A Geometric Approach to Differential Forms [Paperback]

David Bachman (Author)

Book Description

ISBN-10: 0817644997 | ISBN-13: 978-0817644994 | Publication Date: August 30, 2006 | Edition: 1

This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.

Editorial Reviews

Review

From the reviews: "[The author's] idea is to use geometric intuition to alleviate some of the algebraic difficulties...The emphasis is on understanding rather than on detailed derivations and proofs. This is definitely the right approach in a course at this level." —MAA Reviews "This book is intended as an elementary introduction to the notion of differential forms, written at an undergraduate level. … The book certainly has its merits and is very nicely illustrated … . It should be noted that the material, which has been tested already in the classroom, aims at three potential course tracks: a course in multivariable calculus, a course in vector calculus and a course for more advanced undergraduates (and beginning graduates)." —Frans Cantrijn, Mathematical Reviews, Issue 2007 d "Students! Read this book if you want to get acquainted with differential forms! This is a kind of unusual book, which is really nice to read for a beginner … a lot of examples and numerous meaningful ‘experiments’ to get the real meaning behind the scenes, that otherwise should be digged from formal proofs. … I recommend the book not only for students, but also for teachers!" (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

From the Back Cover

The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the undergraduate level. The book begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The author approaches the subject with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. This facilitates the development of differential forms without assuming a background in linear algebra. Throughout the text, emphasis is placed on applications in 3 dimensions, but all definitions are given so as to be easily generalized to higher dimensions. A centerpiece of the text is the generalized Stokes' theorem. Although this theorem implies all of the classical integral theorems of vector calculus, it is far easier for students to both comprehend and remember. The text is designed to support three distinct course tracks: the first as the primary textbook for third semester (multivariable) calculus, suitable for anyone with a year of calculus; the second is aimed at students enrolled in vector calculus; while the third targets advanced undergraduates and beginning graduate students in physics or mathematics, touching on more advanced topics such as Maxwell's equations, foliation theory, and cohomology. Containing excellent motivation, numerous illustrations and solutions to selected problems in an appendix, the material has been tested in the classroom along all three potential course tracks.

Product Details

• Paperback: 149 pages
• Publisher: Birkhäuser Boston; 1 edition (August 30, 2006)
• Language: English
• ISBN-10: 0817644997
• ISBN-13: 978-0817644994
• Product Dimensions: 9.1 x 6 x 0.6 inches
• Shipping Weight: 9.6 ounces
• Average Customer Review: 3.7 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #1,085,509 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

59 of 63 people found the following review helpful:

5.0 out of 5 stars A Very Accessable Intro to Forms, March 10, 2007

By 

Cl3ccooh (Pullman, WA) - See all my reviews

This review is from: A Geometric Approach to Differential Forms (Paperback)

I highly recommend this text for anyone looking for a "gentle" introduction to forms and manifolds.

When learning a topic, I believe that it is important to develop both computational proficiency and a deep conceptual understanding. I have come to understand that manipulating symbols is not sufficient and that whenever possible, understanding the underlying geometry is critical.

For whatever reason, I struggled to understand forms from other sources. (Maybe I was too focused on the algebra of the wedge product.) However, Bachmann's exposition was easy to follow and very insightful. It was a revelation that all integrands are not differential forms. Also, I had read elsewhere that forms are a basis for the tangent space of a manifold. I could say the words but they contained little meaning for me. Within the first couple of days with Bachmannn's book, this as well as some other basic ideas became crystal clear. I particularly liked that he at times presents more than one geometric interpretation of an concept.

Anyone who has already seen some vector calculus and now wants a very quick introduction to forms with a minimum time investment can benefit greatly from this text. In total, I spent about a month reading, A Geometric Approach to Differential Forms and I am now confident that I am ready to tackle more advanced texts on the topic.

