(Ebook PDF) Electricity and Magnetism for Mathematicians A Guided Path from Maxwell’s Equations to Yang Mills 1st Edition by Thomas Garrity ISBN 9781316235089, 1316235084 full chapters

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ISBN 10:1316235084

ISBN 13:9781316235089

Author: Thomas A. Garrity

Table of Contents:

1. A Brief History
   1.1 Pre-1820: The Two Subjects of Electricity and Magnetism
   1.2 1820–1861: The Experimental Glory Days of Electricity and Magnetism
   1.3 Maxwell and His Four Equations
   1.4 Einstein and the Special Theory of Relativity
   1.5 Quantum Mechanics and Photons
   1.6 Gauge Theories for Physicists: The Standard Model
   1.7 Four-Manifolds
   1.8 This Book
   1.9 Some Sources

2. Maxwell’s Equations
   2.1 A Statement of Maxwell’s Equations
   2.2 Other Versions of Maxwell’s Equations
   2.2.1 Some Background in Nabla
   2.2.2 Nabla and Maxwell
   2.3 Exercises

3. Electromagnetic Waves
   3.1 The Wave Equation
   3.2 Electromagnetic Waves
   3.3 The Speed of Electromagnetic Waves Is Constant
   3.3.1 Intuitive Meaning
   3.3.2 Changing Coordinates for the Wave Equation
   3.4 Exercises

4. Special Relativity
   4.1 Special Theory of Relativity
   4.2 Clocks and Rulers
   4.3 Galilean Transformations
   4.4 Lorentz Transformations
   4.4.1 A Heuristic Approach
   4.4.2 Lorentz Contractions and Time Dilations
   4.4.3 Proper Time
   4.4.4 The Special Relativity Invariant
   4.4.5 Lorentz Transformations, the Minkowski Metric, and Relativistic Displacement
   4.5 Velocity and Lorentz Transformations
   4.6 Acceleration and Lorentz Transformations
   4.7 Relativistic Momentum
   4.8 Appendix: Relativistic Mass
   4.8.1 Mass and Lorentz Transformations
   4.8.2 More General Changes in Mass
   4.9 Exercises

5. Mechanics and Maxwell’s Equations
   5.1 Newton’s Three Laws
   5.2 Forces for Electricity and Magnetism
   5.2.1 $F=q(E+v\times B)$
   5.2.2 Coulomb’s Law
   5.3 Force and Special Relativity
   5.3.1 The Special Relativistic Force
   5.3.2 Force and Lorentz Transformations
   5.4 Coulomb + Special Relativity + Charge Conservation = Magnetism
   5.5 Exercises

6. Mechanics, Lagrangians, and the Calculus of Variations
   6.1 Overview of Lagrangians and Mechanics
   6.2 Calculus of Variations
   6.2.1 Basic Framework
   6.2.2 Euler-Lagrange Equations
   6.2.3 More Generalized Calculus of Variations Problems
   6.3 A Lagrangian Approach to Newtonian Mechanics
   6.4 Conservation of Energy from Lagrangians
   6.5 Noether’s Theorem and Conservation Laws
   6.6 Exercises

7. Potentials
   7.1 Using Potentials to Create Solutions for Maxwell’s Equations
   7.2 Existence of Potentials
   7.3 Ambiguity in the Potential
   7.4 Appendix: Some Vector Calculus
   7.5 Exercises

8. Lagrangians and Electromagnetic Forces
   8.1 Desired Properties for the Electromagnetic Lagrangian
   8.2 The Electromagnetic Lagrangian
   8.3 Exercises

9. Differential Forms
   9.1 The Vector Spaces Λk(Rn)Lambda^k(mathbb{R}^n)Λk(Rn)
   9.1.1 A First Pass at the Definition
   9.1.2 Functions as Coefficients
   9.1.3 The Exterior Derivative
   9.2 Tools for Measuring
   9.2.1 Curves in R3mathbb{R}^3R3
   9.2.2 Surfaces in R3mathbb{R}^3R3
   9.2.3 kkk-Manifolds in Rnmathbb{R}^nRn
   9.3 Exercises

