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Apostol T.M. — Calculus (vol 1)
Apostol T.M. — Calculus (vol 1)

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Название: Calculus (vol 1)

Автор: Apostol T.M.

Аннотация:

An introduction to the calculus, with an excellent balance between theory and technique. Integration is treated before differentiation — this is a departure from most modern texts, but it is historically correct, and it is the best way to establish the true connection between the integral and the derivative. Proofs of all the important theorems are given, generally preceded by geometric or intuitive discussion. This Second Edition introduces the mean-value theorems and their applications earlier in the text, incorporates a treatment of linear algebra, and contains many new and easier exercises. As in the first edition, an interesting historical introduction precedes each important new concept.


Язык: en

Рубрика: Математика/Анализ/Учебники по элементарному анализу/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 1967

Количество страниц: 686

Добавлена в каталог: 14.10.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$e$ (base of natural logarithm), computation of      281
$e$ (base of natural logarithm), definition of      231
$e$ (base of natural logarithm), irrationality of      282
$\pi$ (pi), computation of      285 (Exercise 10)
$\pi$ (pi), definition of      91
Abel's partial summation formula      407
Abel's test for convergence      408
Abel, Niels Henrik      407
Abscissa      48
Absolute convergence of series      406
Absolute maximum and minimum      150
Absolute values      41 363
Acceleration      160 521
Acceleration in polar coordinates      541
Acceleration, normal and tangential components of      527
Addition formulas for the sine and cosine      96
Additive properly of arc length      532
Additive properly of area      59
Additive properly of averages      119 (Exercise 13)
Additive properly of convergent series      385
Additive properly of derivatives      164
Additive properly of finite sums      40
Additive properly of the integral      66 67 80 514
Additive properly of the supremum and infimum      27
Additive properly of volume      112
Additive properly of work      115
Alternating series      403
Analytic geometry      48 471
Analytic model of Euclidean geomelry      471
Angles in a Euclidean space      564
Angles in n-space      458
Angles, radian measurement of      102
Angular acceleration      545 (Exercise 19)
Angular speed      522 545
Angular velocity      545 (Exercise 19)
Antiderivative (primitive)      205
Appolonius      498
Approximations by polynomials      272—304 575
Approximations by trigonometric polynomials      575
Approximations in a Euclidean space      574
Arbitrary constant      211 307
Arbogast, Louis      171
Arc cosine      254
Arc length as an integral      534
Arc length function      533
Arc length in polar coordinates      544 (Exercise 4)
Arc length, definition of      530 531
Arc sine      253
Arc tangent      255
Archimedean property of real numbers      26
Archimedes      2—9 26
Area and similarity transformations      92
Area in polar coordinates      110
Area of a radial set      110
Area of a region between two graphs      88
Area of an ordinate set      75
Area, axiomatic definition of      57—59
Argument of a complex number      363
Arithmetic mean      46 117
Associative law for addition of numbers      18 359
Associative law for addition of vectors      447
Associative law for composition of functions      141 584
Associative law for multiplication of numbers      18 359
Associative law for union and intersection of sets      14
Associative law in a linear space      551 552
Asymptote      190
Asymptote of a hyperbola      506
Asymptotically equal      396
Average      46 149
Average of a function      117—119
Average, rate of chance      160
Average, velocity      157
Average, weighted      118
axes      48 197
Axiom completeness (continuity)      25
Axiom for a linear space      551 552
Axiom for area      58 59
Axiom for the real-number system      17—25
Axiom for volume      112
Axiom, field      17
Axiom, least-Upper-bound      25
Axiom, order      20
Axiomatic development of area      57—59
Axiomatic development of inner products      561
Axiomatic development of the real-number system      17—25
Axiomatic development of vector algebra      551 552
Axiomatic development of volume      111—112
Barrow, Isaac      157
Base of logarithms      232
Basis      327 466
Bernoulli, differential equalion      312
Bernoulli, inequality      46 (Exercise 14)
Bernoulli, Johann      235 292 305 331
Bernoulli, polynomials      225 (Exercise 35)
Bernstein's theorem      437
Bernstein, Sergei      437
Bessel functions      443 (Exercise 10)
Binary scale      393
Binomial coefficient      44 383 442
Binomial series      377 441
Binomial theorem      44 (Exercise 4) 378 442
Boisano, Bernard      143
Bolyai, Johann      474
Bolzano's theorem      143
Boole, George      11
Bound, greatest lower      25
Bound, least upper      24
Bound, upper and lower      23—25
Bounded function      73
Bounded sequence      381
Bounded set of real numbers      23
Boundedness of continuous functions      150
Brouncker, William      377 390
Calculation of $e$      281
Calculation of $\pi$      285 (Exercise 10)
Calculation of logarithms      240-242
Calculation of square roots      444 (Exercise 20 21)
Calculus, fundamental theorems of      202 205 515
Cantor, Georg      11 17
Cardano, Hieronimo      3
Cartesian equation      49 475 494
Cauchy — Schwarz inequality      42 452 563
Cauchy's mean-value formula      186
Cauchy's remainder in Taylor's formula      284
Cauchy, Augustin-Louis      3 42 127 172 186 284 368 378 397 399 411 452
Cavalieri Solid      111
Cavalieri's principle      111 112
Cavalieri, Bonavertura      3 111
Cayley, Arthur      446
Center of mass      118
Centrifugal      522
Centripetal      522
Chain rule      174 514
