Synopses & Reviews
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.
From the reviews: "...This book is recommended reading for beginning graduate students and other newcomers in this field. The authors of each chapter are world leading experts in their fields and the chapters are well written and organized. In spite of the introductory level of each chapter, many references are given which will help the interested reader go deeper into the subject of interest. Moreover, an appendix written by the editors gives a summary of the existing Clifford algebra software for symbolic computations."--Mathematical Reviews "This text contains a set of lectures presented by P. Lounesto, Introduction to Clifford Algebras; I. Porteous, Mathematical Structure of Clifford Algebras; J. Ryan, Clifford Analysis; W.E. Baylis, Applications of Clifford Algebras in Physics; J.M. Selig, Clifford Algebras in Engineering; T. Branson, Clifford Bundles and Clifford Algebras; and an appendix by R. Ablamowicz and G. Sobczyk describing software for solving different kinds of problems involving computations with Clifford algebras. Each one of the lectures is a jewel and will be appreciated [by] newcomers wanting an introduction to a rapidly developing field as well [as] by practitioners, [who] for pleasure certainly will enjoy reading the texts of those well-known experts...."--Zentralblatt Math "The book under review contains the series of lectures on Clifford Geometric Algebras ... . the principal aim of the book to provide beginning graduate students in mathematics and physics and other who are new-interested with no prior knowledge in the secrets of Clifford algebras has been achieved. ... the presented material subsumes research activities and some of the recent scientific advances of this theory, and it is an useful learning tool for scientists and engineers from academia and industry." (Clementina D. Mladenova, Journal of Geometry and Symmetry in Physics, Issue 3, 2005)
Advances in technology over the last 25 years have created a situation in which workers in diverse areas of computerscience and engineering have found it neces sary to increase their knowledge of related fields in order to make further progress. Clifford (geometric) algebra offers a unified algebraic framework for the direct expression of the geometric ideas underlying the great mathematical theories of linear and multilinear algebra, projective and affine geometries, and differential geometry. Indeed, for many people working in this area, geometric algebra is the natural extension of the real number system to include the concept of direction. The familiar complex numbers of the plane and the quaternions of four dimen sions are examples of lower-dimensional geometric algebras. During "The 6th International Conference on Clifford Algebras and their Ap plications in Mathematical Physics" held May 20--25, 2002, at Tennessee Tech nological University in Cookeville, Tennessee, a Lecture Series on Clifford Ge ometric Algebras was presented. Its goal was to to provide beginning graduate students in mathematics and physics and other newcomers to the field with no prior knowledge of Clifford algebras with a bird's eye view of Clifford geometric algebras and their applications. The lectures were given by some of the field's most recognized experts. The enthusiastic response of the more than 80 partici pants in the Lecture Series, many of whom were graduate students or postdocs, encouraged us to publish the expanded lectures as chapters in book form."
The subject of Clifford (geometric) algebras offers a unified algebraic framework for the direct expression of the geometric concepts in algebra, geometry, and physics. This bird's-eye view of the discipline is presented by six of the world's leading experts in the field; it features an introductory chapter on Clifford algebras, followed by extensive explorations of their applications to physics, computer science, and differential geometry. The book is ideal for graduate students in mathematics, physics, and computer science; it is appropriate both for newcomers who have little prior knowledge of the field and professionals who wish to keep abreast of the latest applications.
This text, written by established mathematicians and physicists, provides a systematic, unified exposition of Clifford (geometric) algebras. Beginning with an introductory chapter, the book covers the mathematical structure of Clifford algebras and the basic concepts of Clifford analysis, and then provides a detailed examination of the many applications of Clifford algebras to differential geometry, physics, computer vision and robotics. No prior knowledge of the subject is assumed. The book 's breadth will appeal to graduate students and researchers in mathematics, physics, and engineering.Contents: P. Lounesto, Introduction to Clifford Algebras; I. Porteous, Mathematical Structure of Clifford Algebras; J. Ryan, Clifford Analysis; W. Baylis, Applications of Clifford Algebras in Physics; J. Selig, Clifford Algebras in Engineering; T. Branson, Clifford Bundles and Clifford Algebras; R. Ablamowicz and G. Sobczyk, Appendix: Software for Clifford (Geometric) Algebras
Table of Contents
Preface (Rafal Ablamowicz and Garret Sobczyk) * Lecture 1: Introduction to Clifford Algebras (Pertti Lounesto) * 1.1 Introduction * 1.2 Clifford algebra of the Euclidean plane * 1.3 Quaternions * 1.4 Clifford algebra of the Euclidean space R3 * 1.5 The electron spin in a magnetic field * 1.6 From column spinors to spinor operators * 1.7 In 4D: Clifford algebra Cl4 of R4 * 1.8 Clifford algebra of Minkowski spacetime * 1.9 The exterior algebra and contractions * 1.10 The Grassmann-Cayley algebra and shuffle products * 1.11 Alternative definitions of the Clifford algebra * 1.12 References * Lecture 2: Mathematical Structure of Clifford Algebras (Ian Porteous) * 2.1 Clifford algebras * 2.2 Conjugation * 2.3 References * Lecture 3: Clifford Analysis (John Ryan) * 3.1 Introduction * 3.2 Foundations of Clifford analysis * 3.3 Other types of Clifford holomorphic functions * 3.4 The equation Dkƒ = 0 * 3.5 Conformal groups and Clifford analysis * 3.6 Conformally flat spin manifolds * 3.7 Boundary behavior and Hardy spaces * 3.8 More on Clifford analysis on the sphere * 3.9 The Fourier transform and Clifford analysis * 3.10 Complex Clifford analysis * 3.11 References * Lecture 4: Applications of Clifford Algebras in Physics (William E. Baylis) * 4.1 Introduction * 4.2 Three Clifford algebras * 4.3 Paravectors and relativity * 4.4 Eigenspinors * 4.5 Maxwell's equation * 4.6 Quantum theory * 4.7 Conclusions * 4.8 References * Lecture 5: Clifford Algebras in Engineering (J.M. Selig) * 5.1 Introduction * 5.2 Quaternions * 5.3 Biquaternions * 5.4 Points, lines, and planes * 5.5 Computer vision example * 5.6 Robot kinematics * 5.7 Concluding remarks * 5.8 References * Lecture 6: Clifford Bundles and Clifford Algebras (Thomas Branson) * 6.1 Spin Geometry * 6.2 Conformal Structure * 6.3 Tractor constructions * 6.4 References * Appendix (Rafal Ablamowicz and Garret Sobczyk) * 7.1 Software for Clifford algebras * 7.2 References * Index