Dubrovin B.A., Fomenko A.T., Novikov S.P. — Modern Geometry - Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups and Fields :: Электронная библиотека попечительского совета мехмата МГУ

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Dubrovin B.A., Fomenko A.T., Novikov S.P. — Modern Geometry - Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups and Fields

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Название: Modern Geometry - Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups and Fields

Авторы: Dubrovin B.A., Fomenko A.T., Novikov S.P.

Аннотация:

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 1991

Количество страниц: 492

Добавлена в каталог: 01.06.2008

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Предметный указатель

Acceleration      38
action      318
Action of a gravitational field      395 420
Action of an electromagnetic field      388—390
Action, truncated      346
Addition rule for velocities      59
Affine group      27 38
Affine transformation      27 33
Area functional      380 401
Automorphism of an algebra      431
Automorphism, inner      431
Beltrami's equation      111
Beltrami's operator      111
Beltrami's pseudosphere      314
Bianchi's classification of 3-dimensional Lie algebras      230
Bianchi's identities      315 448
Bilinear form      17 37
Bravais lattice      178
Broken vector field      375
Cartan structure equations      315
Cartan's formula      243
Catenoid      86
Cauchy — Bunjakovskii inequality      19
Cauchy — Riemann equations      103 262
Cell, primitive      177
Cell, symmetric      180
Cell, Wigner — Seitz      see “Symmetric cell”
Characteristic class      456
Charge conjugation      442
Christoffel symbols      283
Christoffel's formula      293
Circle group      99
Circulation      256
Clairaut's theorem      324
Classical matrix Lie algebras      124 129 218
Classical transformation groups      120 218
Co-ordinate transformation      5 23
Co-ordinate, complex analytic      104
Co-ordinate, linear      5
Co-ordinates, affine      284
Co-ordinates, Cartesian      2
Co-ordinates, conformal      108
Co-ordinates, cyclic      323
Co-ordinates, cylindrical      7
Co-ordinates, Euclidean      10
Co-ordinates, polar      7
Co-ordinates, pseudo-spherical      22
Co-ordinates, rectangular Cartesian      7
Co-ordinates, spherical      7
Commutation      216
Commutator of operators      213
Commutator of vector fields      213
Complex analytic co-ordinate change      104
Complex analytic function      103
complex curve      106
Complex form of rank k      271
Complex gradient      107
Complex Implicit Function Theorem      107
Complex Inverse Function Theorem      105
Conformal map      136
Conjugate Dirac equation      440 441
Conjugate points of a geodesic      376
Connexion      283 see also “Gauge field”
Connexion compatible with a metric      292
Connexion, affine      see “Euclidean connexion”
Connexion, Cartan      500
Connexion, curvature form of      447
Connexion, Euclidean      283
Connexion, symmetric      285
Connexion, torsion-free      see “Symmetric connexion”
Connexion, trivial      447
Conservation of energy      322
Conservation of momentum      323 326
Coriolis force      343
Cosmological constant      424
Covariant differentiation      283 446
Covariant of spinors with respect to a metric      451
Covector      15 148
Critical function      see “Stationary function”
Cross product      43
Crystal lattice      176
Crystal, basic translations of      176
Crystal, basis for      179
Crystal, classification of group of motions of      189 195
Crystal, point group of      184
Crystal, primitive vectors of      176
Crystal, space group of      182
Curl      240
Current four-vector      240
Curvature normal, Euler's formula for      82
Curvature of a curve      39 43
Curvature, Gaussian      77 81 84 257—259 307 400—401
Curvature, mean      77 81
Curvature, normal      79 81
Curvature, principal      77 81
Curvature, radius of      39 43
Curvature, Riemannian      301
Darboux' Theorem      355
Darboux' vector      49
Diffeomorphism      207
Differential form      162 164
Differential, exterior product of      167
Differential, of a form      see “Exterior derivative”
Differential, of a general tensor      see “Gradient of a tensor”
Differential, of a mapping      207
Differential, scalar-valued gauge-invariant      456
Dilation      25
Dirac's and the current vector      440
Dirac's and the energy-momentum tensor      440
Dirac's as Euler — Lagrange equation      440
Dirac's equation      440
Dirac's in an electromagnetic field      441
Dirac's in terms of semi-spinors      441
Directional derivative      161 288
Dirichlet functional      406
Dirichlet region      see “Symmetric cell”
Discrete group      183
Divergence of a vector field      240 278
Eigenvalues of a pair of quadratic forms      79
Eigenvalues of a rank-two tensor      202
Einstein's as Euler — Lagrange equations      399—400
Einstein's equations for the space-time metric      309 423—424
Einstein's postulate      329 420
electric field      197 389
Electric strength      389
Electrical conductivity      195
Electromagnetic (tensor) field      197 205 345 389
Electromagnetic stress tensor      205
Electromagnetic waves      205
Electromagnetic, canonical forms of      201
Electromagnetic, energy density of      205
Electromagnetic, four-potential of      389
Electromagnetic, inclusion of      344 419—420 421
Electromagnetic, invariants of      201
Electromagnetic, scalar potential of      389
Electromagnetic, vector-potential of      389
Element of area      73 249
Element of volume      166 250
Elementary Euclidean geometry      1
Energy      320
Energy density      205 386
Energy level of an extremal      345
Energy-momentum of a charge-free electromagnetic field      393
Energy-momentum of a dense isotropic medium      429
Energy-momentum tensor      204 309 385
Energy-momentum vector      329
Equilibrium equation for deformed elastic plate      411
Euclidean metric in various co-ordinates      20
Euclidean plane, isometries of      27
Euclidean space      9

