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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

Dual basis      39
Dual bundle      417 482
Dual Hodge *      362
Dual space      39
Eigenvalue of a quadratic form      63 209
Eight-Fold Way      649
Einstein equations      296 316 317 Einstein Wheeler's
Einstein geodesic assumption      292 297
Einstein tensor G      315
Electric field $\mathbf{E}$       119
Electric field $\mathbf{E}$ and topology      123 378 381
Electric field $\mathbf{E}$ , 1-form $\mathscr{E}$       120
Electric field $\mathbf{E}$ , 2-form $\ast\mathscr{E}$       121
Electromagnetic bundle      441
Electromagnetic connection      440
Electromagnetic field strength $F^{2}$       197
Electromagnetic Lagrangian      308
Electromagnetic stress-energy-momentum tensor      308
Electromagnetic vector potential 1-form $A^{1}$       199
Electromagnetism and Maxwell's equations in curved spacetime      366—367
Electromagnetism and Maxwell's equations on projective space      164
Electromagnetism and Maxwell's equations on the 3-sphere      163
Electromagnetism and Maxwell's equations on the 3-torus      122
Electromagnetism and Maxwell's equations, existence and uniqueness      378 387
Embedded submanifold      27
Energy density      316
Energy momentum tensor      295
Energy momentum vector      195
Energy of a path      274
Energy of deformation      623—626
Energy, hypersurface      148 Energy hypersurface invariant
Energy, internal      179
Energy, rest      195
Energy, total      148 196
entropy      183
Entropy, empirical      185
Equations of motion      144
Equations of motion, relativistic      303
Equilibrium equations      630—632
Euclidean metric in quantum fields      551
Euler characteristic      423 426
Euler equations of fluid flow      144
Euler integrability condition      166
Euler principle of least action      281
Exact form      156
Exact sequence      598—600 Exact sequence homology
EXP      284
Exponential map for a Lie group      399 403
Extension theorem      592
Exterior algebra      68
Exterior covariant differential $\nabla$       250 430
Exterior covariant differential $\nabla$ of a form section of a vector bundle      488
Exterior covariant divergence $\nabla^{\ast}$       545
Exterior differential d      73 Exterior differential d coordinate
Exterior differential d, spatial $\mathbf{d}$       141
Exterior form      66 Exterior form and vector analysis
Exterior power operation      588
Exterior product      67 Exterior product and determinants
Face      335
Faraday's law      121
Fermat's principle      297
Fermion      658
Fiber      49 415
Fiber bundle      451 594
Fiber coordinate      416
Fiber over p      416
Fiber space      593
Field strength      64
Flamm paraboloid      321
Flow generated by a vector field      32 33
Flow generated by a vector field by invariant fields      408
Flow generated by a vector field by Lie bracket      129
Flow generated by a vector field, straightened      35
Fluid flow      30 143—145
Fluid flow, magnetohydrodynamic      145
Foliation      173
Force, classical      195
Force, Lorentz      119
Force, Minkowski      195
Form and pseudo-form      122
Form of type Ad      489 490
Form with values in a Lie algebra      475 477
Form, bi-invariant      561—563
Form, Cartan      562
Form, Cauchy stress form      618
Form, closed      156
Form, exact      156
Form, exterior      66
Form, first fundamental      202
Form, harmonic      370
Form, heat 1-form      179
Form, integration of      95—102 Form integration
Form, invariant      395
Form, Maurer — Cartan      476
Form, normal      376
Form, p-form      41
Form, pull-back      77—82
Form, second fundamental      204 309 Form second and
Form, stress, Cauchy      618 Form stress Piola
Form, tangential      376
Form, vector bundle-valued      429
Form, vector-valued, $d\mathbf{r}$ and $d\mathbf{S}$       203 248
Form, volume      86 88
Form, work 1-form      179
Frame $\mathbf{e}$       243
Frame $\mathbf{e}$ of sections      417
Frame $\mathbf{e}$ , change of      253
Frame $\mathbf{e}$ , coordinate      243
Frame $\mathbf{e}$ , orthonormal      255
Frame bundle      453
Frobenius chart      167
Frobenius theorem      170
Frobenius — Schur relations      664
Functional derivative      307
Fundamental group $\pi_{1}$       567—569 578
Fundamental theorem of algebra      215
Fundamental vector field      455
Galloway's theorem      578
Gauge bundle      490
Gauge field      255 536
Gauge invariance      441 449 533—536
Gauge particles, gluons      540 Gauge particles mesons
Gauge principle      537
Gauge transformation      255 490 Gauge global
Gauss curvature      207
Gauss equations      229 310 311—314 Gauss relativistic
Gauss formula for variation of area      225
Gauss law      121
Gauss lemma      286
Gauss linking or looping integral      218
Gauss normal map      208 215 260
Gauss theorema egregium      231 317—318
Gauss — Bonnet theorem      215 323 462 Gauss
Gaussian coordinates      284
Gell-Mann, Gell-Mann matrices      540 652
Gell-Mann, Gell-Mann/Okuba mass formula      659
General linear group GL(n)      254 391
General relativity      291—322
Generalized momentum      55
Generalized velocity      50
Geodesic      233 271—274
Geodesic circle      287
Geodesic completeness      564
Geodesic curvature $\kappa_{g}$       235 239
Geodesic equation      235
Geodesic in a bi-invariant metric      563
Geodesic J. Bernoulli's theorem      234
Geodesic null      303
Geodesic polar coordinates      287
Geodesic stability      324 326

