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| Авторизация |
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| Apostol T.M. — Calculus (Volume 2): Multi-Variable Calculus and Linear Algebra with Applications |
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| Предметный указатель |
Abel's formula for Wronskian determinants 162
Abel, Niels Henrik 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
Approximate integration by Euler's summation formula 613 615
Approximate integration by Simpson'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 random 478
Attraction, gravitational 335
Average density 374
Average of a function 374
Average rate of change 253
Axioms for a determinant function 73
Axioms for a linear space 3
Axioms for an inner product 14
Axioms for probability 474 506 511
Ball in n-space 244
Basis 12
Bayes' formulas 549 Exercise
Bernoulli trials 495
Bernoulli, Daniel 182
Bernoulli, Jakob 469 495
Bessel differential equation 182
Bessel functions of first kind 186
Bessel functions of second kind 188
Bessel functions, graphs of 186
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 fhnction 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 — Hamilton theorem 203
Cayley, Arthur 202
Center of mass 373 374 431
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
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
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
Cotes, Roger 612 Exercise
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 in n-space 324
Curve, closed 334 379
Curve, Jordan 379
Curve, piecewise smooth 324
Cylindrical coordinates 409
D'Alembert, Jean 288
de Fermat, Pierre 469
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 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, directional 254
Derivative, partial 254
| Derivative, total 258 270
Determinants 71
Determinants, axiomatic definition of 73
Determinants, differentiation of 80 Exercise
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, first-order linear 143
Differential equations, homogeneous 146
Differential equations, nth-order linear 145
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
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
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(s) in polar coordinates 397
Double integral(s) of a bounded function 357
Double integral(s) of a step function 355
Double integral(s), applications of 374
Double integral(s), evaluation by repeated integration 358 367
Double integral(s), existence of 358 363
Double integral(s), transformation formula for 394
Doubly connected region 391 Exercise
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
Euler's gamma function 184 413 620 Exercises 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(s) for determinants 90
Existence theorem(s) for first-order linear differential equations 143
Existence theorem(s) for first-order linear systems of differential equations 213 219 220
Existence theorem(s) for first-order nonlinear systems of differential equations 229
Existence theorem(s) for implicit functions 237
Existence theorem(s) for integral equations 239
Existence theorem(s) for interpolation polynomials 580
Existence theorem(s) for linear nth-order differential equations 147
Existence theorem(s) for linear second-order differential equations 143
Existence theorem(s) for potential functions 339
Existence theorem(s) for Taylor polynomials 572
Existence theorem(s) for vector fields with prescribed curl 448
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 with constraints 314
Extremum, second-derivative test for 312
Factorial polynomials (factorial nth powers) 592
Feller, William 568
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 Exercises 9
Gauss — Jordan process for calculating determinants 78
Gauss — Jordan process for solving linear equations 61
Gauss' theorem (divergence theorem) 457
Gauss, Karl Friedrich 61 78 457
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 — Schmidt process 22
Gram, Jorgen Pedersen 22
Green's formula 387 Exercise
Green's theorem for multiply connected regions 387
Green's theorem for simply connected regions 380
Green, George 379
Hadamard matrices 69 Exercise
Hadamard, Jacques 69
Hamilton, William Rowan 202
Harmonic function 445
Heat equation 292 Exercise
Hermite differential equation 178 Exercise
Hermite, Charles 15 114 122 178
Hermitian matrix 122
Hermitian operator 115
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