# Multiforms, Dyadics, and Electromagnetic Media

ISBN: 978-1-119-05238-8

Feb 2015

416 pages

Select type: O-Book

## Description

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

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

2.1 Mapping Vectors and One-Forms, 21

2.1.2 Double-Bar Product || 23

2.2 Mapping Multivectors and Multiforms, 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

2.3.1 Contraction Identities 32

2.3.2 Special Cases 33

2.3.3 More General Rules 35

2.3.4 Cayley–Hamilton Equation 36

2.5 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

2.6.3 Inverse Rules for Metric Dyadics 48

Problems, 49

3.1 Cayley–Hamilton Equation, 54

3.1.1 Coefficient Functions 55

3.1.2 Determinant of a Bidyadic 57

3.3 Hehl–Obukhov Decomposition, 61

3.4 Example: Simple Antisymmetric Bidyadic, 64

3.5 Inverse Rules for Bidyadics, 66

3.5.3 3D Expansions 73

Problems, 74

4.1 Orthogonality Conditions, 79

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

5.2.2 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

5.5.3 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

8.1 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

9.1.1 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

9.3.1 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