This is a weird book review. It starts with a confession and a referral:
I do not believe that I can do any better than Warwick Tucker, who wrote a detailed review of the second edition of this book for the Monthly . In that review Tucker gives an exhaustive summary of the book in its earlier reincarnation. Since the third edition retains all the strengths of the second edition, readers will in fact find an accurate evaluation of the current book in Tucker's October 2003 review.
The good news is that if your institution subscribes to JSTOR or you have bought JSTOR access with your MAA membership, you can access Tucker's review online. Nonetheless, I will still attempt to provide a self contained (though admittedly a much shorter) review of the book here.
In many colleges and universities, students take a basic multivariable calculus course right after two semesters of single variable calculus. In such courses, students are not expected to have any prior knowledge of linear algebra. Therefore many calculus textbooks introduce (what some would claim is) just the right amount of vector analysis to make things work right. Nonetheless, without a full blown excursion into linear algebra and then a quick trip back to calculus recovering the derivative as a linear transformation, many parts of multivariable calculus (eg. the chain rule, the Jacobian) have to remain quite incomprehensible or at least somewhat ad hoc.
The book under review is one solution to this problem. It is clear that John H. Hubbard and Barbara Burke Hubbard have written a text for a very particular type of student, one who accepts that reading and learning mathematics will no longer be a plug-and-chug activity, one who is intrigued by connections of mathematical ideas with one another, and more specifically, one who is looking forward to learning a vast amount of mathematics. In short, the book is a guide for a pretty tough course. It can be used for a two semester sequence which integrates multivariable calculus and linear algebra quite seamlessly, and which along the way introduces mathematical proof, the all-powerful tool of mathematical thinking. However it also includes so many details (and proofs) in basic analysis, single and multivariable, that it can even be used for more advanced students in a one semester analysis course.
When reading this book, I constantly was aware of the fact that I would have benefited immensely if I gotten my hands on it when I was an undergraduate. It is clear to me that the authors have put their hearts and souls into this project. The book has many details sprinkled in, many anecdotes, many personal opinions about how one does mathematics; any student interested in mathematics would find it a valuable experience to even flip through its pages randomly. I was especially excited about the last chapter where the natural framework of differential forms is developed and applied to the theory of electromagnetism.
The second edition was definitely more than a good enough book; one may ask why there is a third edition at all or why a reader should consider the third edition instead of the second. There are two answers. The first is of course that the authors have made many improvements. In fact several earlier typos are corrected and, more substantively, three new sections are added to the text. One is on eigenvectors and eigenvalues, which are now introduced in a way independent of the determinant and thus made more amenable to computation. This agrees with the book's general emphasis on computationally effective algorithms. The second new section includes rules for computing Taylor polynomials, and the third, mentioned above, is on electromagnetism.
The second reason to consider this third edition may be quite surprising to those who are used to new editions of famous calculus texts. The third edition of Vector Calculus, Linear Algebra and Differential Forms, a Unified Approach is published by a small publishing company, Matrix Editions, which specializes in books of "serious mathematics, written with the reader in mind." This book is certainly basic mathematics written for a serious reader. Most significantly, the reader will be happy with the price tag: an eight hundred page textbook for less than $80 is a bargain these days. This edition is, in fact, significantly cheaper than the second edition.
I was very impressed with the depth, clarity and ambition of this book. It respects its readers, it assumes that they are intelligent and naturally curious about beautiful mathematics. Then it provides them with all the tools necessary to learn multivariable calculus, linear algebra and basic analysis. I definitely recommend the book to anyone who is planning to teach or learn multivariable calculus.
Gizem Karaali is assistant professor of Mathematics at Pomona College.