Electromagnetic Theory and Computation
File : pdf, 1.8 MB, 288 pages
TOC
Introduction
Chapt 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
Chapt 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
Chapt 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
Chapt 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
Chapt 5. Computing Eddy Currents on Thin Conductors with Scalar Potentials
5A Introduction
5B Potentials as a Consequence of Amphere’s Law
5C Governing Equations as a Consequence of Faraday’s Law
5D Solution of Governing Equations by Projective Methods
5E Weak Form and Discretization
Chapt 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
Chapt 7. A Paradigm Problem
7A The Paradigm Problem
7B The Constitutive Relation and Variational Formulation
7C Gauge Transformations and Conservation Laws
7D Modified 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
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