Atnaujinkite slapukų nuostatas

Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to YangMills [Minkštas viršelis]

4.11/5 (33 ratings by Goodreads)
(Williams College, Massachusetts)
  • Formatas: Paperback / softback, 294 pages, aukštis x plotis x storis: 228x151x15 mm, weight: 400 g, Worked examples or Exercises; 2 Plates, color; 81 Line drawings, unspecified
  • Išleidimo metai: 19-Jan-2015
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107435161
  • ISBN-13: 9781107435162
Kitos knygos pagal šią temą:
Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to YangMillsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 294 pages, aukštis x plotis x storis: 228x151x15 mm, weight: 400 g, Worked examples or Exercises; 2 Plates, color; 81 Line drawings, unspecified
  • Išleidimo metai: 19-Jan-2015
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107435161
  • ISBN-13: 9781107435162
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781107435162.html
Raktažodžiai:
Almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell's equations, which have led to some of the most important mathematical discoveries of the past thirty years. This text introduces some of the mathematical wonders of these equations.

This text is an introduction to some of the mathematical wonders of Maxwell's equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell's equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell's equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell's equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.

Daugiau informacijos

Maxwell's equations have led to many important mathematical discoveries. This text introduces mathematics students to some of their wonders.
List of Symbols
xi
Acknowledgments xiii
1 A Brief History
1(8)
1.1 Pre-1820: The Two Subjects of Electricity and Magnetism
1(1)
1.2 1820--1861: The Experimental Glory Days of Electricity and Magnetism
2(1)
1.3 Maxwell and His Four Equations
2(1)
1.4 Einstein and the Special Theory of Relativity
2(1)
1.5 Quantum Mechanics and Photons
3(1)
1.6 Gauge Theories for Physicists: The Standard Model
4(1)
1.7 Four-Manifolds
5(2)
1.8 This Book
7(1)
1.9 Some Sources
7(2)
2 Maxwell's Equations
9(8)
2.1 A Statement of Maxwell's Equations
9(3)
2.2 Other Versions of Maxwell's Equations
12(2)
2.2.1 Some Background in Nabla
12(2)
2.2.2 Nabla and Maxwell
14(1)
2.3 Exercises
14(3)
3 Electromagnetic Waves
17(10)
3.1 The Wave Equation
17(3)
3.2 Electromagnetic Waves
20(1)
3.3 The Speed of Electromagnetic Waves Is Constant
21(4)
3.3.1 Intuitive Meaning
21(1)
3.3.2 Changing Coordinates for the Wave Equation
22(3)
3.4 Exercises
25(2)
4 Special Relativity
27(29)
4.1 Special Theory of Relativity
27(1)
4.2 Clocks and Rulers
28(3)
4.3 Galilean Transformations
31(1)
4.4 Lorentz Transformations
32(11)
4.4.1 A Heuristic Approach
32(3)
4.4.2 Lorentz Contractions and Time Dilations
35(1)
4.4.3 Proper Time
36(1)
4.4.4 The Special Relativity Invariant
37(1)
4.4.5 Lorentz Transformations, the Minkowski Metric, and Relativistic Displacement
38(5)
4.5 Velocity and Lorentz Transformations
43(2)
4.6 Acceleration and Lorentz Transformations
45(1)
4.7 Relativistic Momentum
46(2)
4.8 Appendix: Relativistic Mass
48(4)
4.8.1 Mass and Lorentz Transformations
48(3)
4.8.2 More General Changes in Mass
51(1)
4.9 Exercises
52(4)
5 Mechanics and Maxwell's Equations
56(14)
5.1 Newton's Three Laws
56(2)
5.2 Forces for Electricity and Magnetism
58(2)
5.2.1 F = q(E + υ × B)
58(1)
5.2.2 Coulomb's Law
59(1)
5.3 Force and Special Relativity
60(2)
5.3.1 The Special Relativistic Force
60(1)
5.3.2 Force and Lorentz Transformations
61(1)
5.4 Coulomb + Special Relativity + Charge Conservation = Magnetism
62(3)
5.5 Exercises
65(5)
6 Mechanics, Lagrangians, and the Calculus of Variations
70(18)
6.1 Overview of Lagrangians and Mechanics
70(1)
6.2 Calculus of Variations
71(7)
6.2.1 Basic Framework
71(2)
6.2.2 Euler-Lagrange Equations
73(4)
6.2.3 More Generalized Calculus of Variations Problems
77(1)
6.3 A Lagrangian Approach to Newtonian Mechanics
78(5)
6.4 Conservation of Energy from Lagrangians
83(2)
6.5 Noether's Theorem and Conservation Laws
85(1)
6.6 Exercises
86(2)
7 Potentials
88(10)
7.1 Using Potentials to Create Solutions for Maxwell's Equations
88(1)
7.2 Existence of Potentials
89(2)
7.3 Ambiguity in the Potential
91(1)
7.4 Appendix: Some Vector Calculus
91(4)
7.5 Exercises
95(3)
8 Lagrangians and Electromagnetic Forces
98(5)
8.1 Desired Properties for the Electromagnetic Lagrangian
