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

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Springer Science & Business Media, Aug 31, 1987 - Mathematics - 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

Geometric Algebra
1
11 Axioms Definitions and Identities
3
12 Vector Spaces Pseudoscalars and Projections
16
13 Frames and Matrices
27
14 Alternating Forms and Determinants
33
15 Geometric Algebras of PseudoEuclidean Spaces
41
Differentiation
44
22 Multivector Derivatives Differentials and Adjoints
53
47 Complex Numbers and Conform Transformations
180
Differential Geometry of Vector Manifolds
188
51 Curl and Curvature
189
52 Hypersurfaces in Euclidean Space
196
53 Related Geometries
201
54 Parallelism and Projectively Related Geometries
203
55 Conformally Related Geometries
210
56 Induced Geometries
220

23 Factorization and Simplicial Derivatives
59
Linear and Multilinear Functions
63
31 Linear Transformations and Outermorphisms
66
32 Characteristic Multivecton and the CaylcyHamilton Theorem
71
33 Eigenblades and Invariant Spaces
75
34 Symmetric and Skewsymmetric Transformations
78
35 Normal and Orthogonal Transformations
86
36 Canonical Forms for General Linear Transformations
94
37 Metric Tensors and Isometries
96
38 Isometrics and Spinors of PseudoEuclidean Spaces
102
39 Linear Multivector Functions
111
310 Tensors
130
Calculus on Vector Manifolds
137
41 Vector Manifolds
139
42 Projection Shape and Curl
147
43 Intrinsic Derivatives and Lie Brackets
155
44 Curl and Pseudoscalar
162
45 Transformations of Vector Manifolds
165
46 Computation of Induced Transformations
173
The Method of Mobiles
225
62 Mobiles and Curvature
230
63 Curves and Comoving Frames
237
64 The Calculus of Differential Forms
240
Directed Integration Theory
249
72 Derivatives from Integrals
252
73 The Fundamental Theorem of Calculus
256
74 Antidcrivatives Analytic Functions and Complex Variables
259
75 Changing Integration Variables
266
76 Inverse and Implicit Functions
269
77 Winding Numbers
272
78 The GaussBonnet Theorem
276
Lie Groups and Lie Algebras
283
82 Computation
291
83 Classification
296
References
305
Index
309
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About the author (1987)

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.