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

24 customer reviews

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Book by John H. Hubbard


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Product Details

  • Hardcover
  • Publisher: Matrix Editions
  • Language: English
  • ISBN-10: 0971576653
  • ISBN-13: 978-0971576650
  • Product Dimensions: 10 x 8 x 1.5 inches
  • Shipping Weight: 3.2 pounds
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (24 customer reviews)
  • Amazon Best Sellers Rank: #293,616 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

66 of 68 people found the following review helpful By M. Feldman on May 14, 2004
Format: Hardcover
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.
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37 of 38 people found the following review helpful By Jun Zhou Zhang on May 25, 2005
Format: Hardcover
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.
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34 of 35 people found the following review helpful By A Customer on February 14, 2002
Format: Hardcover
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.
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23 of 23 people found the following review helpful By A Customer on February 5, 2000
Format: Hardcover
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.
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