- Acceleration of curve, 205, 283, 497
- Adjoint of linear map, 65
- Alternating map, 136
- Annihilator of linear map, 20
- Area of regular surface in
, 551
- Atlas, 230, 338, 450
- Ball, 193
- Basis for topological space, 185
- Basis for vector space, 7
- ordered, 9
- orthogonal, 62
- orthonormal, 62
- Bijective map, 613
- Bilinear function, 45
- negative definite, 46
- nondegenerate, 46
- Boost, 591
- Boundary chart, 450
- Boundary of set in topological space, 184
- Boundary of smooth manifold with boundary, 451
- Boundary point, 450
- Bounded function, 192
- Bounded set in Euclidean space, 197
- Bump function, 342
- Cardinality of set, 614
- Cartan's identity, 424
- Cauchy–Binet identity, 41
- classical, 42
- Cauchy–Schwarz inequality, 51, 77
- of basis, 31, 48, 100, 148
- of coordinate basis, 239, 349
- of coordinate frame, 372, 384
- of dual coordinate basis, 355
- of dual coordinate frame, 378
- Change of basis matrix, 31
- Change of variables theorem, 220
- Characteristic polynomial
- of linear map, 82
- of matrix, 81
- Christoffel symbol, 269
- Christoffel symbols, 467
- Classical differential operators, 221
- Classical divergence theorem, 586
- Classical Green's theorem, 584
- Classical line integral, 583
- Classical Stokes's theorem, 585
- Classical surface integral, 583
- Closed cell, 218
- Closed set in topological space, 183
- Closed upper half‐space, 449
- Closure of set in topological space, 184
- Codifferential, 562
- Codomain, 612
- Cofactor of matrix, 38
- Commutativity of
- Lie derivative and exterior derivative, 420, 425
- pullback and exterior derivative, 394, 408
- Compact set in topological space, 191
- Compact support
- of differential form, 432
- of function, 192
- Compatible connection, 519
- Component
- of covector field, 377
- of differential form, 388
- of Riemann curvature tensor, 539
- of tensor field, 384
- of vector, 27
- of vector field, 209, 270, 284, 371
- Cone‐shaped, 17
- Connected component, 189
- Connected set in topological space, 189
- Connected topological space, 189
- Connection, 463
- compatible, 519
- Consistent atlas, 288
- Consistent bases, 158
- Consistent charts, 426
- Consistent coordinate bases, 426
- Consistent coordinate frames, 288
- Consistent multicovectors, 160
- Consistent orientation forms, 428
- Consistent smooth atlas, 426
- Constant speed curve, 519
- Content, 218, 220
- Continuity equation, 598
- Continuous map, 186, 189
- Contraction of tensor, 106, 107
- Contraction of tensor field, 405, 406
- Coordinate covector field, 377
- Coordinate domain, 230, 337
- Coordinate frame, 238, 371
- consistent, 288
- Coordinate map, 230, 337
- Coordinate unit normal vector field, 263
- Coordinate vector field, 371
- Countable set, 614
- Covariant derivative, 275, 464
- Covariant derivative on curve, 282, 473
- Covector, 17
- Covector field, 376
- components of, 377
- local coordinate expression of, 377
- Covector field characterization theorem, 381
- Cross product, see vector product
- Curl, 221, 572
- Curvature tensor, 488
- Curve, 205, 237, 358
- acceleration of, 205, 283, 497
- d'Alembert operator, 576
- Derivation, 373
- Derivative, see covariant derivative, see covariant derivative on curve
- second order Euclidean, 213
- Diffeomorphism invariance
- of boundary, 452
- of integral, 433
- of length, 256, 519
- of line integral, 436
- Difference of sets, 611
- Differentiable, 199
- Differential, 351
- local coordinate expression of, 388
- path‐independent, 437
- pullback of, 402
- semi‐Riemannian manifolds, 561–578
- Differentiating under integral sign, 220
- Dimension
- of topological manifold, 338
- of vector space, 8
- Direct sum, 15
- Disconnected set in topological space, 189
- Disconnected topological space, 189
- Disjoint sets, 611
- Disk, 197
- Distance function, 193
- for semi‐Riemannian manifolds, 571
- Domain, 612
- Domain of integration, 219
- Double dual, 17
- Dual basis, 18
- Dual coordinate basis, 354
- Dual coordinate frame, 377
- Dual frame, 377
- Dual scalar product space, 72
- Dual vector space, 17
- Eigenvalue, 82
- Eigenvector, 82
- Einstein summation convention, 6
- Einstein tensor, 603
- Einstein field equations, 607
- Electromagnetic field, 598
