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

by John H. Hubbard (Author)

Editorial Reviews

Product Description
Using a dual-presentation that is rigorous and comprehensive--yet exceptionally "student-friendly" in approach--this text covers most of the standard topics in multivariate calculus and a substantial part of a standard first course in linear algebra. It focuses on underlying ideas, integrates theory and applications, offers a host of pedagogical aids, and features coverage of differential forms. There is an emphasis on numerical methods to prepare students for modern applications of mathematics.

From the Author
Several readers have complained about the lack of a student solution manual. One now exists, published by Matrix Editions. Errata for the book are posted on the book web site (URL given in the book). The most recent posting was Feb. 29, 2002. Readers who wish to be notified by e-mail when new errata are posted can sign up via the web site or e-mail the authors (address given in the book).

What's new in the second edition (the one with the pale yellow cover now being sold):

The main change is that we introduce a new approach to Lebesgue integration. In addition, the second edition has approximately 270 additional exercises and 50 additional examples. We have added pictures of mathematicians and more historical notes. There are now end-of-section exercises, as well as review exercises for Chapters 1--6. Some useful formulas are listed on the back cover.

More difficult material from Chapter 0 was moved to the Appendix. The inverse and implicit function theorems have been rewritten. In Chapter 3 we simplified the definition of a manifold, and we now begin with the general case and discuss curves and surfaces as examples. Similarly, in Chapter 5, we eliminated the separate sections on arc length and surface area; we now have one section on volume of manifolds.

In Chapter 6, we rewrote the discussion of orientation and changed the definition of a piece-with-boundary of a manifold, to make it both simpler and more inclusive. --This text refers to an out of print or unavailable edition of this title.

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Product Details

• Hardcover: 687 pages
• Publisher: Prentice Hall; 1 edition (September 21, 1998)
• Language: English
• ISBN-10: 0136574467
• ISBN-13: 978-0136574460
• Product Dimensions: 9.5 x 8.2 x 1.2 inches
• Shipping Weight: 3 pounds
• Average Customer Review: 4.2 out of 5 stars See all reviews (18 customer reviews)
• Amazon.com Sales Rank: #1,104,910 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

42 of 43 people found the following review helpful:

5.0 out of 5 stars A Must Text, May 14, 2004

By	Mark Feldman (St. Louis, MO USA) - See all my reviews

This review is from: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition) (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.

20 of 20 people found the following review helpful:

5.0 out of 5 stars Revolutionize the way calculus is taught, May 25, 2005

By	Jun Zhou Zhang (Coram, NY USA) - See all my reviews (REAL NAME)

This review is from: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition) (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.

18 of 18 people found the following review helpful:

5.0 out of 5 stars 2nd edition much improved, May 15, 2002

By A Customer

This review is from: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition) (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.