This book applies the four-dimensional formalism with an extended toolbox of operation rules, allowing readers to define more general classes of electromagnetic media and to analyze EM waves that can exist in them
* End-of-chapter exercises
* Formalism allows readers to find novel classes of media
* Covers various properties of electromagnetic media in terms of which they can be set in different classes
Inhaltsverzeichnis
Preface xi
1 Multivectors and Multiforms 1
1. 1 Vectors and One-Forms, 1
1. 1. 1 Bar Product | 1
1. 1. 2 Basis Expansions 2
1. 2 Bivectors and Two-Forms, 3
1. 2. 1 Wedge Product 3
1. 2. 2 Basis Expansions 4
1. 2. 3 Bar Product 5
1. 2. 4 Contraction Products and 6
1. 2. 5 Decomposition of Vectors and One-Forms 8
1. 3 Multivectors and Multiforms, 8
1. 3. 1 Basis of Multivectors 9
1. 3. 2 Bar Product of Multivectors and Multiforms 10
1. 3. 3 Contraction of Trivectors and Three-Forms 11
1. 3. 4 Contraction of Quadrivectors and Four-Forms 12
1. 3. 5 Construction of Reciprocal Basis 13
1. 3. 6 Contraction of Quintivector 14
1. 3. 7 Generalized Bac-Cab Rules 14
1. 4 Some Properties of Bivectors and Two-Forms, 16
1. 4. 1 Bivector Invariant 16
1. 4. 2 Natural Dot Product 17
1. 4. 3 Bivector as Mapping 17
Problems, 18
2 Dyadics 21
2. 1 Mapping Vectors and One-Forms, 21
2. 1. 1 Dyadics 21
2. 1. 2 Double-Bar Product || 23
2. 1. 3 Metric Dyadics 24
2. 2 Mapping Multivectors and Multiforms, 25
2. 2. 1 Bidyadics 25
2. 2. 2 Double-Wedge Product
2. 2. 3 Double-Wedge Powers 28
2. 2. 4 Double Contractions and 30
2. 2. 5 Natural Dot Product for Bidyadics 31
2. 3 Dyadic Identities, 32
2. 3. 1 Contraction Identities 32
2. 3. 2 Special Cases 33
2. 3. 3 More General Rules 35
2. 3. 4 Cayley-Hamilton Equation 36
2. 3. 5 Inverse Dyadics 36
2. 4 Rank of Dyadics, 39
2. 5 Eigenproblems, 41
2. 5. 1 Eigenvectors and Eigen One-Forms 41
2. 5. 2 Reduced Cayley-Hamilton Equations 42
2. 5. 3 Construction of Eigenvectors 43
2. 6 Metric Dyadics, 45
2. 6. 1 Symmetric Dyadics 46
2. 6. 2 Antisymmetric Dyadics 47
2. 6. 3 Inverse Rules for Metric Dyadics 48
Problems, 49
3 Bidyadics 53
3. 1 Cayley-Hamilton Equation, 54
3. 1. 1 Coefficient Functions 55
3. 1. 2 Determinant of a Bidyadic 57
3. 1. 3 Antisymmetric Bidyadic 57
3. 2 Bidyadic Eigenproblem, 58
3. 2. 1 Eigenbidyadic C 60
3. 2. 2 Eigenbidyadic C+ 60
3. 3 Hehl-Obukhov Decomposition, 61
3. 4 Example: Simple Antisymmetric Bidyadic, 64
3. 5 Inverse Rules for Bidyadics, 66
3. 5. 1 Skewon Bidyadic 67
3. 5. 2 Extended Bidyadics 70
3. 5. 3 3D Expansions 73
Problems, 74
4 Special Dyadics and Bidyadics 79
4. 1 Orthogonality Conditions, 79
4. 1. 1 Orthogonality of Dyadics 79
4. 1. 2 Orthogonality of Bidyadics 81
4. 2 Nilpotent Dyadics and Bidyadics, 81
4. 3 Projection Dyadics and Bidyadics, 83
4. 4 Unipotent Dyadics and Bidyadics, 85
4. 5 Almost-Complex Dyadics, 87
4. 5. 1 Two-Dimensional AC Dyadics 89
4. 5. 2 Four-Dimensional AC Dyadics 89
4. 6 Almost-Complex Bidyadics, 91
4. 7 Modified Closure Relation, 93
4. 7. 1 Equivalent Conditions 94
4. 7. 2 Solutions 94
4. 7. 3 Testing the Two Solutions 96
Problems, 98
5 Electromagnetic Fields 101
5. 1 Field Equations, 101
5. 1. 1 Differentiation Operator 101
5. 1. 2 Maxwell Equations 103
5. 1. 3 Potential One-Form 105
5. 2 Medium Equations, 106
5. 2. 1 Medium Bidyadics 106
5. 2. 2 Potential Equation 107
5. 2. 3 Expansions of Medium Bidyadics 107
5. 2. 4 Gibbsian Representation 109
5. 3 Basic Classes of Media, 110
5. 3. 1 Hehl-Obukhov Decomposition 110
5. 3. 2 3D Expansions 112
5. 3. 3 Simple Principal Medium 114
5. 4 Interfaces and Boundaries, 117
