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Apostol T.M. — Calculus (vol 2)
Apostol T.M. — Calculus (vol 2)

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Название: Calculus (vol 2)

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: 1969

Количество страниц: 696

Добавлена в каталог: 14.10.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel, Niels Henrik      162
Abel’s formula for Wronskian determinants      162
Absolute maximum      304
Absolute minimum      304
Adams, John Couch      571
Adjoint of a matrix      122
Angles in a real Euclidean space      18
Annihilators      151
Annihilators, method of      163
Annihilators, table of      166
Approximate integration by Cotes' rule      612 (Exercise 11)
Approximate integration by Euler's summation formula      613 615
Approximate integration by Sympson's rule      608 609
Approximate integration by the trapezoidal rule      604
Approximations by interpolation polynomials      579
Approximations by Legendre polynomials      29
Approximations by Taylor polynomials      576
Approximations by trigonometric polynomials      29
Approximations in a Euclidean space      28
Approximations, successive      223
Archimedes      571
Area as a line integral      383
Area cosine principle      426
Area of a plane region      368
Area of a surface      424
At randome      478
Attraction, gravitational      335
Average density      374
Average of a funcnion      374
Ball in n-space      244
Basis      12
Bayes' formulas      549 (Exercise 6)
Bernoulli trials      495
Bernoulli, Daniel      182
Bernoulli, Jakob      469 495
Bessel differential equation      182
Bessel functions, graphs of      186
Bessel functions, of first kind      186
Bessel functions, of second kind      188
Bessel, Friedrich Wilhelm      182
Binomial coefficients      481
Binomial distribution      521
Binomial distribution, approximation by normal      538
Boolean algebra of sets      471
Borel set      510
Borel, Emile      510
Boundary of a set      245
Boundary point      245
Bounded function      357
Boundedness of continuous functions      319
Bridge hands      481
Caque, J.      223
Cartesian product      244
Cauchy distribution      530
Cauchy functional equation      528
Cauchy — Schwarz inequality      16
Cauchy, Augustin — Louis      16 142 191 530
Cayley Arthur      202
Cayley-Hamilton theorem      203
Center of mass      373 374
Central limit property      567
Central limit theorem      566
Centroid      374
Chain rule for derivatives of matrix functions      194
Chain rule for derivatives of scalar fields      264 273
Chain rule for derivatives of vector fields      272
Change of variable in a double integral      394
Change of variable in a line integral      327
Change of variable in a surface integral      433
Change of variable in an n-fold integral      408
Characteristic equation      150
Characteristic polynomial      103 149
Chebyshev inequality      563
Chebyshev integration formulas      611
Chebyshev polynomials      596
Chebyshev, Pafnuti Liwowich      563 596 611
Christoffel numbers      610 (Exercise 6)
Circulation      330 462
Clockwise (negative) direction      389
Closed curve      334 379
Closed set      246
Closure axioms      3
Coefficient matrix      58
Cofactor      87
Cofactor matrix      92
Coin-tossing games      506 558
Column matrix (column vector)      51
Combinatorial analysis      481
Complement of a set      246
Complex Euclidean space      15
Complex linear space      4
components      13
Composite functions      37 194 250
Composite functions, continuity of      250
Composite functions, differentiation of      194 264 272
Composition of transformations      37
Compound experiments      492
Conditional probability      486
Connected set      332
Conservative field      329
Constant-coefficient derivative operators      148
Constant-coefficient difference operators      595 (Exercise 8)
Content zero      364 406
Continuity of scalar and vector fields      247
Continuous distribution      525
Continuous random variable      525
Continuously differentiable scalar field      261
Contraction constant      235
Contraction operator      234
Convex combination      377
Convex set      344
Coordinates cylindrical      409
Coordinates polar      274 397
Coordinates spherical      293 410 414
Cotes' rule      612 (Exercise 11)
Cotes, Roger      612 (Exercise 11)
Countable set      502
Countably additive set function      506
Countably infinite set      502
Counterclockwise (positive) direction      389
Cramer's rule      93
Cramer, Gabriel      93
Cramer, Harald      520
Curl of a vector field      441
Curve closed      334 379
Curve in n-space      324
Curve Jordan      379
Curve piecewise smooth      324
Cylindrical coordinates      409
D'Alembert, Jean      288
De Moivre, Abraham      469 567
Del operator V      259
Density function      525
Density function for Cauchy distributions      531
Density function for exponential distributions      533
Density function for normal distributions      535
Density function for uniform distributions      527
Dependent sets      9
Derivative and continuity      260
Derivative directional      254
Derivative of a matrix function      193
Derivative of a scalar field with respect to a vector      253
Derivative of a vector field with respect to a vector      270
Derivative partial      254
Derivative total      258 270
Determinants      71
Determinants, axiomatic definition of      73
Determinants, differentiation of      80 (Exercise 6)
Determinants, expansion formulas for      86
Diagonal matrix      48 96
Diagonal quadratic form      127
Diagonalizing matrix      112
Difference of two sets      246
Difference operator A      590
Differentiable scalar field      258
Differentiable vector field      270
Differential equations      142
