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📗 Book Details
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English |
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Pages |
300 |
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Format |
PDF |
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Size |
3.71 MB |
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📜 Book Contents
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 × 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
Contents vii.
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 k (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 R3 .
9.2.2 Surfaces in R3 .
9.2.3 k-manifolds in Rn .
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 (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.
viii Contents.
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 ) = x f and p( f ) = −id f /dx.
12.3.3 S(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 Rn.
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 Rk.
17.2.2 Definition of a Vector Bundle.
17.3 Principal Bundles 219.
17.4 Cylinders and M¨obius Strips.
17.5 Tangent Bundles .
17.5.1 Intuitions .
17.5.2 Tangent Bundles for Parametrically Defined
Manifolds .
17.5.3 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 247.
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 263.
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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