Customer Reviews

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition)

5 star:		(12)
4 star:		(3)
3 star:		(1)
2 star:		(2)
1 star:		(1)

 

 

I have used this book to teach gifted high school students about the following topics: the implicit function theorem, manifolds, and differential forms. With the Hubbards' approach, even students without a course in linear algebra actually get it! Not only do they understand the material, but they also become amazingly enthusiastic when they begin to see the unifying effect of understanding differential forms.

This is the only text that I have seen that really makes forms clear. It does so by taking the time to carefully, but rigorously, explain them in a "classical" setting. One of the reasons forms are so difficult to grasp is that while some things, such as the exterior derivative and the work form of a function, can be seen as natural objects (when explained well), the connection between these objects and calculating with forms using coordinates is not so easy to make clear. The Hubbards' do make these ideas clear - even when presenting topics as hard as orientation.

Unfortunately, most of us had to wait till graduate school to see forms - usually, in a more abstract setting. By then, we probably didn't have time to sit, calculate, and make clear connections. This text makes that later transition, for those in math, much easier. It also makes physics easier. The Hubbards' make that point by showing that the electric field shouldn't really be a field, but a two form. Any book that lets one explain that - and much more - to high school students, which I do, should be a part of every multivariable calculus course.

Finally, I should note that this book contains much, much more than manifolds, the implicit function theorem and differential forms. But, even if that were all it contained, it would fully be worth the price.

In summary, this book opens the door to new worlds that most students never get to see clearly. What a present to us all.

This is the textbook used for the math 223/224 Theoretical Calculus and Linear Algebra sequence in Cornell University. The book is designed for prospective math students. Although the book mainly follows a rigorous development of the theories of multi-dimensional calculus, the mathematical machinery used in developing the theories is immensely broad, especially in linear algebra. The book covers most of the standard topics in a first semester linear algebra course and touches on many other areas of mathematics such as, real and complex analysis, set theory, differential geometry, integration theory, measure theory, numerical analysis, probability theory, topology, etc. The highlight of the book is its introduction of differential forms to generalize the fundamental theorems of vector calculus. The author is not the first one who follows this path. There are many other books written before this one that have similar approach, such as Calculus On Manifolds by Spivak, which was written 40 years ago and was too old to suit modern students.
The author tries hard to retain rigor and present to the readers as many examples and applications as possible. Often he tries to cover a broad range of mathematics and digresses a little. The book more or less touches on most of the areas of undergraduate mathematics curriculum and does not go into depth. It sometimes gives me the impression that the book is almost like a survey of undergradute math. The book is also not error-free. There are many typos and some technical errors. If you buy this book, make sure to get the errata from the author's website.

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.

First of let me state that I own the 2nd edition of this book, and that I am a doctoral student in computational physics.

After borrowing this book from the library, I read it cover to cover. I then bought a copy for myself to use as a reference. I learned a lot about the foundations of mathematics that I had not learned as a physics student. The book is very clearly written and actually enjoyable to read, with many examples, applications, and historical notes. The proofs were easy for me to follow.

Although the book is mainly concerned with multivariate calculus and linear algebra, it touches on many interesting and important matehmatical topics from set theory, topology, differential geometry, fractals, chaos, and analysis. It also provides an appendix that gives proofs for 25 of theorems that are considered harder to prove than is expected for a text of this level.

I also appreciated that the notation is thoroughly modern. (A glossary to the notation is given on the inside cover, with references to where in the book that you can find the full definition and explanation.) This may well be a drawback for many people, but for me it was very helpful because I now have an easier time reading papers on the more mathematical side of physics. Another modern aspect of this text is the introduction of differential forms, which are becoming essential to theoreticians in many branches of physics (quantum field theory, string theory, classical mechanics, and general relativity).

Lastly, this is a book on "pure mathematics", so if you are only interested in applied math, you will not like this book.

For me, it's been a great investment!

Although I am a graduate student in Mathematics, 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.

I've read sections of both the first & second editions and the second has numerous minor changes that make it a much better book. The changes are not major--the content and order are almost identical. However, places where the explanations were unclear or difficult frequently have new diagrams or helpful comments in the margins. A few topics that were too difficult or digressions have been moved to appendices or omitted. It remains a challenging book, intended for honors students, but is now a reasonable alternative to Apostol or a sequel to Spivak.

when I took Hubbard's Math 223-224 classes. I was one of the original group of students when the manuscript and class were being tried out. I must say the class was excruciating, even more than the students (who wrote the reviews below) could imagine. The primary source of pain was the incomprehensible manuscript. There were not only typos all over, but the layout was not as nice as in the published version. Actually, the book was changed _a lot_. The harder proofs in the appendix used to be in the main body of the manuscript, and the original appendix had even harder proofs that were cut out eventually.

