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Praise from Readers

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

John H. Hubbard and Barbara Burke Hubbard

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Comments inspired by various editions of the book

"...the clearest, cleanest, most coherent, readable coverage of forms that I have ever encountered (I have a number of books collecting dust in my library). I would have been spared countless hours of frustration had I known of the existence of your book." — Joe Downs

"Presently going thru Chapter 6 -- it's remarkable how it illuminates what has gone before. Lewis Robinson M.D.

I looked up the introduction to the Jacobian because I am going over the proof that the normal distribution pdf integrates to unity with my single-variable BC Calc students (who have now taken the AP). Your presentation of the Jacobian, taking the reader back to single-variable and redefining that derivative to create a form that can be naturally extended, is just marvelous. The style of presentation is also remarkable: it treats the reader like an intelligent and thoughtful person but also explains everything using fundamental principles and clear language. I am truly smitten." — Daniel Bettendorf, The Roxbury Latin School

"All of the chapters I have worked through thus far in my three weeks with the book have been a pure delight. As a recent computer science graduate whom is now wading into more specialized maths, I felt very uncomfortable with my foundations, and this text has been the perfect response! — Mike Rooney

"Your book contains a lot of interesting  and challenging problems that helped me a lot to prepare for my PhD Analysis Qualifying exam. This is definitely one of the best textbooks I have ever seen in Analysis. I haven't yet read the Linear Algebra part, but I am pretty sure it is at the level of the Analysis part." — Zayd Ghoggali

I truly value the approach you take to try to foster true understanding (including the amazingly helpful margin notes that dot each chapter), the obvious deep thought you put into the order in which all the material is presented, and the clear desire it seems the authors have to truly share their expertise in the subject in a way most likely for the target audience to fully benefit from it." — Ross Lipsky, Valley Stream South High School Mathematics Department

"I am having a great time reading the third edition of your book (and attempting all the problems).  It's quite user friendly without being sappy... I particularly enjoyed problem A1.5 - I thought I'd never see how Peano did it.   Part c. is worth the answer book alone. - Lewis Robinson, retired M. D. indulging a lifelong taste for mathematics

"A great book. I love books that cut through the smog and show that math is not all that hard. It is a bonus when they are entertaining. " — Dr. Ralph Kelsey, Ohio University

Although I finished a Ph.D. in electrical engineering several years ago, I found your textbook so insightful that I want to work through the problems. I rank your book among the top 10 technical books I've ever read. Your use of the margin notes is extremely useful'' — Jose Unpingco, Los Angeles.

"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 gold mine of information not available in my other texts." — Professor Thomas Tredon, Lord Fairfax Community College

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 book Vector Calculus, Linear Algebra and Differential Forms: a Unified Approach is 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.'' — Luís Serra, Professor of Genetics at the University of Barcelona.

"A marvelous book. What a great idea to combine all those topics!''— George Fegan, chair, Department of Applied Mathematics, Santa Clara University.

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.

"I bought your book because I was interested in a down to earth, elementary exposition of differential forms, which shows one how to do practical calculations with these objects. I later found out that it is full of other wonderful, hands-on explanations of many things I had already learned but found a little unsatisfactory. For instance, as far as I remember, in no other textbook I consulted have I seen a statement of the inverse function theorem that say something about the "size" of the domain... Your book always has an eye on the practical implications of the concepts developed while never slipping off into the unrigorous, as it is all too often with "practical" books. — Nikolas Akerblom, graduate student in theoretical physics, Hannover, Germany

"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

"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. This is a much nicer approach than one will find in texts such as Rudin's Principals of Mathematical Analysis. I highly recommend Hubbard's book to anyone wishing to learn differential forms.'' — Review posted at Feb. 21, 2002.

"I am currently using the book as part of a small team at Microsoft informally investigating Quantum Computing.... we are recapitulating modern physics in the language of the Exterior Calculus, and we find your book to be the best all-around introduction to the subject (the others are either too abstract to furnish intuition or too applied to furnish rigor).

For instance, your book is the first and only one I have seen that motivates a basis k-form in n-space procedurally: concatenate k n-column-vectors in a matrix, strike out all rows but the k mentioned in the indices of the k-form, and calculate the determinant. All other presentations I have seen start either with axiomitization of the wedge product, or with study of the generic permutation symbols, or with oddball "eggcrate" metaphors, or some other equally sidelong approach that is both very time-consuming and ultimately leaves one unequipped to do anything with k-forms other than wonder why. This one single aspect of your book makes it worthwhile and, at least for me, provides an absolute keystone for deeper understanding.

...if these topics were taught to physicists out of your book rather than through the standard physics curriculum, much time and aggravation could be saved. I see no reason for students to study vector/tensor calculus and linear algebra separately, then NEVER formally study differential forms away from applications, THEN FINALLY have to try to unlearn them ALL and relearn them together when they can all be learnt at once right the first time through your approach.'' — Brian Beckman, Microsoft.

"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 Euclidean 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 on Feb. 14, 2002.

"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.

"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.

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