An introduction to multivectors, dyadics, and differential forms for electrical engineersWhile physicists have long applied differential forms to various areas of theoretical analysis, dyadic algebra is also the most natural language for expressing electromagnetic phenomena mathematically. George Deschamps pioneered the application of differential forms to electrical engineering but never completed his work. Now, Ismo V. Lindell, an internationally recognized authority on differential forms, provides a clear and practical introduction to replacing classical Gibbsian vector calculus with the mathematical formalism of differential forms.In Differential Forms in Electromagnetics, Lindell simplifies the notation and adds memory aids in order to ease the reader's leap from Gibbsian analysis to differential forms, and provides the algebraic tools corresponding to the dyadics of Gibbsian analysis that have long been missing from the formalism. He introduces the reader to basic EM theory and wave equations for the electromagnetic two-forms, discusses the derivation of useful identities, and explains novel ways of treating problems in general linear (bi-anisotropic) media.Clearly written and devoid of unnecessary mathematical jargon, Differential Forms in Electromagnetics helps engineers master an area of intense interest for anyone involved in research on metamaterials.

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.

Preface

xi

1 Multivectors

1

(34)

1.1 The Grassmann algebra

1

(4)

1.2 Vectors and dual vectors

5

(4)

1.2.1 Basic definitions

5

(1)

1.2.2 Duality product

6

(1)

1.2.3 Dyadics

7

(2)

1.3 Bivectors

9

(8)

1.3.1 Wedge product

9

(1)

1.3.2 Basis bivectors

10

(2)

1.3.3 Duality product

12

(2)

1.3.4 Incomplete duality product

14

(1)

1.3.5 Bivector dadics

15

(2)

1.4 Multivectors

17

(13)

1.4.1 Trivectors

17

(1)

1.4.2 Basis trivectors

18

(1)

1.4.3 Trivector identities

19

(2)

1.4.4 p-vectors

21

(1)

1.4.5 Incomplete duality product

22

(1)

1.4.6 Basis multivectors

23

(2)

1.4.7 Generalized bac cab rule

25

(5)

1.5 Geometric interpretation

30

(5)

1.5.1 Vectors and bivectors

30

(1)

1.5.2 Trivectors

31

(1)

1.5.3 Dual vectors

32

(1)

1.5.4 Dual bivectors and trivectors

32

(3)

2 Dyadic Algebra

35

(48)

2.1 Products of dyadics

35

(11)

2.1.1 Basic notation

35

(2)

2.1.2 Duality product

37

(1)

2.1.3 Double-duality product

37

(1)

2.1.4 Double-wedge product

38

(1)

2.1.5 Double-wedge square

39

(2)

2.1.6 Double-wedge cube

41

(3)

2.1.7 Higher double-wedge powers

44

(1)

2.1.8 Double-incomplete duality product

44

(2)

2.2 Dyadic identities

46

(9)

2.2.1 Gibbs' identity in three dimensions

48

(1)

2.2.2 Gibbs' identity inn dimensions

49

(1)

2.2.3 Constructing identities

50

(5)

2.3 Eigenproblems

55

(4)

2.3.1 Left and right eigenvectors

55

(1)

2.3.2 Eigenvalues

56

(1)

2.3.3 Eigenvectors

57

(2)

2.4 Inverse dyadic

59

(9)

2.4.1 Reciprocal basis

59

(1)

2.4.2 The inverse dyadic

60

(2)

2.4.3 Inverse in three dimensions

62

(6)

2.5 Metric dyadics

68

(5)

2.5.1 Dot product

68

(1)

2.5.2 Metric dyadics

68

(1)

2.5.3 Properties of the dot product

69

(1)

2.5.4 Metric in multivector spaces

70

(3)

2.6 Hodge dyadics

73

(10)

2.6.1 Complementary spaces

73

(1)

2.6.2 Hodge dyadics

74

(1)

2.6.3 Three-dimensional Euclidean Hodge dyadics

75

(3)

2.6.4 Two-dimensional Euclidean Hodge dyadic

78

(1)

2.6.5 Four-dimensional Minkowskian Hodge dyadics

79

(4)

3 Differential Forms

83

(22)

3.1 Differentiation

83

(8)

3.1.1 Three-dimensional space

83

(3)

3.1.2 Four-dimensional space

86

(3)

3.1.3 Spatial and temporal components

89

(2)

3.2 Differentiation theorems

91

(3)

3.2.1 Poincaré's lemma and de Rham's theorem

91

(1)

3.2.2 Helmholtz decomposition

92

(2)

3.3 Integration

94

(5)

3.3.1 Manifolds

94

(2)

3.3.2 Stokes' theorem

96

(1)

3.3.3 Euclidean simplexes

97

(2)

