The
following review of the third edition is reproduced with permission
from MAA Reviews, which holds the copyright. Emphasis added.
"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 fiind an
accurate evaluation of the current book in Tucker's
October 2003 review.
...
Nonetheless, I will still attempt to provide a self contained (though
admittedly 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 aniy prior
knowledge of linear algebra. Therefore many calculus
textbooks introduce (what some wold 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 a a plugandchug 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
allpowerful 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 had 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.
Ii 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, assistant professor of Mathematics at Pomona College
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Review of
2nd edition from the Mathematical Association of America Monthly
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