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| Efimov A.V. — Mathematical analysis: advanced topics. Part 2. Application of some methods of mathematical and functional analysis |
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| Предметный указатель |
Abelian group 153
Accessory conditions 227
action 257
Affine transformation 125
Alternance, Chebyshev's 298
Alternance, Valle-Poussin's 298
Application of contraction mapping principle to solution of equations 140
Application of Galerkin's method directly to solving boundary-value problems 270
Application of Hilbert-Schmidt theorem 190—202
Application of interpolation to problems of numerical differentiation and integration 304—338
Application of Ostrogradsky — Gauss formula to a special kind of the vector field 86
Application of Schauder's theorem 163—172
Application of Tailor's formula to the difference of equations 228
Arzela's theorem 148
Axiom of the metric 122
Banach — Steinhaus theorem 307
Bernstein's theorem 298
Bolzano — Weierstrass theorem 145
Brouwer's theorem 164
Cauchy — Buniakowski inequality 117
Cauchy's sequence 133
Chebyshev polynomials of the first kind 301
Chebyshev's theorem 29
Circulation of vector field 51
Compact sets 144—152
Completely ordered set 168
Computation of curl in Cartesian coordinates 54
Computational error 285
Computing the norm of a self-adjoint operator 183
Condition(s), accessory 227
Condition(s), field potentiality 67—71
Condition(s), isoperimetric 204
Condition(s), natural boundary 247—248
Condition(s), transversality 249
Construction problems 290
Continuity equation 104
Continuous operator 131 156
contour 11
Contraction mapping principle 139
Convergence in norm 154
Coordinate, axes 15
Coordinate, basis 15
Coordinate, line 15
Coordinate, surface 15
Coordinate, system 15
Cubic spline 317
Curl 53
Curl, computation of, in an orthogonal curvilinear coordinate system 58
Curl, computation of, in Cartesian coordinates 54
Curve(s), orientation of 10
Curve(s), orientation of, negative 10
Curve(s), orientation of, positive 10
Curve(s), oriented 10
Curve(s), piecewise smooth 11
Curve(s), smooth 9
Difference scheme(s), explicit 364
Difference scheme(s), implicit 364
Difference scheme(s), stable 363
Differential operations of the second order 84
Direct methods in the calculus of variations 263
Directional derivative 21
Distance between the elements 122
Divergence 45—51
Divergence, computing of, in Cartesian coordinates 46—47
Divergence, computing of, in orthogonal curvilinear coordinates 50—51
Divergence, notion of 45
Divergence, of a field at a point 46
Divergence, properties of 47—48
Divided difference(s), first-order 311
Divided difference(s), n-order 311
Divided difference(s), second-order 311
Domain, multiply connected 59
Domain, simple 32
Domain, simply connected 59
Domain, superficially simply connected 60
Element of best approximation 289
Equation(s) of mathematical physics 104—110
Equation(s) of string vibrations 259
Equation(s), continuity 104
Equation(s), Euler's 215
Equation(s), Euler's, particular cases of 217—222
Equation(s), Fredholm symmetric integral 194
Equation(s), Fredholm's integral 174
Equation(s), heat 108
Equation(s), Laplace's 88
Equation(s), membrane motion 260
Equation(s), Poisson's 94 254
Error(s) of the method 285
Error(s), absolute 283
Error(s), computational 285
Error(s), due to approximate calculations 283—288
Error(s), due to arithmetic operations 285
Error(s), irreducible 283
Error(s), relative 283
Exact methods 339
Expanding the values of an operator into a series 190
Extremal(s) 216 252
Extremum, necessary condition for 212
Extremum, strong 212
Extremum, weak 212
Field(s) 19
Field(s) of a point source 79
Field(s) of sources and sinks 79
Field(s), central-symmetric 48
Field(s), nonstationary 20
Field(s), scalar 19
Field(s), scalar, examples of 19
Field(s), stationary 20
Field(s), vector 19
Field(s), vector lines of 79
Flux of a vector field 36—45
Flux of a velocity vector 39
Formula(s) for numerical differentiation 318
Formula(s) in vector form 48—50
Formula(s), complicated quadrature 325
Formula(s), cubature 333
Formula(s), Gaussian 330
Formula(s), Green's 32 86
Formula(s), Green's, first 86
Formula(s), Green's, second 86
Formula(s), Green's, third 87
Formula(s), interpolation quadrature 324
Formula(s), Ostrogradsky — Gauss 41—45
Formula(s), quadrature 323
Formula(s), rectangular 324
Formula(s), Simpson's 324
Formula(s), Stokes' 59
Formula(s), trapezoidal 324
Function(s) of a point source 98
Function(s), Green's, of the Dirichlet problem 97
Function(s), harmonic 87—88
Function(s), harmonic, properties of 91
Function(s), integral representation of 88
Function(s), net 362
Functional(s), dependent on higher-order derivatives 224—227
Functional(s), extremum of 208 211
Functional(s), extremum of, strong 212
Functional(s), extremum of, weak 212
Functional(s), increment of 208
Functional(s), linear 207
Functional(s), maximum of 211
Functional(s), minimum of 211
Functional(s), natural boundary conditions for 247
Functional(s), variation of 208
