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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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| 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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