Ebook: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
- Tags: Theoretical Mathematical and Computational Physics, Algebra, Geometry, Group Theory and Generalizations
- Series: Fundamental Theories of Physics 5
- Year: 1984
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xviii
Geometric Algebra....Pages 1-43
Differentiation....Pages 44-62
Linear and Multilinear Functions....Pages 63-136
Calculus on Vector Manifolds....Pages 137-187
Differential Geometry of Vector Manifolds....Pages 188-224
The Method of Mobiles....Pages 225-248
Directed Integration Theory....Pages 249-282
Lie Groups and Lie Algebras....Pages 283-304
Back Matter....Pages 305-314
Content:
Front Matter....Pages i-xviii
Geometric Algebra....Pages 1-43
Differentiation....Pages 44-62
Linear and Multilinear Functions....Pages 63-136
Calculus on Vector Manifolds....Pages 137-187
Differential Geometry of Vector Manifolds....Pages 188-224
The Method of Mobiles....Pages 225-248
Directed Integration Theory....Pages 249-282
Lie Groups and Lie Algebras....Pages 283-304
Back Matter....Pages 305-314
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