Electricity and magnetism for mathematicians: a guided path from Maxwell's equations to Yang-Mills
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Cambridge University Press
2015
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027943150&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references and index |
| Umfang: | XIV, 282 S. Ill., graph. Darst. |
| ISBN: | 9781107078208 9781107435162 |
Internformat
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| 245 | 1 | 0 | |a Electricity and magnetism for mathematicians |b a guided path from Maxwell's equations to Yang-Mills |c Thomas A. Garrity |
| 264 | 1 | |a New York, NY |b Cambridge University Press |c 2015 | |
| 300 | |a XIV, 282 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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|---|---|
| DE-BY-TUM_katkey | 2107403 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040008061165 |
| _version_ | 1821934067567820800 |
| adam_text | Titel: Electricity and magnetism for mathematicians
Autor: Garrity, Thomas A
Jahr: 2015
Contents
List of Symbols page 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
v
Vi
Contents
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 Metrie,
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 + vxB) 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
Contents
vii
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 A^M ) 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 IR3 109
9.2.2 Surfaces in R3 111
9.2.3 fc-manifolds in K 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 A(R ) 123
10.5 The ? Operator with the Minkowski Metrie 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
142
142
149
153
153
155
157
159
160
163
163
164
170
172
172
176
176
179
184
186
186
187
193
195
197
201
201
201
201
203
203
205
206
206
212
212
Contents
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) = —idf/dx
12.3.3 S(M) Is Not a Hilbert Space
12.4 Caveats: On Lebesgue Measure, Types of Convergence,
and Different Bases
12.5 Exercises
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
Quantum Mechanics of Harmonie Oscillators
14.1 The Classical Harmonie Oscillator
14.2 The Quantum Harmonie Oscillator
14.3 Exercises
Quantizing Maxwell s Equations
15.1 OurApproach
15.2 The Coulomb Gauge
15.3 The Hidden Harmonie Oscillator
15.4 Quantization of Maxwell s Equations
15.5 Exercises
Manifolds
16.1 Introduction to Manifolds
16.1.1 Force = Curvature
16.1.2 Intuitions behind Manifolds
16.2 Manifolds Embedded in R
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
Contents
ix
17 Vector Bundles 214
17.1 Intuitions 214
17.2 Technical Definitions 216
17.2.1 The Vector Space K* 216
17.2.2 Definition of a Vector Bündle 216
17.3 Principal Bundles 219
17.4 Cylinders and Möbius 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(M2) 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 Symmetrie 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
X
Contents
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 page 234
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| isbn | 9781107078208 9781107435162 |
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| spellingShingle | Garrity, Thomas A. 1959- Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills Feldtheorie (DE-588)4016698-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4016698-3 (DE-588)4012248-7 |
| title | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills |
| title_auth | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills |
| title_exact_search | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills |
| title_full | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills Thomas A. Garrity |
| title_fullStr | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills Thomas A. Garrity |
| title_full_unstemmed | Electricity and magnetism for mathematicians a guided path from Maxwell's equations to Yang-Mills Thomas A. Garrity |
| title_short | Electricity and magnetism for mathematicians |
| title_sort | electricity and magnetism for mathematicians a guided path from maxwell s equations to yang mills |
| title_sub | a guided path from Maxwell's equations to Yang-Mills |
| topic | Feldtheorie (DE-588)4016698-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Feldtheorie Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027943150&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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