Characteristic equation      327
circle      49 521
Circle of convergence      428
Circular helix      523
circular motion      521
Class of sets      14
Closed interval      60
Closure axioms      551
Coefficient matrix      605
Column matrix (column vector)      592 598
Commutative law for addition of numbers      18 359
Commutative law for addition of vectors      447
Commutative law for dot products      451
Commutative law for inner products      561
Commutative law for multiplication of numbers      18 359
Commutative law for union and intersection of sets      14
Commutative law in a linear space      551
Comparison tests for convergence of improper integrals      418
Comparison tests for convergence of series      394—396
Comparison theorem for integrals      67 81
Complement of a sel      14
Complex Euclidean space      562
Complex function      368
Complex linear space      552
complex numbers      358—373
Complex vector space      468
Composite function      140 584
Composite function, continuity of      141
Composite function, differentiation of      174 514
Composition of transformations      584
Concave function      122 189
Congruence of sels      58
Conic sections      497—507
Conjugate complex number      364
Constant function      54
Continuous functions, definition of      130 369 513
Continuous functions, integrability of      153
Continuous functions, theorems on      132 141—154
Contour lines      197
Convergence of improper integrals      416 418
Convergence of sequences      379
Convergence of series      384—425
Convergence, pointwise      422
Convergence, tests for      394—408
Convergence, uniform      424
Convex function      122 189
Convex set      112
Coordinates, cylindrical      543
Coordinates, polar      108 540
Coordinates, rectangular      48 197
Copernican theory      545
Cosine function, continuity of      134 139
Cosine function, differential equation for      323
Cosine function, differentiation of      162
Cosine function, integration of      100 207
Cosine function, power series for      436
Cosine function, properties of      96
Cotangent function      103
Cramer's rule      491
Critical point      188
Cross product (vector product)      483
Curvature      537
Curve, definition of      517
Curve, length of      529—535
Curve, nonrectifiable      530 536
Curve, rectifiable      530
Cycloid      536 (Exercise 20)
Cylindrical coordinates      543
Damped vibrations      335
Dandelin, Germinal P.      498
De Moivre's theorem      371 (Exercise 5)
Decimal expansion of real numbers      30 393
Decreasing function      76
Decreasing sequence      381
Dedekind, Richard      17
Deductive systems      8
Definite integral, definition of      73
Definite integral, properties of      80 81
Dependence, linear      463 557
Derivatives and continuity      163
Derivatives of complex-valued functions      369
Derivatives of higher order      160 200
Derivatives of vector-valued functions      513
Derivatives, functions of one variable      160
Derivatives, functions of several variables      199-201
Derivatives, notations for      160 171 172 199 200
Derivatives, partial      199—201
Derivatives, theorems on      164
Descartes, Rene      48 446
Determinant      486
Difference of functions      132
Difference of real numbers      18
Difference of sets      14
Difference of vectors      447
Difference quotient      157 159 517
Differential equations      305—357
Differential equations, first-order linear      308
Differential equations, homogeneous first-order      347—350
Differential equations, power-series solutions of      439-443
Differential equations, second-order linear      322
Differential equations, separable      345
Dimension of a linear space      559
Direction field      343
Directrix of conic sections      500
Dirichlet's test for convergence      407
Dirichlet, Peter Gustav Lejeune      407
Discontinuity, infinite      131
Discontinuity, jump      131
Discontinuity, removable      131
Disjoint sets      14
Distance between two planes      495
Distance between two points      364 462
Distance from a point to a line      476 477
Distance from a point to a plane      494
Distributive law for cross products      483
Distributive law for inner products      451 561
Distributive law for numbers      18 359
Distributive law for set operations      16 (Exercise 10)
Distributive law in a linear space      552
Divergent improper integral      416 418
Divergent scries      384
Divergent sequence      379
Division of functions      55
Division of numbers      18 360
Domain of a function      50 53 196 512 578
Dot product (inner product)      451 469 562
Duodecimal scale      393
Earth      545
Eccentricity of conic sections      500
Electric circuits      317 336
Element of a determinant      486
Element of a matrix      592 598
Element of a set      11
Elementary function      282
ellipse      498 500 506
Elliptic integral      535 (Exercise 17)
Elliptic reflector      519
Empty set      13
Endpoints of an interval      60
Envelope      342
Equality of complex numbers      358
Equality of functions      54
Equality of sets      12
Equality of vectors      447 468
Equipotential lines      351
Error in Taylor's formula      278 280
Euclid      9 471
Euclidean geometry      9 471
Euclidean space      472 561
Euler Leonard      231 377 396 405 420
Euler's constant      405
Even function      84
Even integer      28 (Exercise 10)
Exhaustion property of area      59
Exhaustion, method of      2—8
Existence theorems      308 323
Exponential function complex-valued      367
Exponential function, definition of      242
Exponential function, derivative of      243
Exponential function, integral of      246
Exponential function, power series for      436
Extreme-value theorem for continuous functions      151
Extremum, definition of      182
Extremum, tests for      182 188 189
Factorials      44 52
Family of curves      341 351
Fermat, Pierre de      3 156
1 2 3
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