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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition), by John H. Hubbard, Barbara Burke Hubbard
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Using a dual presentation that is rigorous and comprehensive—yet exceptionaly reader-friendly in approach—this book covers most of the standard topics in multivariate calculus and an introduction to linear algebra. It focuses in underlying ideas, integrates theory and applications, offers a host of learning aids, features coverage of differential forms, and emphasizes numerical methods that highlight modern applications of mathematics. The revised and expanded content of this edition includes new discussions of functions; complex numbers; closure, interior, and boundary; orientation; forms restricted to vector spaces; expanded discussions of subsets and subspaces of R^n; probability, change of basis matrix; and more. For individuals interested in the fields of mathematics, engineering, and science—and looking for a unified approach and better understanding of vector calculus, linear algebra, and differential forms.
- Sales Rank: #907771 in Books
- Published on: 2001-09-15
- Ingredients: Example Ingredients
- Original language: English
- Number of items: 1
- Dimensions: 9.63" h x 1.38" w x 8.38" l,
- Binding: Hardcover
- 800 pages
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.
From the Back Cover
Using a dual presentation that is rigorous and comprehensive—yet exceptionaly reader-friendly in approach—this book covers most of the standard topics in multivariate calculus and an introduction to linear algebra. It focuses in underlying ideas, integrates theory and applications, offers a host of learning aids, features coverage of differential forms, and emphasizes numerical methods that highlight modern applications of mathematics. The revised and expanded content of this edition includes new discussions of functions; complex numbers; closure, interior, and boundary; orientation; forms restricted to vector spaces; expanded discussions of subsets and subspaces of R^n; probability, change of basis matrix; and more. For individuals interested in the fields of mathematics, engineering, and science—and looking for a unified approach and better understanding of vector calculus, linear algebra, and differential forms.
About the Author
John H. Hubbard (BA Harvard University, PhD University of Paris) is professor of mathematics at Cornell University and at the University of Provence in Marseilles he is the author of several books on differential equations. His research mainly concerns complex analysis, differential equations, and dynamical systems. He believes that mathematics research and teaching are activities that enrich each other and should not be separated.
Barbara Burke Hubbard (BA Harvard University) is the author of The World According to Wavelets, which was awarded the prix d'Alembert by the French Mathematical Society in 1996.
Most helpful customer reviews
0 of 0 people found the following review helpful.
Five Stars
By Lady Nigella
Really unique approach in unifying fundamentals of real analysis, linear algebra and differential forms.
16 of 17 people found the following review helpful.
Excellent approach to vectors, forms and things linear and integrable
By Georg Essl
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.
10 of 10 people found the following review helpful.
Excellent text for beginners and more advanced students
By A Customer
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).
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