- Empty set, 611
- Equality of mixed partial derivatives, 202
- Equivalence class, 614
- Equivalence relation, 614
- Euclidean derivative, 209
- Euclidean inner product, 76
- Euclidean norm, 76
- Euclidean space, 76
- Euler's rotation theorem, 83
- Existence of
- coordinate basis, 347, 354
- covariant derivative on a curve, 472
- maximal geodesic, 498
- maximal integral curve, 443
- orthonormal frame, 525
- outward‐pointing vector field, 456, 554
- parallel frame on a curve, 484
- parallel vector field on a curve, 479
- volume form, 550
- volume multicovector, 164
- Exponential map, 503
- Extended smooth, 207
- Exterior derivative, 378, 407
- Exterior of set in topological space, 184
- Extreme value theorem, 192
- Field, 623
- Finite‐dimensional vector space, 8
- First Bianchi identity, 308, 491, 540
- First fundamental form, 257
- Flat map, 51, 114, 526, 528
- Form, see differential form
- Form derivation, 417
- Frame on curve, 370
- Frame on open set, 368
- smooth, 202, 340
- strictly decreasing, 614
- strictly increasing, 614
- strictly negative, 614
- strictly positive, 614
- Fundamental theorem of semi‐Riemannian manifolds, 522
- Future‐directed curve, 519
- Future‐directed vector, 93, 94
- Gauge transformation, 599
- Gauss curvature, 297
- Gauss map, 289
- Gauss's equation, 307
- General linear group, 43
- Geodesic, 497
- length maximizing in
, 549
- length minimizing in
, 547
- length minimizing on Riemannian manifolds, 579
- Geodesic equations, 498
- Geodesic variation, 510
- Gradient, 221, 244, 262, 566
- Gram–Schmidt process, 79
- Graph of function, 241
- Great circle, 483
- Group, 617
- Group isomorphism, 618
- Half‐space, see closed upper half‐space
- Height function, 242
- Hesse operator, 573
- Hodge star, 178, 561
- Homeomorphism, 189
- Homeomorphism invariance of dimension, 197
- Hyperbolic space, 264, 331
- Hyperquadrics, 264
- Hypersurface, 361
- of
with xy‐plane in 
, 226, 321
- of
with 
, 386
- of
with Der(M), 374
- of
with 
, 
, 381
- of
with 
, 37
- of
with 
, 452
- of
with 
, 504
- of
with 
, 357
- of
with 
, 363
- of
with V, 347, 352
- of
with 
, 354
- of
with V, 19
- of metric, 260, 515, 518
- of regular surface in
, 260
- of scalar product space, 63
- of semi‐Riemannian manifold, 515
- of semi‐Riemannian submanifold, 518
- Induced connection, 559
- Induced covariant derivative, 559
- Induced Euclidean derivative, 266
- Injective map, 613
- Inner product, 75
- Inner product space, 75
- Integrable function, 219
- Integral
- Integral curve, 443
- Interior chart, 450
- Interior multiplication, 148, 423
- Interior of set in topological space, 184
- Interior point, 450
- Intermediate value theorem, 190
- Intersection of sets, 611
- Intrinsic, 270
- Inverse map theorem, 208, 357
- Inward‐pointing vector, 454
- Inward‐pointing vector field, 455
- Isometry, 535
- Isomorphism, see group isomorphism, linear isomorphism, module isomorphism
- Iterated integral, 220
- Jacobi field, 509
- Jacobi's equation, 509
- Jacobi's identity, 214, 287, 375, 491, 626
- Jacobian determinant, 204
- Jacobian matrix, 203, 403
- Joins, 358
- Kernel, 11, 624
- Koszul's formula, 523
- Kronecker's delta, 614
- Laplace operator, 221, 575
- Laplace–de Rham operator, 576
- Latitude curve, 243, 364
- Length of curve, 256, 519
- Level set of function, 244
- Levi‐Civita connection, 522
- Lie algebra, 626
- Lie bracket, 213, 285, 375
- Lie derivative, 419
- Light cone, 85
- Light set, 49
- Lightlike curve, 256, 519
- Lightlike subspace, 85
- Lightlike vector, 47
- Line integral, 435
- Linear combination, 6
- Linear isometry, 68
- Linear isomorphism, 10, 626
- Linear map, 10
- adjoint of, 65
- annihilator of, 20
- rank of, 11
- representation of, 28, 29
- Linearly dependent, 7
- Linearly independent, 6
- Local coordinate expression
- of covector field, 377
- of differential form, 388
- of tensor field, 384
- of vector field, 271, 284, 371
- Local coordinates, 337
- Local isometry, 535
- Longitude curve, 243, 364
- Lorentz group, 587
- Lorentz hypersurface, 518
- Lorentz manifold, 516
- Lorentz scalar product, 84
- Lorentz submanifold, 518
- Lorentz vector space, 84
- summary of properties of, 90
- Lorenz gauge, 599
