|First-Year Calculus Notes|
|summary||Would make a good concise refresher.|
The author provides this book in PDF format. As far as I can tell from the somewhat ambiguous notice on his web page, the book is intended to be licensed under the GPL copyleft license. That warms my heart as an open-source enthusiast, but it's slightly strange, for a couple of reasons. First, the GPL is a software license, and is less suitable as a copyleft license for books than the GFDL or a CC license. Also, the source code of the book isn't available (it appears to have been done in LaTeX), which I think makes it legally impossible under the GPL to redistribute the book, whereas the author's intent in GPL-ing it was presumably to make it freely distributable. Just as I was in the process of submitting this review to Slashdot, the author replied to an e-mail I'd sent him about this, and it sounds like he's interested in clearing up this issue, and really does want his book to be free as in speech.
This is a lively and very readable treatment of basic calculus. At 70 pages, it's a welcome antidote to the usual bloated textbooks, and the topics that are included match up pretty well with my own opinions of what it's really vital for a student to know after taking a calculus course. The tone is conversational without being condescending or cutesy, and the author almost always explains why he's introducing something, rather than just throwing it at the reader. (An unfortunate exception is the opening section on inequalities.) There is no attempt at rigor whatsoever, which I consider to be a feature, not a bug. Applications are discussed, although not enough for my taste (and I have to suppress my gag reflex every time I see a calculus book that insists on presenting the acceleration of gravity in non-metric units).
Although the book comes with some of the paraphernalia of a complete college textbook, such as homework problems, it's probably not the kind of book that another professor could just adopt as a stand-alone text, nor would I recommend it for someone learning calculus on her own for the first time. The title suggests that the author had in mind more of a memory aid, or a way to keep students from having to scribble madly in their notebooks for an hour and a half at a stretch. It lacks an index and illustrations, and there are some misfeatures in terms of organization: the chapters aren't numbered, and the homework problems are scattered around where they're hard to find. In some cases it sounds as though the first time a word or concept is used, he's assuming the reader has already heard it defined. I would, however, recommend this book to someone who needs to refresh her memory of calculus, and doesn't want to spend hours wading through epsilons and deltas to get to the highlights. It might also be a good option for the student who is completely broke, and needs a reference to use in place of an officially required text that carries an exploitative price tag. Although there are other calculus textbooks that can be downloaded without paying, this is the only one I'm aware of that follows the typical order of topics, and is also (AFAICT) copylefted, so that we can be assured it needn't evaporate if the author signs a publishing contract, or loses interest in maintaining his web site.
|Difference Equations to Differential Equations: An Introduction to Calculus|
|summary||Takes too long to get there.|
Like Garrett's text, this one appears to have been done in LaTeX, is licensed under the GPL, and appears to suffer from the same legal problems, because it's not available in source form.
The book is well written, and seems to have been well designed for practical classroom use. The approach is visual and intuitive, and there are lots and lots of graphs and numerical calculations. I felt, however, that it took a long time to get going, and the idiosyncratic selection of topics might make it difficult to use at many schools. Although the very first page gives a nice clear explanation of what calculus is about, we then have to wait until about page 136 to learn any calculus. I say "about" because of the inconvenient way in which the book is split up into 54 separate PDF files, each of which has page numbers starting from 1. I had to estimate page number 136 by weighing part of the book on a postal scale. Related to this problem is the fact that the book has no index or table of contents.
The book uses many numerical examples, which gives it a modern feeling . After all, calculus was invented by Newton and Leibniz because they needed to do calculations in closed form, but nowadays it's more natural to solve many problems on a computer, using a spreadsheet or a programming language. The book has a problem, however, in integrating the computer stuff with the didactic parts and the homework problems. No indication is given of how the numerical examples were actually computed. The author may consider it a trivial task to set up a spreadsheet or write a ten-line program in Python or Mathematica, but it's not so trivial for many students, and they will need extensive guidance from elsewhere to be able to carry out such computations for themselves. This makes the text incomplete in practical terms: any instructor wanting to use it would have to come up with extensive support materials to go with it. It also contributes to my sense that the book lacks focus. Students have a hard enough time learning the basic concepts and techniques of integration and differentiation, but to use this book, they would also have to learn about computer programming and difference equations. Adding to the bloat is the author's tendency to discuss every possible pathological case before moving on to the main event. It's a little like a parent trying to explain sex to his child, but feeling obliged to explain foot fetishes before getting on with where babies come from.
The examples that students are expected to do numerically also presuppose quite a bit of resourcefulness and insight. For instance, one of the homework problems asks the student to sum the series 4(1-1/3+1/5-1/7+...) numerically, adding up "...a sufficient number of terms to enable you to guess the value of the sum," which turns out to be pi. The trouble is that over 600 terms are required to get the sum to settle down in the second decimal place, which is about the minimum I'd want to see to convince me it was pi. Pity the poor student who first tries 10 terms on a calculator, then 50 terms on a spreadsheet, and then finally realizes he's going to need to write a Python program to get the job done. Of course, some students might enjoy the process, but my experience (teaching college science majors taking introductory physics) is that the majority don't consider computers to be fun.
|Lectures on Calculus|
|summary||Not for consumption by mere students.|
This book is from a set of lectures on calculus given by visiting professor Evgeny Shchepin at Uppsala University in 2001. The first obstacle potential readers will encounter is that the book is provided in PostScript format, with hideous bitmapped type 3 fonts embedded. This makes it virtually impossible to view the book on a monitor in any legible representation, although it looks fine when you print it out. The typical Windows or MacOS user will give up long before that point. This is a shame, because it's not at all difficult these days to get LaTeX to output Adobe Acrobat files that are viewable on virtually any computer, and are legible on the screen. There is no index, and virtually no graphs or other figures.
