Author: Carlos Polanco

Exterior Calculus: Theory and Cases

eBook: US \$49 Special Offer (PDF + Printed Copy): US \$90
Printed Copy: US \$65
ISBN: 978-981-4998-79-6 (Print)
ISBN: 978-981-4998-78-9 (Online)
Year of Publication: 2021
DOI: 10.2174/97898149987891210101

Introduction

Exterior calculus is a branch of mathematics which involves differential geometry. In Exterior calculus the concept of differentiations is generalized to antisymmetric exterior derivatives and the notions of ordinary integration to differentiable manifolds of arbitrary dimensions. It therefore generalizes the fundamental theorem of calculus to Stokes' theorem.

This textbook covers the fundamental requirements of exterior calculus in curricula for college students in mathematics and engineering programs. Chapters start from Heaviside-Gibbs algebra, and progress to different concepts in Grassman algebra. The final section of the book covers applications of exterior calculus with solutions.

Readers will find a concise and clear study of vector calculus and differential geometry, along with several examples and exercises. The solutions to the exercises are also included at the end of the book.

This is an ideal book for students with a basic background in mathematics who wish to learn about exterior calculus as part of their college curriculum and equip themselves with the knowledge to apply relevant theoretical concepts in practical situations.

Preface

This Exterior Calculus ebook has been designed for third-year students of Sciences, as it contains the fundamentals related to Geometric algebra or Grassmann algebra oriented to Calculus. Without any doubt, this algebra has important implications in Science and Engineering. Here, the reader will find a clear presentation of the Geometric algebra on a plane and in space, as well as the extension of all its operators in ℝn . In order to make the comprehension of this important algebra easier, some examples and completely solved exercises are included.

The ebook thoroughly examines the elements of Geometric algebra G over the Real field and these operators: inner product, outer product, and geometric product, their components, and their geometric representation, as well as their properties and the rigid transformations on the plane and in space. It also reviews the differentiation and the integration over Geometric algebra, including the line integral and surface integral. The Green, Stokes and Gauss theorems are also studied in detail and the Theorem of Fundamental Calculus is generalized.

The author hopes the reader interested in the study of the fundamentals of Exterior calculus, finds useful the material presented here and that the students that start studying this field find this information motivating. The author would like to acknowledge the Faculty of Sciences at Universidad Nacional Autónoma de México for support.

Not applicable.

CONFLICT OF INTEREST

The author declares no conflict of interest regarding the contents of each of the chapters of this ebook.

Carlos Polanco
Faculty of Sciences