Preface......Page 9
Road map:......Page 14
1.1 Vector Spaces......Page 15
1.2 Duality......Page 18
1.3 Inner Products and Spacetime......Page 22
1.4 Linear Maps and Tensors......Page 26
1.5 Complex Linear Spaces......Page 30
1.6 Comments and References......Page 34
Road map:......Page 36
2.1 Multivectors......Page 37
2.2 The Grassmann Cone......Page 47
2.3 Mapping Multivectors......Page 53
2.4 Oriented Measure......Page 57
2.5 Multicovectors......Page 60
2.6 Interior Products and Hodge Stars......Page 64
2.7 Mappings of Interior Products......Page 74
2.8 Anticommutation Relations......Page 76
2.9 The Plücker Relations......Page 80
2.10 Comments and References......Page 82
Road map:......Page 85
3.1 The Clifford Product......Page 86
3.2 Complex Numbers and Quaternions......Page 94
3.3 Abstract Clifford Algebras......Page 101
3.4 Matrix Representations......Page 105
3.5 Comments and References......Page 114
Road map:......Page 116
4.1 Isometries and the Clifford Cone......Page 117
4.2 Infinitesimal Rotations and Bivectors......Page 124
4.3 Euclidean Rotations......Page 128
4.4 Spacetime Rotations......Page 136
4.5 Fractional Linear Maps......Page 145
4..6 Mappings of the Celestial Sphere......Page 153
4.7 Comments and References......Page 161
Road map:......Page 163
5.1 Complex Representations......Page 164
5.2 The Complex Spinor Space......Page 171
5.3 Mapping Spinors......Page 177
5.4 Abstract Spinor Spaces......Page 182
5.5 Comments and References......Page 193
Road map:......Page 195
6.1 Domains and Manifolds......Page 196
6.2 Fourier Transforms......Page 201
6.3 Partial Differential Equations......Page 207
6.4 Operator Theory......Page 210
6.5 Comments and References......Page 216
Road map:......Page 218
Highlights:......Page 219
7.1 Exterior and Interior Derivatives......Page 220
7.2 Pullbacks and Pushforwards......Page 225
7.3 Integration of Forms......Page 233
7.4 Vector Fields and Cartan's Formula......Page 244
7.5 Poincaré's Theorem......Page 248
7.6 Hodge Decompositions......Page 251
7.7 Comments and References......Page 262
Road map:......Page 264
Highlights:......Page 265
8.1 Monogenic Multivector Fields......Page 266
8.2 Spherical monogenics......Page 274
8.3 Hardy Space Splittings......Page 286
8.4 Comments and References......Page 292
Road map:......Page 294
Highlights:......Page 295
9.1 Wave and Spin Equations......Page 296
9.2 Dirac Equations in Physics......Page 300
9.3 Time-Harmonic Waves......Page 312
9.4 Boundary Value Problems......Page 318
9.5 Integral Equations......Page 328
9.6 Boundary Hodge Decompositions......Page 336
9.7 Maxwell Scattering......Page 341
9.8 Comments and References......Page 348
Road map:......Page 351
10.1 Nilpotent operators......Page 352
10.2 Half-Elliptic Boundary Conditions......Page 358
10.3 Hodge Potentials......Page 362
10.4 Bogovskiı and Poincaré Potentials......Page 370
10.5 Cech Cohomology......Page 375
10.6 De Rham Cohomology......Page 380
10.7 Comments and References......Page 389
Road map:......Page 391
11.1 Tangent Vectors and Derivatives......Page 393
11.2 Multivector Calculus on Manifolds......Page 398
11.3 Curvature and Bivectors......Page 406
11.4 Conformal Maps and ON-Frames......Page 413
11.5 Weitzenböck Identities......Page 416
11.6 Spinor Bundles......Page 421
11.7 Comments and References......Page 429
Road map:......Page 431
12.1 Fredholm Dirac Operators......Page 433
12.2 Normal Coordinates......Page 439
12.3 The Chern–Gauss–Bonnet Theorem......Page 442
12.4 The Atiyah–Singer Index Theorem......Page 449
12.5 Comments and References......Page 456
Bibliography......Page 458
Index......Page 465