The main question in my mind is for whom this book was written. This deep, dark forest of mathematical symbols, interspersed with ungrammatical English, is meant to follow the historical development of the subject, but it never makes it clear why the historical route is the right one to follow. There are many seemingly pointless digressions.
Is it possible that this book was meant for young people taking their first calculus course? The presence of end-of-chapter homework problems would seem to imply that it was. If so, I feel sorry for them. Although it's cute that the author manages to develop integrals before limits, and derivatives only at the very end, I somehow doubt that real, live students would read this book and exclaim, "We sure are lucky to be learning calculus using this novel order of topics!" Most of the problems begin with the words "Prove that...," and neither the text nor the problems give any of the standard applications to biology, economics, physics, etc.
|Elementary Calculus: An Approach Using Infinitesimals|
|author||Jerome H. Keisler|
|summary||I wish I'd learned calculus from it!|
Textbooks are usually unoriginal, because most teachers are conservative in their choices. They get used to teaching a subject a certain way, and don't want to change. This is a calculus textbook with a very unusual approach. It was published in 1976, and evidently was successful enough, despite its idiosyncracy, to justify a second edition a decade later. Its publisher, however, eventually allowed it to go out of print. The copyright has reverted to the author, and he has made it available in digital form on his web site. The digital book consists of pages scanned in from a printed copy and assembled into an Acrobat file, so it's a big download, and you can't do some things with it, such as searching the text for a particular word.
The title leaves no doubt that the book is different. Whereas most textbooks these days define derivatives and integrals in terms of limits, this one uses infinitesimals. The real numbers are generalized to make a number system called the hyperreal numbers, which include infinitesimally small numbers as well as infinitely large ones. Essentially, this represents a return to the way Newton and Leibniz originally conceptualized the calculus, but with more rigor.
I don't know about other people, but when I learned calculus, I got very uneasy when we got to the Leibniz notation. My teacher said that dy/dx wasn't really one number divided by another, but rather an abbreviation for the limit of the quantity y/x. That wasn't so bad, but what really made me queasy was when he then suggested that you could usually get the right answer by treating these dx and dy thingies as if they were numbers. The scary part was that word "usually." What was legal and what wasn't? How many sizes of infinitesimals were there? Was it legal to say that 1/dx was infinite? What operations would lead to paradoxes? What about proofs that used infinite numbers to show that 1=2? The wonderful thing about this book is that you end up knowing exactly what you can and can't do with infinities and infinitesimals, and you get to use the Leibniz notation in all its intuitively appealing glory. For instance, the chain rule really can be proved simply by writing (dz/dy)(dy/dx)=dz/dx, simply canceling the dy's.
It would be interesting to see how students reacted to this book when learning calculus from scratch. I suspect that they'd have an easier time with many of the concepts like implicit differentiation, which seems so awkward in the traditional approach, but they might be scared a little by the initial development of the hyperreal number system. The book develops the hypperreal system axiomatically, which left me yearning for more of a constructive method. Then again, we develop the rational and real numbers axiomatically in high school, so maybe it's not such a big issue. My initial unease was cleared up by a few crucial examples:
- If H and K are infinite, then H-K may be infinite or finite -- it depends on which infinite numbers H and K are.
- If H is infinite, then (2H+1)/(H+1) isn't equal to 2, but it differs infinitesimally from 2.
- (H+1)1/2-(H-1)1/2 is infinitesimal.
I confess, however, to a little residual indigestion at the way the author develops the integral. He introduces finite Reimann sums first, and gives several numerical examples. But next, instead of taking the limit of sums with more and more terms, he takes the finite sum with n terms, and replaces n with an infinite integer. Instant vertigo!
This is a wonderful, original textbook, and I hope it remains free on the web forever -- it's not copylefted, so unfortunately it may disappear if the author stops maintaining his web site.
|The Calculus Bible|
|summary||Incomplete, and badly written.|
I'm reviewing this book in February of 2004. It's clearly not a finished product, and I'm not sure whether or not the author is still actively working on it. The book is available from the Brigham Young University math department's server, but the author isn't on the department's list of faculty, which makes me think he may have moved on to another job and abandoned the book. It's provided as a PDF file. There is no copyright page and no licensing agreement, so it's hard to know the book's real legal status.
The path through the topics is pretty standard for an introductory calculus course: a review of functions and trigonometry, followed by limits, differentiation, and integration. There is a good selection of problems, although to my taste as a physicist far too few are applied to anything useful. There is a table of contents, but no index. There are no illustrations; sprinkled throughout the text are little placeholders for graphs that just say "graph."
Although the problems I've referred to so far are ones that could be fixed if the author continued to work on the book, I feel that there are some more fundamental problems with this text that will not go away unless it is extensively rewritten. The style is extremely dry, and moreover the author has a habit of introducing concepts without any explanation or preparation. A symptom of this is that the student is expected to grind through the first hundred pages without any clear statement about what calculus is, what it's good for, or even whether the initial chapters are calculus (they're not). Equal prominence is given to topics that I would consider vital (the fundamental theorem of calculus) and others that I would label as trivial (tabulations of facts) or esoteric (the Dedekind cut property).
The Leibniz notation, dy/dx, is given with only this explanation "To emphasize the fact that the derivatives are taken with respect to the independent variable x, we use the following notation, as is customary..." Huh? So are these dx and dy things numbers? Is dy/dx the quotient of them?
Even if the missing graphs were included, the approach would still be relentlessly symbolic, rather than visual. For instance, integration by parts is introduced without ever giving its geometric interpretation.