#48: Electromagnetic Theory and Computation: A Topological Approach

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MSRI Publications -- Volume 48

Electromagnetic Theory and Computation: A Topological Approach

By Paul W. Gross and P. Robert Kotiuga

Preface iii

Introduction 1

Outline of Book 3

Chapter 1 From Vector Calculus to Algebraic Topology 7
1A Chains, Cochains and Integration 7
1B Integral Laws and Homology 10
1C Cohomology and Vector Analysis 15
1D Nineteenth-Century Problems Illustrating the First and Second Homology Groups 18
1E Homotopy Versus Homology and Linking Numbers 25
1F Chain and Cochain Complexes 28
1G Relative Homology Groups 32
1H The Long Exact Homology Sequence 37
1I Relative Cohomology and Vector Analysis 41
1J A Remark on the Association of Relative Cohomology Groups with Perfect Conductors 46

Chapter 2 Quasistatic Electromagnetic Fields 49
2A The Quasistatic Limit Of Maxwell's Equations 49
2B Variational Principles For Electroquasistatics 63
2C Variational Principles For Magnetoquasistatics 70
2D Steady Current Flow 80
2E The Electromagnetic Lagrangian and Rayleigh Dissipation Functions 89

Chapter 3 Duality Theorems for Manifolds With Boundary 99
3A Duality Theorems 99
3B Examples of Duality Theorems in Electromagnetism 101
3C Linking Numbers, Solid Angle, and Cuts 112
3D Lack of Torsion for Three-Manifolds with Boundary 117

Chapter 4 The Finite Element Method and Data Structures 121
4A The Finite Element Method for Laplace's Equation 122
4B Finite Element Data Structures 127
4C The Euler Characteristic and the Long Exact Homology Sequence 138

Chapter 5 Computing Eddy Currents on Thin Conductors with Scalar Potentials 141
5A Introduction 141
5B Potentials as a Consequence of Ampére's Law 142}
5C Governing Equations as a Consequence of Faraday's Law 147
5D Solution of Governing Equations by Projective Methods 147
5E Weak Form and Discretization 150

Chapter 6 An Algorithm to Make Cuts for Magnetic Scalar Potentials 159
6A Introduction and Outline 159
6B Topological and Variational Context 161
6C Variational Formulation of the Cuts Problem 168
6D The Connection Between Finite Elements and Cuts 169
6E Computation of 1-Cocycle Basis 172
6F Summary and Conclusions 180

Chapter 7 A Paradigm Problem 183
7A The Paradigm Problem 183
7B The Constitutive Relation and Variational Formulation 185
7C Gauge Transformations and Conservation Laws 191
7D Modified Variational Principles 196
7E Tonti Diagrams 207

Mathematical Appendix: Manifolds, Differential Forms, Cohomology, Riemannian Structures 215
MA-A Differentiable Manifolds 216
MA-B Tangent Vectors and the Dual Space of One-Forms 217
MA-C Higher-Order Differential Forms and Exterior Algebra 220
MA-D Behavior of Differential Forms Under Mappings 223
MA-E The Exterior Derivative 226
MA-F Cohomology with Differential Forms 229
MA-G Cochain Maps Induced by Mappings Between Manifolds 231
MA-H Stokes' Theorem, de Rham's Theorems and Duality Theorems 232
MA-I Existence of Cuts Via Eilenberg--MacLane Spaces 240
MA-J Riemannian Structures, the Hodge Star Operator and an Inner Product for Differential Forms 243
MA-K The Operator Adjoint to the Exterior Derivative 249
MA-L The Hodge Decomposition and Ellipticity 252
MA-M Orthogonal Decompositions of p-Forms and Duality Theorems 253

Bibliography 261

Summary of Notation 267

Examples and Tables 273

Index 275