Quaternions and Cayley Numbers: Algebra and Applications

Overview

This monograph is an accessible account of the normed algebras over the real field, particularly the quaternions and the Cayley numbers. The application of quaternions to spherical geometry and to mechanics is considered and the relation between quaternions and rotations in 3- and 4-dimensional Euclidean space is fully developed. The algebra of complexified quaternions is described and applied to electromagnetism and to special relativity. By looking at a 3-dimensional complex space we explore the use of a ...
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Paperback (Softcover reprint of the original 1st ed. 1997)
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Overview

This monograph is an accessible account of the normed algebras over the real field, particularly the quaternions and the Cayley numbers. The application of quaternions to spherical geometry and to mechanics is considered and the relation between quaternions and rotations in 3- and 4-dimensional Euclidean space is fully developed. The algebra of complexified quaternions is described and applied to electromagnetism and to special relativity. By looking at a 3-dimensional complex space we explore the use of a quaternion formalism to the Lorentz transformation and we examine the classification of electromagnetic and Weyl tensors. In the final chapter, extensions of quaternion algebra to the alternative non-associative algebra of Cayley numbers are investigated. The standard Cayley number identities are derived and their use in the analysis of 7- and 8-dimensional rotations is studied. Appendices on Clifford algebras and on the use of dynamic computation in Cayley algebra are included.
Audience: This volume has been written at a level suitable for final year and postgraduate students.
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Product Details

  • ISBN-13: 9789401064347
  • Publisher: Springer Netherlands
  • Publication date: 12/31/2013
  • Series: Mathematics and Its Applications (closed) Series
  • Edition description: Softcover reprint of the original 1st ed. 1997
  • Pages: 242
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.54 (d)

Table of Contents

1 Fundamentals of Linear Algebra.- 1.1 Integers, Rationals and Real Numbers.- 1.2 Real Numbers and Displacements.- 1.3 Groups.- 1.4 Rings and Fields.- 1.5 Linear Spaces.- 1.6 Inner Product Spaces.- 1.7 Algebras.- 1.8 Complex Numbers.- 2 Quaternions.- 2.1 Inventing Quaternions.- 2.2 Quaternion Algebra.- 2.3 The Exponential Form and Root Extraction.- 2.4 Frobenius’ Theorem.- 2.5 Inner Product for Quaternions.- 2.6 Quaternions and Rotations in 3- and 4-Dimensions.- 2.7 Relation to the Rotation Matrix.- 2.8 Matrix Formulation of Quaternions.- 2.9 Applications to Spherical Trigonometry.- 2.10 Rotating Axes in Mechanics.- 3 Complexified Quaternions.- 3.1 Scalars, Pseudoscalars, Vectors and Pseudovectors.- 3.2 Complexified Quaternions: Euclidean Metric.- 3.3 Complexified Quaternions: Minkowski Metric.- 3.4 Application of Complexified Quaternions to Space-Time.- 3.5 Quaternions and Electromagnet ism.- 3.6 Quaternionic Representation of Bivectors.- 3.7 Null Tetrad for Space-time.- 3.8 Classification of Complex Bivectors and of the Weyl Tensor.- 4 Cayley Numbers.- 4.1 A Common Notation for Numbers.- 4.2 Cayley Numbers.- 4.3 Angles and Cayley Numbers.- 4.4 Cayley Number Identities.- 4.5 Normed Algebras and the Hurwitz Theorem.- 4.6 Rotations in 7-and 8-Dimensional Euclidean Space.- 4.7 Basis Elements for Cayley Numbers.- 4.8 Geometry of 8-Dimensional Rotations.- Appendix 1 Clifford Algebras.- Appendix 2 Computer Algebra and Cayley Numbers.- References.
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