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Zorich V. — Mathematical Analysis
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Название: Mathematical AnalysisАвтор: Zorich V. Аннотация: This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, integral transforms, and distributions. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books.
The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 681Добавлена в каталог: 19.02.2014Операции: Положить на полку |
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Предметный указатель
Metric spaces, direct product 7 Metric, Chebyshev 3 Metric, discrete 2 8 Metric, integral 4 Metric, Riemannian 267 Milnor, J. 335 Minimum, local 87 Minkowski, H. 121 126 194 494 Modulus of an elliptic integral 411 Moebius band 168 175 180 184 327 330 Moebius, A. 168 327 330 Moment of a function 404 Moment, dipole 299 496 Moment, multipole 299 Morse's lemma 633 637 641 Morse, A. 149 Morse, M. 633 637 641 Multilinear transformation 49 Multiple integral with variable singularity 476 478 Multiple integral, depending on a parameter 476—497 Multiple integral, improper 151 Multiple integral, integration by parts 255 Multiple integral, iterated 127—129 Multiplication of generalized functions 475 Multipole 299 Multipole moment 299 Multipole potential 299 n-dimensional disk 179 322 Nabla (Hamilton operator) 262 265 Natural frequencies 518 Natural oscillations 518 Natural parametrization 73 Necessary condition for uniform convergence 374 Neighborhood 13 Neighborhood in a metric space 5 6 9 Neighborhood in a topological space 11 Neighborhood of a germ of functions 11 Newton — Leibniz formula 573 Newton, I. 38 237 278 288 306 307 408 573 Norm in a vector space 42 589 Norm of a transformation 52 Norm of a vector 42 589 nth moment 404 Null-series, Men'shov's 531 Numbers, Bernoulli 635 Numbers, Bernoulli, generating function 635 One-parameter group 350 Open set in a metric space 5 Open set in a topological space 9 Operational calculus 466 Operator of field theory 260 Operator of field theory in curvilinear coordinates 265—274 Operator, differential 486 Operator, differential, adjoint 486 Operator, differential, self-adjoint 437 Operator, differential, transpose 486 Operator, Hamilton (nabla) 262 265 Operator, integral 568 Operator, Laplace 264 265 273 493 Operator, nilpotent 71 Operator, symmetric 519 Operator, translation 449 Operator, translation-invariant 449 Orbit of a point 324 335 Order (degree) of a differential form 195 Orientation class 328 Orientation class of atlases of a surface 175 Orientation class of coordinate systems 171 172 Orientation class of frames 171 Orientation of a domain of space 172 Orientation of a manifold 328 Orientation of a surface 171—177 181 Orientation of the boundary of a surface 181 Orientation, induced on the boundary of a manifold 331 Orientation, opposite to a given orientation 172 Oriented space 171 Orienting frame 172 Orthogonal vectors 499 519 Orthogonality with a weight 522 Orthogonalization 503—504 Oscillation of a mapping 30 Oscillation of a mapping at a point 32 Oscillations, natural 518 Ostrogradskii, M. 237 242 249 278 286 489 496 overtones 518 p-cycle 358 p-form 197 Parallelepiped, coordinate 107 Parameter domain 161 320 366 Parameter set 366 Parameters, Lame 268 275 Parseval's equality 520 529 569 592 Parseval, M. 520 529 547 569 592 Partition of an interval 108 Partition of an interval with distinguished points 108 Partition of unity 148 331—333 Partition of unity, k-smooth 332 Partition of unity, subordinate to a covering 332 Partition, locally finite 189 Paths, homotopic 294 Paths, tangent 347 Period of an integral 297 Period, over a cycle 360 Phase 560 634 Phase characteristic 562 Phase function 634 Phase, stationary 634 Picard, E. 34 38 Piecewise continuous function 536 Piecewise continuously differentiable function 536 Planar flow 310 Plancherel's Theorem 590 Plancherel, M. 590 Plane, projective 326 Plane-parallel flow 310 Poincare, H. 250 296 353 356 602 610 Point of a metric space 1 Point, boundary 6 Point, boundary, (of a manifold) 321 Point, boundary, (of a surface) 178 Point, boundary, in a topological space 13 Point, critical 149 Point, exterior 6 Point, exterior in a topological space 13 Point, interior 6 13 Point, interior in a topological space 13 Point, limit 5 7 13 Poisson bracket 348 Poisson integral 459 588 Poisson kernel 472 Poisson's equation 298 304 493 Poisson, S. 224 298 304 348 433 442 459 472 493 567 580 588 593 610 Polar coordinates 166—167 Polynomial, trigonometric 526 Polynomials, Bernoulli 555 Polynomials, Chebyshev 525 Polynomials, Chebyshev — Laguerre 