A Geometric Approach to Differential Forms

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.

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)

“The book is dedicated to a very interesting, but rather difficult area involving differential forms, integration and the Stokes theorem. … In this book, as the title suggests, the author tries to get geometrical imagination involved as much as possible, together with keeping everything correct. … may serve as a good introduction to multivariable calculus and parametrizations. … can be recommended for all readers who want to gain deeper understanding of the area.” (Pavel Kůs, Applications of Mathematics, Vol. 56 (2), 2011)