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| Borisenko A.I., Tarapov I.E. — Vector and Tensor Analysis with Applications |
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| Предметный указатель |
Acceleration field of a moving fluid 165—166
Angular velocity 45
Antisymmetrization 107
Arc length 32 86
Archimedes’ law 208
Aris, R. 72 96 97 204 206
Axial vector 18 109
Barotropic flow 233
Basis 9
Basis vectors 9 85
Basis vectors, direct transformation of 13
Basis vectors, direct transformation of, coefficients of 13
Basis vectors, inverse transformation of 13
Basis vectors, inverse transformation of, coefficients of 13
Basis, left-handed 21
Basis, local 85
Basis, orthogonal 34
Basis, reciprocal 24
Basis, right-handed 21
Bernoulli’s law 234
Biharmonic equation 232
Biot — Savart law 214
Bipolar coordinates 101
Bound vector 3
Boyanovitch, D. 240
Buck, R. C 140
Cartesian tensors 61
Center of charge 48
Center of mass 54
Characteristic directions 109
Characteristic equation 114
Characteristic plane 119
Characteristic values 109
Characteristic vectors 109
Christoffel symbols 187—190
Christoffel symbols of the first kind 187
Christoffel symbols of the second kind 188
Circulation 136 235 236
Collinearity condition 43
Collision of particles 49—51
components 15
Components, contravariant 28
Components, covariant 28
Components, physical 30
Constitutive relations 228
Contravariant components 28
Contravariant components, relation of, to covariant components 31
Control surface 210
Coordinate curve 11 84
Coordinate curve, positive direction along 85
Coordinate surface 84
Coordinate system(s), nonorthogonal 13
Coordinate system(s), oblique 10
Coordinate system(s), orthogonal 11 86
Coordinate system(s), rectangular 10ff
Coordinate system(s), rectangular, transformation of 38—39
coordinates 11
Coordinates, curvilinear 11 82—88
Coordinates, cylindrical 82
Coordinates, generalized 82
Coordinates, generalized, basis of 85
Coordinates, orthogonal 11 86
Coordinates, polar 11 12
Coordinates, polar, generalized 12
Coordinates, spherical 83
Covariant components 28
Covariant components, relation of, to contravariant components 31
Covariant derivative 186
Covariant differentiation 185—196
Covariant differentiation of tensors 190—191
Covariant differentiation of vectors 185—187
Cross product see “Vector product”
Curl 161—164
Current density 226
Curvature 175
Curvilinear coordinates 11 86
Cylindrical coordinates 82
Deformation tensor 2 70—72
Del operator see “Operator  ”
Deviators 122
Dielectric tensor 100
Differential operators 168ff
Differential operators in generalized coordinates 192—196
Differential operators in orthogonal curvilinear coordinates 171—174
Dipole moment 47
Directional derivative of a scalar field 147—148
Directional derivative of a vector field 164—165
Dirichlet problem 222
Dissipation function 127
Divergence 155—161 193—194
Divergence of the velocity field of a fluid 157—158
Divergence theorem 157
Dot product see “Scalar product”
Dummy indices 34 91
Dyad 93
d’Alembertian operator 230
Efflux 154
Eigenvalues 109
Eigenvectors 109
Electric displacement 110 228
electric field 19 110 226
Electromagnetic waves 244
Electromagnetic waves, plane monochromatic 245
Electromotive force 226
Equation of continuity 159
Eulerian angles 46
Faraday’s law of induction 226
Finite rotations 4—5
Fluid contour 240
Fluid dynamics 203—211
Fluid motion, equations of 203—208
Flux density 209
Free energy 133
Free vector 2
Frenet — Serret formulas 176
Fundamental theorem of vector analysis 223
Gauss’ Theorem 137—139
Gradient 92 147 150—151 192—193
Green’s formulas 201
Green’s Theorem 139—140
Harmonic functions 219
Harmonic functions, properties of 220—222
Helmholtz’s theorem 249
Higher-order tensors 76—77
Higher-order tensors in generalized coordinates 90
Hodograph 35
Homogeneous form of degree 3 76
Homogeneous linear form 76
Homogeneous quadratic form 76
Ideal fluid 207
Infinitesimal rotations 5 44—45
Influx 154
Inhomogeneous wave equation 230
Integral theorems 196—203
Integral theorems, related to Gauss’ theorem 197—198
Integral theorems, related to Stokes’ theorem 198—200
Invariance of tensor equations 81—82
Invariants of a tensor 121
Isotropic tensor 96
Kibel, I. A. 240
Kochin, N. E. 240
Kronecker delta 39
Kutta — Joukowski theorem 235—236
Laplace’s equation 219
Laplacian (operator) 169 194—196
Laplacian field 219
Left-handed basis 21
Level surface 146
