Multiforms, Dyadics, and Electromagnetic Media

Ismo V. Lindell is a Professor Emeritus in the Department of Radio Science and Engineering, in the School of Electrical Engineering at the Aalto University, Finland. Dr. Lindell has received many honors in the course of his career, including his recognition as an IEEE Fellow in 1990 for his contributions to electromagnetic theory and for the development of education in electromagnetics in Finland. Dr. Lindell has authored or co-authored 3 books in English, authored or co-authored 10 books in Finnish, and published several hundred articles in professional journals, conference proceedings, and contributed chapters to other books.

• Preface xi1 Multivectors and Multiforms 11.1 Vectors and One-Forms, 11.1.1 Bar Product | 11.1.2 Basis Expansions 21.2 Bivectors and Two-Forms, 31.2.1 Wedge Product ∧ 31.2.2 Basis Expansions 41.2.3 Bar Product 51.2.4 Contraction Products ⌋ and ⌊ 61.2.5 Decomposition of Vectors and One-Forms 81.3 Multivectors and Multiforms, 81.3.1 Basis of Multivectors 91.3.2 Bar Product of Multivectors and Multiforms 101.3.3 Contraction of Trivectors and Three-Forms 111.3.4 Contraction of Quadrivectors and Four-Forms 121.3.5 Construction of Reciprocal Basis 131.3.6 Contraction of Quintivector 141.3.7 Generalized Bac-Cab Rules 141.4 Some Properties of Bivectors and Two-Forms, 161.4.1 Bivector Invariant 161.4.2 Natural Dot Product 171.4.3 Bivector as Mapping 17Problems, 182 Dyadics 212.1 Mapping Vectors and One-Forms, 212.1.1 Dyadics 212.1.2 Double-Bar Product || 232.1.3 Metric Dyadics 242.2 Mapping Multivectors and Multiforms, 252.2.1 Bidyadics 252.2.2 Double-Wedge Product ∧∧2.2.3 Double-Wedge Powers 282.2.4 Double Contractions ⌊⌊ and ⌋⌋ 302.2.5 Natural Dot Product for Bidyadics 312.3 Dyadic Identities, 322.3.1 Contraction Identities 322.3.2 Special Cases 332.3.3 More General Rules 352.3.4 Cayley–Hamilton Equation 362.3.5 Inverse Dyadics 362.4 Rank of Dyadics, 392.5 Eigenproblems, 412.5.1 Eigenvectors and Eigen One-Forms 412.5.2 Reduced Cayley–Hamilton Equations 422.5.3 Construction of Eigenvectors 432.6 Metric Dyadics, 452.6.1 Symmetric Dyadics 462.6.2 Antisymmetric Dyadics 472.6.3 Inverse Rules for Metric Dyadics 48Problems, 493 Bidyadics 533.1 Cayley–Hamilton Equation, 543.1.1 Coefficient Functions 553.1.2 Determinant of a Bidyadic 573.1.3 Antisymmetric Bidyadic 573.2 Bidyadic Eigenproblem, 583.2.1 Eigenbidyadic C− 603.2.2 Eigenbidyadic C+ 603.3 Hehl–Obukhov Decomposition, 613.4 Example: Simple Antisymmetric Bidyadic, 643.5 Inverse Rules for Bidyadics, 663.5.1 Skewon Bidyadic 673.5.2 Extended Bidyadics 703.5.3 3D Expansions 73Problems, 744 Special Dyadics and Bidyadics 794.1 Orthogonality Conditions, 794.1.1 Orthogonality of Dyadics 794.1.2 Orthogonality of Bidyadics 814.2 Nilpotent Dyadics and Bidyadics, 814.3 Projection Dyadics and Bidyadics, 834.4 Unipotent Dyadics and Bidyadics, 854.5 Almost-Complex Dyadics, 874.5.1 Two-Dimensional AC Dyadics 894.5.2 Four-Dimensional AC Dyadics 894.6 Almost-Complex Bidyadics, 914.7 Modified Closure Relation, 934.7.1 Equivalent Conditions 944.7.2 Solutions 944.7.3 Testing the Two Solutions 96Problems, 985 Electromagnetic Fields 1015.1 Field Equations, 1015.1.1 Differentiation Operator 1015.1.2 Maxwell Equations 1035.1.3 Potential One-Form 1055.2 Medium Equations, 1065.2.1 Medium Bidyadics 1065.2.2 Potential Equation 1075.2.3 Expansions of Medium Bidyadics 1075.2.4 Gibbsian Representation 1095.3 Basic Classes