The Geometry of Physics: An Introduction (Second Edition)

Cover......Page 1
Copyright......Page 5
Contents......Page 8
Preface to the Second Edition......Page 20
Preface to the Revised Printing......Page 22
Preface to the First Edition......Page 24
I Manifolds, Tensors, and Exterior Forms......Page 28
1.1. Submanifolds of Euclidean Space......Page 30
1.2. Manifolds......Page 38
1.3. Tangent Vectors and Mappings......Page 49
1.4. Vector Fields and Flows......Page 57
2.1. Covectors and Riemannian Metrics......Page 64
2.2. The Tangent Bundle......Page 75
2.3. The Cotangent Bundle and Phase Space......Page 79
2.4. Tensors......Page 85
2.5. The Grassmann or Exterior Algebra......Page 93
2.6. Exterior Differentiation......Page 100
2.7. Pull-Backs......Page 104
2.8. Orientation and Pseudoforms......Page 109
2.9. Interior Products and Vector Analysis......Page 116
2.10. Dictionary......Page 121
3.1. Integration over a Parameterized Subset......Page 122
3.2. Integration over Manifolds with Boundary......Page 131
3.3. Stokes\'s Theorem......Page 137
3.4. Integration of Pseudoforms......Page 141
3.5. Maxwell\'s Equations......Page 145
4.1. The Lie Derivative of a Vector Field......Page 152
4.2. The Lie Derivative of a Form......Page 159
4.3. Differentiation of Integrals......Page 165
4.4. A Problem Set on Hamiltonian Mechanics......Page 172
5.1. A More General Stokes\'s Theorem......Page 182
5.2. Closed Forms and Exact Forms......Page 183
5.3. Complex Analysis......Page 185
5.4. The Converse to the Poincare Lemma......Page 187
5.5. Finding Potentials......Page 189
6.1. The Frobenius Integrability Condition......Page 192
6.2. Integrability and Constraints......Page 199
6.3. Heuristic Thermodynamics via Caratheodory......Page 205
II Geometry and Topology......Page 216
7.1. Curvature and Special Relativity......Page 218
7.2. Electromagnetism in Minkowski Space......Page 223
8.1. The First and Second Fundamental Forms......Page 228
8.2. Gaussian and Mean Curvatures......Page 232
8.3. The Brouwer Degree of a Map: A Problem Set......Page 237
8.4. Area, Mean Curvature, and Soap Bubbles......Page 248
8.5. Gauss\'s Theorema Egregium......Page 255
8.6. Geodesics......Page 259
8.7. The Parallel Displacement of Levi-Civita......Page 263
9.1. Covariant Differentiation......Page 268
9.2. The Riemannian Connection......Page 273
9.3. Cartan\'s Exterior Covariant Differential......Page 274
9.4. Change of Basis and Gauge Transformations......Page 280
9.5. The Curvature Forms in a Riemannian Manifold......Page 282
9.6. Parallel Displacement and Curvature on a Surface......Page 286
9.7. Riemann\'s Theorem and the Horizontal Distribution......Page 290
10.1. Geodesics and Jacobi Fields......Page 296
10.2. Variational Principles in Mechanics......Page 302
10.3. Geodesics, Spiders, and the Universe......Page 311
11.1. Heuristics of Einstein\'s Theory......Page 318
11.2. Tensor Analysis......Page 325
11.3. Hilbert\'s Action Principle......Page 330
11.4. The Second Fundamental Form in the Riemannian Case......Page 336
11.5. The Geometry of Einstein\'s Equations......Page 342
12 Curvature and Topology: Synge\'s Theorem......Page 350
12.1. Synge\'s Formula for Second Variation......Page 351
12.2. Curvature and Simple Connectivity......Page 356
13.1. Singular Chains and Their Boundaries......Page 360
13.2. The Singular Homology Groups......Page 369
13.3. Homology Groups of Familiar Manifolds......Page 374
