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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
Group, Lie 72 336 Group, one-parameter 350 Group, topological 72 336 Haar system 525 Haar, A. 525 Hadamard, J. 375 415 Hamilton operator (nabla) 262 265 Hamilton, W. 262 265 Hardy, G. 395 Harmonic analysis 560 Harmonic frequency 560 Harmonic function 286 304 Harmonic function, conjugate 310 Harmonic polynomials 524 Hausdorff space 12 14 Hausdorff, F. 11 13 heat capacity 223 Heat capacity, molecular 223 heat engine 224 Heat equation 302 583 Heaviside function 469 491 497 Heaviside, O. 466 469 491 497 Heisenberg, W. 573 Helmholtz, H. 302 Herinite, Ch. 45 524 Hermite polynomials 524 Hermitian form 45 Hermitian form, nondegenerate 45 Hermitian form, nonnegative 45 Hilbert's fifth problem 336 Hilbert, D. 47 336 Hoelder, O. 126 Homeomorphism 31 Homologous cycles 358 Homology 358 Homology group 298 356 Homotopic paths 294 Homotopy 293 Homotopy formula 360 Homotopy group 298 Homotopy identity 352 Hurwitz, A. 550 Hypergeometric equation 393 Hypergeometric series 392 Identity, approximate 455 Identity, homotopy 352 Identity, Jacobi 72 Improper integral 151 Improper integral depending on a parameter, Abel — Dirichlet test 421 Improper integral depending on a parameter, Cauchy criterion 422 Improper integral depending on a parameter, continuity 423—426 Improper integral depending on a parameter, limiting passage 423—426 Improper integral depending on a parameter, uniform convergence 431 Improper integral depending on a parameter, Weierstrass' M-test 431 Improper integral with variable singularity 478—483 Improper integral, depending on a parameter 407 Induced orientation on the boundary of a surface 181 Inequality, Bessel's 508—509 Inequality, Bessel's, for trigonometric system 534 Inequality, Brunn — Minkowski 121 Inequality, Cauchy — Bunyakovskii 46 452 505 Inequality, Clausius 225 Inequality, Hoelder's 126 Inequality, isoperimetric 194 550—551 Inequality, Minkowski's 44 126 494 Inequality, Minkowski's, generalized 494 Inequality, Steklov's 556 Inequality, triangle 1 Inequality, Wirtinger's 556 Inertia 306 Inner product 45—47 351 Inner product of a field and a form 351 Instantaneous axis of rotation 70 integral 108 Integral metric 4 Integral of a differential form, over a surface 215 218 Integral of a form on a manifold 343—345 Integral of a function over a surface 227 231 Integral operator 568 Integral representation of the partial sum of a Fourier series 531 Integral transform 568 Integral, Bernoulli 309 Integral, canonical, asymptotics 633 Integral, Cauchy 309 Integral, Darboux, lower 114 Integral, Darboux, upper 114 Integral, depending on a parameter, continuity 408—409 Integral, depending on a parameter, differentiation 409—412 Integral, depending on a parameter, integration 413 Integral, depending on a paramter 407—415 Integral, Dirichlet 286 436 447 570 610 Integral, double 109 Integral, elliptic 393 Integral, elliptic, complete, of first kind 393 411 435 Integral, elliptic, complete, of second kind 393 411 Integral, Euler — Poisson 433 442 567 580 610 Integral, Eulerian 437—448 Integral, Eulerian, first kind 437 Integral, Eulerian, second kind 437 Integral, Fourier 561 564 567 571 578 585 588 634 Integral, Fourier, asymptotics 635 639 Integral, Fourier, multiple 641 Integral, Fresnel 436 447 610 Integral, Gauss' 253 Integral, improper, differentiation with respect to a parameter 426—428 Integral, improper, integration with respect to a parameter 429—434 Integral, iterated 127—129 Integral, Laplace 572 612 634 Integral, Laplace, asymptotics 635 638 Integral, Lebesgue 395 Integral, line 212 Integral, multiple 109 Integral, multiple, depending on a parameter 476—497 Integral, multiple, with variable singularity 481 Integral, over a chain 358 Integral, over a set 117 Integral, over a singular cube 358 Integral, Poisson 459 588 Integral, Raabe's 446 Integral, Riemann 395 Integral, Riemann, over a set 117 Integral, Riemann, over an interval 108 Integral, surface, of first kind 231 Integral, surface, of second kind 232 Integral, triple 109 Integration by parts in a multiple integral 255 Integration of an integral depending on a parameter 413 Integration with respect to a parameter 429—434 Interchange of a series 391 Interchange of differentiation and passage to the limit 388—393 Interchange of integrals 127 Interchange of integrals, improper 429—434 Interchange of integrals, proper 127 Interchange of integration and passage to the limit 386—388 Interchange of limiting passages 381 Interior point 6 13 Interval in 107 Isobar 222 