Fundamentals of the theory of function approximation 288—293
Geodesies 231
Gradient in a scalar field 22
Gradient in orthogonal curvilinear coordinate system 23
Hamiltonian operator 55
Harmonic 88
| Hausdorff's criterion 146
Hausdorff's theorem 147
Heat equation 108
Hilbert — Schmidt theorem 191
Hodograph 9
Hoelder's inequality 116
Inequality(ies), auxiliary 114
Inequality(ies), Cauchy — Buniakowski 117
Inequality(ies), Cauchy's 118
Inequality(ies), Hoelder's 116
Inequality(ies), Minkowski's 118
Inequality(ies), Schwarz's 117
Jackson's theorem 298
Jordan measure 128
Kernel(s), Fredholm's iterated 172
Lagrangian multipliers 228
Lame's coefficients 17
Laplace's equation 88
Laplace's equation, vector field 84
Laplacian (operator) 85
Level lines (surfaces) 20
Limit of a sequence 122
Line integral(s) of the first kind 28
Line integral(s) of the second kind 28
Line(s) level 21
Line(s) of force 25
Line(s) vector 25
Linear manifold 155
Local bases 15
Matrix(ces), banded 342
Matrix(ces), characteristic equation of 346
Matrix(ces), poorly conditioned 340
Matrix(ces), stable 340
Matrix(ces), symmetric 345
Matrix(ces), unstable 340
Measure, Jordan 128
Measure, Lebesgue 128
Method(s) of simple iteration 344
Method(s) of successive substitution 344
Method(s), Adams' 359
Method(s), direct, in the calculus of variations 263
Method(s), exact 339
Method(s), Galerkin's 270
Method(s), Gauss elimination 340
Method(s), Gauss elimination, backward procedure of 341
Method(s), Gauss elimination, forward procedure of 341
Method(s), general orthogonalization 124
Method(s), iteration, for solving Fredholm's equation 172
Method(s), iterative 339 344
Method(s), Kantorovich's 275
Method(s), Monte Carlo 336
Method(s), net-point 362
Method(s), Newton's, for solving systems of equations 352
Method(s), Nikolsky 326
Method(s), of finding the potential 72
Method(s), optimization 330
Method(s), Ritz' 264—270
Method(s), Runge — Kutta 356
Method(s), Seidel's 345
Method(s), sweep 344
Metric spaces 121—138
Metric spaces, complete 133
Metric spaces, isometric 138
Moebius strip 13
n-dimensional Euclidean space 124
nabla 55
Neumann's series 174
Newton's interpolation polynomial 312
Norm 153
Numerical differentiation 318—322
Numerical integration 322—325
Operator(s), characteristic value of 185
Operator(s), completely continuous 157
Operator(s), continuous 131 156
Operator(s), eigenfunction of 185
Operator(s), eigenvalue of 185
Operator(s), Fredholm's 159 172
Operator(s), monotonically increasing (decreasing) 168
Operator(s), norm of 180
Operator(s), self-adjoint 181
Orientation(s) of a curve 10
Orientation(s) of a surface 13
Orientation(s), negative 10
Orientation(s), opposite 10
Orientation(s), positive 10
Polynomial(s) of best approximation 293
Polynomial(s), construction of 299
Polynomial(s), generalized 304
Polynomial(s), Hermite interpolation 304
Polynomial(s), interpolation 304
Polynomial(s), Lagrange's interpolation 309
Polynomial(s), Newton's interpolation 312
Polynomial(s), Newton's interpolation for equal intervals 313
Potential of an electrostatic field 75
Potential, methods of finding 72
Potential, vector 82—84
Principal error term 328
Problem(s) of error estimate in an approximate solution 112
Problem(s) of the choice of a method for an exact or approximate solution 112
Problem(s) of the existence of the best approximation element 289
Problem(s) of the line of quickest descent 203
Problem(s) with differential constraints 233
Problem(s), boundary-value 94
Problem(s), construction 290
Problem(s), Dirichlet 94
Problem(s), interpolation 304 314
Problem(s), isoperimetric 234—239
Problem(s), Neumann 95
Problem(s), solution existence 112
Problem(s), solution stability 112
Problem(s), solution uniqueness 112
Problem(s), Sturm — Liouville 197
Problem(s), third boundary-value 262
Problem(s), uniqueness 289
Problem(s), variation 203
Procedure, backward sweep 344
Procedure, forward sweep 344
Reciprocity principle 240
Rectangular formula 324
Ritz' method 264—270
Runge — Kutta method 356
Schauder's theorem 163
Sequence (s) of coordinate functions 265
Sequence (s) of nested closed balls 133
Sequence (s), Cauchy's 133
Sequence (s), minimizing 264
Solving nonlinear equations 350—354
Solving the Dirichlet problem with the aid of Green's function 96
Space(s) of continuous functions 127
Space(s) of continuously differentiable functions 204
Space(s) of convergent sequences 129—130
Space(s) of pth power integrable functions 128
Space(s) of sequences with convergent series 130
Space(s) of type B 154
Space(s), Banach 154
Space(s), Hilbert 178
Space(s), Hilbert, coordinate 131
Space(s), Hilbert, functional 129
Space(s), infinite dimensional 155
Space(s), Metric 121—138
Space(s), n-dimensional Euclidean 124
Space(s), n-dimensional vector 124
Space(s), normed linear 153
Space(s), separable 151
Stokes' theorem 61
Surface integral of the second type 37
Surface(s), orientation of 13
Surface(s), orientation of, negative 13
Surface(s), orientation of, positive 13
Surface(s), oriented 13
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