A word of caution, in the book's Preface, it is suggested that there are three possible tracks one can take with this text. In addition to an upper division track that focuses on forms and manifolds, one is a vector calculus track and another is a multi-variable calculus track. In either of the latter two cases, if that is your main interest, I would recommend a text like Marsden's Vector Calculus. It encompasses a broader base of material and it is also very well written.

9 of 9 people found the following review helpful:

4.0 out of 5 stars A good place to start and objective accomplished, December 6, 2008

By 

Ricardo Avila (Santiago, Chile) - See all my reviews
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This review is from: A Geometric Approach to Differential Forms (Paperback)

I am a graduate physics student who as such, has got a prior and long exposure to vector calculus and I was searching for a good intuitive exposition to the subject of differential forms. I also had this objective: To finally understand how the fundamental theorem of calculus, Green's theorem on the plane, Gauss Theorem of the divergence and the stoke theorem of the curl in vector calculus all arise and where diferent faces of ONE SINGLE FORMULA, namely: The "generalized" Stoke's Theorem of differential forms. I must say before anything else that after reading this book the objective was accomplished.
I have found this text to be a very nice introduction to differential forms. I read it in just two weeks starting from chapter 3 to 9 (The book has 9 chapters and an Appendix), I didn't bother with the first two chapters which are a review of multivariable calculus (Calculus III).
The chapters are as follow:
1-Multivariable calculus, 2-Parameterizations, 3-Introduction to forms, 4-Forms, 5-Differential forms, 6-Differentiation of forms, 7-Stokes' Theorem, 8-Applications, 9-Manifolds, A-Non-linear forms
I list now some of the good features that I have found about this book:
i)-The author does a very clear presentation of each topic and gives plenty of intuitive explanations.
ii)-It is suited for undergraduates.
iii)-The book and therefore, the chapters are short making it an even easier reading.
The bad features (reasons for why I gave it only four stars):
iv)-Chapter 9 is different from all previous chapters, is harder and explanations aren't clear, the only drawback of the book (or perhaps is just me). To cite an example of this: The definition of a pull-back of a differential form or the section on quotient spaces.
But nevermind all in all, a great place to start.

8 of 8 people found the following review helpful:

5.0 out of 5 stars A very nice introduction to forms, May 25, 2009

By 

Me (Los Angeles, Ca) - See all my reviews

Amazon Verified Purchase

This review is from: A Geometric Approach to Differential Forms (Paperback)

This, and another book I will mention shortly, is where I learned about differential forms. The author spells out the fundamentals of differential forms in a very friendly and very geometric and intuitive way. I'm one of those people who managed to collect a ton of books on the subject and couldn't really made sense of forms - that is, until I started this one (Flanders, Bishop & Goldberg, do Carmo, Cartan, etc... sound familiar?).

First of all, it must be noted - this is a very simplified version of what Valdimir Arnold covers in his book titled "Mathematical Methods of Classical Mechanics" - the other book where I learned about forms. Bachman takes chapter 7 of Arnold's book and translates it into "english" ... or math that the rest of us can understand (Arnold's book is even cited in the bibliography of Bachman's book). Arnold can be a little confusing at first - this book is a very very welcome addition to my library and a very welcome companion to Arnold's book.

All the problems in this book either have answers in the back or enough hints for you to get through it painlessly. It took me about 3 - 4 days of non-stop reading to get through all the problems (I was on summer break when going through this book).

As one reviewer mentioned - this isn't a thorough book on forms, you won't learn all the algebraic details. You will get a hint of it's application to manifold calculus - for these I might recommend Morita's book titled "Geometry of Differential Forms". For the physicist I might recommend (after this one and Arnold's book) Frankel's "The Geometry of Physics", which goes into much more. I also am enjoying Schutz's "Geometrical Methods of Mathematical Physics" after learning what forms are and how to use them - the former is shorter and gets to the gist much quicker and not entirely as rigorous.

After reading this book, and then Arnold's, I felt gypped! This stuff is so simple and an almost obvious extension of multivariable calculus! Why are people complicating forms so much?