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Zorich V. — Mathematical Analysis
Zorich V. — Mathematical Analysis

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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.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 681

Добавлена в каталог: 19.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\delta$-function      281
$\delta$-neighborhood      5 11
$\tau_{1}$ space      14
$\tau_{2}$ space      14
$\theta$-formula      593
$\varepsilon$-grid      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 $('A \rho\chi\iota\mu\eta\delta\eta\varsigma')$      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 $\mathbb{R}^{n}$      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 $C^{(k)}$      325
Atlas, analytic      325
Atlas, orienting      174 328
Atlas, smooth      325
Averaging of a function using a kernel      494
Ball in $\mathbb{R}^{n}$, volume      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 $\mathbb{R}^{n}$      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 ($\delta$-function)      281 449 455 461 464 469 473 485 496 559 591 593
Delta function ($\delta$-function), shifted      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 $C^{(k)}$      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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