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Geometric Multivector Analysis

From Grassmann to Dirac

  • Andreas Rosén
Textbook

Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Andreas Rosén
    Pages 1-22
  3. Andreas Rosén
    Pages 23-71
  4. Andreas Rosén
    Pages 73-103
  5. Andreas Rosén
    Pages 105-151
  6. Andreas Rosén
    Pages 153-184
  7. Andreas Rosén
    Pages 185-207
  8. Andreas Rosén
    Pages 209-254
  9. Andreas Rosén
    Pages 255-284
  10. Andreas Rosén
    Pages 285-341
  11. Andreas Rosén
    Pages 343-382
  12. Andreas Rosén
    Pages 383-422
  13. Andreas Rosén
    Pages 423-449
  14. Back Matter
    Pages 451-465

About this book

Introduction

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.

Keywords

Clifford algebra exterior algebra spinors Hodge decompositions dirac equations boundary value problems, index theorems

Authors and affiliations

  • Andreas Rosén
    • 1
  1. 1.Department of Mathematical SciencesChalmers University of Technology and the University of GothenburgGothenburgSweden

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-31411-8
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-31410-1
  • Online ISBN 978-3-030-31411-8
  • Series Print ISSN 1019-6242
  • Series Online ISSN 2296-4894
  • Buy this book on publisher's site
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