Szekeres P. — A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Szekeres P. — A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

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Название: A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Автор: Szekeres P.

Аннотация:

Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2004

Количество страниц: 600

Добавлена в каталог: 05.10.2005

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Предметный указатель

Double pendulum      471
Dual electromagnetic tensor      247
Dual Maxwell 2-form      502
Dual space      90 283
Dummy indices      81
Dust      551
Dynamical variable      475
d’Alembertian      248
Eddington — Finkelstein coordinates      546
Effective action      572
Eigenvalue      100 351
Eigenvalue multiplicity      102
Eigenvector      100 351
Einstein elevator      535
Einstein tensor      527
Einstein’s field equations      537
Einstein’s gravitational constant      537
Einstein’s principle of relativity      229
Einstem — Cartan theory      518
electric field      246
Electrodynamics      246
Electromagnete field      246
Electromagnetic 4-potential      247
Electron spin      373
Elementary divisors      105
Embedded submanifold      429
Embedding      429
Empirical entropy      461
Empirical temperature      458
Empty set      5
Energy operator      379
Energy-stress tensor      252
Energy-stress tensor of electromagnetic field      253
Ensemble      401
Ensemble canonical      401
Ensemble grand canonical      406
Ensemble microcanonical      401
entropy      403
Eotvos experiment      535
Equal operators      357
Equality of sets      4
Equation of continuity      245
Equivalence class      8
Equivalence relation      8
Euclidean geometry      19
Euclidean group      580
Euclidean space      53
Euclidean transformation      54
Euclidean vector space      127
Euler characteristic      497
Euler — Lagrange equations      467
Even parity      394
Event      54 230 536
Event horizon      546
Exponential map      566
Exponential operator      350
Extended real line      287
Exterior algebra      164 209
Exterior derivative      448
Exterior product      163 208
External direct sum      67
Extremal curve      467
Factor group      47
Factor space      8
Family of sets      4
Fermi — Dirac statistics      397
Fermion      397
Field      60
Finer topology      261
Finite set      4 13
Flow      433
Focal basis of vector fields      422
Focal one-parameter group of transformations      434
Focal one-parameter group of transformations, generated by a vector field      434
Focus stable      118
Focus unstable      118
Four-dimensional Gauss theorem      251
Fourier series      338
Fourier transform      320
Fourier transform inverse      320
Fourier’s integral theorem      320
Free action      50
Free associative algebra      182
Free indices      81
Free vector space      178
Friedmann equation      551
Frobenius theorem      441
Frobenius theorem expressed in differential forms      455
Fubini’s Theorem      306
Function      10
Function, analytic      410
Function, differentiable      410
Function, real differentiable      415
Fundamental n-chain      489
Galilean group      54
Galilean space      54
Galilean transformation      55 229
Gauge transformation      248 542
Gauss theorem      491
General linear group on a vector space      65
General linear groups      37
General linear groups, complex      39
General relativity      536—552
Generalized coordinates      468
Generalized momentum      473
Generalized thermal force      460
Geodesic      508
Geodesic affine parameter along      509
Geodesic coordinates      520—522
Geodesic deviation, equation of      522—524
Geodesic null      536
Geodesic timelike      536
Geodesics curves of stationary length      519
Graded algebra      164
Gradient of 4-tensor field      244
Gram — Schmidt orthonormalization      129
Graph of a map      269
Grassmann algebra      160 166
Gravitational redshift      548
Green’s Theorem      491
Group      27
Group of components      279
Group, abelian      28
Group, cyclic      29
Group, cyclic, generator of      29
Group, finite      29
Group, order of      29
Group, simple      46
Hamilton — Jacobi equation      480
Hamiltonian      475
Hamiltonian operator      379
Hamiltonian symmetry      393
Hamiltonian vector field      475
Hamilton’s equations      477
Hamilton’s principle      469
Harmonic gauge      542
Harmonic oscillator quantum      384 402
Heat 1-form      459
Heat added to a system      459
Heaviside step function      316
Heckmann — Schucking solutions      583
Heisenberg picture      380
Heisenberg uncertainty relation      372
Henon map      22
Hermite polynomials      338
Hermitian operator eigenvalues      351
Hermitian operator eigenvectors      351
Hermitian operator spectrum      355
Hermitian operator, complete      352
Hilbert Lagrangian      554

