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

The way this book presents Vector Calc. is as good if not better than Tom Apostol. I could not recommend another even if I could. But hey... That is my take.

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!

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.

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.

Although I am a graduate student in Mathematics, I foundHubbard's "undergraduate" text to be extremely helpful.Hubbard combines an intuitive heuristic approach appropriatefor undergraduates with a thoroughly rigorous set of proofsappropriate for graduate students. I found his discussion ofdifferential forms particularly helpful. He provides anexcellent intuitive motivation for the definitions, and thenhe follows this with a mathematically sound treatment of thetopic. This is a much nicer approach than one will find intexts such as Rudin's Principals of Mathematical Analysis.I highly recommend Hubbard's book to anyone wishing to learndifferential forms.

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.