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Lectures on Clifford (geometric) algebras and applications
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Lectures on Clifford (geometric) algebras and applications

Lectures on Clifford (geometric) algebras and applications

Rafal Ablamowicz, Garret Sobczyk

238 pages, parution le 09/12/2003

Résumé

The subject of Clifford (geometric) algebras offers a unified algebraic framework for the direct expression of the geometric concepts underlying the mathematical theories of linear and multilinear algebra, projective and affine geometries, and differential geometry. This bird's-eye view of Clifford (geometric) algebras and their applications is presented by six of the world's leading experts in the field.

Key topics and features of this systematic exposition:

  • An Introductory chapter on Clifford Algebras by Pertti Lounesto
  • Ian Porteous (Chapter 2) reveals the mathematical structure of Clifford algebras in terms of the classical groups
  • John Ryan (Chapter 3) introduces the basic concepts of Clifford analysis, which extends the well-known complex analysis of the plane to three and higher dimensions
  • William Baylis (Chapter 4) investigates some of the extensive applications that have been made in mathematical physics, including the basic ideas of electromagnetism and special relativity
  • John Selig (Chapter 5) explores the successes that Clifford algebras, especially quaternions and bi-quaternions, have found in computer vision and robotics
  • Tom Branson (Chapter 6) discusses some of the deepest results that Clifford algebras have made possible in our understanding of differential geometry
  • Editors (Appendix) give an extensive review of various software packages for computations with Clifford algebras including standalone programs, on-line calculators, special purpose numeric software, and symbolic add-ons to computer algebra systems

This text will serve beginning graduate students and researchers in diverse areas---mathematics, physics, computer science and engineering; it will be useful both for newcomers who have little prior knowledge of the subject and established professionals who wish to keep abreast of the latest applications.

Contents

  • Preface (Rafal Ablamowicz and Garret Sobczyk)
  • Lecture 1: Introduction to Clifford Algebras (Pertti Lounesto)
    • Introduction
    • Clifford algebra of the Euclidean plane
    • Quaternions
    • Clifford algebra of the Euclidean space R3
    • The electron spin in a magnetic field
    • From column spinors to spinor operators
    • In 4D: Clifford algebra Cl4 of R4
    • Clifford algebra of Minkowski spacetime
    • The exterior algebra and contractions
    • The Grassmann - Cayley algebra and shuffle products
    • Alternative definitions of the Clifford algebra
    • References
  • Lecture 2: Mathematical Structure of Clifford Algebras (Ian Porteous)
    • Clifford algebras
    • Conjugation
    • References
  • Lecture 3: Clifford Analysis (John Ryan)
    • Introduction
    • Foundations of Clifford analysis
    • Other types of Clifford holomorphic functions
    • The equation Dkf = 0
    • Conformal groups and Clifford analysis
    • Conformally flat spin manifolds
    • Boundary behavior and Hardy spaces
    • More on Clifford analysis on the sphere
    • The Fourier transform and Clifford analysis
    • Complex Clifford analysis
    • References
  • Lecture 4: Applications of Clifford Algebras in Physics (William E. Baylis)
    • Introduction
    • Three Clifford algebras
    • Paravectors and relativity
    • Eigenspinors
    • Maxwell's equation
    • Quantum theory
    • Conclusions
    • References
  • Lecture 5: Clifford Algebras in Engineering (J.M. Selig)
    • Introduction
    • Quaternions
    • Biquaternions
    • Points, lines, and planes
    • Computer vision example
    • Robot kinematics
    • Concluding remarks
    • References
  • Lecture 6: Clifford Bundles and Clifford Algebras (Thomas Branson)
    • Spin Geometry
    • Conformal Structure
    • Tractor constructions
    • References
  • Appendix (Rafal Ablamowicz and Garret Sobczyk)
    • Software for Clifford algebras
    • References
  • Index

Caractéristiques techniques

  PAPIER
Éditeur(s) Birkhäuser
Auteur(s) Rafal Ablamowicz, Garret Sobczyk
Parution 09/12/2003
Nb. de pages 238
Format 15,5 x 23,5
Couverture Broché
Poids 345g
Intérieur Noir et Blanc
EAN13 9780817632571
ISBN13 978-0-8176-3257-1

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