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| Schutz B. — Geometrical Methods in Mathematical Physics |
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| Предметный указатель |
 1
Accelerated observer 71
Adiabatic 182 183
Adjoint transformation 107
Affine connection 76 201—222 218 221
Affine connection, linearity 204
Affine connection metric
Affine connection, torsion 207
Affine parameter 209 218
Analytic function 9 10
Analytic manifold 26
Angular momentum operators 85
Angular momentum operators as Killing vector fields
Annihilate 154
Annulling a form 120 153 166
Antiderivation 134
Antisymmetric tensor 115
Area 113
Area tensor
Atlas 25
Automorphism 107
Axial eigenvalue 90 91
Axial harmonics, scalar 90
Axial harmonics vector
Axial symmetry 87 89
Axial
Axial vector 126
Basis 14
Basis Cartesian
Basis transformation 60
Basis transformation matrix
Basis, dual 55
Basis for one-forms
Basis, transformation of 60
Basis-invariance 63
Betti number 152
Bianchi identities 212
big bang 199
Bijection 7
Boundary of a region 144
Canonical energy 173
Canonical momentum 173 221
Canonical transformation 168
Caratheodory's theorem 165
Cartan, E. 113
Chart 24
Christoffel symbols 205 218
Christoffel
Closed form 138
Closed form locally
Closed form on a sphere 150
Closed form sufficient
Closed ideal 163
Closed set of forms 158
Cofactor 18
Cohomology 139 142 150
Cohomology
Commutator 44
Commutator of covariant derivatives
Compatibility of connection and differential structure 204
Compatibility of connection and metric
components see "Individual types of tensor"
Composition of two maps 7
Configuration space 174
Congruence 43 151
Connection one-form 220
Connection, metric 216
Conservation law 89 149 158 163 171 173 182
Conservative 168
Conserved quantities and Killing vector fields 171
Continuity, function 7 8
Continuity group
Continuity, one-form 52
Continuity space
Continuum mechanics 58
Contraction 50 56 59
Contraction
Coordinate transformation 31 62 255
coordinates 23 24
Coordinates curvilinear
Copernican Principle 189
Cosmological principle 18 9
Cosmology 161 136—199
Cosmology big
Cosmology, closed 198 199
Cosmology expanding
Cosmology, isotropic 187 189
Cosmology open
Cosmology, standard model 199
Cotangent bundle 53 174 175
Covariant derivative 184 203
Covariant commutator
Covariant derivative, Jacobi identity for 212
Covariant derivative Leibniz
Cross-product 125 126
Cross-product triple
Cross-section 38 53 220
Curl 136 176
Curvature 68 214
Curve 30 153
Curve congruence
Curve, geodesic 208
Curve parameter
Density, scalar 129
Density tensor
Diffeomorphism 30 73 92 188
Differentiability class of a function 8
Differentiability class of a one-form
Differential form 68 117
Differential annulling
Differential form, field 120
Differential form independent
Differential form, sectioning 120
Differential forms, annihilate of 154
Differential forms, closed ideal 163
Differential forms closed
Dirac bra and ket 51
Dirac delta function 51
Direct product 59
Directional derivation 33 53
Directional derivative 33 53
Discrete set 2
Distance function 1 3—5
Distance on a manifold 68 70
Distribution (of vector fields) 83
Distribution (over functions) 51
Divergence 137 176 196
Divergence
Divergence theorem 147
Domain of an operator 11
Dual 186 196
Dual double
Duality of vectors and one-forms 50
Eigenvalue 19 65 96 99
Eigenvector 19
Einstein summation convention 56 62
Electromagnetism 175—181
Electromagnetism as a gauge theory
Electromagnetism, one-form potential 180 221 219
Electromagnetism, plane waves 181
Electromagnetism polarization
Electromagnetism, vector potential 180
entropy 163 166 167 182 183
Equation of continuity 149
Equation of state 163
Equivalence class 150
Equivalence relation 150
Ertel's theorem 185
Euclidean space 15 65 79 121 161 176 182 197 198 214 218
Euclidean vector algebra 68 125 131 132
Euclidean vector calculus 70 136 137 142 147 148 182
Exact form 138 163 170
Exact
expansion 186
Exponential map 167 210 215
Exponentiation of an operator 43
Exponentiation of covariant derivative
Exterior derivative 134
Exterior derivative commutes
Faraday tensor 176 221
Fiber bundle 35 36 40 53 183 220
Fiber base
Fiber bundle, fiber 36 40
Fiber global
Fiber bundle, globally trivial 38
Fiber bundle locally
Fiber bundle, structure group 40
Fixed-point theorem 39 151 196
Flat manifold 172 214
Fluid dynamics 149
Fluid, multicomponent 164
Fluid perfect
Fluid, single-component 163
Foliation 81 108 188
Foliation leaf
Frame bundle 42
Frobenius' Theorem 81 82 153—155 163 165 166
Function 5 30
Function space 51
Function, analytic 9 10
Function
Fundamental theorem of calculus 134
Galilean spacetime 182 183 218
Gauge curvature two-form 221
Gauge theories
Gauge-covariant derivative 220
Gauss' theorem 147 148
Geodesic curve 208
Geodesic curve affine
Geodesic deviation 213 214
Geodesic equation 208