13.4. De Rham\'s Theorem......Page 382
14.1. The Hodge Operators......Page 388
14.2. Harmonic Forms......Page 395
14.3. Boundary Values, Relative Homology, and Morse Theory......Page 402
III Lie Groups, Bundles, and Chern Forms......Page 416
15.1. Lie Groups, Invariant Vector Fields, and Forms......Page 418
15.2. One-Parameter Subgroups......Page 425
15.3. The Lie Algebra of a Lie Group......Page 429
15.4. Subgroups and Subalgebras......Page 434
16.1. Vector Bundles......Page 440
16.2. Poincare\'s Theorem and the Euler Characteristic......Page 448
16.3. Connections in a Vector Bundle......Page 455
16.4. The Electromagnetic Connection......Page 462
17.1. Fiber Bundles and Principal Bundles......Page 478
17.2. Coset Spaces......Page 483
17.3. Chern\'s Proof of the Gauss- Bonnet-Poincare Theorem......Page 487
17.4. Line Bundles, Topological Quantization, and Berry Phase......Page 492
18.1. Forms with Values in a Lie Algebra......Page 502
18.2. Associated Bundles and Connections......Page 508
18.3. r-Form Sections of a Vector Bundle: Curvature......Page 515
19.1. The Groups SO(3) and SU(2)......Page 518
19.2. Hamilton, Clifford, and Dirac......Page 524
19.3. The Dirac Algebra......Page 531
19.4. The Dirac Operator d in Minkowski Space......Page 538
19.5. The Dirac Operator in Curved Space-Time......Page 542
20.1. Noether\'s Theorem for Internal Symmetries......Page 550
20.2. Weyl\'s Gauge Invariance Revisited......Page 558
20.3. The Yang-Mills Nucleon......Page 564
20.4. Compact Groups and Yang-Mills Action......Page 568
20.5. The Yang-Mills Equation......Page 572
20.6. Yang-Mills Instantons......Page 577
21.1. Bi-invariant Forms on Compact Groups......Page 588
21.2. The Fundamental Group and Covering Spaces......Page 594
21.3. The Theorem of S.B. Myers: A Problem Set......Page 603
21.4. The Geometry of a Lie Group......Page 607
22.1. Chern Forms and Winding Numbers......Page 610
22.2. Homotopies and Extensions......Page 618
22.3. The Higher Homotopy Groups \\pi_k(M)......Page 623
22.4. Some Computations of Homotopy Groups......Page 632
22.5. Chern Forms as Obstructions......Page 635
A.a. The Classical Cauchy Stress Tensor and Equations of Motion......Page 644
A.b. Stresses in Terms of Exterior Forms......Page 645
A.c. Symmetry of Cauchy \'s Stress Tensor in Rn......Page 647
A.d. The Piola-Kirchhoff Stress Tensors......Page 649
A.e. Stored Energy of Deformation......Page 650
A.f. Hamilton\'s Principle in Elasticity......Page 653
A.g. Some Typical Computations Using Forms......Page 656
A.h. Concluding Remarks......Page 662
B.a. Chain Complexes......Page 663
B.b. Cochains and Cohomology......Page 665
B.c. Transpose and Adjoint......Page 666
B.d. Laplacians and Harmonic Cochains......Page 668
B.e. Kirchhoff\'s Circuit Laws......Page 670
C.a. Flavored Quarks......Page 675
C.b. Interactions of Quarks and Antiquarks......Page 677
C.c. The Lie Algebra of SU(3)......Page 679
C.d. Pions, Kaons, and Etas......Page 680
C.e. A Reduced Symmetry Group......Page 683
C.f. Meson Masses......Page 685
D.a. Hyperelastic Bodies......Page 687
D.b. Isotropic Bodies......Page 688
D.c. Application of Schur\'s Lemma......Page 689
D.d. Frobenius- Schur Relations......Page 691
D.e. The Symmetric Traceless 3x3 Matrices Are Irreducible......Page 693
E.a. The Topology of Conjugacy Orbits......Page 697
E.b. Application of Bott\'s Extension of Morse Theory......Page 700
References......Page 706
Index......Page 710