Korábbi, hasonló című írásomban1 említett könyv2 előszavában idézte a szerző a könyv szellemi elődjét: David Hestenes & Garret Sobczyk, Clifford Algebra to Geometric Calculus – A Unified Language forMathematics and Physics3 . Ez a kötet viszonylag régen, 1984-ben jelent meg először, de én csak most figyeltem fel rá, és szereztem be az 1987-es újabb kiadását.
A két említett könyvet csak együtt érdemes értékelni, így később fogok írni róluk.
A most említett könyv tartalomjegyzéke a következő:
Table of Contents
Preface vii
Introduction xi
Symbols and Notation xv
Chapter1 / Geometric Algebra 1
1-1. Axioms, Definitions and Identities 3
1-2. Vector Spaces, Pseudoscalars and Projections 16
1-3. Frames and Matrices 27
1-4. Alternating Forms and Determinants 33
1-5. Geometric Algebras of PseudoEuclidean Spaces 41
Chapter2 / Differentiation 44
2-1. Differentiation by Vectors 44
2-2. Multivector Derivative, Differential and Adjoints 53
2-3. Factorization and Simplicial Derivatives 59
Chapter3 / Linearand MultilinearFunctions 63
3-1. Linear Transformations and Outermorphisms 66
3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem 71
3-3. Eigenblades and Invariant Spaces 75
3-4. Synunetric and Skew-synunetric Transformations 78
3-5. Normal and Orthogonal Transformations 86
3-6. Canonical Forms for General Linear Transformations 94
3-7. Metric Tensors and Isornetries 96
3-8. Isometries and Spinors of PseudoEuclidean Spaces 102
3-9. Linear Multivector Functions 111
3-10. Tensors 130
Chapter4 / Calculus on Vector Manifolds 137
4-1. Vector Manifolds 139
4-2. Projection, Shape and Curl 147
4-3. Intrinsic Derivatives and Lie Brackets 155
4-4. Curl and Pseudoscalar162
4-5. Transformations of Vector Manifolds 165
4-6. Computation of Induced Transformations 173
4-7. Complex Numbers and Conformal Transformations 180
Chapter5 / Differential Geometry of Vector Manifolds 188
5·1. Curl and Curvature189
5-2. Hypersurfaces in Euclidean Space 196
5-3. Related Geometries 201
5-4. Parallelism and Projectively Related Geometries 203
5-5. Conformally Related Geometries 210
5-6. Induced Geometries 220
Chapter6 /The Method of Mobiles 225
6-1. Frames and Coordinates 225
6-2. Mobiles and Curvature 230
6-3. Curves and Comoving Frames 237
6-4. The Calculus of Differential Forms 240
Chapter7 / Directed Integration Theory 249
7-1. Directed Integrals 249
7-2. Derivatives from Integrals 252
7-3. The Fundamental Theorem of Calculus 256
7-4. Antiderivatives, Analytic Functions and Complex Variables 259
7-5. Changing Integration Variables 266
7-6. Inverse and Implicit Functions 269
7-7. Winding Numbers 272
7-8. The Gauss-Bonnet Theorem 276
Chapter8 / Lie Groups and Lie Algebra 283
8-1. General Theory 283
8-2. Computation 291
8-3. Classification 296
References 305
Index 309
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2 Garret Sobczyk, New Foundations in Mathematics – The Geometric Concept of Number