- Lowering an index, 53
- Manifold version, 383
- Map, 612
- onto, 613
- orientation‐preserving, 161, 432
- orientation‐reversing, 161
- change of basis, 31
- cofactor of, 38
- column, 24
- column rank of, 32
- identity, 24
- inverse of, 25
- invertible, 25
- rank of, 33
- row, 23
- row rank of, 32
- square, 23
- submatrix of, 25
- symmetric, 24
- trace of, 27
- transpose of, 23
- special orthogonal group, 43
- Maximal flow, 444
- Maximal geodesic, 498
- Maxwell's equations, 598
- Mean value theorem for integrals, 221
- Metric, 258, 515
- Metric contraction, 127, 533
- Metric space, 193
- Minkowski scalar product, 85
- Minkowski space, 85
- Mobius band, 291
- Module, 624
- Module homomorphism, 624
- Module isomorphism, 624
- Multi‐index notation, 615
- Multicovector scalar product space, 153
- Musical isomorphisms, 75
- mv, see manifold version
- Neighborhood, 183
- normal, 505
- Nonorthochronous, 94
- Nonvanishing, 209, 252, 376, 614
- Norm, 57, 76, 195
- Normal coordinates, 507
- Normal neighborhood, 505
- Normal vector, 258, 518
- Normal vector field, 258, 518
- Normed vector space, 195
- Nowhere‐vanishing, 209, 252, 376, 614
- Nullity, 11
- Open set in topological space, 183
- Open submanifold, 363
- Ordinary contraction, 108, 128, 533
- Orientability criterion
- for hypersurfaces, 552
- for smooth manifolds, 430
- Orientable regular surface in
, 288
- Orientable smooth manifold, 426
- Orientation
- Orientation multicovector, 159
- Orientation of boundary
- Orientation of Euclidean space, 155
- clockwise, 155
- counterclockwise, 155
- Orientation of hypersurface, 551–556
- induced by vector field, 431
- Orientation of regular surface in
- opposite, 289
- Orientation of smooth manifold, 426
- induced by atlas, 426
- induced by orientation form, 428
- Orientation of vector space, 158
- induced by orientation multicovector, 160
- induced by vector, 161
- Orientation‐preserving map, 161, 432
- Orientation‐reversing map, 161
- Oriented regular surface in
, 288
- Oriented smooth manifold, 426
- Oriented vector space, 158
- Orthochronous, 94
- Orthogonal basis, 62
- Orthogonal complement, 59
- Orthogonal group, 43
- Orthogonal projection, 60
- Orthogonal transformation, see linear isometry
- Orthogonal tuple, 62
- Orthogonal vector fields, 517
- Orthogonal vectors, 48
- Orthonormal basis, 62
- Orthonormal frame, 525
- Orthonormal tuple, 62
- Outward‐pointing vector, 454
- Outward‐pointing vector field, 455
- Overlapping charts, 230, 337
- Overlapping sets, 611
- Parallel metric, 519
- Parallel tensor field, 476
- Parallel translation, 480
- Parallel vector field, 479
- Parametrized surface, 226, 364
- partial covariant derivative on, 476
- Partial covariant derivative, 476
- Partial derivative, 200, 201, 345, 350
- Partition of set, 614
- Partition of unity, 344
- Past‐directed curve, 519
- Past‐directed vector, 93, 94
- Permutation, 618
- Perp of subspace, 49
- Poincaré half‐plane, 516
- Poincaré's lemma, 440
- Pointwise orientation, 426
- Polar coordinates, 239
- Polarization identity, 49
- Positively‐oriented chart, 426
- Positively‐oriented frame, 525
- Positively‐oriented vector space, 158
- Product manifold, 340
- Product of sets, 611
- Product of vector subspaces, 17
- Projection map, 187
- Proper linear isometry, 589
- Proper orthochronous Lorentz group, 590
- Pseudosphere, 264, 330
- Pullback
- of covariant tensor, 103
- of covariant tensor field, 399
- of covector, 19
- of covector field, 394
- of differential form, 402
- of function, 390
- of multicovector, 145
- of vector field, 391
- Punctured plane, 397
- Pushforward
- of function, 389
- of vector field, 391
- Quadratic form, 45
- Quadratic function, 48
- Radial geodesic, 503
- Radius function, 242
- Raising an index, 75
- Rank
- of contraction, 108
- of linear map, 11
- of matrix, 33
- of matrix, column, 32
- of matrix, row, 32
- of tensor, 97
- Rank–nullity theorem, 11
- Regular surface, 230
- Regular surface in
, 260
- orientable, 288
- oriented, 288
- oriented by atlas, 289
- of covariant tensor field, 531
- of covector field, 380
- of linear map, 28, 29
- of Riemann curvature tensor, 532, 539
- of tensor, 104, 124
- of tensor field, 385