525 Polynomials, harmonic 524 Polynomials, Hermite 524 Polynomials, Legendre 504 516 623 Positive direction of circuit 176 Potential field 288 Potential of a field 288 Potential, dipole 299 Potential, multipole 299 Potential, quadrupole 299 Potential, scalar 296 Potential, single-layer 493 Potential, vector 295 296 Potential, vector, of a magnetic field 296 Potential, velocity 309 Power series 375 Power series, differentiation of 391 Principal value (Cauchy) of an integral 155 Principle, Cavalieri's 132 Principle, contraction mapping 36 Principle, D'Alembert's 306 Principle, Dirichlet's 286 Principle, fixed-point 34 38 Principle, localization, for a Fourier series 532 Principle, localization, for a Laplace integral 615 Principle, Picard — Banach 34 38 Principle, stationary phase 633 634 640 Principle, uncertainty 590 Probability error function 610 Probability error function, asymptotics 628 Problem, asymptotic 596 Problem, brachistochrone 92 94 Problem, curve of most rapid descent 92 Problem, Luzin's 531 Problem, Riemann 530 Problem, shortest-time 92 Problem, Sturm — Liouville 523 Process, adiabatic 223 Process, quasi-static 222 Product of generalized functions 475 Product of manifolds 320 Product of topological spaces 13 Product, exterior 315 342 Product, inner 45—47 351 Product, inner, of a field and a form 351 Product, inner, of functions 522 Product, tensor 313 Projective line 326 Projective plane 326 Projective plane, real 326 Pulse, rectangular 587 Pulse, triangular 588 Pythagoras 193 505 Quadrupole 299 Quadrupole potential 299 Quantities of the same order 598 Raabe's integral 446 Raabe, J. 446 Rademacher system 525 Rademacher, H. 525 Range of a chart 161 320 Rapidly decreasing function 575 Rectangular pulse 587 Restriction of a form to a submanifold 207 350 Riemann sum 108 Riemann — Lebesgue lemma 532 638 Riemann, B. 23 107—109 310 395 447 530 532 638 Riemannian metric 267 Rule, Leibniz 134 409 453 Sample function 588 Sard's lemma 149 150 Sard's theorem 236 Sard, A. 149 236 Savart, F. 235 Scalar potential 296 Schmidt, E. 503 Schwartz space 592 Schwartz, L. 464 591 592 Schwarz boot 194 Schwarz, H. 193 Separable metric space 15 Separation of points by functions 401 Separation of variables 517—520 Sequence, asymptotic 601 Sequence, Cauchy 21 Sequence, convergent 21 Sequence, convergent at a point 363 Sequence, convergent on a set 363 Sequence, convergent, uniformly 368 Sequence, fundamental 21 Sequence, monotonic, of functions 376 Sequence, nondecreasing, of functions 376 Sequence, nonincreasing, of functions 376 Series of functions 366 Series, asymptotic 601 602 Series, asymptotic, general 601 Series, asymptotic, in the sense of Erdelyi 610 Series, asymptotic, in the sense of Poincare 602 Series, asymptotic, power 606—609 Series, continuity of sum 384 Series, Dirichlet 380 Series, Fourier 499—526 Series, Fourier in a general orthogonal system 499—500 Series, Fourier of generalized functions 558 Series, Fourier, multiple 557 Series, Fourier, partial sum 531 Series, Fourier, pointwise convergence 527 Series, Fourier, rate of convergence and smoothness 540 Series, hypergeometric 392 Series, power 375 Series, Stirling's 636 Series, trigonometric 526—560 Series, uniformly convergent, Cauchy criterion 378 Set of area zero 190 Set of content zero 120 Set of convergence 363 367 Set of first category 28 Set of measure zero (Jordan) 120 Set of measure zero (Lebesgue) 117 120 121 Set of second category 28 Set of volume zero 120 Set, admissible 117 Set, bicompact 15 Set, Cantor 23 Set, closed, in a metric space 5 6 Set, closed, in a topological space 13 Set, compact 15 Set, conditionally compact 18 Set, everywhere dense 12 Set, Jordan measurable 119 Set, nowhere dense 28 Set, open, in a metric space 5 Set, open, in a topological space 9 Set, parameter 366 Set, relatively compact 18 Set, totally bounded 18 Shannon, C. 585 Sigual, band-limited 585 Sigual, spectrum of 560 Simply connected domain 293 Sine integral 610 Sine, cardinal 588 Single layer 482 Single-layer potential 493 Singular cube 357 Singular cube, boundary of 357 Smooth structure 326 Smooth structures, isomorphic 335 Sobolev, S. 464 Solenoidal field 295 Solution, fundamental 469—470 490 Solution, fundamental, of the Laplacian 490 Space of distributions 463 Space of fundamental functions 484 Space of generalized functions 463 484 Space of tempered distributions 591 Space of test functions 463 484 Space, Banach 43 Space, complete 21 Space, connected 19 Space, cotangent to a manifold 340 Space, Euclidean 47 Space, Hausdorff 12 14 Space, Hermitian (unitary) 47
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