 81 ), the derivative can be DEFINED by the inequality |f(x+h)-f(x)-f'(x)h|<=Kh^2, where K is a constant that does not depend on x or h. This definition is a natural generalization of differentiation of polynomials.

Let us see why. For a polynomial f(x), f(x+h) is a polynomial (of the same degree) in h, with coefficients depending on x, i.e., f(x+h)=f(x)+f'(x)h+R(x,h)h^2, f'(x) is the derivative of f at x and R is a polynomial in x and h, and as such, it is bounded by a constant K for any finite range of x and h (the derivative of a polynomial f can be DEFINED as the coefficient of the term linear in h in the expression f(x+h), so we need no limits here).

We can rewrite our definition as |(f(x+h)-f(x))/h-f'(x)|<=K|h|, and observe that it agrees with the classical definition, although it is stronger. The corresponding notion is called "uniform Lipschitz differentiability." We can also rewrite our inequality as |(f(x)-(f(a))/(x-a)-f'(a)|<=K|x-a| and, by interchanging x and a and observing that (f(x)-f(a))/(x-a)=(f(a)-f(x))/(a-x) conclude that |f'(x)-f'(a)|<=2K|x-a|, which means that f' is Lipschitz, i.e., a uniform Lipschitz derivative must be a Lipschitz function. So we end up with a streamlined calculus of Lipschitz functions, which is enough for most applications.

If you consider Lipschitz estimates too restrictive, take any modulus of continuity m(h), such as the square root of h, for example, and use the estimate |(f(x+h)-f(x))/h-f'(x)|<=Km(h)|h| instead (see page 18 of the book). Assuming these stronger definitions, all the practical results of calculus can be proved in a very elementary manner, see http://www.mathfoolery.org/talk-2004.pdf for more details.

One may ask whether we are missing anything of the classical theory, and the answer is a resounding "NO," if we stick with continuously differentiable functions defined on intervals, because then our inequality |(f(x+h)-f(x))/h-f'(x)|<=Km(h)|h| will hold true with m being the modulus of continuity for f', i.e., the minimal increasing concave function m, such that |f'(x)-f'(a)|<=m(|x-a|).

So, what does the subtitle of the of the book really mean? It means that most of the technicalities developed in the 19th century are not necessary to understand calculus, that it is in fact much simpler and straight forward. In my opinion, the first 20 pages of the book describe the basic ideas of calculus much better than the average 1200 page calculus text, where these ideas are obscured by cumbersome and unnecessary technicalities that are usually not well explained.