Levi-Civita T. — The Absolute Differential Calculus (Calculus of Tensors) :: Электронная библиотека попечительского совета мехмата МГУ

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Levi-Civita T. — The Absolute Differential Calculus (Calculus of Tensors)

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Название: The Absolute Differential Calculus (Calculus of Tensors)

Автор: Levi-Civita T.

Аннотация:

Great 20th-century mathematician's classic work on material necessary for mathematical grasp of theory of relativity. Thorough treatment of introductory theories provides basics for discussion of fundamental quadratic form and absolute differential calculus. Final section deals with physical applications. 1926 ed.

Язык:

Рубрика: Математика/Анализ/Тензорный анализ, формы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1927

Количество страниц: 450

Добавлена в каталог: 12.04.2005

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Предметный указатель

Relative motion      313 316
Relativity and Newtonian theory, differences      377
Relativity, composition of velocities in      317
Relativity, general, and Poisson’s equation      386 387
Relativity, general, postulates of      364
Relativity, invariance in      322
Relativity, kinematics of      311 316
Relativity, metrics, qualities of      325
Relativity, metrics, statical      326
Relativity, metrics, stationary      326
Relativity, postulates of      364
Relativity, principle of      311
Relativity, restricted      300
Relativity, Special Theory of      300
Reversibility of light propagation      365
Reversible motion      327
Reversible motion, transformation      3 7 61
Ricci      14 148 182 183 199 200 235 257 263 268 271 278 372 380 385 389 411
Ricci’s coefficients of rotation      268
Ricci’s lemma      148 152
Ricci’s symbols      199 372 389 411 426
Ricci’s tensor      199
Ricci’s tensor, linear invariant of      200 380
Riemann      172 175 176 177 178 179 180 181 182 184 193 195 196 200 209 215 216 219 220 221 224 225 228 232 233 234 236 240 242 243 244 247 257 261 278 374 376 378 379 384 389 414 417 426 430 436 437
Riemann — Christoffel tensor in $V_{4}$       372
Riemann — Christoffel tensor in $V_{4}$ , 20 components of      372 (see also “Riemann’s symbols”)
Riemannian curvature of $V_{n}$       195—198
Riemann’s symbols      172
Riemann’s symbols and conformal representation      228 246
Riemann’s symbols and Euclidean metric manifold      242—246
Riemann’s symbols of first kind      176 179—182
Riemann’s symbols of second kind      175 177 178
Riemann’s symbols, Bianchi’s identities in      182
Rigid motion in any manifold      408
Roemer      307
Roemerian units      307
Rotation, Ricci’s coefficients of      268
Rotor of vector      161
Saturation (of indices)      see “Contraction”
Sbrana      414
Scalar product of vectors      98 126 152
Schouten      172 198 231
schur      235
Schur’s Theorem      235
Schwarzschild      423
Schwarzschild’s solution of gravitational equations      419—423
Schwarzschild’s solution, extensions of      439
Second covariant derivatives      184
Second differential parameter      154 393
Second fundamental form of $V_{n}$       252
Section of $V_{3}$       201
Section of manifold      163
Sereni      383
Sets of orthogonal directions      205
Sets of simple systems      74 156
Sets, reciprocal      74
Seven      22 171 196
Severi’s theorem      171
Shift, spectral      400
Signals, light, and coefficients of $ds^{2}$       364—366
Simultaneity      290 311
Sirius, spectrum of Companion of      402
Sobral expedition      407
Solution of differential equations      36
Solution of gravitational equations, Schwarzschild’s      419—423
Solution of gravitational first approximation deduced from      425
Solutions, rigorous, of gravitational equations      437
Space of constant curvature, extension of      426 427 428
Space of constant curvature, gravitational equations in      428
Space, metric of, and Newtonian potential      391
Space, non-Euclidean      391
Space-time      290
Space-time metric, and energy tensor      374
Space-time with assigned Newtonian field      392
Space-time, an Einsteinian, ds’2 for      392
Space-time, co-ordinate transformations      290
Space-time, De Sitter’s      429—435
Space-time, De Sitter’s, constant negative curvature of      436
Space-time, Einstein’s and De Sitter’s, case including      435
Space-time, Einstein’s cylindrical      429
Space-time, Einstein’s, curvature of space in      438
Spatially uniform metrics      425
Spectral displacement      400
Sphere, geodesic, in $F_{3}$       409 410
Spherical symmetry and gravitational equations      419
Spherical symmetry, metrics with      408—414
St. John      402
Statical $ds^{2}$       326 327 371 377 378 392
Statical field      400
Statical metrics      326 327
Stationary metrics      326 327
Stationary metrics, Fermat’s principle for      340
Stress      344
Stress and bilinear form      345
Stress in spatially symmetrical metrics      425 429 430 436 437 438
Stress tensor and equations of motion      351
Stress tensor in classical theory      344
Stress tensor in generalized co-ordinates      346
Stress tensor, divergence of      344 346 354
Stress tensor, interpretation of divergence      346
Stress, force absorbed in      349
Stress, kinetic      351 356 358
Stress, normal      345
Struik      231