3.4 Affine transformations

99

(6)

3.4.1 Transformation of differential forms

99

(2)

3.4.2 Three-dimensional rotation

101

(1)

3.4.3 Four-dimensional rotation

102

(3)

4 Electromagnetic Fields and Sources

105

(18)

4.1 Basic electromagnetic quantities

105

(2)

4.2 Maxwell equations in three dimensions

107

(3)

4.2.1 Maxwell-Faraday equations

107

(2)

4.2.2 Maxwell Ampere equations

109

(1)

4.2.3 Time-harmonic fields and sources

109

(1)

4.3 Maxwell equations in four dimensions

110

(4)

4.3.1 The force field

110

(2)

4.3.2 The source field

112

(1)

4.3.3 Deschamps graphs

112

(1)

4.3.4 Medium equation

113

(1)

4.3.5 Magnetic sources

113

(1)

4.4 Transformations

114

(4)

4.4.1 Coordinate transformations

114

(2)

4.4.2 Affine transformation

116

(2)

4.5 Super forms

118

(5)

4.5.1 Maxwell equations

118

(1)

4.5.2 Medium equations

119

(1)

4.5.3 Time-harmonic sources

120

(3)

5 Medium, Boundary, and Power Conditions

123

(40)

5.1 Medium conditions

123

(15)

5.1.1 Modified medium dyadics

124

(1)

5.1.2 Bi-anisotropic medium

124

(1)

5.1.3 Different representations

125

(2)

5.1.4 Isotropic medium

127

(2)

5.1.5 Bi-isotropic medium

129

(1)

5.1.6 Uniaxial medium

130

(1)

5.1.7 Q-medium

131

(4)

5.1.8 Generalized Q-medium

135

(3)

5.2 Conditions on boundaries and interfaces

138

(7)

5.2.1 Combining source-field systems

138

(3)

5.2.2 Interface conditions

141

(1)

5.2.3 Boundary conditions

142

(1)

5.2.4 Huygens' principle

143

(2)

5.3 Power conditions

145

(6)

5.3.1 Three-dimensional formalism

145

(2)

5.3.2 Four-dimensional formalism

147

(1)

5.3.3 Complex power relations

148

(1)

5.3.4 Ideal boundary conditions

149

(2)

5.4 The Lorentz force law

151

(4)

5.4.1 Three-dimensional force

152

(2)

5.4.2 Force-energy in four dimensions

154

(1)

5.5 Stress dyadic

155

(8)

5.5.1 Stress dyadic in four dimensions

155

(2)

5.5.2 Expansion in three dimensions

157

(1)

5.5.3 Medium condition

158

(2)

5.5.4 Complex force and stress

160

(3)

6 Theorems and Transformations

163

(18)

6.1 Duality transformation

163

(9)

6.1.1 Dual substitution

164

(1)

6.1.2 General duality

165

(4)

6.1.3 Simple duality

169

(1)

6.1.4 Duality rotation

170

(2)

6.2 Reciprocity

172

(2)

6.2.1 Lorentz reciprocity

172

(1)

6.2.2 Medium conditions

172

(2)

6.3 Equivalence of sources

174

(7)

6.3.1 Nonradiating sources

175

(1)

6.3.2 Equivalent sources

176

(5)

7 Electromagnetic Waves

181

(32)

7.1 Wave equation for potentials

181

(7)

7.1.1 Electric four-potential

182

(1)

7.1.2 Magnetic four-potential

183

(1)

7.1.3 Anisotropic medium

183

(2)

7.1.4 Special anisotropic medium

185

(1)

7.1.5 Three-dimensional equations

186

(1)

7.1.6 Equations for field two forms

187

(1)

7.2 Wave equation for fields

188

(7)

7.2.1 Three-dimensional field equations

188

(1)

7.2.2 Four-dimensional field equations

189

(2)

7.2.3 Q-medium

191

(2)

7.2.4 Generalized Q-medium

193

(2)

7.3 Plane waves

195

(6)

7.3.1 Wave equations

195

(2)

7.3.2 Q-medium

197

(2)

7.3.3 Generalized Q-medium

199

(2)

7.4 TE and TM polarized waves

201

(5)

7.4.1 Plane-wave equations

202

(1)

7.4.2 TE and TM polarizations

203

(1)

7.4.3 Medium conditions

203

(3)

7.5 Green functions

206

(7)

7.5.1 Green function as a mapping

207

(1)

7.5.2 Three-dimensional representation

207

(2)

7.5.3 Four-dimensional representation

209

(4)

References

213

(6)

Appendix A Multivector and Dyadic Identities

219

(10)

Appendix B Solutions to Selected Problems

229

(20)

Index

249

(6)

About the Author

255

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