**
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