Differential forms in electromagnetics

Title:

Personal Author:

Lindell, Ismo V.

Series:

IEEE Press series on electromagnetic wave theory

Publication Information:

Hoboken, N.J. : Wiley-Interscience, 2004

ISBN:

9780471648017

Subject Term:

Electromagnetism -- Mathematics

Differential forms

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010070233	QC60.4.M37 L56 2004	Open Access Book	Book	Searching... Unknown

Summary

An introduction to multivectors, dyadics, and differential forms for electrical engineers

While 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.

Author Notes

Ismo V. Lindell is a professor of electromagnetic theory at the Helsinki University of Technology Department of Electrical and Communication Engineering.

Preface	p. xi
1 Multivectors	p. 1
1.1 The Grassmann algebra	p. 1
1.2 Vectors and dual vectors	p. 5
1.2.1 Basic definitions	p. 5
1.2.2 Duality product	p. 6
1.2.3 Dyadics	p. 7
1.3 Bivectors	p. 9
1.3.1 Wedge product	p. 9
1.3.2 Basis bivectors	p. 10
1.3.3 Duality product	p. 12
1.3.4 Incomplete duality product	p. 14
1.3.5 Bivector dyadics	p. 15
1.4 Multivectors	p. 17
1.4.1 Trivectors	p. 17
1.4.2 Basis trivectors	p. 18
1.4.3 Trivector identities	p. 19
1.4.4 p-vectors	p. 21
1.4.5 Incomplete duality product	p. 22
1.4.6 Basis multivectors	p. 23
1.4.7 Generalized bac cab rule	p. 25
1.5 Geometric interpretation	p. 30
1.5.1 Vectors and bivectors	p. 30
1.5.2 Trivectors	p. 31
1.5.3 Dual vectors	p. 32
1.5.4 Dual bivectors and trivectors	p. 32
2 Dyadic Algebra	p. 35
2.1 Products of dyadics	p. 35
2.1.1 Basic notation	p. 35
2.1.2 Duality product	p. 37
2.1.3 Double-duality product	p. 37
2.1.4 Double-wedge product	p. 38
2.1.5 Double-wedge square	p. 39
2.1.6 Double-wedge cube	p. 41
2.1.7 Higher double-wedge powers	p. 44
2.1.8 Double-incomplete duality product	p. 44
2.2 Dyadic identities	p. 46
2.2.1 Gibbs'identity in three dimensions	p. 48
2.2.2 Gibbs'identity in n dimensions	p. 49
2.2.3 Constructing identities	p. 50
2.3 Eigenproblems	p. 55
2.3.1 Left and right eigenvectors	p. 55
2.3.2 Eigenvalues	p. 56
2.3.3 Eigenvectors	p. 57
2.4 Inverse dyadic	p. 59
2.4.1 Reciprocal basis	p. 59
2.4.2 The inverse dyadic	p. 60
2.4.3 Inverse in three dimensions	p. 62
2.5 Metric dyadics	p. 68
2.5.1 Dot product	p. 68
2.5.2 Metric dyadics	p. 68
2.5.3 Properties of the dot product	p. 69
2.5.4 Metric in multivector spaces	p. 70
2.6 Hodge dyadics	p. 73
2.6.1 Complementary spaces	p. 73
2.6.2 Hodge dyadics	p. 74
2.6.3 Three-dimensional Euclidean Hodge dyadics	p. 75
2.6.4 Two-dimensional Euclidean Hodge dyadic	p. 78
2.6.5 Four-dimensional Minkowskian Hodge dyadics	p. 79
3 Differential Forms	p. 83
3.1 Differentiation	p. 83
3.1.1 Three-dimensional space	p. 83
3.1.2 Four-dimensional space	p. 86
3.1.3 Spatial and temporal components	p. 89
3.2 Differentiation theorems	p. 91
3.2.1 Poincare's lemma and de Rham's theorem	p. 91
3.2.2 Helmholtz decomposition	p. 92
3.3 Integration	p. 94
3.3.1 Manifolds	p. 94
3.3.2 Stokes'theorem	p. 96
3.3.3 Euclidean simplexes	p. 97
3.4 Affine transformations	p. 99
3.4.1 Transformation of differential forms	p. 99
3.4.2 Three-dimensional rotation	p. 101
3.4.3 Four-dimensional rotation	p. 102
