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

$ds^{2}$ and gravitation      375
$ds^{2}$ for an Einsteinian space-time      392
$ds^{2}$ for assigned Newtonian field      388—392
$ds^{2}$ for single point mass      419—423
$ds^{2}$ in mechanics      293
$ds^{2}$ in mechanics, invariance of      294
$ds^{2}$ same for dynamics and light      336 337
$ds^{2}$ , approximately pseudo-Euclidean      383
$ds^{2}$ , coefficients of, and gravitational experiments      367
$ds^{2}$ , coefficients of, by experiment      363—368
$ds^{2}$ , covariance of coefficients of      90
$ds^{2}$ , discriminant of coefficients of      90
$ds^{2}$ , expression for      88
$ds^{2}$ , generalization of      320
$ds^{2}$ , indefinite      141
$ds^{2}$ , invariance of      308 311
$ds^{2}$ , qualitative properties of coefficients of      369
$ds^{2}$ , statical      326 327 371 377 378 392
$ds^{2}$ , statical, Einstein tensor in      380 381
$ds^{2}$ , statical, Riemann’s symbols in      379
$ds^{2}$ , ten functions involved in      322
$ds^{2}$ , vanishing, for light      332 338
$ds^{2}$ , vanishing, variational principle for      340
$G_{ik}$ (Einstein tensor) in statical $ds^{2}$       380
$G_{ik}$ (Einstein tensor), linear invariant G of      380 381
$\epsilon$ -systems      158
Action in mechanics      331
Action, stationary      324 331
Addition of tensors      75
Almansi      390
Amaldi      272 324 334 397 398
Angle between in $V_{n}$       123—126
Angle between two directions      92
Angular metric      123
Antisymmetrical systems      66
Area, element of      99
Associated tensors      95
Associated tensors with respect to quadric      96
Associated vectors      140
Atom, vibration of      400
Attracting mass, spherical      419—423
Attraction on planet      369
Autoparallelism of geodesic      104 140
BACH      439
Beltrami      238
Bending of light rays      402 403—408
Bianchi      90 137 139 140 182 183 205 209 212 235 257 286 373 429
Bianchi’s derived vectors      137—140
Bibliography      see “Preface”
Bibliography of solutions of Schwarzschild type      439
Bilinear covariant      18 20 21
Bilinear form      66 70
Binary form, curvature of      414—419
Birkhoff      439
Blascljke      208
Bompiani      105 198
Bromwich      205
c, meaning of constant      335
Campbell      408
Canonical form of quadratic      206 281
Canonical system as to given congruence      278
Carpanese      272
Cartan      14
Cartesian co-ordinates      61
Cartesian co-ordinates, locally      164 167 171 202
Central forces      397
Change of co-ordinates      61
Change variables      2 61
Change variables, general      80
Characteristic of envelope      101
Characteristic of matrix      9 39 87 250
Characteristic of surfaces      47
Chazy      439
Christoffel      109 110 111 112 118 121 133 139 140 165 166 171 172 182 186 218 220 221 222 228 229 284 376 378 379 384 413
Christoffel’s symbols of first kind      109 111
Christoffel’s symbols of second kind      110 111
Christoffel’s symbols, derivatives of $a_{ik}$ in terms of      111
Christoffel’s symbols, derivatives, determinant a in terms of      112
Christoffel’s symbols, vanishing of      121
Cisotti      362
Classical mechanics, correction to      392
Clifford      374
Co-ordinate hypersurfaces, angle between      128
Co-ordinate lines, angle between      128
Co-ordinate lines, moments of      98
Co-ordinate lines, parameters of      98
Co-ordinates, Cartesian      61
Co-ordinates, curvilinear      61 87
Co-ordinates, in space-time      290
Co-ordinates, locally Cartesian      164—171 202
Co-ordinates, locally geodesic      164 167 171
Co-ordinates, Pluecker’s      68
Coefficient $g_{00}$ , second approximation to      392—394
Coefficients of $ds^{2}$ by experiment      363—368
Coefficients of $ds^{2}$ , qualitative properties of      369
Coefficients of quadratic form, covariance of      73
Commutation of second derivatives      273
Commutation, rule      184
