Calculus in the First Three Dimensions

The inevitable question that arises when facing a Dover reprint of a 1967 calculus book is: where would the book slot into today’s curriculum? And if the answer is “nowhere,” a follow-up question might naturally be “how does this book serve for self-study?”

Of course it is hard to approach this book, with one’s shoulders and head battered by the 1200+ page, “new comma? new edition!” calculus textbook herd, without crying out “calculus in about 600 pages? C’est incroyable!” in a happy if incredulous voice — but such a miracle is to no good if the book does not serve a purpose.

Stein’s text is unusual in that it steps up to its title right away and introduces integrals by peppering the reader (right along that \(z\)-axis between book and face) with Riemann sums in 1, 2 ... and 3 dimensions — lots and lots of Riemann sums, in fact: so many that this reader was apt to beg for mercy and tell the writer that his point had been made.

That said, the exposition is clear and concise. Overall, Stein’s book has personality to spare even if the exercises (in the early chapters, at least) seem to belabor a point. There is a sly, dry wit in evidence. My favorite from the early chapters is “[a]t this point we should no more interpret the symbol \(\frac{dy}{dx}\) as a quotient than the symbol 8 as two zeroes.” You likely won’t fall over laughing, but it’s a neat reminder that there is a human back there.

The structure of the book is also intriguing: Part I (Chapters 1–9) presents “the core of the calculus”; Part II (Chapters 10–20) is “topics in the calculus” and Part III (chapters 21–26) covers “further applications of the calculus.” The applications include things like “business management and economics” and “the basic equation of rocket propulsion,” covering a range sure to include plenty topics of real interest.

Part I, the “core,” takes less than 300 pages, which seems a miracle; however, it’s useful to consult the author’s “Instructor’s Manual” (downloadable from Dover Publications) and note that things we typically consider basic (or “core”), like second derivatives, are deferred until Part II’s “topics.” Chapter 10 is, in fact, called “The higher derivatives.” Stein has reshuffled the deck — or perhaps modern authors have reshuffled Stein’s deck.

All this is to say that neither the order nor the style of presentation is 2000-something norm, but what do you expect from a book published in the 1960s? The author’ development is dense (I almost wrote ‘rigorous,’ but that has a different meaning) in that the exercises mesh tightly with what is demonstrated in the text and often with each other: many exercises depend directly upon techniques learned from preceding exercises. Because of this, Stein’s text encourages a serious, sit-down engagement rather than a scattershot “oh, I’ll just do these five problems before playing video games” approach. The latter is eminently possible with today’s doorstops: it is part of their design.

Is this density and interdependence in any way a problem? It could be: if a student just can’t solve one of the problems, the link in the learning chain is at least temporarily broken. The solutions in the Instructor’s Manual skip some problems and are, for the most part, fairly telegraphic.

Readers intending to use this book for self-study should also note that answers are provided for the first few problems but the rest can only be found in the Instructor’s Manual noted above: there is no “Answers to Odd-Numbered Problems” here.

This then is a fine text in applied calculus, with the applications growing (a nice thing for a book on calculus) in intensity and number through the book’s three parts — until Part III which, as we have seen, is all applications. Whether this is a calculus book for you, then, depends on how you feel about that kind of structure and focus. I liked it quite a lot. 

Glenn Becker is a staff member at the Harvard-Smithsonian Center for Astrophysics in Cambridge, MA, where he toils in the data archive of the Chandra X-Ray telescope. He is a “reborn astronomy and mathematics fellow traveler” who spent far too many years getting advanced degrees in theater, only to ultimately abandon the entire discipline.