4 Electromagnetic Fields and Sources	p. 105
4.1 Basic electromagnetic quantities	p. 105
4.2 Maxwell equations in three dimensions	p. 107
4.2.1 Maxwell-Faraday equations	p. 107
4.2.2 Maxwell-Ampere equations	p. 109
4.2.3 Time-harmonic fields and sources	p. 109
4.3 Maxwell equations in four dimensions	p. 110
4.3.1 The force field	p. 110
4.3.2 The source field	p. 112
4.3.3 Deschamps graphs	p. 112
4.3.4 Medium equation	p. 113
4.3.5 Magnetic sources	p. 113
4.4 Transformations	p. 114
4.4.1 Coordinate transformations	p. 114
4.4.2 Affine transformation	p. 116
4.5 Super forms	p. 118
4.5.1 Maxwell equations	p. 118
4.5.2 Medium equations	p. 119
4.5.3 Time-harmonic sources	p. 120
5 Medium, Boundary, and Power Conditions	p. 123
5.1 Medium conditions	p. 123
5.1.1 Modified medium dyadics	p. 124
5.1.2 Bi-anisotropic medium	p. 124
5.1.3 Different representations	p. 125
5.1.4 Isotropic medium	p. 127
5.1.5 Bi-isotropic medium	p. 129
5.1.6 Uniaxial medium	p. 130
5.1.7 Q-medium	p. 131
5.1.8 Generalized Q-medium	p. 135
5.2 Conditions on boundaries and interfaces	p. 138
5.2.1 Combining source-field systems	p. 138
5.2.2 Interface conditions	p. 141
5.2.3 Boundary conditions	p. 142
5.2.4 Huygens' principle	p. 143
5.3 Power conditions	p. 145
5.3.1 Three-dimensional formalism	p. 145
5.3.2 Four-dimensional formalism	p. 147
5.3.3 Complex power relations	p. 148
5.3.4 Ideal boundary conditions	p. 149
5.4 The Lorentz force law	p. 151
5.4.1 Three-dimensional force	p. 152
5.4.2 Force-energy in four dimensions	p. 154
5.5 Stress dyadic	p. 155
5.5.1 Stress dyadic in four dimensions	p. 155
5.5.2 Expansion in three dimensions	p. 157
5.5.3 Medium condition	p. 158
5.5.4 Complex force and stress	p. 160
6 Theorems and Transformations	p. 163
6.1 Duality transformation	p. 163
6.1.1 Dual substitution	p. 164
6.1.2 General duality	p. 165
6.1.3 Simple duality	p. 169
6.1.4 Duality rotation	p. 170
6.2 Reciprocity	p. 172
6.2.1 Lorentz reciprocity	p. 172
6.2.2 Medium conditions	p. 172
6.3 Equivalence of sources	p. 174
6.3.1 Nonradiating sources	p. 175
6.3.2 Equivalent sources	p. 176
7 Electromagnetic Waves	p. 181
7.1 Wave equation for potentials	p. 181
7.1.1 Electric four-potential	p. 182
7.1.2 Magnetic four-potential	p. 183
7.1.3 Anisotropic medium	p. 183
7.1.4 Special anisotropic medium	p. 185
7.1.5 Three-dimensional equations	p. 186
7.1.6 Equations for field two-forms	p. 187
7.2 Wave equation for fields	p. 188
7.2.1 Three-dimensional field equations	p. 188
7.2.2 Four-dimensional field equations	p. 189
7.2.3 Q-medium	p. 191
7.2.4 Generalized Q-medium	p. 193
7.3 Plane waves	p. 195
7.3.1 Wave equations	p. 195
7.3.2 Q-medium	p. 197
7.3.3 Generalized Q-medium	p. 199
7.4 TE and TM polarized waves	p. 201
7.4.1 Plane-wave equations	p. 202
7.4.2 TE and TM polarizations	p. 203
7.4.3 Medium conditions	p. 203
7.5 Green functions	p. 206
7.5.1 Green function as a mapping	p. 207
7.5.2 Three-dimensional representation	p. 207
7.5.3 Four-dimensional representation	p. 209
References	p. 213
Appendix A Multivector and Dyadic Identities	p. 219
Appendix B Solutions to Selected Problems	p. 229
Index	p. 249
About the Author	p. 255

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents