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ОбложкаGarrity T.A. Electricity and magnetism for mathematicians: a guided path from Maxwell's equations to Yang-Mills. - New York: Cambridge university press, 2015. - xiv, 282 p.: ill. - Bibliogr.: p.275-278. - Ind.: p.279-282. - ISBN 978-1-107-43516-2
Шифр: (И/В31-G22) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
List of Symbols ................................................ xi
Acknowledgments .............................................. xiii

1  A Brief History .............................................. 1
   1.1  Pre-1820: The Two Subjects of Electricity and
        Magnetism ............................................... 1
   1.2  1820-1861: The Experimental Glory Days of
        Electricity and Magnetism ............................... 2
   1.3  Maxwell and His Four Equations .......................... 2
   1.4  Einstein and the Special Theory of Relativity ........... 2
   1.5  Quantum Mechanics and Photons ........................... 3
   1.6  Gauge Theories for Physicists: The Standard Model ....... 4
   1.7  Four-Manifolds .......................................... 5
   1.8  This Book ............................................... 7
   1.9  Some Sources ............................................ 7
2  Maxwell's Equations .......................................... 9
   2.1  A Statement of Maxwell's Equations ...................... 9
   2.2  Other Versions of Maxwell's Equations .................. 12
        2.2.1  Some Background in Nabla ........................ 12
        2.2.2  Nabla and Maxwell ............................... 14
   2.3  Exercises .............................................. 14
3  Electromagnetic Waves ....................................... 17
   3.1  The Wave Equation ...................................... 17
   3.2  Electromagnetic Waves .................................. 20
   3.3  The Speed of Electromagnetic Waves Is Constant ......... 21
        3.3.1  Intuitive Meaning ............................... 21
        3.3.2  Changing Coordinates for the Wave Equation ...... 22
   3.4  Exercises .............................................. 25
4  Special Relativity .......................................... 27
   4.1  Special Theory of Relativity ........................... 27
   4.2  Clocks and Rulers ...................................... 28
   4.3  Galilean Transformations ............................... 31
   4.4  Lorentz Transformations ................................ 32
        4.4.1  A Heuristic Approach ............................ 32
        4.4.2  Lorentz Contractions and Time Dilations ......... 35
        4.4.3  Proper Time ..................................... 36
        4.4.4  The Special Relativity Invariant ................ 37
        4.4.5  Lorentz Transformations, the Minkowski Metric,
               and Relativistic Displacement ................... 38
   4.5  Velocity and Lorentz Transformations ................... 43
   4.6  Acceleration and Lorentz Transformations ............... 45
   4.7  Relativistic Momentum .................................. 46
   4.8  Appendix: Relativistic Mass ............................ 48
        4.8.1  Mass and Lorentz Transformations ................ 48
        4.8.2  More General Changes in Mass .................... 51
   4.9  Exercises .............................................. 52
5  Mechanics and Maxwell's Equations ........................... 56
   5.1  Newton's Three Laws .................................... 56
   5.2  Forces for Electricity and Magnetism ................... 58
        5.2.1  F = q(E + v × B) ................................ 58
        5.2.2  Coulomb's Law ................................... 59
   5.3  Force and Special Relativity ........................... 60
        5.3.1  The Special Relativistic Force .................. 60
        5.3.2  Force and Lorentz Transformations ............... 61
   5.4  Coulomb + Special Relativity + Charge Conservation =
        Magnetism .............................................. 62
   5.5  Exercises .............................................. 65
6  Mechanics, Lagrangians, and the Calculus of Variations ...... 70
   6.1  Overview of Lagrangians and Mechanics .................. 70
   6.2  Calculus of Variations ................................. 71
        6.2.1  Basic Framework ................................. 71
        6.2.2  Euler-Lagrange Equations ........................ 73
        6.2.3  More Generalized Calculus of Variations
               Problems ........................................ 77
   6.3  A Lagrangian Approach to Newtonian Mechanics ........... 78
   6.4  Conservation of Energy from Lagrangians ................ 83
   6.5  Noether's Theorem and Conservation Laws ................ 85
   6.6  Exercises .............................................. 86
7  Potentials .................................................. 88
   7.1  Using Potentials to Create Solutions for Maxwell's
        Equations .............................................. 88
   7.2  Existence of Potentials ................................ 89
   7.3  Ambiguity in the Potential ............................. 91
   7.4  Appendix: Some Vector Calculus ......................... 91
   7.5  Exercises .............................................. 95
8  Lagrangians and Electromagnetic Forces ...................... 98
   8.1  Desired Properties for the Electromagnetic Lagrangian .. 98
   8.2  The Electromagnetic Lagrangian ......................... 99
   8.3  Exercises ............................................. 101
9  Differential Forms ......................................... 103
   9.1  The Vector Spaces Λk(fig.2n) .............................. 103
        9.1.1  A First Pass at the Definition ................. 103
        9.1.2  Functions as Coefficients ...................... 106
        9.1.3  The Exterior Derivative ........................ 106
   9.2  Tools for Measuring ................................... 109
        9.2.1  Curves in fig.23 ................................... 109
        9.2.2  Surfaces in fig.23 ................................. 111
        9.2.3  k-manifolds in fig.2n .............................. 113
   9.3  Exercises ............................................. 115
10 The Hodge *Operator ........................................ 119
   10.1 The Exterior Algebra and the * Operator ............... 119
   10.2 Vector Fields and Differential Forms .................. 121
   10.3 The ? Operator and Inner Products ..................... 122
