# A Geometric Approach to Differential Forms

• David Bachman
Book

1. Front Matter
Pages i-xvii
2. Pages 1-16
3. Pages 17-25
4. Pages 27-32
5. Pages 33-48
6. Pages 49-73
7. Pages 75-81
8. Pages 83-97
9. Pages 99-106
10. Pages 107-118
11. Back Matter
Pages 119-133

### Introduction

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.

### Keywords

algebra calculus differential equation differential forms equation manifold mathematics nonlinear forms theorem variable

#### Authors and affiliations

• David Bachman
• 1
1. 1.Department of MathematicsPitzer CollegeClaremontUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-0-8176-4520-5
• Copyright Information Birkhäuser Boston 2006
• Publisher Name Birkhäuser Boston
• eBook Packages Mathematics and Statistics
• Print ISBN 978-0-8176-4499-4
• Online ISBN 978-0-8176-4520-5