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

Fresnel’s convection coefficient      319
Fresnel’s formula for velocity of light in moving media      318—320
Function, alternate      35
Function, implicit      3
Function, uniform      14
Functional matrix, characteristic of      9 39 87 250
Functions of position      80 83
G, linear invariant of Einstein tensor      380 381
Galilean systems      349
Galilean systems, force, stress and divergence in      349
Gauss      99 100 414
Gauss, on intrinsic geometry      99
Gaussian curvature      172
Gaussian curvature of $F_{2}$       193
General relativity, concept of      294
General relativity, postulates of      364
Generalization of Lagrangian function      322—324
Generalization of metric of $F_{4}$       320
Geodesic curvature      135—137
Geodesic deviation      208—220
Geodesic deviation, Jacobi’s formula for      219
Geodesic excess      197
Geodesic manifold      162
Geodesic principle      337
Geodesic principle in $V_{4}$       341
Geodesic principle, Einstein’s      328 331
Geodesic sphere as horizon      434
Geodesic surface      164
Geodesic triangle      197
Geodesic, co-ordinates locally      164 167 171
Geodesic, definition of      103 128
Geodesic, motion of particle in      326
Geodesies and trajectories      324 326 331
Geodesies autoparallelism of      104 140
Geodesies differential equations of      131—135
Geodesies in rigid motions      408
Geodesies in space of constant curvature      428
Geodesies Lagrange’s equations for      208 331 332 341 367 413
Geodesies near given geodesic      208
Geodesies of zero length      330—334 337
Geometrical optics      334
Geometrical optics according to Einstein      335—338
Goursat      14
Gravitation with point mass      419—423
Gravitation, modification of Newton’s law of      397
Gravitation, not absorbed in energy tensor      375
Gravitational constant      386
Gravitational equations and the facts      387
Gravitational equations for spherical symmetry      419
Gravitational equations for statical $ds^{2}$       381
Gravitational equations in space of constant curvature      428
Gravitational equations, Einstein’s      376
Gravitational equations, modified by cosmological term      438
Gravitational equations, rigorous solutions of      437
Gravitational equations, solution of      419—423
Gravitational experiments and $ds^{2}$       367
Gravitational field and spectral lines      400—402
Gravitational field, path of light in      403—408
Gravitational forces, as privileged      374
Gravitational tensor      371 372
Gravitational tensor, divergence of      372
Hamilton      287 288 291 293 294 296 298 301 320 322 351 385
Hamilton’s principle      287
Hamilton’s principle, Einstein’s form of      291
Hamilton’s principle, modified      294—298 301 322—324 351
Hessenberg      146
Hoelder      160
Horizon, in De Sitter’s space-time      434
Hyperspherical representation      258
Hypersurface      121
Hypersurface, hyperspherical representation of      258
Hypersurfaces in Euclidean space      249 253
Hypersurfaces, parallel      251
Immersion of $V_{n}$ in Euclidean space      121
Indefinite $ds^{2}$       141
Independence of functions      5 8—10
Inertia, index of      299
Inertia, principle of, in relativity      298
Inner multiplication of tensors      79
Integral of differential equations      36 37
Integral, general      40 42 43 45 50
Integral, independent      40 42
Integral, principal      38 39 49
Intrinsic geometry of surface      99
Invariance and Hamilton’s principle      291
Invariance in relativity      322
Invariance of $ds^{2}$       308 311
Invariance, m-fold system      69
Invariance, simple system      67
Invariance, transformation by      62
Invariant, derivatives of      83
Invariant, quadratic form      73 84
Isotropic manifolds      232
jacobi      209 219 220
Jacobi on geodesies      208
Jacobian systems of equations      52 53
Jacobians      2 (see also “Determinant Junctional”)
Jeans      383
Kasner      439
Kepler      377
Kinematics of modified      303
Kinematics of rigid systems      301
Kinematics, Galilean      318
Kinematics, relativity      311 316
Kummer      286
Kummer on congruences      286
Lagrange      289 295 413
Lagrange and geodesies      208 331 332 341 367 413
Lagrangian binomials      289
Lagrangian equations      289 331 332 341 367 413
Lagrangian parameters      288
Lamb      397
Laplace      394
Laplace’s operator      394
Laue      439
Law of gravitation, modifications of      397
Levi-Civita      104 139 172 272 286 324 334 381 383 394 397 398 429 439
Light in gravitational field, frequency of      400—402
Light, constancy of velocity of      335
Light, path in gravitational field      403—408
Light, path of, as trajectory      403
Light, propagation of, reversible      365
Light, rays and trajectories      343
Light, signals      364
Lipka      136 159
Local time      290 311 312
Longo      439
Lorentz      300 301 306 310 311 312 313 315 316 318 352 353 354
Lorentz transformation      300 308 310 316
Lorentz transformation, invariance for      352 353 354
Lorentz transformation, most general      313
Lorentz translation      316
Love      344
Maclaurin      166
Majorana      335
Manifold      1
Manifold, Euclidean      121
Manifold, geodesic      162
Manifold, metric      119
Manifold, n-dimensional      119
Manifold, sections of      162
Manifolds of constant curvature      236 238 240 246
Manifolds of constant, their mutual applicability      249
Manifolds, isotropic      232
Marcolongo      300 308
Mass and energy      294 298
Mass and metric of $F_{4}$       328
Mass and velocity      295
Matrices, functional      8—12
Matter, mean cosmic density of      439