98(1)
8.2 The Electromagnetic Lagrangian
99(2)
8.3 Exercises
101(2)
9 Differential Forms
103(16)
9.1 The Vector Spaces Λk(Rn)
103(6)
9.1.1 A First Pass at the Definition
103(3)
9.1.2 Functions as Coefficients
106(1)
9.1.3 The Exterior Derivative
106(3)
9.2 Tools for Measuring
109(6)
9.2.1 Curves in R3
109(2)
9.2.2 Surfaces in R3
111(2)
9.2.3 k-manifolds in Rn
113(2)
9.3 Exercises
115(4)
10 The Hodge * Operator
119(11)
10.1 The Exterior Algebra and the * Operator
119(2)
10.2 Vector Fields and Differential Forms
121(1)
10.3 The * Operator and Inner Products
122(1)
10.4 Inner Products on Λ(Rn)
123(2)
10.5 The * Operator with the Minkowski Metric
125(2)
10.6 Exercises
127(3)
11 The Electromagnetic Two-Form
130(12)
11.1 The Electromagnetic Two-Form
130(1)
11.2 Maxwell's Equations via Forms
130(1)
11.3 Potentials
131(1)
11.4 Maxwell's Equations via Lagrangians
132(4)
11.5 Euler-Lagrange Equations for the Electromagnetic Lagrangian
136(3)
11.6 Exercises
139(3)
12 Some Mathematics Needed for Quantum Mechanics
142(21)
12.1 Hilbert Spaces
142(7)
12.2 Hermitian Operators
149(4)
12.3 The Schwartz Space
153(6)
12.3.1 The Definition
153(2)
12.3.2 The Operators q(ƒ) = xƒ and p(ƒ) = --idƒ/dx
155(2)
12.3.3 S(R) Is Not a Hilbert Space
157(2)
12.4 Caveats: On Lebesgue Measure, Types of Convergence, and Different Bases
159(1)
12.5 Exercises
160(3)
13 Some Quantum Mechanical Thinking
163(13)
13.1 The Photoelectric Effect: Light as Photons
163(1)
13.2 Some Rules for Quantum Mechanics
164(6)
13.3 Quantization
170(2)
13.4 Warnings of Subtleties
172(1)
13.5 Exercises
172(4)
14 Quantum Mechanics of Harmonic Oscillators
176(10)
14.1 The Classical Harmonic Oscillator
176(3)
14.2 The Quantum Harmonic Oscillator
179(5)
14.3 Exercises
184(2)
15 Quantizing Maxwell's Equations
186(15)
15.1 Our Approach
186(1)
15.2 The Coulomb Gauge
187(6)
15.3 The "Hidden" Harmonic Oscillator
193(2)
15.4 Quantization of Maxwell's Equations
195(2)
15.5 Exercises
197(4)
16 Manifolds
201(13)
16.1 Introduction to Manifolds
201(2)
16.1.1 Force = Curvature
201(1)
16.1.2 Intuitions behind Manifolds
201(2)
16.2 Manifolds Embedded in Rn
203(3)
16.2.1 Parametric Manifolds
203(2)
16.2.2 Implicitly Defined Manifolds
205(1)
16.3 Abstract Manifolds
206(6)
16.3.1 Definition
206(6)
16.3.2 Functions on a Manifold
212(1)
16.4 Exercises
212(2)
17 Vector Bundles
214(18)
17.1 Intuitions
214(2)
17.2 Technical Definitions
216(3)
17.2.1 The Vector Space Rk
216(1)
17.2.2 Definition of a Vector Bundle
216(3)
17.3 Principal Bundles
219(1)
17.4 Cylinders and Mobius Strips
220(2)
17.5 Tangent Bundles
222(8)
17.5.1 Intuitions
222(2)
17.5.2 Tangent Bundles for Parametrically Defined Manifolds
224(1)
17.5.3 T(R2) as Partial Derivatives
225(2)
17.5.4 Tangent Space at a Point of an Abstract Manifold
227(1)
17.5.5 Tangent Bundles for Abstract Manifolds
228(2)
17.6 Exercises
230(2)
18 Connections
232(25)
18.1 Intuitions
232(1)
18.2 Technical Definitions
233(7)
18.2.1 Operator Approach
233(4)
18.2.2 Connections for Trivial Bundles
237(3)
18.3 Covariant Derivatives of Sections
240(3)
18.4 Parallel Transport: Why Connections Are Called Connections
243(4)
18.5 Appendix: Tensor Products of Vector Spaces
247(6)
18.5.1 A Concrete Description
247(1)
18.5.2 Alternating Forms as Tensors
248(2)
18.5.3 Homogeneous Polynomials as Symmetric Tensors
250(1)
18.5.4 Tensors as Linearizations of Bilinear Maps
251(2)
18.6 Exercises
253(4)
19 Curvature
257(6)
19.1 Motivation
257(1)
19.2 Curvature and the Curvature Matrix
258(2)
19.3 Deriving the Curvature Matrix
260(1)
19.4 Exercises
261(2)
20 Maxwell via Connections and Curvature
263(4)
20.1 Maxwell in Some of Its Guises
263(1)
20.2 Maxwell for Connections and Curvature
264(2)
20.3 Exercises
266(1)
21 The Lagrangian Machine, Yang-Mills, and Other Forces
267(8)
21.1 The Lagrangian Machine
267(1)
21.2 U(1) Bundles
268(1)
21.3 Other Forces
269(1)
21.4 A Dictionary
270(2)
21.5 Yang-Mills Equations
272(3)
Bibliography 275(4)
Index 279
Thomas A. Garrity is the William R. Kenan, Jr Professor of Mathematics at Williams College, where he was the director of the Williams College Project for Effective Teaching for many years. He has written a number of research papers and has authored or coauthored two other books, All the Mathematics You Missed (But Need to Know for Graduate School) and Algebraic Geometry: A Problem Solving Approach. Among his awards and honors is the MAA Deborah and Franklin Tepper Haimo Award for outstanding college or university teaching.