Euclidean, arc length in      10
Euclidean, scalar product in      10
Euler angles      125
Euler — Lagrange equations      318 320 324
Euler — Lagrange for area functional      402
Euler — Lagrange in several dimensions      383
Euler — Lagrange, full system of      430
Extended complex plane      106
Exterior algebra      167
Exterior algebra derivative of a form      242
Extremal      320 381
Extremal function      see “Stationary function”
Family of matrices      35 47
Fermat's principle      347
Flat and curved space-times      394
Flux      255 256
force      320
Forms normal to a curve      45
Four-momentum      328
Free energy of deformed elastic plate      408
Functional      318
Galilean group      37 38
Galilean transformation      37 52
Galileo's principle of relativity      36
Gauge field      446
Gauge group      446
Gauge transformation      415 446
Gauss equations      313
Gauss map      257
Gauss — Bonnet theorem      401
Gauss — Ostrogradskii formula      263
Gaussian curvature      77 81 84 307 400—401
Gaussian in conformal co-ordinates      112
Gaussian of a space-like surface      95
General Stokes formula      261
Generalized gradient      278 see also “Covariant derivative”
Generators of the action of a group      225
Geodesic      291
Gradient of a general tensor      239
Gradient system      349
Gravitational field as metric      420
Gravitational field, energy density of      431
Gravitational field, weak      330 421
Gravitational wave-packet      431
Green's formula      261
Group      23
Group of conformal transformations      142
Group, affine      27 38
Group, general linear      27 96
Group, Lorentz      55
Group, orthogonal      33 34
Group, Poincare      53
Group, special linear      97
Group, special orthogonal      33 34
Hamilton — Jacobi equation      365
Hamilton — Jacobi, truncated      367
Hamilton's equations      338
Hamiltonian      338
Hamiltonian in the Keplerian problem      357
Hamiltonian of a free relativisitic particle      419
Hamiltonian system      350
Hermitian form      97
Hermitian scalar product      see “Hermitian form”
hessian      77
Hilbert action of a gravitational field      395
Holomorphic form      272
Homomorphism      55
Homothety      see “Dilation”
Hooke's law      156
Huygens' principle      369
Hypersurface      65
Idempotent      432
Idempotent, one-dimensional      432
Implicit function theorem      62 65
Independence of a variable (for polynomials)      102
Inertial frame      37
Instantaneous moving co-ordinate frame      341
Integral curve of a vector field      208
Integral function of a Hamiltonian system      355
Integral of a field      212
Integral of motion      see “Integral function”
Integral of the first kind      251 253
Integral of the second kind      250 252
Integral with respect to anti-commuting variables      168
Integral, double      74
Integral, multiple      248
Inverse for complex analytic co-ordinate transformations      105
Inverse function theorem      6
Isometries of Euclidean 3-space      31
Isometries of the sphere      89
Isometry      24
Isometry group      24
Isometry, direct      see “Proper isometry”
Isometry, proper      28 56
Isomorphism      25
Isothermal co-ordinates      see “Conformal co-ordinates”
Isotropic cone      50
Isotropic curve      417
Isotropic medium      156
Isotropic vector      50 417
Jacobi field of a Lagrangian      376
Jacobi operator of a Lagrangian      373
Jacobi's identity      44 60 216
Jacobi's minimality condition      376
Jacobian      6
Jacobian matrix      6 104
Jacobian, complex      104
Kepler's third law      37
Killing form      228 453
Klein — Gordon equation      414
Klein — Gordon operator      439
Kronecker delta      5
Lagrange surface in extended phase space      363—364
Lagrange surface in phase space      363 365
Lagrange surface, conical      368
Lagrangian      320 379
Lagrangian of a free relativistic particle      326—327 414
Lagrangian of mutual interaction of field and gauge field      443—446
Lagrangian, non-singular      338
Lagrangian, spherically symmetric      333
Lagrangian, strongly non-singular      338
Lagrangian, translation-invariant      331
Lagrangians for STR      327 417—420
Lagrangians of a gauge field alone      453
Laplace operator      103
Laplace — Runge — Lenz vector      357
Laplace's equation      85
Least action, principle of      326
Left-invariant field on a group      226
Legendre transformation      338
Length      10 18
Length and independence of parametrization      13
Levi — Civita tensor      165
Lie algebra      216
Lie derivative of a general tensor      211
Lie of a covector      214
Lie of a rank-two tensor      214
Lie of a scalar      212
Lie of a vector      213
Lie of the volume element      215
Lie, derivation of      216
Lie, of the symplectic group      361
Lie, structural constants of      229
Light cone      see “Isotropic cone”
Light curve      see “Isotropic curve”
Light vector      see “Isotropic vector”
Line integral      251 256
Linear combination of tensors      153
Linear fractional transformation      99 106
Linear transformations of the plane      25

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