Matter, total quantity of      427 428
Mattioli      350
Maximum and minimum      128
Maxwell      383
Maxwell’s theory      383
Mayer      22 25 106
Mayer’s method of integration      25
Mechanical equivalence, a theorem of      394
Mechanics in covariant equations      348 349
Mechanics of continuous systems      347 352
Mechanics with any co-ordinates      347
Mechanics, classical, correction to      291—294 320 392
Mechanics, generalized      320—324
Metric of $V_{4}$ and physical phenomena      374
Metric of space-time and energy tensor      383
Metric of, generalization of      320
Metric, angular      123
Metric, pseudo-Euclidean      299 360
Metrical elements of figure      100
Metrics in conformal representation      229
Metrics with spherical symmetry      408—414
Metrics, different, covariant derivatives for      222
Metrics, different, for same $V_{n}$       220
Metrics, different, Riemann’s symbols for      224
Metrics, relativity, qualities of      325
Metrics, relativity, statical      326
Metrics, relativity, stationary      326
Metrics, spatially uniform      425
Michelson      306 319 335
Michelson — Morley experiment      335
Minimum time, principle of      341 (see “Fermat’s Principle”)
Mixed system, or tensor      70 71
Mixed systems of total differential equations      29—33
Molecular action, system with no      360—363
Moments of co-ordinate lines      98
Moments of direction      92 120
Moments of direction, covariance of      92 120
Moments of direction, relation connecting      92 120
Momentum      295
Morera      22 25
Morera’s method of integration      22—25
Morley      306 319 335
Motion, Einsteinian, of planets      396
Multilinear form      66 69 83
Multiplication of tensors      76
Munari      316
Myller      106
Nebulae, density of      439
Newton      375 377 397
Newtonian equations      287 377
Newtonian field, assigned, space-time for      388—392
Newtonian motion, differences from Einsteinian      377
Newtonian potential      375
Newtonian potential and $ds^{2}$       336 369
Normal congruence      263 275 277 285
normal form of differential equations      36
Nuyens      439
Operator $\Delta$ , properties of      176
Operator, linear      33—37 48 84
Optics, geometrical      334
Orbit, equation of      397
Orthogonal directions, sets of      205
Padova      182
Palatini      146 408 439
Parallel displacement      103
Parallel displacement along a geodesic      103 104
Parallel displacement, angles unchanged by      103 114
Parallel displacement, cyclic      173 186
Parallel displacement, cyclic, of vector      192
Parallel displacement, cyclic, Peres’s formula for      193
Parallel, ambiental      171
Parallelism      102
Parallelism and curvature      193—198
Parallelism and infinitesimal displacement      104
Parallelism with respect to surface      102
Parallelism, Angle of      198
Parallelism, differential definition of      105
Parallelism, equations of      110—112
Parallelism, extension of notion of      137
Parallelism, intrinsic character of      106
Parallelism, intrinsic equations of      107
Parallelism, invariance of      110
Parallelism, symbolic equation of      107
Parallelogram rule for vectors      117
Parallels, kinematical construction of      102 104
Parameter of family of surfaces      45
Parameter, first differential      231 418
Parameter, second differential      154 393 418
Parameters and moments, relation of      92 125
Parameters of co-ordinate lines      98
Parameters of direction      91 120
Parameters of direction, contravariance of      91 120
Parameters of direction, relation connecting      91 120
Parameters, Lagrangian      288
Parametric equations of surface      86
Path of light, in gravitational field      402 403—408
Peres      172 193
Peres’s formula      193
Perihelion, displacement of      396 398
Perihelion, displacement of Mercury      399
Perihelion, displacement of other planets      400
Perihelion, displacement of, formula for      398
Permutability $(d\delta=\delta d)$       116
Perot      402
Persico      102
Perturbations, Newtonian      399
Pfaff      14
Pfaffian      13 20 26 161 174
Pfaffian as invariant      81 82
Pfaffian systems      14
Physical phenomena and metric of $V_{4}$       374
Planet, motion of      369
Planets, Einsteinian motion of      396
Planets, motion of, discrepancies in      322
Pluecker      68
poisson      35 53 56 375 376 377 386 387
Poisson’s equation and Einstein’s theory      386 387
Poisson’s equation for potential      375 377
Poisson’s parentheses      35 36
Postulates of general relativity      364
Potential and $ds^{2}$       336 369
Potential and metric of space      391
Potential, Newtonian      287 292 297 322 323 369 375 377 388 394 396 400 403
Potentials, 10 gravitational      375
Principe expedition      407
Product of tensors      76
Pseudo-Euclidean $ds^{2}$       325 376
Pseudo-Euclidean every metric locally      360
Pseudo-Euclidean metric      299 360 383
Pseudo-Euclidean metric and versors      329
Quadratic differential form, invariant      84
Quadratic form      66
Quadratic form with non-vanishing discriminant      90
Quadratic form, $ds^{2}$ , character of      120 (see also “ $ds^{2}$ ”)
Quadratic form, covariance of coefficients of      73
Quadratic form, definite      90
Quadratic form, invariant      73
Quadratic forms of class i      253
Quadratic forms of class zero      242
Quadratic forms, Euclidean      242
Quadratic forms, Riemann’s symbols for      242—246
Quadratic forms, theory of      205
Quadratic, canonical form of      205 281
Radioactivity      297
Radius of universe      439
Reciprocal elements in determinants      54 55 80 81
Reciprocal tensors      95
Refracting medium, space as      402
Refraction of light      334
Refractive index      334

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