## Reader Comments

Vector Calculus, Linear Algebra, and Differential Forms:

A Unified Approach

John H. Hubbard and Barbara Burke Hubbard

ISBN 0-13-041408-5

Published by Pearson Education (Prentice Hall)

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Second Edition:The book's chief asset is its overall structure and philosophy --- it does things right. It is the unique tactic of engaging rather than insulting the students' intelligence that makes the book great. -- Professor Robert Ghrist, Georgia Institute of Technology

A marvelous book. What a great idea to combine all those topics! -- George Fegan, chair, Department of Applied Mathematics, Santa Clara University ``I am a Professor of Genetics at the University of Barcelona. My research work is focused on evolutionary biology, and I teach

population genetics and evolution. Thus, my students require a good background on mathematics.Your bookVector Calculus, Linear Algebra and Differential Forms: a Unified Approachis fantastic. It has helped me a lot. Some insights and observations (brilliant, from a pedagogical point of view) are rarely found, if ever, in other books. My warmest congratulations.'' - Luis Serra, Professor of Genetics at the University of Barcelona.A beautiful book for undergrads and grads alike Although I am a graduate student , I found Hubbard's ``undergraduate'' text to be extremely helpful. Hubbard combines an intuitive heuristic approach appropriate for undergraduates with a thoroughly rigorous set of proofs appropriate for graduate students.

I found his discussion of differential forms particularly helpful. He provides an excellent intuitive motivation for the definitions, and then he follows this with a mathematically sound treatment of the topic..... I highly recommend Hubbard's book to anyone wishing to learn differential forms.- Review posted on amazon.com Feb. 21, 2002

``Thanks for your great text! It explains everything in great detail and provides adequate examples. It is also very intuitive.''- Chi Lam Mak, Yale undergraduate. ``This is a fantastic textbook. It seems to attack directly every hurdle I always got stuck on in the past and explain it like someone is watching a movie. Somehow it changes one's whole view of analysis.'' - Harry Hirsch (who has made many contributions to the errata page). This book is unique in several ways: it covers an immense amount of material, much of which is never presented in books aimed at this level. The underlying idea of the authors is to present constructive proofs, which has the great benefit of providing the reader with the ability to actually compute quantities appearing in the theorems.

As an example, the Inverse Function Theorem is proved using Newton's method, which relies on Kantorovich's Theorem, and thus actually gives an explicit size of the domain on which the inverse exists. The book also contains a very nice section on Lebesgue integrals, a topic which is usually reserved for graduate level courses. The construction is to my knowledge completely new, and does not rely on sigma-algebras, but utilizes only elementary mathematics. Another nice feature is that the book considers abstract spaces at an early stage. Thus the reader is presented with the idea of computing derivatives of functions acting on e.g. matrix-spaces, as opposed to the usual Euclidian spaces.

The concluding treatment on differential forms brings a lot of the introduced ideas together and completes the picture by a thorough treatment on integration over manifolds.

This book can be studied at several levels. For a first year honours course, one may skip the trickiest proofs, which appear in the appendix. More advanced readers may choose to study constructions and details of selected theorems and proofs. Anyone who buys this book will have a solid companion for many years ahead. -- Review posted at amazon.com on Feb. 14, 2002

First Edition:The authors condense in less than 600 pages an incredible amount of classical material. Most of it is presented in a very original way, many times very different from classical presentations (e.g., Stokes's formula, Lebesgue dominated convergence for Riemann integrals....) The book compiles material scattered over the mathematical literature and is an excellent reference book. It is the best book that I know for freshmen with a taste for mathematics. The presentation, pictures, anecdotes and historical comments make it extremely enjoyable, not only for the student but also for the professor. A must-have that will become a classic. - Professor Ricardo Perez-Marco, UCLA Department of Mathematics

Amazing book! This is one of the best written math books I have seen. The authors write in a clear and engaging style which makes the reader understand the beauty of math. After you read this text you can put this on your bookcase besides other classics such as Spivak's

Calculus. Let's hope that the sequel will appear in the near future. -A reader from Toronto, Canada

Never before had I even considered contacting the author just to tell him/her how much I loved the book. Your unified approach is a very original, unique, and effective teaching method. There's much more for the student to think about (hence more scratch work to be done on the side), but it's well worth the effort! Your clear and concise presentation of topics coupled with penetrating insights offered at key moments make reading (and learning) the subject matter a most enjoyable experience! -- Vincent Chang

When I compared this text to other texts that friends of mine have used in similar classes at various other universities, I found one of two things to be true. Either my friends owned a copy of Hubbard's text or they owned a rather dull, uninspired, possibly outdated text. In the latter case, I was then able to understand why I often hear complaints that math is a ``cold,'' ``esoteric,'' ``dry,'' or ``soulless'' subject. --A Cornell student

The book is a wonderful combination of explanations using simple terms and a presentation of the multivariable and linear algebra concepts in a more rigorous mathematical sense.--A reader from Ithaca, NY

As the title suggests, this "unified approach" is is a very unique and effective teaching method of presenting three subject areas (that are normally taught as two or three individual classes) in a single text! The authors do a magnificent job of showing and stressing the interconnectedness among vector calculus, linear algebra, and differential forms; so for those readers expecting a bland and disjoint presentation, you'll be in for a very pleasant surprise!

...The authors' clear and concise presentation of topics, coupled with penetrating insights offered at key moments (in the form of side-notes, footnotes, remarks, inserts, margin notes, etc.) make reading (and learning) the subject matter a most enjoyable experience! This reader wishes that this textbook was available when he was taking vector calculus and linear algebra! For those who have this book, be on the lookout for the sequel. -- A reader from Sunnyvale, CA