Lectures on Clifford (geometric) algebras and applications
Rafal Ablamowicz, Garret Sobczyk
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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