Customer Reviews

A Geometric Approach to Differential Forms

5 star:		(2)
4 star:		(3)
3 star:		(1)
2 star:		(0)
1 star:		(1)

 

 

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.

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.

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?

I think the usefulness of this book will depend a lot on the reader's background and goals.

The subject of differential forms was one of the gaps in my otherwise strong math background. More than once I started reading a differential geometry text and found myself bogged down in definitions. I chose this book in the hope of being quickly "brought on board". I was not disappointed -- in a few weeks, I finally understood what the generalized Stokes Theorem was.

In my case I had a background in multivariate calculus, so skipped the initial chapters. It is not clear to me how useful this book would be to someone without that background.

I feel there is one big point that the author does not adequately emphasize: a large part of the motivation for differential forms is their independence of coordinate system. The large number of numerical examples, while quite helpful, tend to obscure this point. On the other hand, to elaborate this point might have involved so much formalism as to lose me like the other books did.

DO NOT buy the Kindle edition of this book. You will be wasting your money. The mathematical fonts are bitmapped and almost unreadable. Amazon needs to fix this problem. Buy the print edition.

The Kindle edition is indeed not ideal, but I would still recommend it. The math fonts are not perfect, but I was able to read it just fine. Occationally there are formulas that are not clear, but I was able to interpret them. I read this entire book on an IPOD touch with no real problem, so it should be better on larger devices.

The book itself (in any format) is good and I recommend it. Personally I used it to get a better physical understanding of differential forms to aid in my study of differential geometry. The book delivers on its promise to provide clear descriptions and explanations. It does provide a real geometric (and physical) understanding of the subject.

Easy to read, but not too deep in theory or algebraic properties of differntial forms. Interesting for many exercices to solve ( with solutions !) Useful to grasp an intuitive approach to the concept, but if you are seeking a thoroughtly book on the subject this is not the book.