Complete system of partial differential equations      52 53
Complete system of total differential equations      15—18
Composition of tensors      79
Composition of velocities, in relativity      317
Compounded tensor      79
Conformal representation      229 246
Conformal representation of $V_{3}$       411
Conformal representation of Einsteinian in Euclidean space      423
Congruence, canonical system as to      278
Congruence, geodesic      262 274
Congruence, geodesic, curvature of      275
Congruence, normal      263 275 285
Congruence, of curves      46 47
Congruence, solenoidal geodesic      363
Congruences in Euclidean space      282
Congruences, set of normal      277
Congruences, sets of      265
Constant $\kappa$ , in Einstein’s equations      387
Constant, gravitational      386
Continuity, equation of      347 349—351 360 363
Continuity, equation of, molecular      361
Continuous system in covariant equations      348 349
Continuous system in terms of energy tensor      351
Continuous system, mechanics of      347 352
Continuous system, relativistic equations for      359
Contraction of bodies in motion      313
Contraction of tensors      77—79
Contravariance of the $a^{ik}$ ’s      92
Contravariance, m-fold system      69—71
Contravariance, simple system      67 81
Contravariance, transformation by      67 69—71
Contravariant differentiation      149
Corbellini      104
Cosmological interest, solutions of      426
Cosmological term, Einstein’s      438 439
Covariance, m-fold system      69—71
Covariance, simple system      67 68 82
Covariance, transformation by      64 67—71
Covariant, bilinear      18 20 21
Covariant, derivative      146
Covariant, derivative, second      184
Covariant, differentiation      144 149
Covariant, differentiation, of invariant      147
Covariant, differentiation, of vector      147
Covariant, simple system, typical      82
Covariant, simple systems, sets of      74
Cramer      53 54 68 in
Cramer’s Rule      54 55
Curl of vector      161
Curvature of a $V_{3}$       201
Curvature of calculated      413
Curvature of formula for      203

Curvature of space, in Einstein’s space-time      429 438
Curvature, calculated      413
Curvature, constant, canonical forms for manifolds of      238 240
Curvature, Gaussian      172
Curvature, Gaussian, of ordinary surface      193
Curvature, geodesic      135—137
Curvature, geodesic, of congruence      275
Curvature, hyperspherical      258
Curvature, hyperspherical, formula for      260
Curvature, lines of      286
Curvature, locally constant      235
Curvature, manifolds of      236 246
Curvature, mean, of $V_{3}$       372
Curvature, mean, of $V_{n}$       258
Curvature, mean, of physical space      382
Curvature, principal, in $V_{3}$       204
Curvature, Riemannian, of $V_{n}$       195
Curvature, space of constant      425—439
Curvatures, calculation of      414—419
Curvilinear co-ordinates      61 87
Cylindrical space-time, Einstein’s      429
Darboux      208
de Donder      439
de Sitter      429 435 436 437
De Sitter’s space-time      429—435
De Sitter’s space-time, constant negative curvature of      436
Dei      278
Density of energy      350 351
Density of momentum      350
Density of nebulae      439
Density, mean cosmic, of matter      426 439
Derivative of vector      139 140
Derivative, covariant      146
Derivatives of determinant a      112
Derivatives, transformation of      85
Determinant, a, derivatives of, in terms of Christoffel’s symbols      112
Determinant, functional      2 4—12
Determinant, reciprocal elements in      54 55 80 81
Developable, circumscribed      101
Developable, surfaces      100 ior
Differential equations, linear normal form of      36
Differential equations, linear partial      33—61
Differential equations, total      13—33
Differential parameter, first      231
Differential parameter, second      154 393
Differential parameters, in special case      418
Differential total      13 64
