I have never met, or had any other kind of contact with, Jerry Shurman, the author of the book now under review, but despite this lack of familiarity, I would be willing to bet that he is an excellent teacher. He has certainly written an excellent book, one which reflects a considerable amount of time and effort spent thinking about the best way to present this material to an undergraduate audience.
The subject matter of the text, as the title indicates, is calculus in n-dimensional Euclidean space, done rigorously and precisely enough to go from the typical three-semester calculus sequence to analysis. Both differential and integral calculus are discussed; in fact, except for an introductory chapter, there are two parts to the book, with these as the subject headings.
The book begins with a review chapter. The basics of one-variable real analysis are briefly and quickly reviewed, except for Taylor series, which are discussed more slowly and in somewhat more detail. In one of the book’s rare departures from clarity, the completeness of the real number system is initially described as “every binary search sequence… converges to a unique limit”. I suspect that many readers may not be familiar with the concept of a “binary search sequence”, a term which is not defined in the book; fortunately, however, the author quickly gives other characterizations of completeness (namely that every nonempty set that is bounded from above has a least upper bound, and that every bounded monotonic sequence converges) that should be more familiar to students.
After this introductory chapter, Part I of the book, on differential calculus, begins. There are four chapters in this part. The first reviews the basic vector space structure, topology and geometry of Euclidean spaces, and the second is a fairly thorough account of linear mappings from one Euclidean space to another, covering, among other things, the connection between mappings and matrices, and determinants and their geometric interpretation.
Even in these two early chapters, we see a pet theme of the author at work: he likes, whenever feasible, to work with axioms that characterize a given topic. So, for example, after defining the dot product in terms of coordinates, he immediately deduces the properties that characterize a general inner product, and from that point on works from those properties rather than the actual coordinate definition of an inner product. Likewise, the notion of a determinant is introduced by first specifying axioms for a determinant function, deducing things from those axioms, and then proving that there exists a unique determinant function satisfying those properties. I first saw this approach to determinants many years ago in Hoffman and Kunze’s Linear Algebra, was very impressed with it, and am pleased that a modern book takes steps to make this approach more readily available again. I think the author’s idea of axiomatically characterizing things whenever possible is a very sound one pedagogically, and works well throughout the book.
The remaining two chapters in Part I of the text discuss the derivative of a function from one Euclidean space to another, and the inverse and implicit function theorems (along with Lagrange multipliers, done as an application of the implicit function theorem). I particularly liked the way in which the derivative was defined: slowly and gradually by first introducing “big O” and “little o” notation and then rethinking the definition of the familiar derivative of one-variable calculus in a way that makes the general definition more natural and well-motivated.
The remaining four chapters comprise Part II of the text and discuss integral calculus in several variables. The integral is first defined for functions defined on a box in n-space; since a box is a natural generalization of an interval, the definition tracks that of the ordinary integral studied in single-variable calculus and analysis. It is then explained how this definition can be generalized to functions defined on more general subsets of n-space. The chapter ends with a discussion of Fubini’s theorem and the change-of-variables theorem.
The next, relatively short, chapter is a fairly technical one, on how (using convolutions) a continuous function can be approximated by a smooth one. As the author notes, readers who are willing to accept this result without proof can skip this chapter.
Chapter 8 discusses parametrized curves (as a prelude to the introduction of surfaces) and can be viewed as an introduction to classical differential geometry. It discusses such familiar concepts as curvature, torsion, the Frenet frame, and so forth. In an unusual feature, the text also discusses some of the impossible Euclidean compass-straightedge constructions (trisecting an arbitrary angle and doubling a cube), proves that they are impossible (developing the field of constructible numbers in the process) and then shows how, with the aid of some specific curves (specifically, the conchoid and cissoid) in addition to the compass and straightedge, one can perform these constructions. This is material that is not found in many undergraduate textbooks.
Finally, the last chapter of the book introduces differential forms and gives both a statement and proof of what I grew up calling the generalized Stokes’ theorem (referred to here as the General Fundamental Theorem of Integral Calculus), which includes as special cases the fundamental theorem of single-variable calculus, Green’s theorem, and the classical Stokes’ theorem. The author’s approach to differential forms is down-to-earth and well-motivated; he begins by stating that we are looking for “objects to integrate over surfaces” and then discusses differential forms “syntactically and operationally”; i.e., he first tells the students what a differential k-form looks like, and then explains how these can be viewed as functions that associate, via integration, a real number to every k-surface in n-dimensional space.
The material in this text is, of course, found in other undergraduate texts, but it is not easy to find other books that exhibit such care and clarity in the presentation of the subject. The author’s writing style is clear and easy to follow, but, more than that, it is exceptionally well-motivated and contains some useful pedagogical ideas. In addition, throughout the book, the author notes issues that are likely to cause trouble to beginning students, and takes the time and effort to single them out and discuss them thoroughly.
There are lots of exercises, many of them quite illuminating. Some are computational, some call for proofs, some require the production of examples, and others ask the student to think critically about something that they’ve just read. (Example: “Which points of the proof of Proposition 6.8.4 are sketchy? Fill in the details.”) And when was the last time you saw an author end a preface to a book with some exercises? This struck me as an excellent idea, giving the students a chance to start thinking about the material even before formally starting the text. (There is a password- protected solutions manual for some, but not all, of the exercises on the book’s website.)
In an effort to make this book as accessible as possible, the author does not attempt to discuss matters in the most general possible context. For example, although the author mentions metric spaces in passing, they are not used here; certain ideas are given definitions (for compactness, for example, “closed and bounded”) that work for Euclidean spaces but not for general metric spaces. These omissions, though reasonable (it is much easier to say that a space is compact if it is closed and bounded, rather than to try and motivate the much less intuitive open cover definition), may limit the value of this book as a text for an upper-level analysis course in which the instructor wants to explore some of these ideas.
A few other quibbles: as the preceding discussion of the content of the book should make clear, a lot of mathematics is covered here, quite likely more material than can be covered in a single semester. A dependency chart, or even a description by the author of how he covers the book in his own course, might be of benefit to an instructor who wants to get to differential form and Stokes’ theorem in one semester.
Additionally, there is no bibliography or list of references at all. A book like this one leads naturally to other topics — topology, differential geometry, etc. A list of possible sources for a reader to follow up with, particularly an annotated list, would have been helpful, I think. Likewise, a few pages at the end of the text in a kind of “where do we go from here?” section may have helped put some of the ideas of this text in context; for example, the notion of a manifold could have been mentioned as a generalization of the idea of a surface.
If these, however, are the worst things that a reviewer can think of to say about a book, then it must be doing something right. This book does a lot of things right. It is highly recommended.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University