Geodesic submanifold      310 Geodesic submanifold total
Geodesic, closed      281 284
Geodesy      252
Gluons      540
Gradient vector      45
Grassmann algebra      see also "Exterior algebra"
Grassmann algebra, manifold      459
Green's reciprocity      647
Green's theorem      368
Group, $\mathbb{R}, \mathbb{Z}, \mathbb{Z}_{2}$       336
Group, boundary      344
Group, chain      337
Group, cycle      344
Group, de Rham      356
Group, exact sequence      598
Group, homology      345
Group, homomorphism      337 398
Group, homotopy      596
Group, quotient      345
H. Cartan's formula      135
Haar measure      397 541
Hadamard's lemma      126
Hairy sphere      423
Hamilton's equations      147
Hamilton's principle      154 275 Hamilton's
Hamilton, on composing rotations      499
Hamiltonian      147
Hamiltonian flow      148
Hamiltonian operator      439
Hamiltonian relativistic      196
Hamiltonian vector field      148
Harmonic cochain      641
Harmonic field      376
Harmonic form      370
Harmonic form in a bi-invariant metric      564
Hawking singularity theorem      579
Heat 1-form      179
Helicity      145
Helmholtz decomposition      372
Hermitian adjoint $^{\dag}$       392
Hermitian line bundle      466
Hessian matrix      383
Hilbert action principle      308
Hilbert space inner product      361
Hilbert variational approach      305—308 368
Hodge * operator      362
Hodge codifferential d*      364
Hodge decomposition      372 388
Hodge theorem      371
Hodge theorem for normal forms      381
Hodge theorem for tangential forms      377
Holomorphic      158
Holonomic constraint      175
Holonomy      259
Homeomorphism      13
Homogeneous space      458
Homologous      345
Homology group      345—355
Homology group, relative      379 Homology group sequence
Homomorphism      337 398
Homomorphism, algebra      78
Homomorphism, boundary      338 601
Homomorphism, induced      337
Homotopically critical point      382
Homotopy      591
Homotopy and homology      603
Homotopy groups $\pi_{k}$       596—598
Homotopy groups $\pi_{k}$ and covering spaces      605
Homotopy groups $\pi_{k}$ of spheres      597 598
Homotopy groups $\pi_{k}$ , computation of      605—608
Homotopy, covering homotopy      592
Homotopy, free homotopy class      282 283
Homotopy, sequence for a bundle      600—603
Hopf bundle      473 474
Hopf map and fibering      606 675
Hopf theorem      427
Hopf — Rinow theorem      564
Horizontal distribution      263—266 481
Hurewicz Theorem      603
Hypercharge      654
Hyperelastic      626
Hypersurface      6
Hypersurface, 1- and 2-sided      84
Hypersurface, parallel      286
Immersion      169 173
Implicit function theorem      5
Incidence matrix      645
Inclusion map      79
Index of a vector field      see also "Kronecker index"
Index of a vector field of a section      466
Index theorem      465
Indicator      315
Infinitesimal generator      399
Instanton      550
Instanton, winding number      556 560
Integrability condition      166 170 174
Integrable constraint      175
Integrable distribution      167
Integral curve      31
Integral manifold      166
Integrating factor      183
Integration of forms      96—109
Integration of pseudoforms      114—117
Integration over manifolds      104—109
Interaction      534
Interior product      89
Intersection number      219
INTRINSIC      234
Intrinsic, derivative      235
Invariant form      395
Invariant vector field      395
Invariant volume form      397
Inverse function theorem      29
Inverse image      12
Involution      167
Isometry      230 314
Isometry, fixed set      314
Isometry, invariant      231
Isotopic spin      648 654
Isotropic body      661
Isotropy subgroup      457
j      432
Jacobi determinant      5
Jacobi equation of geodesic variation      273
Jacobi field      129 273 326—329
Jacobi identity      403
Jacobi metric      281
Jacobi principle of least action      281
Jacobi rule for change of variables in an integral      101
Jacobi variational equation      128
Killing field      528
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 $\delta^{I}_{J}$       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

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