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Tables of Contents for Electromagnetism of Continuous Media
Chapter/Section Title
Page #
Page Count
Part 1 Basic Notions in Electromagnetism
1 Electromagnetic fields
3
52
1.1 Historical remarks
3
2
1.2 Basic principles and balance laws
5
4
1.2.1 Local balance equations
6
3
1.3 Dipole moments and balance laws in matter
9
3
1.4 Macroscopic properties and constitutive equations
12
4
1.4.1 Forced current, magnetic current and magnetic charge
14
1
1.4.2 Duality principle
15
1
1.5 Lorentz-invariant form of Maxwell's equations
16
7
1.5.1 The Lorentz force
20
1
1.5.2 Electromagnetic stress and momentum
21
2
1.5.3 Electromagnetism based on three principles
23
1
1.6 Poynting's theorem and balance of energy
23
3
1.7 Balance laws in matter and discontinuous fields
26
6
1.8 Boundary conditions
32
2
1.9 Consequences of the Clausius-Duhem inequality
34
3
1.9.1 Other choices of independent variables
36
1
1.10 Electromagnetic potentials
37
5
1.11 Differential equations for the electromagnetic fields
42
3
1.12 Force, torque and energy of dipoles
45
6
1.13 Electromagnetism of deformable media
51
4
1.13.1 Electrodynamics of moving media
51
1
1.13.2 Electrodynamics of deformable media
52
3
2 Green's functions and retarded potentials
55
46
2.1 Green's formula and distributional solutions
55
3
2.2 The Fourier transform of distributions
58
3
2.3 Green's function for the harmonic oscillator
61
3
2.4 Green's function for the wave equation
64
5
2.4.1 One-dimensional wave equation
65
1
2.4.2 Two-dimensional wave equation
66
1
2.4.3 Three-dimensional wave equation
67
2
2.5 Green's function for the reduced wave equation
69
1
2.6 Green's function for the lossy wave equation
70
3
2.7 Retarded potentials
73
3
2.8 Kirchhoff's solution
76
2
2.9 Integral formulae for the electromagnetic field
78
3
2.10 Electric dipole fields
81
2
2.11 Liénard-Wiechem potentials and point charge radiation
83
6
2.12 Initial-value problem for the wave equation
89
6
2.13 Initial-value problem for the telegraph equation
95
4
2.14 Boundary-value problem for the telegraph equation
99
2
3 Time-harmonic fields
101
45
3.1 Fields and potentials
101
3
3.2 Helmholtz's theorem
104
2
3.3 Energy balance
106
5
3.3.1 Uniqueness
108
1
3.3.2 Energy density
109
2
3.4 Green's functions for Helmholtz's equation
111
2
3.5 Green's tensor
113
2
3.6 Helmholtz's equation and waves
115
8
3.7 Huygens' principle
123
2
3.8 Time-harmonic plane waves
125
3
3.9 Reciprocity
128
2
3.10 Superposition of waves and group velocity
130
7
3.11 Doppler effect, dispersion and convection of light
137
2
3.12 Signal velocity
139
3
3.13 The method of stationary phase
142
4
4 Models of materials with memory
146
49
4.1 A motivation of memory from waves in water
146
2
4.2 Simple models of material behaviour
148
22
4.2.1 Dielectrics
148
2
4.2.2 Debye model of dielectrics
150
3
4.2.3 Bound electrons in a magnetic field
153
2
4.2.4 Radiation damping
155
2
4.2.5 Magnetic materials
157
4
4.2.6 Ferroelectric materials
161
1
4.2.7 Molecular crystals with permanent dipoles
162
1
4.2.8 Metals
163
1
4.2.9 Ionosphere
164
3
4.2.10 Magnetosphere
167
2
4.2.11 A mixture description of plasmas
169
1
4.3 Kinetic approach to a plasma model
170
5
4.4 Restrictions placed by the Clausius-Duhem inequality
175
2
4.5 Causality and Kramers-Kronig relations
177
4
4.6 Approximate consequences of the Kramers-Kronig relations
181
3
4.7 Integral theorems
184
3
4.8 Causality conditions
187
8
Part 2 Thermodynamics and Mathematical Problems
5 Thermodynamics of simple electromagnetic systems
195
65
5.1 Electromagnetic systems
195
4
5.2 Materials with fading memory
