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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach Hardcover – Sep 7 2001


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Hardcover, Sep 7 2001
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Product Details

  • Hardcover: 668 pages
  • Publisher: Prentice Hall; 2 edition (Sept. 7 2001)
  • Language: English
  • ISBN-10: 0130414085
  • ISBN-13: 978-0130414083
  • Product Dimensions: 23.4 x 21.1 x 3.6 cm
  • Shipping Weight: 1.5 Kg
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Bestsellers Rank: #2,159,915 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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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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By J. K. OYoung on March 7 2004
Format: Hardcover
this guy tells u a story before he gives you another story in his proof of theorem, he notation is very hard to follow. His proofs are ugly. is there a better advanced calculus book?
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By A Customer on May 16 2002
Format: Hardcover
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.
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Format: Hardcover
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.
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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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