Differential Forms

✍ Scribed by Victor Guillemin, Peter Haine

• Publisher: WSPC
• Year: 2019
• Tongue: English
• Leaves: 273
• Category: Library

✦ Synopsis

"Guillemin and Haine’s goal is to construct a well-documented road map that extends undergraduate understanding of multivariable calculus into the theory of differential forms. Throughout, the authors emphasize connections between differential forms and topology while making connections to single and multivariable calculus via the change of variables formula, vector space duals, physics; classical mechanisms, div, curl, grad, Brouwer’s fixed-point theorem, divergence theorem, and Stokes’s theorem … The exercises support, apply and justify the developing road map." CHOICE "The book is very well written and could be readable and usable for some undergraduates." zbMATH There already exist a number of excellent graduate textbooks on the theory of differential forms as well as a handful of very good undergraduate textbooks on multivariable calculus in which this subject is briefly touched upon but not elaborated on enough. The goal of this textbook is to be readable and usable for undergraduates. It is entirely devoted to the subject of differential forms and explores a lot of its important ramifications. In particular, our book provides a detailed and lucid account of a fundamental result in the theory of differential forms which is, as a rule, not touched upon in undergraduate texts: the isomorphism between the Čech cohomology groups of a differential manifold and its de Rham cohomology groups. •Authoritative textbook on differential forms for undergraduates •Includes numerous Examples and Exercises for further in-depth understanding on the presented concepts •The first author, Victor Guillemin, is a world-renowned mathematician in the field of symplectic geometry •His co-author, Peter Haine, is a talented doctoral student at MIT under Clark Barwick. His research interests center around homotopy theory, algebraic K-theory and algebraic geometry

✦ Table of Contents

Preface
About the Authors
Contents
Chapter 1. Multilinear Algebra
Chapter 2. The Concept of a Differential Form
Chapter 3. Integration of Forms
Chapter 4. Manifolds and Forms on Manifolds
Chapter 5. Cohomology via Forms
Appendix a. Bump Functions and Partitions of Unity
Appendix b. The Implicit Function Theorem
Appendix c. Good Covers and Convexity Theorems
Bibliography
Index of Notation
Glossary of Terminology