10. The Hodge ∗*∗ Operator
    10.1 The Exterior Algebra and the ∗*∗ Operator
    10.2 Vector Fields and Differential Forms
    10.3 The ∗*∗ Operator and Inner Products
    10.4 Inner Products on Λk(Rn)Lambda^k(mathbb{R}^n)Λk(Rn)
    10.5 The ∗*∗ Operator with the Minkowski Metric
    10.6 Exercises

11. The Electromagnetic Two-Form
    11.1 The Electromagnetic Two-Form
    11.2 Maxwell’s Equations via Forms
    11.3 Potentials
    11.4 Maxwell’s Equations via Lagrangians
    11.5 Euler-Lagrange Equations for the Electromagnetic Lagrangian
    11.6 Exercises

12. Some Mathematics Needed for Quantum Mechanics
    12.1 Hilbert Spaces
    12.2 Hermitian Operators
    12.3 The Schwartz Space
    12.3.1 The Definition
    12.3.2 The Operators $q(f)=xf$ and p(f)=−idfdxp(f) = -i frac{df}{dx}p(f)=−idxdf​
    12.3.3 $\mathcal{S}(\mathbb{R})$ Is Not a Hilbert Space
    12.4 Caveats: On Lebesgue Measure, Types of Convergence, and Different Bases
    12.5 Exercises

13. Some Quantum Mechanical Thinking
    13.1 The Photoelectric Effect: Light as Photons
    13.2 Some Rules for Quantum Mechanics
    13.3 Quantization
    13.4 Warnings of Subtleties
    13.5 Exercises

14. Quantum Mechanics of Harmonic Oscillators
    14.1 The Classical Harmonic Oscillator
    14.2 The Quantum Harmonic Oscillator
    14.3 Exercises

15. Quantizing Maxwell’s Equations
    15.1 Our Approach
    15.2 The Coulomb Gauge
    15.3 The “Hidden” Harmonic Oscillator
    15.4 Quantization of Maxwell’s Equations
    15.5 Exercises

16. Manifolds
    16.1 Introduction to Manifolds
    16.1.1 Force = Curvature
    16.1.2 Intuitions behind Manifolds
    16.2 Manifolds Embedded in Rnmathbb{R}^nRn
    16.2.1 Parametric Manifolds
    16.2.2 Implicitly Defined Manifolds
    16.3 Abstract Manifolds
    16.3.1 Definition
    16.3.2 Functions on a Manifold
    16.4 Exercises

17. Vector Bundles
    17.1 Intuitions
    17.2 Technical Definitions
    17.2.1 The Vector Space Rkmathbb{R}^kRk
    17.2.2 Definition of a Vector Bundle
    17.3 Principal Bundles
    17.4 Cylinders and Möbius Strips
    17.5 Tangent Bundles
    17.5.1 Intuitions
    17.5.2 Tangent Bundles for Parametrically Defined Manifolds
    17.5.3 T(R2)T(mathbb{R}^2)T(R2) as Partial Derivatives
    17.5.4 Tangent Space at a Point of an Abstract Manifold
    17.5.5 Tangent Bundles for Abstract Manifolds
    17.6 Exercises

18. Connections
    18.1 Intuitions
    18.2 Technical Definitions
    18.2.1 Operator Approach
    18.2.2 Connections for Trivial Bundles
    18.3 Covariant Derivatives of Sections
    18.4 Parallel Transport: Why Connections Are Called Connections
    18.5 Appendix: Tensor Products of Vector Spaces
    18.5.1 A Concrete Description
    18.5.2 Alternating Forms as Tensors
    18.5.3 Homogeneous Polynomials as Symmetric Tensors
    18.5.4 Tensors as Linearizations of Bilinear Maps
    18.6 Exercises

19. Curvature
    19.1 Motivation
    19.2 Curvature and the Curvature Matrix
    19.3 Deriving the Curvature Matrix
    19.4 Exercises

20. Maxwell via Connections and Curvature
    20.1 Maxwell in Some of Its Guises
    20.2 Maxwell for Connections and Curvature
    20.3 Exercises

21. The Lagrangian Machine, Yang-Mills, and Other Forces
    21.1 The Lagrangian Machine
    21.2 $U(1)$ Bundles
    21.3 Other Forces
    21.4 A Dictionary
    21.5 Yang-Mills Equations

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