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Apostol T.M. — Calculus (vol 1) |
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Предметный указатель |
(base of natural logarithm), computation of 281
(base of natural logarithm), definition of 231
(base of natural logarithm), irrationality of 282
(pi), computation of 285 (Exercise 10)
(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 281
Calculation of 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
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