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Frankel T. — The geometry of physics: An introduction

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Название: The geometry of physics: An introduction

Автор: Frankel T.

Аннотация:

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd

Год издания: 2004

Количество страниц: 717

Добавлена в каталог: 21.02.2015

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Предметный указатель

$\epsilon_{J}$       67
$\mathbb{Z}_{2}$       336
$\mathscr{D}$       200
$\mathscr{H}$       200
$\surd g$       88
(Cross) section      50 416 466
Absolute temperature      187
Acceleration 4-vector      194
Accessibility      181 182
Accumulation point      106
action      152 274 524
Action, Euclidean      551
Action, first variation of      154
Action, group      454
Action, Hamilton's principle of stationary action      154
Action, Jacobi's principle of least action      281
Action, relativistic      196
AD      486
Ad, bundle      487 489
Ad, connection      487
Adiabatic distribution and leaf      183
Adiabatic process      180
Adjoint      392 640
Adjoint group      486
Adjoint representation      486
Admissible boundary form      378
Admittance matrix      645
Affine connection      242
Affine group of the line A(1)      394
Affine parameter      272
Aharonov — Bohm effect      447—448 554
Aharonov — Susskind and spinors      517
Algebra homomorphism      78
Ampere — Maxwell law      121 163
Annihilator subspace      167
Anticommutator      478
Antiderivation      89 135
Antiquark      649
antisymmetric      66
Associated bundle      482
Associated bundle, connection      483—487
Atiyah — Singer index theorem      465
Atlas      15
Bernoulli's theorem      234
Berry phase      468—472
Berry phase, equation      472
Bertrand — Puiseux and Diguet      288
Betti numbers      157 346
Bi-invariant, connection on a Lie group      580
Bi-invariant, forms on a Lie group      561
Bi-invariant, Riemannian metric and their geodesics      563
Bianchi identities      300 489
Binormal      196
Bochner's theorems      374 530
Bonnet's theorem      229
Boson      658
Bott's version of Morse theory      673 675
Boundary (of a manifold) = edge      106
Boundary conditions, essential or imposed      527
Boundary conditions, natural      527
Boundary group      344
Boundary homomorphism      338 601
Boundary operator      335
Bracket, anticommutator      478
Bracket, commutator      408
Bracket, Lagrange      80 100
Bracket, Lie      126 402 Bracket Lie of
Bracket, Poisson      154
Brillouin and the stress form      635
Brouwer degree      210—213 360
Brouwer degree, fixed point theorem      217
Bump form      107
Bundle, associated      482
Bundle, complex line      433
Bundle, cotangent      52
Bundle, determinant      487
Bundle, dual      482
Bundle, electromagnetic      441
Bundle, fiber      415
Bundle, frame      453
Bundle, gauge      490
Bundle, line      433
Bundle, local trivialization      417
Bundle, monopole      444 473
Bundle, normal      419
Bundle, orientable      611
Bundle, principal      454 481
Bundle, product      418
Bundle, projection      415
Bundle, pull back      622
Bundle, section      50 416 466
Bundle, space      415
Bundle, structure group      433 452
Bundle, tangent      48
Bundle, transition functions      24 254 414
Bundle, trivial      418
Bundle, unit tangent      51
Bundle, vector      413—417
Bundle, volume      488
Canonical form      394
Canonical map      149
Caratheodory's formulation of the second law of thermodynamics      181
Caratheodory's theorem      182
Cartan's, 3-form on a Lie group      566
Cartan's, bi-invariant forms      562
Cartan's, exterior covariant differential      250 430
Cartan's, method for computing curvature      257
Cartan's, structural equations      249
Cartan's, theorem $\pi_{2}(G) = 0$       606
Cauchy equations of motion      618 620 628
Cauchy stress form      619 Cauchy stress form Lie
Cauchy stress form, symmetry of      621
Cauchy — Green tensor      82
Cauchy — Riemann equations      158 159
Center of a Lie algebra      580
Center of a Lie group      565
centralizer      667
Chain complex      636
Chain group      337
Chain group, integer      336
Chain group, simplicial      343
Chain group, singular      333
CHARACTER      665
Characteristic cohomology class      616
Charge form      118
Chern — Simons form      586
Chern — Weil theorem      589
Chern's forms and classes      587—591 Chern's forms and classes as obstructions
Chern's integral      612
Chern's proof of Gauss — Bonnet — Poincare      462—465 553—557
Chern's theorem      615
Chow's theorem      178 187
Christoffel symbols      229
Circulation      144 377
Clairaut's relation      530
Classical force      195
Classical momentum      194
Classical velocity      193
Clifford algebra      500
Clifford embedding      262
Clifford numbers      503
Closed form      156 158
Closed manifold      120
Closed set      11
Closure      106
Coboundary      638
Cochain      638
Coclosed      370
Cocycle      639
Codazzi equation      229 302 311—313 320

