Customer Reviews

Free Calculus: A Liberation from Concepts and Proofs

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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.

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.

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.

A regular calculus assumes a lot of a priori knowledge(such as real numbers, $\delta-\epsilon$). This may make beginners giddy and delay the physics teaching. We abandon the complete theory but jump immediately into the subject:

differential+integral+Fundamental Theorem+Taylor's Theorem.

A general idea: offer intuition ahead (with pictures always) and rigor behind(with a few lines proofs usually), covering both plain language and rigorous proof.

In particular, the differential here is motivated by a trigonometrical measurement, measuring a short curve's height through the "tangent's height" (called by the differential, if the tangent exists) with

measured error =short curve's height-­differential

(see Fig. 1 in the book). We can see, without calculating, from Fig. 1 that the measured error can be arbitrarily small (when shortening the curve or the base) if the curve is "continuous " (in the intuitive sense). What we should calculate is a more important error, the ratio of the measured error to the short curve's length(or base), called by

relative error=(measured error)/base.

This is nothing, from the tangent's construction in Fig. 2, but the visible difference:

secant's slope-tangent's slope

which can be arbitrarily small(when the secant tends to the tangent). So far it is intuitive. From now on, instead of the intuition, we use the property

relative error<<1

to uniquely define the tangent or the differential(see the proof in Page 29) at a point, where we use the shorter notation, <<1, to denote an amount which is less than an arbitrary given number(in absolute value sense) when shortening the base.

We now turn to the integral. This is motivated by a chain of trigonometrical measurements, measuring a long curve's height through the "sum of differentials" (if exists everywhere) with

total error=sum of (measured error)s=total height-sum of differentials

(see Fig. 1) . This is nothing, through a few lines arithmetic, but the average:

(total error/total base)=sum of ((measured error)s/total base)=sum of((relative error)s*bases/total base)=average of (relative error)s

(we use the fact that "sum of coefficients=1"), where the relative errors, and so the average, depends on the partition. If and only if for all partitions when refining,

|averages| <<1

we have

|total height-sum of differentials|=C*|averages|<<1

(where the "sum of differentials" is nothing but so called the Riemann's sum)read as the FT:

total height= integral of differentials

where the discrete Riemann's sum is written as the continuous integral.There are several sufficient conditions to guarantee the condition, "upper of |averages|<<1". The simplest one is

upper of |(relative error)s| <<1

which is nothing but equivalent to the classical "continuously differentialable" condition.

As for TT, it is a chain of (FT)s, see Page VIII with a half page proof (f' in 4th and 5th lines should be corrected by f'' ).

The differential, integral and the FT stated above are written with plain language, an irregular(while still rigorous) way. More regular and eloquent way is using the function language, see Michael's review (November 28, 2008) and http://mathfoolery.org/talk-2004.pdf, or the book, for more details. If these expositions are understandable and rigorous, a liberated calculus for beginners would be possible. I want to apologize for the irregular English(with wrong grammar) and the irregular mathematics(a mish-mash of plain language and rigorous reasoning) which makes the book ambiguous, cryptic or not eloquent. More critics are welcome!