of Media, 1105.3.1 Hehl–Obukhov Decomposition 1105.3.2 3D Expansions 1125.3.3 Simple Principal Medium 1145.4 Interfaces and Boundaries, 1175.4.1 Interface Conditions 1175.4.2 Boundary Conditions 1195.5 Power and Energy, 1235.5.1 Bilinear Invariants 1235.5.2 The Stress–Energy Dyadic 1255.5.3 Differentiation Rule 1275.6 Plane Waves, 1285.6.1 Basic Equations 1285.6.2 Dispersion Equation 1305.6.3 Special Cases 1325.6.4 Plane-Wave Fields 1325.6.5 Simple Principal Medium 1345.6.6 Handedness of Plane Wave 135Problems, 1366 Transformation of Fields and Media 1416.1 Affine Transformation, 1416.1.1 Transformation of Fields 1416.1.2 Transformation of Media 1426.1.3 Dispersion Equation 1446.1.4 Simple Principal Medium 1456.2 Duality Transformation, 1456.2.1 Transformation of Fields 1466.2.2 Involutionary Duality Transformation 1476.2.3 Transformation of Media 1496.3 Transformation of Boundary Conditions, 1506.3.1 Simple Principal Medium 1526.3.2 Plane Wave 1526.4 Reciprocity Transformation, 1536.4.1 Medium Transformation 1536.4.2 Reciprocity Conditions 1556.4.3 Field Relations 1576.4.4 Time-Harmonic Fields 1586.5 Conformal Transformation, 1596.5.1 Properties of the Conformal Transformation 1606.5.2 Field Transformation 1646.5.3 Medium Transformation 165Problems, 1667 Basic Classes of Electromagnetic Media 1697.1 Gibbsian Isotropy, 1697.1.1 Gibbsian Isotropic Medium 1697.1.2 Gibbsian Bi-isotropic Medium 1707.1.3 Decomposition of GBI Medium 1717.1.4 Affine Transformation 1737.1.5 Eigenfields in GBI Medium 1747.1.6 Plane Wave in GBI Medium 1767.2 The Axion Medium, 1787.2.1 Perfect Electromagnetic Conductor 1797.2.2 PEMC as Limiting Case of GBI Medium 1807.2.3 PEMC Boundary Problems 1817.3 Skewon–Axion Media, 1827.3.1 Plane Wave in Skewon–Axion Medium 1847.3.2 Gibbsian Representation 1857.3.3 Boundary Conditions 1877.4 Extended Skewon–Axion Media, 192Problems, 1948 Quadratic Media 1978.1 P Media and Q Media, 1978.2 Transformations, 2008.3 Spatial Expansions, 2018.3.1 Spatial Expansion of Q Media 2018.3.2 Spatial Expansion of P Media 2038.3.3 Relation Between P Media and Q Media 2048.4 Plane Waves, 2058.4.1 Plane Waves in Q Media 2058.4.2 Plane Waves in P Media 2078.4.3 P Medium as Boundary Material 2088.5 P-Axion and Q-Axion Media, 2098.6 Extended Q Media, 2118.6.1 Gibbsian Representation 2118.6.2 Field Decomposition 2148.6.3 Transformations 2158.6.4 Plane Waves in Extended Q Media 2158.7 Extended P Media, 2188.7.1 Medium Conditions 2188.7.2 Plane Waves in Extended P Media 2198.7.3 Field Conditions 220Problems, 2219 Media Defined by Bidyadic Equations 2259.1 Quadratic Equation, 2269.1.1 SD Media 2279.1.2 Eigenexpansions 2289.1.3 Duality Transformation 2299.1.4 3D Representations 2319.1.5 SDN Media 2349.2 Cubic Equation, 2359.2.1 CU Media 2359.2.2 Eigenexpansions 2369.2.3 Examples of CU Media 2389.3 Bi-Quadratic Equation, 2409.3.1 BQ Media 2419.3.2 Eigenexpansions 2429.3.3 3D Representation 2449.3.4 Special Case 245Problems, 24610 Media Defined by Plane-Wave Properties 24910.1 Media with No Dispersion Equation (NDE Media), 24910.1.1 Two Cases of Solutions 25010.1.2 Plane-Wave Fields in NDE Media 25510.1.3 Other Possible NDE Media 25710.2 Decomposable Media, 25910.2.1 Special Cases 25910.2.2 DC-Medium Subclasses 26310.2.3 Plane-Wave Properties 267Problems, 269Appendix A Solutions to Problems 273Appendix B Transformation to Gibbsian Formalism 369Appendix C Multivector and Dyadic Identities 375References 389Index 395