Hilbert space      330—334
Hilbert space finite dimensional      134
Hilbert space of states      369
Hodge dual      220—227
Hodge star operator      223
Homeomorphic      263
Homeomorphism      263
Homogeneous manifold      573
Homologous cycles      497
Homology spaces (groups)      497
Homomorphism of groups      40
Ideal gas      458
Ideal left      152
Ideal right      152
Ideal two-sided      152
Idempotent      205
Identity element      27
identity map      12
Ignorable coordinate      473
Image of a linear map      70
Immersion      429
Impossible event      293
Inclusion map      12
Independent events      293
Independent set of points in $\mathbb{R}^n$       493
Indexing set      4
Indiscrete topology      261
Indistinguishable particles      395
Induced      267
Induced topology      259
Induced vector field      575
Inertial frame      228 230 232
Infinite set      13
Infinitesimal generator      170 390
Injection      11 267
Injective map      11
Inner automorphism      43
Inner measure      300
Inner product      330
Inner product Euclidean      127
Inner product Minkowskian      131
Inner product of p-vectors      221
Inner product on complex spaces      133
Inner product positive definite      127
Inner product real      126
Inner product space, complex      134
Inner product, complex, components of      137
Inner product, components of      128
Inner product, index of inner product      131
Inner product, non-singular      127
Instantaneous rest frame      240
Integral curve of a vector field      433
Integral of a 1-form on the curve      428
Integral of n-form with compact support      484
Interior of a set      260
Interior product      213
Internal energy      459
Intersection of sets      6
Intertwining operator      121
Invanance group of a set of functions      53
Invanance of function under group action      52
Invanance of Lagrangian under local flow      474
Invariant subspace      99 121
Invariants of electromagnetic field      247
Inverse element      27
Inverse image of a set under a mapping      10
Inverse map      11
Is a member of      3
Isentropic      461
Isometry      578
Isomorphic algebras      151
Isomorphic groups      42
Isomorphic vector spaces      64
Isomorphism of groups      42
Isotherm      458
Isotropic space      534
Isotropy group      50
Jacob identity      167 432
Jordan canonical form      113
k-cell      486
k-cell support of      486
k-dimensional distribution      440
k-dimensional distribution integral manifold      440
k-dimensional distribution involutive      441
k-dimensional distribution smooth      440
k-dimensional distribution vector field belongs to      440
k-dimensional distribution vector field lies in      440
Kasner solutions      583
Kernel index notation      196
Kernel of a linear map      70
Kernel of an algebra homomorphism      152
Kernel of homomorphism      47
Ket      369
Killing vector      578
Killing’s equations      578
kinetic energy      468
Kronecker delta      36 83
Lagrange’s equations      469
Lagrangian      469 554
Lagrangian function      465
Lagrangian mechanical system      469
Law of composition      18
Law of composition, associative      18
Law of composition, commutative      18
Lebesgue integrable function      304
Lebesgue integral non-negative measurable function      301
Lebesgue integral over measurable set      302
Lebesgue integral simple functions      301
Lebesgue measure      295—300
Lebesgue measure, non-measurable set      299
Lebesgue — Stieltjes inetgral      356
Left action of group on a set      49
Left translation      51 277 559
Left-invariant differential form      562
Left-invariant vector field      560
Leibnitz rule      418
Levi—Civita symbols      215
Lie algebra      166—177
Lie algebra of a Lie group      561
Lie algebra, commutative      167
Lie algebra, factor algebra      168
Lie algebra, ideal      168
Lie bracket      432
Lie derivative      507
Lie derivative of a tensor field      438
Lie derivative of a vector field      436
Lie derivative, components of      439
Lie group      559
Lie group homomorphism      564
Lie group isomorphism      564
Lie group of transformations      572
Lie subgroup      569
Lift of a curve to tangent bundle      424 465
Light cone      230
Light cone at a point      233
lim inf      291
lim sup      291
LIMIT      255 256 280
Limit point      260
Linear connection      see “Connection”
Linear functional      88
Linear functional on Banach space      283
Linear functional, components of      91
Linear map      63
Linear mapping      35
Linear mapping, matrix of      36
Linear operator      65
Linear operator bounded      344
Linear operator continuous      344
Linear ordinary differential equations      116

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