Line integral 136
Lorentz condition 230
| Magnetic field 19 226
Magnetic induction 228
magnetization 228
Magnetomotive force 227
Marion, J. B. 231
Matrix 64
Maxwell’s equations 226
Metric coefficients 34 87
Metric tensor 32 86
Metric tensor, verification of tensor character of 90
Moment of a force 19 56
Moment of inertia tensor 68—70 110—113
Momentum flux density 209
Momentum theorem 209
Motion under gravitational attraction 53—54
Moving trihedral 175
Multiple-valued potential 213
Multiply connected region 145
Nabla operator see “Operator ”
Navier — Stokes equation 207
Neumann problem 222
Neutral charge system 48
Newton’s Second Law 82
Newton’s second law, invariance of 82
Nutation 46
Ohm’s Law 228
Operator 149 168ff
Orthogonal bases 34
Orthogonal coordinates 11 86
Orthogonal transformation 122
Orthogonal transformation, improper 123
Orthogonal transformation, proper 123
Orthogonality conditions 39
Orthonormality conditions 15
Osculating plane 175
Parallel displacement condition 187
Physical components 30
Poisson’s equation 224
Polar axis 11
Polar vector 18
Polarization 228
Pole 11
Potential 211
Potential field 211
Potential, multiple-valued 213
Poynting’s vector 232
Precession 46
Principal directions 109
Pseudotensor(s) 109 122—126
Pseudotensor(s), components of 124
Pseudotensor(s), unit 126
Quadric surface 64
Radius of curvature 175
Radius of torsion 176
Radok, J. R. M. 240
Rate of deformation tensor 72—75
Reciprocal bases 24
Retarded potentials 244
Ricci’s theorem 191
Right-handed basis 21
Roze, N. V. 240
Scalar fields 145 ff.
Scalar potential 211 228
Scalar product 14
Scalar product in variance of 79
Scalar product, commutativity of 14
Scalar product, distributivity of 15
Scalar triple product 20
Second-order tensors 63—75
Second-order tensors, components of 64
Second-order tensors, transformation law of 64
Shilov, G. E. 123 133
Silverman, R. A. 123
Simply connected region 144
Singular point 152
Sink 154 155 160
Sink, strength of 155 160
Sliding vector 3
Solenoidal field 216
Source 154 155 160
Source, strength of 155 160
Spherical coordinates 83
Spherical trigonometry 41—42
Stokes’ Theorem 137 141—144
Strain tensor see “Deformation tensor”
Stream function 217
streamline 151 217
Stress tensor 1 66—68 110 167
Stress tensor in generalized coordinates 92
Stress tensor, viscous 96
Summation convention 26—27
Symmetrization 107
Tensor analysis 134ff
Tensor ellipsoid 118—120
Tensor field 135 166
Tensor field, continuous 135
Tensor field, flux of 166
Tensor field, homogeneous 135
Tensor field, nonstationary 135
Tensor function of a scalar function 134
Tensor function of a scalar function, derivative of 134
Tensor(s) in generalized coordinate systems 88
Tensor(s), addition of 103
Tensor(s), antisymmetric 106
Tensor(s), antisymmetric part of 107
Tensor(s), antisymmetric, equivalence of, to axial vector 107—109
Tensor(s), Cartesian 61
Tensor(s), characteristic directions of 109
Tensor(s), characteristic equation of 114
Tensor(s), characteristic values of 109
Tensor(s), characteristic vectors of 109
Tensor(s), contraction of 104
Tensor(s), contravariant 91
Tensor(s), contravariant components of 89
Tensor(s), covariant 91
Tensor(s), covariant components of 89
Tensor(s), covariant differentiation of 190—191
Tensor(s), deformation 70—72
Tensor(s), eigenvalues of 109
Tensor(s), eigenvectors of 109
Tensor(s), equations 81
Tensor(s), equations, invariance of 81—82
Tensor(s), first-order 61—63
Tensor(s), invariants of 121
Tensor(s), isotropic 96
Tensor(s), mixed 91
Tensor(s), mixed components of 89
Tensor(s), moment of inertia 68—70
Tensor(s), moment of inertia in two dimensions 110—113
Tensor(s), multiplication of 104
Tensor(s), of order n 59 76
Tensor(s), of order n, components of 59 76
Tensor(s), of order n, components of, transformation law of 76
Tensor(s), physical components of 91
Tensor(s), principal directions of 109
Tensor(s), product of 104
Tensor(s), product of, inner 105
Tensor(s), product of, outer 104
Tensor(s), rate of deformation 72—75
Tensor(s), reduction of, to principal axes 109—120
Tensor(s), second-order 63—75
Tensor(s), stress 66—68 110
Tensor(s), symmetric 105
Tensor(s), symmetric part of 107
Tensor(s), transformation of, under rotation about a coordinate axis 77—81
Tensor(s), two-dimensional 79
Tensor(s), unit 70
Tensor(s), zeroth-order 60—61
Thomson’s theorem 241
Torsion 176
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