199
5
5.2.1 Dielectrics with memory
199
3
5.2.2 Conductors with memory
202
2
5.3 Thermodynamic laws
204
1
5.4 Reversibility
205
3
5.5 Principle of electromagnetic energy dissipation
208
9
5.5.1 Equivalent formulations of the minimum free energy
214
3
5.6 Dielectrics and rate-type electromagnetic materials
217
3
5.7 Thermodynamic restrictions on linear systems
220
6
5.8 Free enthalpy of linear systems with memory
226
16
5.8.1 Free enthalpies and free energies for dielectrics with memory
228
11
5.8.2 Free enthalpies and free energies for conductors with memory
239
3
5.9 Topologies for the history space
242
9
5.9.1 Dielectrics with memory
242
8
5.9.2 Conductors with memory
250
1
5.10 Electric conduction in the ionosphere
251
3
5.10.1 Free energies and free enthalpies
252
2
5.11 Dissipativity at interfaces
254
6
6 Thermoelectromagnetic systems
260
37
6.1 Review of previous approaches
260
4
6.1.1 Electromagnetic systems without memory
263
1
6.2 Simple materials
264
2
6.3 Thermodynamic laws
266
5
6.4 Linear approximation with respect to the temperature
271
3
6.5 Examples of internal pseudo-energies
274
5
6.6 Linear systems with memory
279
4
6.6.1 Thermoelectromagnetic dielectrics
279
3
6.6.2 Free enthalpy of a thermoelectromagnetic dielectric
282
1
6.7 Linear thermoelectromagnetic dielectrics
283
4
6.8 Maximum free enthalpy
287
2
6.9 The discrete spectrum model
289
2
6.10 Thermoelectromagnetic conductors
291
1
6.11 Onsager's reciprocal relations
292
5
6.11.1 Application to thermoelectric phenomena
295
2
7 Existence and uniqueness
297
85
7.1 Some function spaces in electromagnetism
297
7
7.2 Stationary solutions in one-dimensional resonators
304
1
7.3 Stationary solutions in a resonator
305
3
7.4 Stationary solutions in conductors
308
4
7.5 Stationary fields with dissipative boundary conditions
312
2
7.6 Static solutions
314
6
7.7 The quasi-static problem
320
3
7.8 The evolution problem
323
15
7.8.1 Domain of dependence inequality
327
3
7.8.2 Uniqueness theorems
330
2
7.8.3 Existence
332
2
7.8.4 Existence of strict solutions
334
4
7.9 Existence and uniqueness for dielectrics with memory
338
3
7.10 Asymptotic behaviour in dielectrics with memory
341
2
7.11 Domain of dependence for dielectrics with memory
343
2
7.12 Absorbing boundary conditions
345
3
7.13 Existence and uniqueness for absorbing boundaries
348
6
7.14 Asymptotic behaviour for absorbing boundaries
354
4
7.15 A counterexample to asymptotic stability
358
3
7.16 Maxwell's equations as a constrained system
361
5
7.17 Spatial decay estimates
366
5
7.18 Spatial decay for dielectrics with memory
371
2
7.19 Thermoelectromagnetic systems
373
9
7.19.1 Thermoelectromagnetic conductors
374
8
8 Wave propagation
382
45
8.1 Plane waves
382
5
8.2 Linear and circular polarization
387
2
8.3 Reflection-transmission of waves between dielectrics
389
7
8.4 Reflectivity and transmissivity
396
3
8.5 Reflection and transmission between dissipative media
399
6
8.5.1 Upgoing and downgoing waves
402
2
8.5.2 Reflected and transmitted waves
404
1
8.6 Magnetohydrodynamic waves
405
4
8.7 Waves in anisotropic materials
409
8
8.7.1 Anisotropic dielectrics
410
3
8.7.2 Gyrotropic media
413
4
8.8 Plane wavefronts
417
4
8.9 Speed of propagation in materials with memory
421
2
8.10 Decay in materials with memory
423
4
9 Extremum principles
427
54
9.1 Some function spaces
427
2
9.2 Variational formulation for a system of equations
429
4
9.3 Models of electromagnetic media
433
4
9.3.1 Dielectrics
434
1
9.3.2 Conductors
434
1
9.3.3 Materials with memory
434
1
9.3.4 Boundary conditions
435
2
9.4 Extremum principles for static problems
437
6
9.4.1 Electrostatic problem
