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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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281 5 11 14 14 593 16 Abel summation 394 Abel — Dirichlet test 376 377 379 Abel's transformation 376 Abel, N. 376 377 379 334 421 Absolute convergence of a series 374—376 Absolute convergence of an improper integral 420 Absolute convergence of functions 375 Adiabatic 223 Adiabatic constant 224 Adjoint mapping 317 Admissible set 117 Alexander horned sphere 162 Alexander, J. 162 Algebra of forms 313 Algebra of forms, skew-symmetric 314 Algebra of functions 401 Algebra of functions, complex 401 Algebra of functions, real 401 Algebra of functions, self-adjoint 405 Algebra of functions, separating points 401 Algebra, exterior 319 Algebra, graded 314 Algebra, Grassmann 319 Algebra, Lie 72 Almost everywhere 111 Alternation 314 Ampere, A. 234 Amplitude 560 Amplitude modulation 587 Analysis, harmonic 560 Angular velocity 69—70 Approximate identity 455 Archimedes 245 Area as the integral of a form 227—230 Area of a k-dimensional surface 187 229 Area of a piecewise-smooth surface 229 Area of a sphere in 192 444 Area, Minkowski outer 194 Arzela, C. 398 404 Ascoli, G. 398 404 Asymptotic equality 598 Asymptotic equivalence 598 Asymptotic estimate 597 Asymptotic estimate, uniform 611 Asymptotic expansion in the sense of Erdelyi 610 Asymptotic expansion, uniform 611 Asymptotic formula 598 Asymptotic methods 597 Asymptotic problem 596 Asymptotic sequence 601 Asymptotic series 601 602 Asymptotic series in the sense of Erdelyi 610 Asymptotic series in the sense of Poincare 602 Asymptotic series, general 601 Asymptotic series, power 606—609 Asymptotic zero 603 Asymptotics 596 597 Asymptotics of a Bessel function 640 Asymptotics of a Laplace integral 638 Asymptotics of canonical integral 633 Asymptotics of Legendre polynomials 623 Asymptotics of the Fourier integral 635 639 Asymptotics of the gamma function 622 629 Asymptotics of the Laplace integral 635 Asymptotics of the probability error function 628 Atlas of a surface (manifold) 162 320 Atlas of class 325 Atlas, analytic 325 Atlas, orienting 174 328 Atlas, smooth 325 Averaging of a function using a kernel 494 Ball in 444 Ball in a metric space 5 Ball, closed 5 Banach, S. 34 Band-limited signal 585 Base in the set of partitions 108 Base of a topology 10 Basis of a vector space 500 Bernoulli integral 309 Bernoulli numbers 635 Bernoulli numbers, generating function 635 Bernoulli polynomials 555 Bernoulli, D. 309 Bernoulli, Jacob 555 635 Bernoulli, Johann 92 Bessel function 392 410 Bessel function, asymptotics 640 Bessel's equation 392 410 414 434 Bessel's inequality 508—509 Bessel's inequality for trigonometric system 534 Bessel, F. 392 410 414 434 508 beta function 437—439 Bicompact set 15 Biot, J. 235 Borel's formulas 581 Borel, E. 581 Boundary cycle 358 Boundary of a half-space 178 Boundary of a manifold 321 Boundary of a p-dimensional cube 357 Boundary of a surface 178 Boundary point 6 13 321 Boundedness, total 397 Boundedness, uniform 397 Boundedness, uniform, of a family of functions 371 Brachistochrone 92—94 Bracket, Poisson 348 Brouwer, L. 178 242 321 Brunn, H. 121 Bundle of tangent paths 347 Bunyakovskii, V. 46 452 505 Canonical embedding 349 Canonical integral, asymptotics 633 Cantor, G. 23 32 111 Cardinal sine 588 Carleson, L. 531 Carnot cycle 226 Carnot, S. 226 Cartan, E. 250 Category of a set 28 Cauchy integral 309 Cauchy — Bunyakovskii inequality 452 505 Cauchy — Riemann equations 310 Cauchy, A. 21 29 46 155 287 309 310 369 375 378 422 452 505 Cavalieri, B. 132 Cesaro, E. 394 Chain of charts 329 Chain of charts, contradictory 329 Chain of charts, disorienting 329 Chain of singular cubes 357 Change of variable in an integral 135—143 156—158 Characteristic, frequency 562 Characteristic, phase 562 Characteristic, spectral 562 Chart, local 161 320 Chart, parameter domain 161 Chart, range 161 Chart, range of 320 Charts, consistent 174 328 Chebyshev metric 3 Chebyshev polynomials 525 Chebyshev — Laguerre polynomials 525 Chebyshev, P. 3 525 Circulation of a field along a curve 233 277 Clapeyron, E. 222 Class, orientation 328 Clausius, R. 225 Closed ball 5 Closed form 352 Closed set 13 Closed set in a metric space 5 6 Closed set in a topological space 13 Closure 6 13 Coefficient of thermal conductivity 303 Coefficient of thermal diffusivity 304 Coefficients, Fourier 506—508 510 513 519 526 531 536 541 551 553 557 Coefficients, Fourier, extremal property 506 Coefficients, Lame 268 275 Cohomology group 356 