Multiforms, Dyadics, and Electromagnetic Media

1 Multivectors and Multiforms 11
1.1 Vectors and one-forms 11
1.2 Bivectors and two-forms 13
1.3 Multivectors and multiforms 17
1.4 Some properties of bivectors and two-forms 23
2 Dyadics 29
2.1 Mapping vectors and one-forms 29
2.2 Mapping multivectors and multiforms 32
2.3 Dyadic identities 38
2.4 Rank of dyadic 44
2.5 Eigenproblems 46
2.6 Metric dyadic 50
3 Bidyadics 57
3.1 Cayley-Hamilton equation 58
3.2 Bidyadic eigenproblem 62
3.3 Hehl-Obukhov decomposition 64
3.4 Example: simple antisymmetric bidyadic 66
3.5 Inverse rules for bidyadics 68
4 Special Dyadics and Bidyadics 79
4.1 Orthogonality conditions 79
4.2 Nilpotent dyadics and bidyadics 81
4.3 Projection dyadics and bidyadics 82
4.4 Unipotent dyadics and bidyadics 85
4.5 Almost-complex (AC) dyadic 86
4.6 Almost-complex bidyadics 90
4.7 Modified closure relation 91
5 Electromagnetic Fields 99
5.1 Field equations 99
5.2 Medium equations 103
5.3 Basic classes of media 107
5.4 Interfaces and boundaries 112
5.5 Power and energy 117
5.6 Plane waves 122
6 Transformation of Fields and Media 133
6.1 A_ne transformation 133
6.2 Duality transformation 137
6.3 Transformation of boundary conditions 141
6.4 Reciprocity transformation 143
6.5 Conformal transformation 148
7 Basic Classes of Electromagnetic Media 157
7.1 Gibbsian isotropy 157
7.2 The axion medium 165
7.3 Skewon-axion media 168
7.4 Extended skewon-axion media 177
8 Quadratic Media 181
8.1 P media and Q media 181
8.2 Transformations 183
8.3 Spatial expansions 184
8.4 Plane waves 188
8.5 P-axion and Q-axion media 191
8.6 Extended Q media 193
8.7 Extended P media 199
9 Media Defined by Bidyadic Equations 205
9.1 Quadratic equation 206
9.2 Cubic equation 214
9.3 Bi-quadratic equation 218
10 Media Defined by Plane-Wave Properties 225
10.1 Media with no dispersion equation (NDE media) 225
10.2 Decomposable media (DC media) 233
A Solutions to Problems 253
B Transformation to Gibbsian Formalism 339
C Multivector and Dyadic Identities 343