437
2
9.4.2 Magnetostatic problem
439
1
9.4.3 Electromagnetostatic problem
440
2
9.4.4 Electromagnetostatic problem with dissipative boundary conditions
442
1
9.5 Rayleigh's variational principles
443
6
9.5.1 Resonance frequencies
444
3
9.5.2 Wave numbers
447
2
9.6 Stationary formulae and Rayleigh-Ritz procedure
449
3
9.7 Least action principle
452
12
9.7.1 The least-action principle in empty space
452
4
9.7.2 Lagrangians involving the electromagnetic tensor
456
3
9.7.3 Electromagnetic Lagrangian
459
3
9.7.4 Lagrangian for nonlinear dielectrics
462
2
9.8 Functionals for initial-value problems
464
3
9.9 Functionals in the Laplace-transform domain
467
14
9.9.1 Reiss-type principles
470
5
9.9.2 Constrained variational formulations
475
6
Part 3 Nonlinearity and Nonlocality
10 Problems in nonlinear electromagnetism
481
61
10.1 Modelling in nonlinear optics
481
4
10.2 Nonlinear constitutive equations with memory
485
6
10.2.1 Nonlinear models for anisotropic media
486
2
10.2.2 Isotropic dielectrics
488
3
10.3 Volterra series
491
6
10.3.1 Sinusoidal inputs
493
2
10.3.2 Gaussian noise input
495
2
10.4 A model of nonlinear materials
497
2
10.5 Luxembourg effect
499
3
10.6 Plane waves in nonlinear media
502
5
10.6.1 Second harmonic
504
1
10.6.2 Phase conjugation
505
2
10.7 Generation of harmonics through perturbation methods
507
7
10.8 Simple waves
514
4
10.9 Hyperbolic systems and shock formation
518
7
10.10 The Bernoulli equation for the weak-wave amplitude
525
7
10.11 Speed of shocks
532
2
10.12 Shocks in dielectrics with instantaneous response
534
3
10.12.1 Shock evolution and constitutive properties
535
2
10.12.2 Shocks of small amplitude
537
1
10.13 Shocks in dielectrics with memory
537
5
11 Nonlocal electromagnetism and superconductivity
542
39
11.1 Remarks on the entropy inequality
542
2
11.2 Balance laws and second law
544
5
11.3 Dielectric bodies with quadrupoles
549
2
11.4 Nonlocal dielectrics with memory
551
4
11.5 Strictly nonlocal materials
555
2
11.6 Superconductivity
557
7
11.6.1 Basic phenomena
557
3
11.6.2 The London equations
560
2
11.6.3 An improvement of the London theory
562
2
11.7 Two nonlocal models
564
3
11.8 Nonlocal properties of superconductivity
567
2
11.9 Superconductors with memory
569
1
11.10 The London gauge
570
1
11.11 The Ginzburg-Landau theory
571
4
11.12 Quasi-stationary model
575
4
11.13 Evolution model
579
2
12 Magnetic hysteresis
581
32
12.1 Micromagnetics
582
3
12.2 Domain wall moon
585
2
12.2.1 Evaluation of the magnetization at a given field
586
1
12.2.2 Solution for major and minor loops
586
1
12.3 Remarks about mathematical models for hysteresis
587
2
12.4 General requirements
589
6
12.5 Rate-type models
595
7
12.5.1 A Preisach-type model
598
1
12.5.2 Two 'bilinear' models
599
1
12.5.3 The Coleman-Hodgdon model
600
2
12.6 History-rate type models
602
7
12.6.1 An example
604
5
12.7 History-integral type models
609
4
12.7.1 The Bouc model
609
1
12.7.2 The Preisach model
609
4
Appendices
613
41
A Some properties of Bessel functions
613
9
B Fòurier transform and Sobolev spaces
622
9
B.1 Fourier transform
622
5
B.2 Introduction to Sobolev spaces
627
1
B.3 Distributions
628
2
B.4 Fourier transform of distributions
630
1
C Compact operators and eigenfunctions
631
13
C.1 Compact operators
631
7
C.2 Eigenfunctions in spatially homogeneous resonators
638
6
D Differential operators in curvilinear coordinates
644
6
D.1 Differentiation
644
3
D.2 Gradient, divergence, curl and the Laplacian
647
1
D.3 Orthogonal curvilinear coordinates
648
2
E Finite formulation of electromagnetism
650
4
E.1 Field laws in global form
651
3
Bibliography
654
11
Index
665