Lectures on Clifford (geometric) algebras and applications
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書誌事項
Lectures on Clifford (geometric) algebras and applications
Birkhäuser, c2004
- : us
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
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 Dkf = 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 forClifford algebras * 7.2 References * Index
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