Differential equations of first-order linear      143
Differential equations of nth-order linear      145
Differential equations, homogeneous      146
Differential equations, partial      283
Differential equations, power-series solutions of      169 220
Differential equations, systems of      191
Dimension of a linear space      12
Dimensionality theorem      147
Dirac, paul a. m.      108 (Exercise 8)
Directional derivative      254
disconnected set      333
Discontinuity      366 517
Discrete distribution      521
Discrete random variable      521
Discriminant of a quadratic form      134
Disk      244
Distance in a Euclidean space      26
Distribution function of a function of a random variable      541 550
Distribution function of a one-dimensional random variable      515
Distribution function of a two-dimensional random variable      543
Distribution function, binomial      521
Distribution function, Cauchy      530
Distribution function, continuous      525
Distribution function, discrete      521
Distribution function, exponential      533
Distribution function, joint      543
Distribution function, mixed      539
Distribution function, normal      535
Distribution function, normal bivariate      555 (Exercise 9)
Distribution function, poisson      522
Distribution function, standard normal      537
Distribution function, uniform      526
Divergence of a vector field      441 460
Divergence theorem      457
Dot product      14
Double integral in polar coordinates      397
Double integral of a bounded function      357
Double integral of a step function      355
Double integral, applications of      374
Double integral, evaluation by repeated integration      358 367
Double integral, existence of      358 363
Double integral, transformation formula for      394
Doubly connected region      391 (Exercise 3)
Eigenfunction      99
Eigenspace      98
Eigenvalue      97
Eigenvector      97
Element of a linear space      3
Entries of a matrix      45 51
Equipotential lines and surfaces      335
Equivalence of sets      501
Error in Lagrange's interpolation formula      583
Error in linear interpolation      584
Error in Taylor polynomial approximation      573
Error in Taylor's formula      258 270
Euclidean space      15
Euler's constant      618 (Exercise 2)
Euler's gamma function      184 413 620 9)
Euler's summation formula      613 615
Euler's theorem for homogeneous functions      287
Euler, Leonard      142 182 613
Events      476
Exact differential equation      347
Existence theorem for determinants      90
Existence theorem for first-order linear differential equations      143
Existence theorem for first-order linear systems of differential equations      213 219 220
Existence theorem for first-order nonlinear systems of differential equations      229
Existence theorem for implicit functions      237
Existence theorem for integral equations      239
Existence theorem for interpolation polynomials      580
Existence theorem for linear nth-order differential equations      147
Existence theorem for potential functions      339
Existence theorem for Taylor polynomials      572
Existence theorem for vector fields with prescribed curl      448
Existence theorem or linear second-order differential equations      143
Expectation (expected value) of a function of a random variable      559
Expectation (expected value) of a random variable      556
Exponential distribution      533
Exponential matrix      197
Exterior of a Jordan curve      380
Exterior of a set      245
Exterior point      245
Extreme-value theorem for continuous scalar fields      321
Extremum      304
Extremum second-derivative test for      312
Extremum with constraints      314
Factorial polynomials (factorial nth powers)      592
Feller, William      568
Fermat, Pierre de      469
Finite additivity      470
Finite sample space      473
Finite set      502
Finitely additive measure      472
Finitely additive set function      470
Fixed point of an operator      233
Fixed-point theorem for contraction operators      235
Fixed-point theorem for contraction operators, applications of      237
Flow integral      330
Flux density      432 461
Forward difference operator A      590
Fourier coefficients      29
Fourier, Joseph      29
Frechet, Rene Maurice      258
Frobenius' method      180
Frobenius, Georg      181
Fuchs, Lazarus      223
Function space      5
Functional equation for the gamma function      185
Functional equation of Cauchy      528
Functions of random variables      541 550
Fundamental theorems for line integrals first      338
Fundamental theorems for line integrals second      334
Fundamental vector product of a surface      420
Galle, Johann      571
Gamma function      184 413 620
Gauss' theorem (divergence theorem)      457
Gauss, Karl Friedrich      61 78 457
Gauss-Jordan process for calculating determinants      78
Gauss-Jordan process for solving linear equations      61
General solution of a linear differential equation      148 156
General solution of a system of linear equations      60
Gombaud, Antoine (Chevalier de Mere)      469
Gradient      259
Gram, Jorgen Pedersen      22
Gram-Schmidt process      22
Green's formula      387 (Exercise 8)
Green's theorem for multiply connected regions      387
Green's theorem for simply connected regions      380
Green, George      379
Hadamard matrices      69 (Exercise 10)
Hadamard, Jacques      69
Hamilton, William Rowan      202
Harmonic function      445
Heat equation      292 (Exercise 1)
Hermite differential equation      178 (Exercise 4)
Hermite, Charles      15 114 122 178
Hermitian matrix      122
Hermitian operator      115
Hermitian symmetry      15 114
Herschel, William      571
Hesse, Ludwig Otto      308
Hessian matrix      308
Homogeneous function      287
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