Vector Analysis [Hardcover]

Klaus Janich (Author), Leslie D. Kay (Author)

Book Description

ISBN-10: 0387986499 | ISBN-13: 978-0387986494 | Publication Date: February 16, 2001 | Edition: 1

This book presents modern vector analysis and carefully describes the classical notation and understanding of the theory. It covers all of the classical vector analysis in Euclidean space, as well as on manifolds, and goes on to introduce de Rham Cohomology, Hodge theory, elementary differential geometry, and basic duality. The material is accessible to readers and students with only calculus and linear algebra as prerequisites. A large number of illustrations, exercises, and tests with answers make this book an invaluable self-study source.

Editorial Reviews

Review

From the reviews: "The present book is a marvelous introduction in the modern theory of manifolds and differential forms. The undergraduate student can closely examine tangent spaces, basic concepts of differential forms, integration on manifolds, Stokes theorem, de Rham-cohomology theorem, differential forms on Riemannian manifolds, elements of the theory of differential equations on manifolds (Laplace-Beltrami operators). Every chapter contains useful exercises for the students."¿ZENTRALBLATT MATH "Within the ambit of the MMath there is increasing need for good source material for reading courses in the 4th year. This is just such a source. It is extremely well written. The exercises are well thought out and, for instance, each chapter ends with a section that discusses rather than solves the exercises. It is very user friendly … . The book is quite well priced and is one to consider seriously for library purchase. … I enjoyed working my way through it immensely." (Tim Porter, The Mathematical Gazette, Vol. 86 (506), 2002) "‘Vector analysis’ or ‘vector calculus,’ as it is sometimes known, is one of the most fascinating subjects in the undergraduate mathematics curriculum. It also is one of the subjects that has the largest number of dramatically different incarnations. … Klaus Jänich’s Vector Analysis is about differential manifolds, differential forms, and integration on manifolds. The approach is quite sophisticated, but the author does try to be more helpful to readers than the typical advanced mathematics text." (MAA online, Mathematical Association of America, November, 2004) "This is a text on calculus on manifolds … for readers who know only the calculus of several variables … . each chapter contains tests and exercises. The exercises are well selected and enhance the description of the text, but one of the special features of this book is the tests. They are in marksheet style. Each problem is easy but appropriate to test the understanding of the reader, so this makes the book suitable for anyone studying the subject independently." (Akira Asada, Mathematical Reviews, Issue 2001 m) "This book is very easily accessible and self contained, clearly recalling at various points the facts from linear algebra which are needed in the progress. This does not mean, though, that we are dealing with a low-aiming text, on the contrary: a big effort is made to let the reader catch a glimpse of the way mathematical results are achieved. … Moreover, the author constantly tries to make the less vigilant reader aware of possible subtle difficulties or not completely straightforward conclusions." (F. Pasquotto, Nieuw Archief voor Wiskunde, Vol. 4 (3), 2003) "When and how to introduce students to surface integration … is difficult to answer. … This book offers a very nice description of the basic conceptual tools needed to explain these topics to mathematics students … . A nice feature is that it contains many figures (making easier an intuitive understanding of the treated topics), exercises with hints, and tests with answers. … It is certainly one of the best books in the field and can be strongly recommended for a general mathematical audience." (European Mathematical Society Newsletter, June, 2002) "The author ties together different approaches to the tangent space of a manifold coming from germs of real-valued functions, smooth curves, and that commonly adopted in physics literature based on Ricci calculus. This enables the reader to move between sources with little difficulty. Each chapter contains a test consisting of multiple-choice questions. The answers are provided and the reader is warned that some of the questions are so obviously simple that a healthy scare will result when they prove not to be so." (Nigel Steele, Times Higher Education Supplement, November, 2002) "This essentially modern text carefully develops vector analysis on manifolds, reinterprets it from the classical viewpoint (and with the classical notation) for three-dimensional Euclidean space, and then goes on to introduce de Rham cohomology and Hodge theory. The material is accessible to an undergraduate student with calculus, linear algebra, and some topology as prerequisites." (L’Enseignement Mathematique, Vol. 47 (1-2), 2001)