- of torsion tensor, 532
- of vector, 27, 28
- of vector field, 373
- Representation map, 104, 530
- Reversed Cauchy–Schwarz inequality, 91
- Reversed triangle inequality, 96
- Ricci curvature tensor, 603
- Riemann curvature tensor, 299, 305, 539
- Riemann's theorem, 545
- Riemannian hypersurface, 518
- Riemannian manifold, 516
- Riemannian submanifold, 518
- Ring, 623
- Rotation, 590
- Scalar curvature, 603
- Scalar product, 57
- Scalar product map, 124, 529
- Scalar product space, 57
- signature of, 63
- Second Bianchi identity, 492, 540
- Second fundamental form, 294
- Second order Euclidean derivative, 213
- Self‐adjoint linear map, 66
- Semi‐Euclidean space, 58
- Semi‐Riemannian manifold, 515
- Semi‐Riemannian submanifold, 518
- of hypersurface, 519
- of permutation, 620
- of real number, 614
- of regular surface in
, 263
- of metric, 515, 518
- of scalar product space, 63
- of semi‐Riemannian manifold, 515
- of semi‐Riemannian submanifold, 518
- Smooth covector field, 376
- Smooth curve, 205, 358
- Smooth extension
- of covector, 378
- of vector, 368
- orientable, 426
- orientation of, 426
- oriented, 426
- Smooth manifold with a connection, 463
- Smooth manifold with boundary, 451
- Smooth map, 205, 341
- Smooth reparametrization, 225, 519
- Smooth structure, 339
- standard, 340
- Smoothly compatible, 339
- Spacelike curve, 256, 519
- Spacelike subspace, 85
- Spacelike vector, 47
- Span, 6
- Special orthogonal group, 43
- Speed of curve, 519
- Sphere, 230
- Standard basis, 9, 10, 33
- Standard coordinate basis, 356
- Standard coordinate frame, 371, 451
- Standard coordinates, 339, 451
- Standard dual coordinate basis, 356
- Standard dual coordinate frame, 378, 451
- Standard orientation, 158, 451
- Standard smooth structure, 340
- Star‐shaped, 17
- Stereographic projection, 235
- Stokes orientation, 456
- Stokes's theorem, 459
- Submanifold with boundary, 452
- Subspace, see vector subspace
- Sum of vector subspaces, 14
- Support of differential form, 432
- Surface integral, 583
- Surface of revolution, 242
- Surjective map, 613
- Sylvester's law of inertia, 60
- Symmetric group, 618
- Symmetric tensor, 134
- Symmetric torsion tensor, 485
- Symmetries of tensors, 129
- Tangent plane, 237
- Tangent space, 345
- Tangent vector, 237, 345
- Tensor, 97
- components of, 99
- contraction of, 106, 107
- contravariant, 97
- covariant, 97
- product, 98
- pullback of covariant, 103
- Tensor derivation, 413
- Tensor field, 382
- components of, 384
- contravariant, 382
- covariant, 382
- local coordinate expression of, 384
- parallel, 476
- pullback of, 399
- rank of, 382
- representation of, 385
- Tensor field characterization theorem, 386
- Theorema Egregium, 308
- Time cone, 91
- Timelike curve, 256, 519
- Timelike subspace, 85
- Timelike vector, 47
- Topological manifold, 338
- Topological manifold with boundary, 450
- Topological subspace, 186
- Topology, 183
- Total covariant derivative, 476
- Total covariant differential, see total covariant derivative
- Trace
- of linear map, 43
- of matrix, 27
- Unit normal vector, 258, 518
- Unit normal vector field, 258, 518
- Unit vector, 48
- Unit vector field, 258, 517
- Upper half‐space, see closed upper half‐space
- orthogonal, 48
- outward‐pointing, 454
- representation of, 27, 28
- along curve, 282
- along regular surface, 252
- along submanifold, 370
- local coordinate expression of, 271, 284, 371
- on parametrized surface, 370
- on regular surface, 252
- on smooth manifold, 367
- orthogonal, 517
- outward‐pointing, 455
- parallel, 479
- pullback of, 391
- pushforward of, 391
- Vector product, 166
- Vector space, 5, 625
- dimension of, 8
- double dual of, 17
- dual of, 17
- orientation of, 158
- oriented, 158
- perp of, 49
- product of, 17
- Velocity of curve, 205, 359, 369
- Volume form, 427, 550
- Volume multicovector, 164
- Volume of semi‐Riemannian manifold, 551
- Zero function, 614
- Zero matrix, 24
- Zero multicovector, 134
- Zero tensor, 97
- Zero vector, 5, 345
- Zero vector space, 5
- Zero vector subspace, 5
..................Content has been hidden....................
You can't read the all page of ebook, please click
here login for view all page.
