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Frankel T. — The geometry of physics: an introduction
Frankel T. — The geometry of physics: an introduction



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Название: The geometry of physics: an introduction

Автор: Frankel T.

Аннотация:

This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should also be of interest to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics. It can be used as a course text or for self study.


Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 688

Добавлена в каталог: 15.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Killing field, equation      529
Kinetic term      535
Kirchhoff's current law (KCL)      644
Kirchhoff's voltage law (KVL)      644
Klein bottle      348
Klein — Gordon equation      502
Kronecker delta, generalized $\delta^{1}_{J}$      67
Kronecker index of a vector field      216
Lagrange bracket { , }      80 100
Lagrange deformation tensor      82 625
Lagrange's equations      147
Lagrange's equations in a curved $M^{3}$      276
Lagrange's equations with electromagnetism      439
Lagrange's equations, tensorial nature      526
Lagrangian      54
Lagrangian for particle in an electromagnetic field      436—439
Lagrangian, Dirac      531
Lagrangian, electromagnetic      308
Lagrangian, significance in special relativity      437
Lambert's formula      290
Laplace operator $\Delta = dd^{*} + d^{*}d$ on forms      368—372
Laplace operator $\Delta = dd^{*} + d^{*}d$ on forms on a 1-form      370
Laplace operator on a cochain      641
Laplace's formula for pressure in a bubble      227
Laplacian $\nabla^{2}$      93 305
Laplacian $\nabla^{2}$ and mean curvature      305
Leaf of a foliation      173
Leaf of a foliation, maximal      173
Levi-Civita connection      242
Levi-Civita equation      297
Levi-Civita parallel displacement      237
Lie algebra g      402
Lie algebra g, Ad invariant scalar product      543
Lie bracket [ , ]      126 402
Lie derivative $\mathfrak{L}_{X}$ of a form      132—138
Lie derivative $\mathfrak{L}_{X}$ of a vector field      125
Lie derivative $\mathfrak{L}_{X}$ of the metric tensor      625
Lie derivative $\mathfrak{L}_{X}$ of the stress form      634 635
Lie group      391—412
Lie group, 1-parameter subgroup      398 405—407 564
Lie group, 1-parameter subgroup on Sl(2, $\mathbb{R}$)      407
Lie group, compact      541
Lie group, compact, averaging over      541
Lie group, compact, bi-invariant forms      561—567
Lie group, connection and curvature of      580
Lie subgroup and subalgebra      410—412
Lifting paths      277
Lifting paths in a bundle      593
Lifting paths in a covering space      574
Lifting spheres      605
Light cone      193
Lightlike      193
Linear functional      38
Linking number      219
Liouville's theorem      148
Local product      49
Local trivialization      417
Lorentz factor      193
Lorentz force      119
Lorentz force, covector      120 197
Lorentz group      504
Lorentz group and spinor representation of Sl(2, $\mathbb{C}$)      509
Lorentz metric      192
Lorentz transformation      46 198
Magnetic field $\mathbf{B}$      119
Magnetic field $\mathbf{B}$ and topology      123 387
Magnetic field $\mathbf{B}$, 1-form $\ast \mathscr{B}$      121
Magnetic field $\mathbf{B}$, 2-form $\mathscr{B}$      120
Magnetohydrodynamics      145
Manifold      13 19
Manifold with boundary      106
Manifold, closed      120
Manifold, complex      21
Manifold, integral      166
Manifold, mechanical      180
Manifold, orientable      83
Manifold, product      15
Manifold, pseudo-Riemannian      45
Manifold, Riemannian      45
Manifold, symplectic      146
Map of manifolds: critical points and values      28
Map of manifolds: regular points and values      28
Map, canonical      149
Map, coordinate      20
Map, differentiable      20
Map, exponential      284 399
Map, geographical      230
Map, inclusion      79
Map, projection      415
Matrix group      394
Maurer — Cartan equations      403 477
Maurer — Cartan form $\Omega$      476
Maximal atlas      15
Maximal torus      393
Maxwell's equations      120—123 198 200 536
Maxwell's equations on a 3-sphere      163
Maxwell's equations on a curved space      366—367
Maxwell's equations on a torus      122
Maxwell's equations on projective space      164
Maxwell's equations, independence of      200
Mayer — Lie system      174
Mean curvature      207 311 529
Mean curvature and divergence      224
Mesons      538
Mesons, Yukawa      540
Metric, conformally related      531
Metric, flat or locally euclidean      263
Metric, Lorentz or Minkowski      192
Metric, potentials      293
Metric, pseudo-Riemannian      45
Metric, Riemannian      45
Metric, spatial      297
Metric, static      292 296
Metric, stationary      291
Metric, tensor      43
Minimal submanifold      311 528
Minimal submanifold, surface      227 305
Minimization of arc length      286
Minkowski electromagnetic field tensor      197
Minkowski force      195
Minkowski metric and space      46 192
Mode, normal      65
Mode, zero      465
Moebius band      18
Momentum, 4-vector      194
Momentum, canonical      439
Momentum, classical      194
Momentum, density      320 322
Momentum, generalized      55
Momentum, kinematical      436
Momentum, operator      439
Monopole bundle      444 473
Morse deformation      47