Product Details

• Hardcover: 304 pages
• Publisher: Springer; 1 edition (February 16, 2001)
• Language: English
• ISBN-10: 0387986499
• ISBN-13: 978-0387986494
• Product Dimensions: 10.2 x 7.3 x 1.2 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 3.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #813,986 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

41 of 43 people found the following review helpful:

5.0 out of 5 stars Another fine text by Klaus Janich, March 4, 2001

By 

Christopher Frenzen (Monterey, CA USA) - See all my reviews

This review is from: Vector Analysis (Hardcover)

Janich's previous texts on topology and linear algebra are very valuable additions to the library of many existing texts on these subjects. His new book on vector analysis is similarly valuable. It deals with manifolds, differential forms, and the generalized Stokes's theorem. This is the mathematical machinery necessary, for example, for mathematical physics and differential geometry. Janich's chapter 10 discusses classical vector analysis, relating Stokes's theorem in its modern form to the classical integrals theorems of vector analysis. He closes the book with a discussion of De Rham cohomology and differential forms on Riemannian manifolds.

Janich's exposition and mathematical taste are, as always, impeccable. Particularly valuable is a discussion in the second chapter which relates three different ways of defining the tangent space to a differentiable manifold. Each chapter also has a multiple choice test with answers in the back of the book which the reader pursuing self study can use to test his/her knowledge of the subject. Exercises close out each chapter as well.

Though this book is a Springer Undergraduate Text in Mathematics, it assumes some knowledge of topology and a fairly thorough knowledge of differential and integral calculus of several variables (the inverse function theorem, the rank theorem, the regular point theorem, and the regular value theorem.) Even readers not extremely well versed with these subjects will enjoy Janich's enthusiasm for his subject and his clarity of exposition. This is a very nice treatment of an important subject!

24 of 26 people found the following review helpful:

2.0 out of 5 stars not good as an intro to forms or manifolds, August 2, 2006

By 

Shawn McDougal - See all my reviews
(REAL NAME)   

This review is from: Vector Analysis (Hardcover)

I borrowed this book from the library and read a few sections to decide whether or not to buy it.

Jänich's writing style is attractive and lively, and he strikes me as the kind of person that would be great to have as a teacher to explain in detail the material that this book tries to cover.

However, the book is not good for a self-study introduction to manifolds or differential forms.

Having studied other texts on the subjects, I was able to follow Jänich's exposition and at the same time see major explanatory holes. The holes were big enough that, if I were truly someone in the intended audience of "university students in their second year" without prior background in differential geometry or multilinear algebra, then i'd really be lost.

The main problem is that Jänich uses lots of advanced terms/concepts without adequately (or even at all) defining them.

For instance, in chapter 3 when he's first explaining differential forms, he says of the alternating k-forms on a vector space V: "Clearly it is a real vector space in a canonical way." Now, I know this because i've seen it spelled out in more detail in other books, which show how k-tuples of basis vectors of V can form just such a canonical basis. However, Jänich does not spell it out at all.

Similarly, in the same section (p. 50) he mentions "the usual dual space". As if someone who is just now learning about forms already knows what "dual space" means!

Then, in section 5.3, Jänich gives a brief overview of some key Lebesgue integration results, since he (rightfully) doesn't assume his audience knows measure theory. Nonetheless, he then uses the term "set of measure zero" without ever defining it!

As another example of the book's lack of detail, in 7.7, as precursor to de Rham cohomology, Jänich breezes through the basics of simplicial homolgy in 3 (!) pages, with nary a picture of a triangulation nor any explanation of how the boundary operator works outside dimension 1. In my own experience grappling with these algebraic topology concepts, the idea that the boundary of the boundary is zero is not clear unless one at least has an understanding of how the boundary operator works. Unfortunately, Jänich provides no such understanding.

I might recommend this book for someone who knew the material already but was interested in a different take on some of the concepts, perhaps for planning their own course.

I would not recommend this book for newcomers to differential forms or manifolds.