Isochore 222 Isometry of metric spaces 24 Isomorphism of normed vector spaces 60 Isomorphism of smooth structures 335 Isoperimetric inequality 194 550—551 Isotherm 222 Iterated integral 127—129 Jacobi identity 72 Jacobi, C. 72 Jacobian of a coordinate change, cylindrical coordinates 269 Jacobian of a coordinate change, general polar 166 Jacobian of a coordinate change, spherical coordinates 269 Jacobian of a coordinate change, triorthogonal coordinates 269 Jordan measure 119 Jordan, C. 119—121 126 130—132 137 144 Joule, G. 224 k-dimensional volume 185 k-path 254 Kantorovich, L. 39 Kelvin, Lord (W. Thomson) 224 237 250 Kernel, Dirichlet 532 557 Kernel, Fejer 557 kernel, Poisson 472 Klein bottle 169 325 Klein, F. 169 325 550 Kolmogorov, A. 530 Kotel'nikov's formula 586 Kotel'nikov, V. 585 Kronrod, A. 236 Krylov, A. 544 Lagrange's theorem 309 Lagrange, J. 39 91 106 287 Laguerre, E. 525 Lame, G. 268 Laplace integral 572 612 634 Laplace integral, asymptotics 635 638 Laplace transform 612 Laplace's equation 304 524 Laplace's method 612—615 Laplace's method, multidimensional 633 Laplace, P. 264 265 273 304 310 493 524 572 612 634 635 638 Laplacian 264 265 273 493 Law of conservation of mass 279 Law, Ampere's 234 Law, Archimedes' 245 Law, Biot — Savart 235 Law, Coulomb's 279 Law, Faraday's 233 Law, Gauss' 286 Law, Newton's 288 306 307 Law, normal distribution 471 Layer, double 496 Layer, single 493 Lebesgue, H. 23 111—114 117 120 121 125 129 136 145 146 149 395 532 570 638 Legendre polynomials 504 516 623 Legendre's formula 446 Legendre, A. 414 446 504 516 623 Leibniz, G. 134 237 278 408 453 573 Lemma on continuity of the inner product 505 Lemma on finite 17 Lemma on orthogonal complement 508 Lemma, Erdelyi's 635 640 641 Lemma, exponential estimate 615 624 637 Lemma, Hadamard's 415 Lemma, Morse's 149 408 633 637 641 Lemma, nested ball 27 Lemma, Poincare's 296 Lemma, Riemann — Lebesgue 532 570 633 Lemma, Sard's 149 Lemma, Watson's 631 635 Lie algebra 72 348 Lie derivative 352 Lie group 72 336 Lie, S. 72 336 348 LIMIT 21 28 Limit function 363 367 Limit of a family of continuous functions 383—386 Limit of a family of functions 370 Limit of a mapping 28 Limit of a sequence 21 Limit of a sequence of functions 363 Limit point 5 7 13 Limiting passage, interchange 381 Limiting passage, under a differentiation sign 388—393 Limiting passage, under an integral sign 386—388 Linear transformation 49 Liouville, J. 523 Local chart 320 Local maximum 87 Local minimum 87 Localization principle for a Fourier series 532 Localization principle for a Laplace integral 615 Locally integrable function 452 Lorentz transformation 595 Lorentz, H. 595 Low-frequency filter 563 Luzin, N. 531 Lyapunov, A. 520 M-test for convergence 374 Manifold 320—325 Manifold of class 325 Manifold with boundary 322 Manifold without boundary 322 Manifold, analytic 325 Manifold, compact 323 Manifold, connected 323 Manifold, contractible 353 Manifold, embedded in 161 Manifold, nonorientable 328 Manifold, orientable 328 Manifold, oriented 328 Manifold, smooth 325 Manifold, topological 325 Mapping of class 325 Mapping, adjoint 317 Mapping, bounded 29 Mapping, continuous 30—33 Mapping, continuous, at a point 30 Mapping, continuously differentiable 75 Mapping, contraction 34 Mapping, derivative 62 Mapping, derivative, of order n 80 Mapping, differentiable, at a point 61—62 Mapping, differentiable, on a set 62 Mapping, homeomorphic 31 Mapping, linear 49 Mapping, multilinear 49 Mapping, partial derivative 70 Mapping, smooth 325 Mapping, tangent 61 339 349 Mapping, ultimately bounded 29 Mapping, uniformly continuous 32 Mass force 306 Maximum, local 87 Maxwell, J. 237 262 263 275 281 286 298 311 Mayer, J. 224 Mean convergence and completeness 545 547 Mean value over a period 560 Mean-square deviation 3 529 Measure of a set (Jordan) 119 Measure of an interval 110 Measure zero 110—111 117 Men'shov, D. 531 Method of Lagrange multipliers 106 Method of least squares (Gauss') 520 Method of separating singularities (Krylov's) 544 Method of tangents (Newton's) 38 Method of tangents, modified (Newton — Kantorovich) 39 Method, asymptotic 597 Method, Fourier 517—520 Method, Laplace's 612—615 Method, Laplace's, multidimensional 633 Method, separation of variables 517—520 Method, stationary phase 633 635 Method, stationary phase, multidimensional 640 Method, stationary phase, one-dimensional 638 Method, Steklov's averaging 520 Metric 1 Metric of mean-square deviation 3 4 Metric of uniform convergence 4 397 399 Metric space 1 Metric space, separable 15
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