Morse equalities      387 428
Morse index      328 384
Morse inequalities      385 386
Morse lacunary principle      388
Morse lemma      384
Morse polynomial      385
Morse theory      382—388
Morse type number      385 604
Multilinear      58
Myers's theorem      576—578
Neighborhood      12
Noether's theorem      527—529
Nomizu's theorem      530
Normal bundle      419 616
Normal coordinates      287 303
Normal derivative      364
Normal map      208
Normal mode      65
Nucleon, Heisenberg      537
Nucleon, Yang — Mills      538
Obstruction cocycle      609—612
One parameter group      31
Open set      11 12
Orientability      83
Orientability and curvature      331
Orientability and homology      349
Orientability and two-sidedness      84
Orientable bundle      611
Orientable manifold      83
Orientable transverse      115
Orientation      82
Orientation of the boundary      110
Orientation, coherent      341
Orientation, transverse      115
Orthogonal group, O(n)      9 392
Orthogonal group, SO(n)      9 392
Osculating plane      191
Paper folding      315
Parallel displacement      237
Parallel displacement, independence of path      260
Parallelizable      252
Parameter, distinguished or affine      272
Parameterized subset      97
Partition of unity      107
Partition of unity and Riemannian metrics      109
Passes peaks and pits      427
Path ordering      555
Pauli algebra      501
Pauli matrices      493
Period of a form      357
Periodic motion      282
Periodic motion for double pendulum      284
Periodic motion for rigid body      331
Pfaffian      167
Phase      448 535
Phase, space      55
Phase, space, extended      151
Physical components      48 630
Piola — Kirchhoff stress forms, first      622
Piola — Kirchhoff stress forms, second      623
Poincare 1-form      56
Poincare 1-form, extended      151
Poincare 2-form      80
Poincare 2-form, extended      151 437
Poincare characteristic      604
Poincare duality      375
Poincare Index Theorem      421—428
Poincare lemma and converse      160
Poincare metric      239 258
Poincare metric, geodesics      274 530
Poincare polynomial      385
Poisson bracket ( , )      154
Poisson equation      293 371
Potential of a closed form      158 160—164
Potential, global vector      443 448
Potential, monopole      444
Potential, singularities      see "Dirac string"
Poynting vector      322
Principal bundle      454 458 481
Principal directions      207 310
Principal normal      191
Principal normal curvatures      207 310
Principle of least action      281
Probability amplitude      447
Projection      49 415
Projection, homomorphism      605
Projective space      16 85
Projective space, $\mathbb{C}P^{n}$      22
Projective space, $\mathbb{R}P^{n}$      16
Projective space, homogeneous coordinates      17
Proper time      193 292
Pseudo-form      86
Pseudo-riemannian      45
Pseudoform      86
Pseudoform, integration of      114—417
Pull-back and integration      102
Pull-back in elasticity      81 622
Pull-back of covariant tensors      53 77 79
Pull-back, bundle      622
Pure gauge      553
Quantization of a gauge field      536
Quantization of a gauge field, topological      261
Quark      540
Quasi-static      179
Quaternion      502
Quotient group      345
Radius of curvature      192 221
Rate of deformation tensor      632—634
Regular points and values      28
Relative boundaries, cycles, and homology groups      379—381
Relative homology sequence      604
Relativistic equations of mass      194
Relativistic equations of motion      303
Reparameterization      101
Representation      481
Representation of a group      481 482
Representation, adjoint, Ad      486
Representation, dual      482
Representation, tensor product      482
Residue of a form      159
Rest mass      194
Retraction      217
Ricci curvature      315 374 577
Ricci identities      302
Ricci tensor $R_{ij}$      295
Riemann sectional curvature K($X \wedge Y$)      313—314
Riemann sphere      21
Riemann Theorem      266
Riemann — Christoffel curvature tensor      229
Riemannian connection      242
Riemannian manifold and metric      45
Riemannian manifold and metric on a surface of revolution      258
Riemannian manifold and metric, bi-invariant      563
Rigid body      9 331
Rotation group SO(n)      392 492
Sard's theorem      29
Scalar curvature R      296
Scalar product      42
Scalar product of Hermitian matrices      494
Scalar product, global      361
Scalar product, nondegenerate      42
Schroedinger's equation      439
Schroedinger's equation in curved space      442
Schroedinger's equation with an electromagnetic field      440 443
Schwarz's formula      228
Schwarzschild solution      320—322
Section      50 416 466
Section, holomorphic      467
Section, p-form section of a vector bundle      488
Sectional curvature      313
Self (anti) dual field      549
Self adjoint      205 317
Serret — Frenet formulas      196 431
Simon connection      472
simplex      333
Simplex, boundary      335
Simplex, face      335
Simplex, ordered      335
Simplex, orientation      336
Simplex, singular      334
Simplex, standard      333
Simplicial complex      343
Simply connected      283 329 595
Singularity of a vector field      422
Skeleton      610
Smooth      7
Soap bubbles and films      226—228
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