Overview
- Develops carefully the foundations of calculus of multivector fields and differential forms
- Explains advanced material in a self-contained and down-to-earth way, e.g. Clifford algebra, spinors, differential forms and index theorems
- Touches on modern research areas in analysis
- Contains a novel treatment of Hodge decompositions based on first-order differential operators
Part of the book series: Birkhäuser Advanced Texts Basler Lehrbücher (BAT)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
Table of contents (12 chapters)
-
Front Matter
-
Back Matter
Keywords
About this book
This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.
The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics.
The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis.
Reviews
“The book is carefully prepared and well presented, and I recommend the book … for students who have just mastered vector calculus and Maxwellian electromagnetism.” (Hirokazu Nishimura, zbMATH 1433.58001, 2020)
Authors and Affiliations
About the author
Andreas Rosén is a Professor at the Chalmers University of Technology and the University of Gothenburg, Sweden. His research mostly concerns Partial Differential Equations, and uses techniques from harmonic analysis and operator theory.
Bibliographic Information
Book Title: Geometric Multivector Analysis
Book Subtitle: From Grassmann to Dirac
Authors: Andreas Rosén
Series Title: Birkhäuser Advanced Texts Basler Lehrbücher
DOI: https://doi.org/10.1007/978-3-030-31411-8
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-31410-1Published: 20 November 2019
Softcover ISBN: 978-3-030-31413-2Published: 20 November 2020
eBook ISBN: 978-3-030-31411-8Published: 09 November 2019
Series ISSN: 1019-6242
Series E-ISSN: 2296-4894
Edition Number: 1
Number of Pages: XIII, 465
Number of Illustrations: 21 b/w illustrations, 8 illustrations in colour
Topics: Linear and Multilinear Algebras, Matrix Theory, Global Analysis and Analysis on Manifolds, Partial Differential Equations, Integral Equations, Differential Geometry