Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics

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Springer Science & Business Media, Dec 6, 2012 - Science - 314 pages
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Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.
 

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Contents

Differentiation
44
Linear and Multilinear Functions
63
Calculus on Vector Manifolds
137
Differential Geometry of Vector Manifolds
188
Chapter 6 The Method of Mobiles
225
Directed Integration Theory
249
Lie Groups and Lie Algebras
283
References
305
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About the author (2012)

David Hesteness is awarded the Oersted Medal for 2002.
The Oersted Award recognizes notable contributions to the teaching of physics. It is the most prestigious award conferred by the American Association of Physics Teachers.

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