The published version is great. I've looked through most of it over the last few years, admittedly from a more advanced viewpoint (no, I didn't buy it, Hubbard gave us free copies). It's very lucid, and the intros to new concepts provide good motivations. I suspect Barbara Hubbard had a great deal to do with how readable it is; she deserves a good deal of credit. I say this, because John Hubbard himself is incomprehensible. His lectures, while sometimes entertaining, were so dense that no one could follow them.

The unique aspect of the book is the 'unified approach.' This works very well at showing the interconnectedness of mathematics. I also like the fact that it is a useful reference. Of course, most of the theorems are proved in the context of Euclidean space, but it is not hard to see how to generalize it.

This is not an easy book though. I found a reviewer's comment that the book is 'incomprehensible to the average Cornell student' very funny. Any math book would be incomprehensible to the average student, whether at Cornell or not. But one should keep in mind that this book is used for the second year honors calc sequence. It is very 'meaty' and not to be delved into lightly. But compared to other books of the same standards, it holds up well. I give it four stars, or maybe up to four and a half.

Lots of people know vectors, even today a lots less know forms. This is a real shame because forms are simple and elementary, yet there are very few sources that introduce forms on a concrete level. The authors introduce the concepts of vectors, integration, and forms on a level that is accessible to a bright and interested high-school student.

Traditionally calculus in higher dimensions taught at an introductory level uses a vectors-only approach. This leads to considerable extra effort to account for two basic things: One is orientation, the second is how to generate higher-dimensional objects from lower-dimensional ones while keeping the same operations intact. For example how can one compute the length of a line, the area of a parallelogram or the volume of a parallelepiped (and higher dimensional version of this) in a linear context?

The answer to this are forms, and if they arise in a differential setting, differential forms. The alternating product (outer product) that calculations with forms bring automatically encode the important property of orientation. At the same time they describe what "area" would be in any dimensions, and if one takes infinitesimal versions of these how to integrate them together to areas of differentiable manifolds.

This book does all this right. It introduces forms in a straight forward way, gives pictures that shows how they look, gives geometric interpretations of computations (like the simple, yet all too rarely taught fact that the determinant of a square matrix is the volume of the vectors making up the matrix). Readers with this knowledge will suddenly have a deep understanding why one gets a determinant when one changes variables in integration!

For anybody who wants to have a good foundation for differential geometry, have a better understanding of vector calculus than most other/older text on the topic contain, or just wanted to know what those forms really are that geometers in more advanced texts just define algebraically, this is at present the best text I know to learn this.

There are other texts (though not too many) that attempt at giving elementary treatments of vector calculus and forms. For example William Burke's "Applied Differential Geometry" is one such text, which also contains graphical representation of forms. By taking a more computational approach the present text does, I think a better job, in clarifying forms in application. Another text would be for example Harvey Flanders' "Differential Forms with Applications to the Physical Sciences". This is a considerably more advanced text than Hubbard's and lacks many elementary foundations and basic geometric properties that Hubbard lays out quite nicely. People interested in electromagnetical applications but also just lots of visual ways of representing forms should check notes of Selfridge, Arnold and Warnick.

I have just two minor remarks. The book is filled with interesting short bios of relevant mathematicians, yet Hermann Grassmann who is primarily responsible (and chronically undercredited) for the introduction of forms is not mentioned in the text.

The second is that I disagree with Hubbard's stance (citing Dieudonne) that multivalued function are meaningless. There are in fact problems that look simpler when multivalued functions are allowed and there are ways to compute with them (branch cuts etc).

But these are minor comments that don't take anything away from this being a great text.

In all this is a beautifully written text on vector calculus, integration and differential forms that I can highly recommend to undergrads yet also graduate students and working colleagues.

I really hope that texts like these will soon be typical for introductory courses on vector calculus and integration, because this is essentially how it should be done... it should be easy to see why after reading the text.

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! This text is suitable for beginning and more advanced students alike. Exercises are clearly marked as basic, intermediate, or more difficult problems. Also, the more difficult proofs are placed in an appendix for the more advanced readers, so that beginners can focus on learning fundamentals without having to bog down in the details of the proof in question. 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! The comments and insights are there for those who need them; those who don't can simply skip them (i.e. no loss of continuity). 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 (that's right, part II).

This book is very helpful in the way that it describes the jist of a theorem or idea in simple terms before stating the actual theorum. Then the theorem is usually stated in a generalized form, which would be difficult to understand without the introduction, but is ideal for people who really want to think about the topic in more depth. 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.