   10.4 Inner Products on Λ(fig.2n) ............................... 123
   10.5 The * Operator with the Minkowski Metric .............. 125
   10.6 Exercises ............................................. 127
11 The Electromagnetic Two-Form ............................... 130
   11.1 The Electromagnetic Two-Form .......................... 130
   11.2 Maxwell's Equations via Forms ......................... 130
   11.3 Potentials ............................................ 131
   11.4 Maxwell's Equations via Lagrangians ................... 132
   11.5 Euler-Lagrange Equations for the Electromagnetic
        Lagrangian ............................................ 136
   11.6 Exercises ............................................. 139
12 Some Mathematics Needed for Quantum Mechanics .............. 142
   12.1 Hilbert Spaces ........................................ 142
   12.2 Hermitian Operators ................................... 149
   12.3 The Schwartz Space .................................... 153
        12.3.1 The Definition ................................. 153
        12.3.2 The Operators q(ƒ) = xƒ and p(ƒ) = -idƒ/dx ..... 155
        12.3.3 S(fig.2) Is Not a Hilbert Space .................... 157
   12.4 Caveats: On Lebesgue Measure, Types of Convergence,
        and Different Bases ................................... 159
   12.5 Exercises ............................................. 160
13 Some Quantum Mechanical Thinking ........................... 163
   13.1 The Photoelectric Effect: Light as Photons ............ 163
   13.2 Some Rules for Quantum Mechanics ...................... 164
   13.3 Quantization .......................................... 170
   13.4 Warnings of Subtleties ................................ 172
   13.5 Exercises ............................................. 172
14 Quantum Mechanics of Harmonic Oscillators .................. 176
   14.1 The Classical Harmonic Oscillator ..................... 176
   14.2 The Quantum Harmonic Oscillator ....................... 179
   14.3 Exercises ............................................. 184
15 Quantizing Maxwell's Equations ............................. 186
   15.1 Our Approach .......................................... 186
   15.2 The Coulomb Gauge ..................................... 187
   15.3 The "Hidden" Harmonic Oscillator ...................... 193
   15.4 Quantization of Maxwell's Equations ................... 195
   15.5 Exercises ............................................. 197
16 Manifolds .................................................. 201
   16.1 Introduction to Manifolds ............................. 201
   16.1.1 Force = Curvature ................................... 201
   16.1.2 Intuitions behind Manifolds ......................... 201
   16.2 Manifolds Embedded in fig.2n .............................. 203
   16.2.1 Parametric Manifolds ................................ 203
   16.2.2 Implicitly Defined Manifolds ........................ 205
   16.3 Abstract Manifolds .................................... 206
   16.3.1 Definition .......................................... 206
   16.3.2 Functions on a Manifold ............................. 212
   16.4 Exercises ............................................. 212
17 Vector Bundles ............................................. 214
   17.1 Intuitions ............................................ 214
   17.2 Technical Definitions ................................. 216
        17.2.1 The Vector Space fig.2k ............................ 216
        17.2.2 Definition of a Vector Bundle .................. 216
   17.3 Principal Bundles ..................................... 219
   17.4 Cylinders and Mцbius Strips ........................... 220
   17.5 Tangent Bundles ....................................... 222
        17.5.1 Intuitions ..................................... 222
        17.5.2 Tangent Bundles for Parametrically Defined
               Manifolds ...................................... 224
        17.5.3 T(fig.22) as Partial Derivatives ................... 225
        17.5.4 Tangent Space at a Point of an Abstract
               Manifold ....................................... 227
        17.5.5 Tangent Bundles for Abstract Manifolds ......... 228
   17.6 Exercises ............................................. 230
18 Connections ................................................ 232
   18.1 Intuitions ............................................ 232
   18.2 Technical Definitions ................................. 233
        18.2.1 Operator Approach .............................. 233
        18.2.2 Connections for Trivial Bundles ................ 237
   18.3 Covariant Derivatives of Sections ..................... 240
   18.4 Parallel Transport: Why Connections Are Called
        Connections ........................................... 243
   18.5 Appendix: Tensor Products of Vector Spaces ............ 247
        18.5.1 A Concrete Description ......................... 247
        18.5.2 Alternating Forms as Tensors ................... 248
        18.5.3 Homogeneous Polynomials as Symmetric Tensors ... 250
        18.5.4 Tensors as Linearizations of Bilinear Maps ..... 251
   18.6 Exercises ............................................. 253
19 Curvature .................................................. 257
   19.1 Motivation ............................................ 257
   19.2 Curvature and the Curvature Matrix .................... 258
   19.3 Deriving the Curvature Matrix ......................... 260
   19.4 Exercises ............................................. 261
20 Maxwell via Connections and Curvature ...................... 263
   20.1 Maxwell in Some of Its Guises ......................... 263
   20.2 Maxwell for Connections and Curvature ................. 264
   20.3 Exercises ............................................. 266
   20.1 21  The Lagrangian Machine, Yang-Mills, and Other
        Forces ................................................ 267
   21.1 The Lagrangian Machine ................................ 267
   21.2 U(l) Bundles .......................................... 268
   21.3 Other Forces .......................................... 269
   21.4 A Dictionary .......................................... 270
   21.5 Yang-Mills Equations .................................. 272

Bibliography .................................................. 275
Index ......................................................... 279
Color plates follow ........................................... 234

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