Number 1, November 1996.
Number 1, November 1996.
Review by Dongming Wang, Institut IMAG, Grenoble (France) wang@leibniz.imag.fr
Clifford algebras, also called geometric algebras because of their capability of expressing geometric ideas, relations and structures, were introduced by W. K. Clifford in 1878 as a generalization of Hamilton's quaternions. They have surprising applications in many areas of mathematics, physics and engineering such as differential geometry, Clifford analysis, quantum mechanics and computer vision. Clifford algebraic formalism and calculus involve extensive treatment and manipulation of multivectors, for which effective means has been provided by modern computing facilities. In particular, mathematical software has become an indispensable device for Clifford algebra applications in research and education.
The book under review is a collection of 20 state-of-the-art contributions on the theory of Clifford algebras and its applications in some of the most interesting areas, where numeric and symbolic computations play a crucial role. It presents for the first time a comprehensive overview of the interaction between Clifford algebras and mathematical computations. With a preface introducing the subject and the structure of the book, the contributions are grouped into four chapters:
1. Verifying and Falsifying Conjectures
2. Differential Geometry, Quantum Mechanics, Spinors and Conformal Group
3. Generalized Clifford Algebras and Number Systems, Projective Geometry and Crystallography
4. Numerical Methods in Clifford Algebras
In addition to areas mentioned above, applications of Clifford algebras in field theory, hypercomplex algebra, function theory and teaching of mathematical physics are also covered. High-level mathematical software systems like Maple, Mathematica, Reduce, Matlab and programming languages like Fortran and C++ are used - with several special-purpose packages developed - to perform the involved computations for such applications and to derive some of the new results. The computer programs are made available on Birkhäuser's web pages ( http://www.birkhauser.com/cgi-win/ISBN/0-8176-3907-1 ). Many of the authors are leading experts on the subject with strong physical background.
The present book provides an excellent source of reference for researchers, educators and students working in the area of Clifford algebras who seek for advanced computing tools and those working on the design and implementation of algorithms and systems for symbolic and numeric computation who want to extend applications to physics and engineering. I recommend the book highly for professionals and students in the disciplines of both Clifford algebra and mathematical computation.
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