Geodesically complete manifold 210
Geometrical object 62
Gl(n, c) 100
GL(n, R) 41 95
GL(n R) acting
Gradient 53
Gradient not
| Grassmann algebra 118 125
Group 11—13
Group abstract
Group, rotation 29 98 99
Group translation
Hamiltonian, equations 167
Hamiltonian equations function
handedness see "Basis"
harmonic oscillator 153 159
Hausdorff property 3
Helmholtz circulation theorem 185
Hermitian conjugate 100
Hilbert space 51 108
Homeomorphism 40
Homogeneous 187 188 191
Homomorphism 102 104
Hypersurface 79 167 178 182 183 188 see
Ideal, closed 163
Ideal complete
Identity transformation 17
Image 6
Index notation 73
Index raising and lowering 68
Indices, antisymmetrized 166
Indices placement
Infinitely differentiable 9
Inner automorphism 107
Inner product 15 51 71
Integrability conditions 138 155
Integration 134
Integration and orientation
Integration of functions 121
Integration, change of variables 9
Integration of forms
Into (map) 7
Invariance of a tensor field 86 see
Inverse function theorem 9 35 49
Inverse image 6
Inverse map 6
inversion 99
Isometry 188 190
Isospin 37
Isotropic 187 189
Jacobi identity for covariant derivatives 212
Jacobi identity for Lie derivatives
Jacobian 9 122 129 132
Jacobian matrix 9 35 63
Killing vector field 88 108 171 188 192 195 216
Killing nonexistence
Killing's equation 216
Klein — Gordon equation 174 219
Kronecker delta 17
Lagrangian function 167 174
Left-invariant vector field 93—95
Leibniz rule for covariant derivative 204
Leibniz rule for exterior derivative 134
Leibniz rule for Lie derivative
Levi-Civita symbol 128
Levi-Civita symbol and determinants
Lie algebra 47 92 93 95 97 101 155 170
Lie Abelian
Lie algebra of invariant vector fields 87
Lie algebra of isometry group
Lie bracket 45 75 157
Lie closure
Lie derivative 68 173 182 184
Lie
Lie derivative of a scalar 76
Lie derivative of a tensor
Lie dragging 73 90 209 213
Lie
Lie group 12 29 87 92 188
Lie abelian
Lie group, adjoint representation 189
Lie group and invariance
Lie subalgebra 192
linear combination 14
Linear independence 14
Linear transformation 16
Linear transformation components
Liouville's theorem 171
Lorentz frame 70 71
Lorentz group 67 192
Lorentz transformation 67 219
Manifold 23
Manifold analytic
Manifold, isotropic 189
Manifold maximally
Manifold, Riemannian 169 201—222
Manifold simply
Many-to-one 6
MAP 5
Map composition
Map, generated by a congruence 73
Map into
Map, inverse 6
Map many-to-one
Map, stereographic 27
Matrix 50
Matrix anti-Hermitian
Matrix, nonsingular 17
Matrix orthogonal
Maxwell identities 164 169
Maxwell's equations 175—181
Metric 36 38 51 54 76 113 169 178 184 201 215
Metric Euclidean
Metric connection 216
Metric dual 128 133
Metric tensor 64 187 214
Metric
Metric tensor field 68 108
Metric local
Metric volume element 132 148 160 193
Metric, Minkowski 66 71
Metric signature
Minkowski metric 66
Minkowski space 70 79 179 214 217 218 219 220
Moebius band 39
Moebius band not
Moebius band, structure group 42
Multilinearity 57
n-tuple 1 15 23
n-vector 125
Neighborhood 1
Neighborhood generalized
Newtonian gravity 187
Norm 14
Norm Euclidean
Normal coordinates 210
Normal one-form 147
O(N) 66 98
O(n)
O(n), dimension of 99
One-form 49
One-form as a tensor
One-form, components 55
One-form coordinate
One-form, dual basis 55
One-form field
One-form, picture of 53
One-parameter subgroup 94 95
One-parameter infinitesimal
One-to-one 6
Onto (map) 7
Open set in 2 3
Operator 10
Operator as a tensor
Operator, extension of 11
Operator multiplicative
Orientability 41 132 175
Orientability internal
Orientation, external 123
Orientation internal
Orthogonal group see "O(n)"
Outer product 59 64
p-delta symbol 130
p-delta symbol contraction
Parallel transport 202 204
Parallel
Parallelism 76 201
Parallelism global
Parallelogram rule 15
Partial differential equations 134 137 152
Partial integrability
Permutation group 12
Phase space 28 168 174
Phase volume
Poincare lemma 140
Poisson bracket 170
Principle of equivalence 218
Principle of mediocrity 189
Principle of minimal coupling 218
Product space 38
Proper distance and time 70
Pseudo-norm 15 71
quantum mechanics 51
Quantum mechanics commutation
Quotient space 150
R 2
Realization 106
Realization faithful
Relativity 38 188
Relativity general
Relativity, special 66 68 70 219
Representation 106
Representation abstract
Restriction of a form 120 153 188
Ricci scalar 217
Ricci tensor 217
Riemann tensor 211 213
Riemann number
Riemannian geometry 76 184 187
Riemannian manifold 169 201—222
Right-invariant vector field 94
Rotation group see "SO(3)"
Rule of linearity 16
scalar 64
Scalar covariant
Sectioning a form 120
Shear 186
Signature 66 68 69
Similarity transformation 19 65
Simply connected manifold 102
Simply connected manifold and cohomology
SO(2) 92
SO(3) 29 98 99 100 111 160 190 192;
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