Sum of tensors      76
Surface vectors      96
Surface, geodesic      164
Surface, intrinsic geometry of      99
Surface, parametric equations of      86
Surfaces, developable      100
Symmetrical double systems      72
Symmetrical systems (tensors)      65
Symmetry, spherical, and gravitational equations      419
Symmetry, spherical, metrics with      408—414
System, mixed      70 71
Systems (tensors) of order m      65
Systems (tensors) of order zero      65
Systems (tensors), antisymmetrical      73
Systems (tensors), double      65
Systems (tensors), m-fold      65
Systems (tensors), symmetrical      65
Tensor      70 71
Tensor with vanishing elements      71
Tensor, Einstein’s      200 371
Tensor, energy      355
Tensor, energy, and equations of motion      359
Tensor, first general definition of      80
Tensor, gravitational      371
Tensor, gravitational, 10 components of      372
Tensor, Riemannian      371 (see also “Riemann’s symbols”)
Tensor, second general definition of      83
Tensor, stress, divergence of      344 346 354
Tensors, addition of      75
Tensors, associated      95 96
Tensors, composition of      79
Tensors, contraction of      77—79
Tensors, inner multiplication of      79
Tensors, multiplication of      76
Tensors, reciprocal      95
Third fundamental form of $V_{n}$       259
Tietze      188
Time, conventional      364
Time, local      290 311 312
Total differential      13
Total differential, equations      13—33
Total differential, equations, complete svstem of      15—18
Trajectories      403
Trajectories and geodesies      324 326
Trajectories and light rays      343
Trajectories in generalized mechanics      324
Trajectories, Einsteinian and Newtonian      395
Trajectories, orthogonal      263
Transformation by contravariance      67 69—71
Transformation by covariance      64 67—71
Transformation by invariance      62
Transformation of derivatives      85
Transformation, affine      304 305
Transformation, formulae of      80
Transformation, homographic      304
Transformation, linear, of differentials      80
Transformations, linear      67
Transformations, Lorentz      300 308 310 316
Transformations, reversible      3 7 61
Transformations, space-time co-ordinate      290
Translation, motion of      305
Trefftz      439
Universe, radius of      439
Vanderlinden      439
Variations, Poincare’s equation of      208
Variety      see “Manifold”
Vector, contravariant and covariant components of      97
Vector, contravariant, in $V_{n}$       120 127
Vector, derivative of      139 140
Vector, determination of, by invariants      266
Vector, product      159
Vector, product of versors      201
Vector, projection of, in $V_{n}$       127
Vector, transformation of      62 63 64
Vectors, equipollence of      103
Vectors, parallel and equal      103
Vectors, scalar product of      98 126
Vectors, surface or tangential      96
Vectors, zero      97
Velocities, absolute and relative      316—319
Velocities, composition of      306
Velocities, composition, according to Einstein      317
Velocity of light      292 306 311 334 335 339 382
Velocity of light, irreversible      340
Velocity of light, law of variation of      339
Velocity of light, non-symmetrical      340
Velocity, earth’s orbital      399
Velocity, large universal constant      291 292 311
Velocity, mass and      295
Versor (unit vector, direction)      92 96 98 102 103 123 125 126 140
Versor (unit vector, direction), in $V_{4}$ , and corresponding vector      329
Versors and pseudo-Euclidean metrics      329
Versors, spacelike      330
Versors, timelike      330
Vibration of atom      400
Volume of curved space      427 428
Weber      14
Weyl      117 439
Whittaker      324
World lines      290 329 352 353
World lines of light      337 364
World lines, parameters and stress tensor      353
World lines, parameters of      353
Zeeman      319
Zero vectors      97

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