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Название: Mathematical Analysis
Автор: 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.
Group, Lie72336 Group, one-parameter350 Group, topological72336 Haar system525 Haar, A.525 Hadamard, J.375415 Hamilton operator (nabla)262265 Hamilton, W.262265 Hardy, G.395 Harmonic analysis560 Harmonic frequency560 Harmonic function286304 Harmonic function, conjugate310 Harmonic polynomials524 Hausdorff space1214 Hausdorff, F.1113 heat capacity223 Heat capacity, molecular223 heat engine224 Heat equation302583 Heaviside function469491497 Heaviside, O.466469491497 Heisenberg, W.573 Helmholtz, H.302 Herinite, Ch.45524 Hermite polynomials524 Hermitian form45 Hermitian form, nondegenerate45 Hermitian form, nonnegative45 Hilbert's fifth problem336 Hilbert, D.47336 Hoelder, O.126 Homeomorphism31 Homologous cycles358 Homology358 Homology group298356 Homotopic paths294 Homotopy293 Homotopy formula360 Homotopy group298 Homotopy identity352 Hurwitz, A.550 Hypergeometric equation393 Hypergeometric series392 Identity, approximate455 Identity, homotopy352 Identity, Jacobi72 Improper integral151 Improper integral depending on a parameter, Abel — Dirichlet test421 Improper integral depending on a parameter, Cauchy criterion422 Improper integral depending on a parameter, continuity423—426 Improper integral depending on a parameter, limiting passage423—426 Improper integral depending on a parameter, uniform convergence431 Improper integral depending on a parameter, Weierstrass' M-test431 Improper integral with variable singularity478—483 Improper integral, depending on a parameter407 Induced orientation on the boundary of a surface181 Inequality, Bessel's508—509 Inequality, Bessel's, for trigonometric system534 Inequality, Brunn — Minkowski121 Inequality, Cauchy — Bunyakovskii46452505 Inequality, Clausius225 Inequality, Hoelder's126 Inequality, isoperimetric194550—551 Inequality, Minkowski's44126494 Inequality, Minkowski's, generalized494 Inequality, Steklov's556 Inequality, triangle1 Inequality, Wirtinger's556 Inertia306 Inner product45—47351 Inner product of a field and a form351 Instantaneous axis of rotation70 integral108 Integral metric4 Integral of a differential form, over a surface215218 Integral of a form on a manifold343—345 Integral of a function over a surface227231 Integral operator568 Integral representation of the partial sum of a Fourier series531 Integral transform568 Integral, Bernoulli309 Integral, canonical, asymptotics633 Integral, Cauchy309 Integral, Darboux, lower114 Integral, Darboux, upper114 Integral, depending on a parameter, continuity408—409 Integral, depending on a parameter, differentiation409—412 Integral, depending on a parameter, integration413 Integral, depending on a paramter407—415 Integral, Dirichlet286436447570610 Integral, double109 Integral, elliptic393 Integral, elliptic, complete, of first kind393411435 Integral, elliptic, complete, of second kind393411 Integral, Euler — Poisson433442567580610 Integral, Eulerian437—448 Integral, Eulerian, first kind437 Integral, Eulerian, second kind437 Integral, Fourier561564567571578585588634 Integral, Fourier, asymptotics635639 Integral, Fourier, multiple641 Integral, Fresnel436447610 Integral, Gauss'253 Integral, improper, differentiation with respect to a parameter426—428 Integral, improper, integration with respect to a parameter429—434 Integral, iterated127—129 Integral, Laplace572612634 Integral, Laplace, asymptotics635638 Integral, Lebesgue395 Integral, line212 Integral, multiple109 Integral, multiple, depending on a parameter476—497 Integral, multiple, with variable singularity481 Integral, over a chain358 Integral, over a set117 Integral, over a singular cube358 Integral, Poisson459588 Integral, Raabe's446 Integral, Riemann395 Integral, Riemann, over a set117 Integral, Riemann, over an interval108 Integral, surface, of first kind231 Integral, surface, of second kind232 Integral, triple109 Integration by parts in a multiple integral255 Integration of an integral depending on a parameter413 Integration with respect to a parameter429—434 Interchange of a series391 Interchange of differentiation and passage to the limit388—393 Interchange of integrals127 Interchange of integrals, improper429—434 Interchange of integrals, proper127 Interchange of integration and passage to the limit386—388 Interchange of limiting passages381 Interior point613 Interval in 107 Isobar222 Isochore222 Isometry of metric spaces24 Isomorphism of normed vector spaces60 Isomorphism of smooth structures335 Isoperimetric inequality194550—551 Isotherm222 Iterated integral127—129 Jacobi identity72 Jacobi, C.72 Jacobian of a coordinate change, cylindrical coordinates269 Jacobian of a coordinate change, general polar166 Jacobian of a coordinate change, spherical coordinates269
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