5. 4. 1 Interface Conditions 117
5. 4. 2 Boundary Conditions 119
5. 5 Power and Energy, 123
5. 5. 1 Bilinear Invariants 123
5. 5. 2 The Stress-Energy Dyadic 125
5. 5. 3 Differentiation Rule 127
5. 6 Plane Waves, 128
5. 6. 1 Basic Equations 128
5. 6. 2 Dispersion Equation 130
5. 6. 3 Special Cases 132
5. 6. 4 Plane-Wave Fields 132
5. 6. 5 Simple Principal Medium 134
5. 6. 6 Handedness of Plane Wave 135
Problems, 136
6 Transformation of Fields and Media 141
6. 1 Affine Transformation, 141
6. 1. 1 Transformation of Fields 141
6. 1. 2 Transformation of Media 142
6. 1. 3 Dispersion Equation 144
6. 1. 4 Simple Principal Medium 145
6. 2 Duality Transformation, 145
6. 2. 1 Transformation of Fields 146
6. 2. 2 Involutionary Duality Transformation 147
6. 2. 3 Transformation of Media 149
6. 3 Transformation of Boundary Conditions, 150
6. 3. 1 Simple Principal Medium 152
6. 3. 2 Plane Wave 152
6. 4 Reciprocity Transformation, 153
6. 4. 1 Medium Transformation 153
6. 4. 2 Reciprocity Conditions 155
6. 4. 3 Field Relations 157
6. 4. 4 Time-Harmonic Fields 158
6. 5 Conformal Transformation, 159
6. 5. 1 Properties of the Conformal Transformation 160
6. 5. 2 Field Transformation 164
6. 5. 3 Medium Transformation 165
Problems, 166
7 Basic Classes of Electromagnetic Media 169
7. 1 Gibbsian Isotropy, 169
7. 1. 1 Gibbsian Isotropic Medium 169
7. 1. 2 Gibbsian Bi-isotropic Medium 170
7. 1. 3 Decomposition of GBI Medium 171
7. 1. 4 Affine Transformation 173
7. 1. 5 Eigenfields in GBI Medium 174
7. 1. 6 Plane Wave in GBI Medium 176
7. 2 The Axion Medium, 178
7. 2. 1 Perfect Electromagnetic Conductor 179
7. 2. 2 PEMC as Limiting Case of GBI Medium 180
7. 2. 3 PEMC Boundary Problems 181
7. 3 Skewon-Axion Media, 182
7. 3. 1 Plane Wave in Skewon-Axion Medium 184
7. 3. 2 Gibbsian Representation 185
7. 3. 3 Boundary Conditions 187
7. 4 Extended Skewon-Axion Media, 192
Problems, 194
8 Quadratic Media 197
8. 1 P Media and Q Media, 197
8. 2 Transformations, 200
8. 3 Spatial Expansions, 201
8. 3. 1 Spatial Expansion of Q Media 201
8. 3. 2 Spatial Expansion of P Media 203
8. 3. 3 Relation Between P Media and Q Media 204
8. 4 Plane Waves, 205
8. 4. 1 Plane Waves in Q Media 205
8. 4. 2 Plane Waves in P Media 207
8. 4. 3 P Medium as Boundary Material 208
8. 5 P-Axion and Q-Axion Media, 209
8. 6 Extended Q Media, 211
8. 6. 1 Gibbsian Representation 211
8. 6. 2 Field Decomposition 214
8. 6. 3 Transformations 215
8. 6. 4 Plane Waves in Extended Q Media 215
8. 7 Extended P Media, 218
8. 7. 1 Medium Conditions 218
8. 7. 2 Plane Waves in Extended P Media 219
8. 7. 3 Field Conditions 220
Problems, 221
9 Media Defined by Bidyadic Equations 225
9. 1 Quadratic Equation, 226
9. 1. 1 SD Media 227
9. 1. 2 Eigenexpansions 228
9. 1. 3 Duality Transformation 229
9. 1. 4 3D Representations 231
9. 1. 5 SDN Media 234
9. 2 Cubic Equation, 235
9. 2. 1 CU Media 235
9. 2. 2 Eigenexpansions 236
9. 2. 3 Examples of CU Media 238
9. 3 Bi-Quadratic Equation, 240
9. 3. 1 BQ Media 241
9. 3. 2 Eigenexpansions 242
9. 3. 3 3D Representation 244
9. 3. 4 Special Case 245
Problems, 246
10 Media Defined by Plane-Wave Properties 249
10. 1 Media with No Dispersion Equation (NDE Media), 249
10. 1. 1 Two Cases of Solutions 250
10. 1. 2 Plane-Wave Fields in NDE Media 255
10. 1. 3 Other Possible NDE Media 257
10. 2 Decomposable Media, 259
10. 2. 1 Special Cases 259
10. 2. 2 DC-Medium Subclasses 263
10. 2. 3 Plane-Wave Properties 267
Problems, 269
Appendix A Solutions to Problems 273
Appendix B Transformation to Gibbsian Formalism 369
Appendix C Multivector and Dyadic Identities 375
References 389
Index 395
