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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
List of contents
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 rfloor and lfloor 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 lfloor lfloor and rfloor rfloor 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
About the author
Ismo V. Lindell, PhD, is a professor of electromagnetic theory at the Helsinki University of Technology, Department of Electrical and Communication Engineering, where he was the founder of the Electromagnetics Laboratory in 1984. Dr. Lindell has received numerous awards, including recognition as an IEEE Fellow for his contributions to electromagnetic theory and for the development of education in electromagnetics in Finland. He is a member of URSI and IEEE, and is the recipient of the IEE Maxwell Premium for both 1997 and 1998, as well as the IEEE S. A. Schelkunoff Best Paper prize in 1987. In addition to two books in English, Dr. Lindell has authored or coauthored ten books in Finnish along with several hundred articles.
Summary
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
