Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Introduction -- 1 From Vector Calculus to Algebraic Topology -- 1A. Chains, Cochains and Integration -- 1B. Integral Laws and Homology -- 1C. Cohomology and Vector Analysis -- 1D. Nineteenth-Century Problems Illustrating the First and Second Homology Groups -- 1E. Homotopy Versus Homology and Linking Numbers -- 1F. Chain and Cochain Complexes -- 1G. Relative Homology Groups -- 1H. The Long Exact Homology Sequence -- 1I. Relative Cohomology and Vector Analysis -- 1J. A Remark on the Association of Relative Cohomology Groups with Perfect Conductors -- 2 Quasistatic Electromagnetic Fields -- 2A. The Quasistatic Limit of Maxwell's Equations -- 2B. Variational Principles For Electroquasistatics -- 2C. Variational Principles for Magnetoquasistatics -- 2D. Steady Current Flow -- 2E. The Electromagnetic Lagrangian and Rayleigh Dissipation Functions -- 3 Duality Theorems for Manifolds With Boundary -- 3A. Duality Theorems -- 3B. Examples of Duality Theorems in Electromagnetism -- 3C. Linking Numbers, Solid Angle, and Cuts -- 3D. Lack of Torsion for Three-Manifolds with Boundary -- 4 The Finite Element Method and Data Structures -- 4A. The Finite Element Method for Laplace's Equation -- 4B. Finite Element Data Structures -- 4C. The Euler Characteristic and the Long Exact Homology Sequence -- 5 Computing Eddy Currents on Thin Conductors with Scalar Potentials -- 5A. Introduction -- 5B. Potentials as a Consequence of Ampère's Law -- 5D. Solution of Governing Equations by Projective Methods -- 5E. Weak Form and Discretization -- 6 An Algorithm to Make Cuts for Magnetic Scalar Potentials -- 6A. Introduction and Outline -- 6B. Topological and Variational Context -- 6C. Variational Formulation of the Cuts Problem -- 6D. The Connection Between Finite Elements and Cuts.

6E. Computation of 1-Cocycle Basis -- 6F. Summary and Conclusions -- 7 A Paradigm Problem -- 7A. The Paradigm Problem -- 7B. The Constitutive Relation and Variational Formulation -- 7C. Gauge Transformations and Conservation Law -- 7D. Modi ed Variational Principles -- 7E. Tonti Diagrams -- Mathematical Appendix: Manifolds, Differential Forms, Cohomology, Riemannian Structures -- MA-A. Differentiable Manifolds -- MA-B. Tangent Vectors and the Dual Space of One-Forms -- MA-C. Higher-Order Differential Forms and Exterior Algebra -- MA-D. Behavior of Differential Forms Under Mappings -- MA-E. The Exterior Derivative -- MA-F. Cohomology with Differential Forms -- MA-G. Cochain Maps Induced by Mappings Between Manifolds -- MA-H. Stokes' Theorem, de Rham's Theorems and Duality Theorems -- MA-I. Existence of Cuts Via Eilenberg-MacLane Spaces -- MA-J. Riemannian Structures, the Hodge Star Operator and an Inner Product for Differential Forms -- MA-K. The Operator Adjoint to the Exterior Derivative -- MA-L. The Hodge Decomposition and Ellipticity -- MA-M. Orthogonal Decompositions of p-Forms and Duality Theorems -- Bibliography -- Summary of Notation -- Examples and Tables -- Index.