Differentials, linear transformation of      80
Differentiation, covariant      144 149
Direction      see “Versor”
Direction, spacelike      356 357
Direction, timelike      356 357
Directions of co-ordinate lines, in $V_{n}$       127
Directions of co-ordinate surfaces, in $V_{n}$       127
Directions, specification of      90
Discrete system, motion incompressible      361
Discrete system, motion of      360—363
Discriminant of $a_{ik}$ ’s      157
Displacement, cyclic      173 186
Displacement, infinitesimal      104
Displacement, parallel      103 (see also “Parallel displacement”)
Displacement, spectral      400
Divergence of double tensor      154
Divergence of gradient of vector      154
Divergence of stress tensor      344
Divergence of vector      153
Dougall      435
Dual variables      68 81
Dupin      204
Eclipse observations      407 408
Eddington      402 439
Einstein      200 287 291 294 298 317 318 322 326 328 371 372 374 375 376 377 378 383 385 396 398 407 429 435 436 437 438
Einsteinian and Newtonian, trajectories      395
Einsteinian motion, law of time in      396
Einsteinian space-time, $ds^{2}$ for an      392
Einstein’s cylindrical space-time      429
Einstein’s form of Hamilton’s principle      291
Einstein’s gravitational equations      376
Einstein’s tensor      200 371
Einstein’s tensor for a $V_{2}$       371
Einstein’s tensor in statical $ds^{2}$       380 381
Einstein’s tensor, divergence of      371
Einstein’s tensor, linear invariant of      371
Electrodynamics of bodies in motion      311 312
Element of area      99
Elongation, relative      305
empty space      392
Empty space, $ds^{2}$ for      382 383
Energy and mass      294 298
Energy and matter      298
Energy and metric of $F_{4}$       328
Energy density      350 351 356 382
Energy field of uniform      426
Energy tensor      355 358
Energy tensor and curvature      374
Energy tensor and electromagnetism      374
Energy tensor and equations of motion      359
Energy tensor and local phenomena      374
Energy tensor and metric of space-time      383
Energy tensor, physical interpretation of      359
Energy tensor, vanishing      382 392
Energy, flux of      350 351 356 357 358
Energy, intrinsic, of matter      297 298
Energy, kinetic and potential      296 324
Envelope of family of planes      101
Equation of orbit      397
Equations in terms of stress tensor      351
Equations of free particle      287
Equations of motion, Einsteinian modification of      351
Equations relativistic, for continuous system      359
Equations total differential      13—33
Equations total differential, integrals of      47
Equations, Einstein’s gravitational      376
Equipollence of vectors      103
Equivalence, mechanical, a theorem of      394
Ether      335
Euclidean manifold      121
Euclidean manifold and Riemann’s symbols      242—246
Euclidean manifold, Christoffel’s symbols in      121
Euclidean metric manifold, condition for      242—246
Euler      44 296 342 432
Experiment and coefficients of $ds^{2}$       363—368
Experiments, gravitational, and $ds^{2}$       367
Experiments, optical, and $ds^{2}$       363
Extension of a field      160
Extension of curved space      426 427 428
Fabry      402
Facet      201
Fermat      335 340 343 402
Fermat’s principle      335 402
Fermat’s principle and geodesies      341—343
Fermat’s principle in relativity      340 341
Fermi      167
Finzi      231
Fizeau      319
Focal directions in congruences      283
Force absorbed in stress system      349 374 375
Force disturbing, in planet’s motion      397
Force in relativity field      328
Force, inversely as cube of distance      397
Form, bilinear      66
Form, invariant      73
Form, linear      67
Form, multilinear      66 83
Form, quadratic      66
Forms of class i      253
Forms of conditions for      257
Frame of reference      335
Frequency of spectral line modified      400
Fresnel      316 318 319 320

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