Codifferential d*      364
Codimension      6
Coefficient group      337
Coefficient group, field      343
Cohomology $H^{p}$       356
Cohomology $H^{p}$ , integral class      615
Commutative diagram      338
Commutator bracket of matrices      408
Compact      13
Completable relative cycle      387
Complex analytic map      158 214
Complex line bundle      433 Complex line bundle connections
Complex manifold      21
Composing rotations      499
Configuration space      9 50
Conformally related metrics      531
Conjugate point      327
Conjugates      667
Connected space      347
Connection      242
Connection on a Lie group      580 Connection on a Lie group flat
Connection on a vector bundle      428—431
Connection on the associated Ad bundle      486
Connection, coefficients of      243 429
Connection, curvature of      244
Connection, electromagnetic      440
Connection, flat      260
Connection, forms $\omega$       249 256
Connection, forms $\omega^{\ast}$ in the frame bundle      462 480
Connection, induced      309
Connection, Levi-Civita or Riemannian      242 245
Connection, Simon      472
Connection, spinor      518—521
Connection, symmetric      245
Connection, torsion of      245
Connection, torsion-free      245
Constraint, holonomic      175
Constraint, nonholonomic      175
continuous      12
Continuum mechanics      617—635 Continuum mechanics equilibrium
Contractible to a point      161
Contraction      89
Contravariant tensor      59
Contravariant vector      23
Coordinate, change of      29
Coordinate, compatible      15
Coordinate, frame      243
Coordinate, homogeneous      17
Coordinate, inertial      192
Coordinate, local      3 4 13
Coordinate, map      20
Coordinate, patch      20
Coset space G/H      456
Coset space G/H, fundamental principle      457
Cotangent space      40
Coupling constant or charge      539
Covariance      430
Covariant components of a tangent vector      43
Covariant constant      267
Covariant derivative $\nabla_{X}$       235 241—244 430 Covariant second
Covariant derivative $\nabla_{X}$ of a tensor      298—299
Covariant differential $\nabla$ , exterior      248
Covariant tensor      58
Covariant vector = covector      41
Covector      41
Covector, transformation law      42
Covering space      569—576
Covering space, associated to a subgroup of $\pi_{1}$       575
Covering space, orientable      573
Covering space, universal      570 Covering space universal covering
Critical manifolds      673
Critical points and values      28 382—387
Critical points and values, homotopically      382 387
Critical points and values, index      384
Critical points and values, inessential      383
Critical points and values, nondegenerate      383
Curl      93
Current, 2-form $\mathbf{j}$       118
Current, 3-form $\mathscr{S}$       199
Current, 3-vector $\mathbf{J}$       119
Current, 4-vector J      199
Current, convective      119
Current, electric, as a chain      644
Curvature and parallel displacement      259—261
Curvature of a connection      243
Curvature of a space curve      191
Curvature of a surface      207
Curvature of a surface of revolution      258
Curvature of the Poincare metric      258
Curvature, extrinsic      318
Curvature, forms $\theta$       251 256 431 Curvature forms
Curvature, forms $\theta$ , $\theta^{\ast}$ on a principal bundle      481
Curvature, forms $\theta$ , of a surface      257 Curvature forms
Curvature, Gauss      207
Curvature, geodesic      235
Curvature, intrinsic      318
Curvature, mean      207
Curvature, principal      207
Curvature, Riemann sectional K(X $\wedge$ Y)      313
Curvature, Riemann tensor      244
Curvature, total      215
Curvature, transformation R(X, Y)      244
Curvature, vector      192 194
Cycle, absolute      344
Cycle, completeable      387
Cycle, group      344
Cycle, relative      379
D'Alembertian $\sqcap$       293 371
de Rham's theorem      355—360
de Rham's vector space $\mathscr{R}^{p}$       356
Deformation, retract      406 506
Deformation, tensor      82
Deformation, theorem      350
Degree of a map      see "Brouwer degree"
Derivation      134
Derivative, covariant      235
Derivative, exterior      73
Derivative, intrinsic      235
Derivative, normal      364
Determinant line bundle      487
Dictionary relating forms and vectors      94
Diffeomorphism      27
Differentiable      20
Differential form      see "Form"
Differential of a function      40
Differential of a map $F_{\ast}$       7 27
Differential, exterior d      73 Differential exterior covariant
Differentiation of integrals      138—143
Dirac (4-component) spinor      513
Dirac adjoint or conjugate spinor      532
Dirac algebra      509
Dirac equation      503
Dirac Lagrangian      531
Dirac matrices      510
Dirac monopole      444 Dirac monopole quantization
Dirac operator      511 514 521 Dirac
Dirac program      502
Dirac representation $\rho$       512
Dirac string      162
Dirichlet's principle      373
Distance from a point to a hypersurface      579
Distribution (of subspaces)      166
Distribution (of subspaces), adiabatic      183
Distribution (of subspaces), horizontal      263
Distribution (of subspaces), integrable      167
Divergence      93 136 304
Divergence of a form      365
Divergence of a symmetric tensor      300
Divergence theorem      139
Divergence, exterior covariant      545

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