A Geometric Approach to Differential Forms by David Bachman

Product Description


The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the sophomore 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 sophomore-level vector calculus; while the third targets advanced undergraduates and beginning graduate students in physics or mathematics, covering 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
Amazon Sales Rank: #26207 in Books
Published on: 2006-08-30
Number of items: 1
Binding: Paperback
133 pages
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
Customer Reviews

A geometric approach to differential forms
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.

A Very Accessable Intro to Forms
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.