Compact set 15 Compact set in a metric space 16 Compact set, elementary 159 Companion trihedral 73 Complete system of vectors 511—516 Completeness of the trigonometric system 545 Completion of a space 24—27 Condition, necessary, for convergence 374 Conditions Dini 535 570 Connected set 19 Consistent charts 174 328 Constant, cyclic 297 360 Content of a set (Jordan) 119 Continuity and passage to the limit 383 Continuity of an improper integral depending on a parameter 423—426 Continuity of an integral depending on a parameter 408—409 Continuous group 72 336 Contribution of a maximum point 616 Contribution of a point to asymptotics 615 Convergence in mean 528 545 Convergence of a family of functions, pointwise 363 367 Convergence of a family of functions, uniform 367 Convergence of a series of vectors 505 Convergence of an improper integral 153 Convergence of an improper integral, Cauchy principal value 155 Convergence of distributions 463 Convergence of generalized functions 463 Convergence of linear functionals, strong (norm) 58 Convergence of test functions 463 Convergence set 363 367 Convergence, absolute 375 Convergence, absolute, of an improper integral 420 Convergence, uniform 397 Convergence, uniform, Cauchy criterion 369—370 Convergence, weak 463 Convolution 449—470 Convolution in 483 Convolution, differentiation 453 Convolution, multidimensional 483—493 Convolution, symmetry 453 Convolution, translation-invariance 453 Coordinate parallelepiped 107 Coordinates of a tangent vector 340 Coordinates, Cartesian 259 261 Coordinates, curvilinear 161 265 Coordinates, cylindrical 268—273 Coordinates, polar 166 Coordinates, spherical 166 268—273 Coordinates, triorthogonal 268—273 Cotangent space to a manifold 340 Coulomb, Ch. 279 288 Covering, locally finite 336 Covering, refinement of another covering 336 Criterion for a field to have a potential 290 Criterion for compactness in a metric space 17 Criterion for continuity of a mapping 31 Criterion, Cauchy 422 Criterion, Cauchy, for uniform convergence 369—370 375 Criterion, Cauchy, for uniform convergence of a series 378 Criterion, Cauchy, for uniform convergence of an integral 418—420 Criterion, Darboux 114—116 129 Criterion, Lebesgue 111—114 117 119 138 145 Critical point 149 Cube, singular 357 Cube, singular, boundary of 357 Curl 203 260 274 Curl, physical interpretation 281—282 Current function 310 Curvature of a curve 73 Cycle of dimension p 358 Cycle, boundary 358 Cycle, Carnot 226 Cycles, homologous 358 Cyclic constant 297 360 Cyclic frequency 560 Cylinder 167 D'Alembert, J. 306 Darboux integral, lower 114 Darboux integral, upper 114 Darboux sum, lower 114 396 Darboux sum, upper 114 Darboux, G. 114—116 120 129 396 de Rham, G. 359 Deformation (of a closed path) 293 Degree (order) of a differential form 195 Delta function ( 281 449 455 461 464 469 473 485 496 559 591 593 Delta function ( 485 Derivation of a ring 352 Derivative mapping 62 Derivative of a mapping 61 62 Derivative of order n 80 Derivative with respect to a vector 81 Derivative, Lie 352 Derivative, partial 70—71 Derivative, second 80 83 Deviation, mean-square 529 Diffeomorphism, elementary 140 Differential equation with variables separable 226 Differential form 197 Differential form of class 342 Differential form of compact support 344 Differential form of order zero 201 Differential form on a manifold 340 Differential form on a smooth surface 207 Differential form, closed 296 352 Differential form, exact 296 352 Differential form, flux 199 Differential form, restriction to a submanifold 350 Differential form, work 198 Differential of a mapping 61 Differential of order n 80 Differential operator 486 Differential operator, adjoint 486 Differential operator, self-adjoint 487 Differential operator, transpose 486 Differential, exterior 342 349 Differential, exterior, of a form 201 342 349 Differential, partial 70 71 Differential, second 80 83 Differential, total 71 Differentiation 61 Differentiation at a point of a manifold 347 Differentiation of a family of functions depending on a parameter 388—393 Differentiation of a Fourier series 544 Differentiation of a generalized function 465—468 Differentiation of a power series 391 Differentiation of a series 391 Differentiation of an integral depending on a parameter 409—412 483 Differentiation of an integral, over a liquid volume 497 Differentiation on a manifold 347 351 Differentiation with respect to a parameter 426—428 Dimension of a manifold 320 Dini conditions 535 Dini, U. 386 430 535 570
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