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| Smirnov V.I. — Higher mathematics. Vol.1 |
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| Предметный указатель |
Abel convergence test 384
Abel integral 515
Abel theorem 386 388
Absolute error 120
Absolute maximum 430
Absolute minimum 430
Absolute value 2
Algebraic equations 480
Algebraic functions 180
Alternating series 327
Analytic method of specifying function 8
Angular coefficient of straight line 17
Angular coefficient of tangent 103
Approximate evaluation by differentials 117
Approximate evaluation by series 339 344 351 356
Approximate evaluation of definite integrals 279
Approximate evaluation of definite integrals, Poncelet's formula 284
Approximate evaluation of definite integrals, rectangle formula 279
Approximate evaluation of definite integrals, Simpson's formula 282
Approximate evaluation of definite integrals, tangent formula 281
Approximate evaluation of definite integrals, trapezoid formula 279
Approximate evaluation of definite integrals, with variable upper limit 286
Approximate evaluation, solution of equations by Newton's method 502
Approximate evaluation, solution of equations by simple interpolation 503
Approximate evaluation, solution of equations by successive approximation 498
Approximate evaluation, solution of equations by tangent method 502
Arc, differential of 164 264 265 417
Arc, length of 261 417
Archimedes, spiral of 196
Area as primitive 222
Area of surface of revolution 271
Area of trapezoids 216
Area, limit of sum of 219
Astroid 193
Asymptote 28 170
Asymptotic point 197
Basic theorem of algebra 481
Bezout's theorem 481
Binomial differential, integration of 512
Binomial series 342
Bounds, of numerical set, upper, lower 85
Cardan's formula 493
Cardioid 192 197 267
Cassini's ovals 200
Catenary 187 266 464
Cauchy formula 152
Cauchy test for convergence of series 320 322
Cauchy test for existence of limit 58
Centre of gravity of arc 275 276
Centre of gravity of plane figure 277
Change of variables, definite integrals 250
complex numbers 444
Complex numbers, amplitude 446
Complex numbers, conjugates of 451
Complex numbers, exponential form of 458
Complex numbers, operations on 444—455
Complex numbers, trigonometric form of 444
Concavity and convexity 166
Continuity of functions 66—72 87 401
Continuity, uniform 70 88
Coordinates, polar 194
Coordinates, rectangular 13
Curvature of arc 166
Cusp 182
Cycloid 187 267
Cylindrical section 269
d'Alembert's test 321
Damped oscillation 138
Darboux integral sums 293—298
Darboux theorem 298
de Moivre's formula 453
Decreasing functions 34 128
Definite integrals 216 218
Definite integrals, evaluation by means of primitive 224
Definite integrals, properties of 237
Definite integrals, relationship to indefinite integrals 222
derivatives 101 102
Derivatives of functions in parametric form 175
Derivatives of higher orders 121 405
Derivatives of implicit functions 164 410
Derivatives, partial 160 402
Derivatives, rules for obtaining 105 163 175 410
Descartes, folium of 177 181
Differences of function 126
Differential(s) 114
Differential(s) equations 117 234
Differential(s) of higher orders 125
Differential(s), geometrical significance of 115
Differential(s), total 162 402
Differential(s), total, of higher orders 408
Differential(s), use in approximations 120
Differentiation of arc 164 265 266 417
Differentiation of integral with respect to upper limit 244
Differentiation of uniformly convergent series 384
Differentiation, rules for 116 163 176 402—414
Discontinuities of functions 64 292
Discontinuities of integrand 245
Domain of definition of function, closed 158 398
Domain of definition of function, open 158 398
Double series 369
e (number) 74
e (number), approximate evaluation of 339
ellipse 257 345 478
Ellipsoid 270 273
Elliptic integrals 516
Empirical formulae 21
Epicycloid 190
Equations of curves 10 15 175 194 417 476
Equations of surfaces 417
Equations of third degree 491
Equations of third degree, solution by trigonometric method 494
Equivalence of infinitely large magnitudes 72
Equivalence of infinitesimals 72
Error, absolute 120
Error, relative 120
Euler formula 458
Euler substitutions 512
Euler theorem 404
Even functions 252
Exponential functions 35 90 111 338 458
Fermat's theorem 146
Finite increments, formula of 150
Functional relationships 8
Functions 7. See also special headings of function
Gauss's test 365 366
Geometric progression 316
Graph of function 15 172
Greatest value of function 140 430
Guldin's theorem 276 277
Harmonic curves 40
Harmonic oscillations in complex form 470 479
Harmonic series 318 326
homogeneous functions 404
Horner's rule 485
Hyperbolic functions 460—463
Hyperbolic spiral 196
Hypergeometric series 367
Hypocycloid 190
Imaginary unity 450
Implicit functions 10 164 410
Improper integral 249
Increasing functions 34 128
Increment of function 19
Increment of several variables 160 161
Increment of variable 18
Increment, partial 160
Increment, total 161—163
Indefinite integrals 213
Indeterminate forms 153 155 358
Infinitely large magnitudes 54
| Infinitesimals 44
Infinitesimals, comparison of 72
Infinitesimals, equivalence of 72
Infinitesimals, orders of 72
Infinitesimals, properties of 46 47
infinity 54
Inflexion, points of 167
Integrability of functions 299
Integral sum 294
Integral, definite 216
Integral, indefinite 213
Integral, test (Cauchy) 324
Integration of binomial differentials 512
Integration of exponential expressions 516—519
Integration of irrational expressions 230 231 282 283
Integration of rational fractions 232 233 508—511
Integration of trigonometric expressions 516—519
Integration of uniformly convergent series 383
Integration, rules for 227—230
Interval, closed, open 5
Inverse functions 31 108
Inverse functions, circular 41 108 111 352
Inverse functions, principal values of 43
Involute of circle 193
Irrational number 81
Isolated point of curve 184
Kepler's equation 59 130
Kummer's test 363
l'Hopital's rule 154 156
Lagrange form of remainder, Taylor's series 336
Lagrange formula 159
Lagrange method of undetermined multipliers 432
Least value of function 140 430
Leibniz's rule 123
Lemniscate 200
Limacon 197
Limit of function 61
Limit of function, double 400
Limit of function, of several variables 400
Limit of variable 49—52
Linear functions 17 20
Logarithm of complex number 469
Logarithm, natural 78
Logarithmic functions 37 93 106 348
Logarithmic scale 37
Logarithmic spiral 196 266 478—479
Maclaurin formula 337
Maclaurin series 337
Many-valued functions 33
Maxima and minima of functions 132—137
Maxima and minima of functions, of several variables 421
Maxima and minima of functions, of several variables, absolute 422 423
Maxima and minima of functions, of several variables, conditional 431
Maxima and minima of functions, rules for finding 133 135
Mean value theorem 242 243
Modulus of complex number 446
Modulus of system of logarithms 79
Multiple zeros of polynomial 383
Multipliers, undetermined, Lagrange method 432
n-dimensional space 399
Natural logarithms 79
Node 181
Normal to curve 184
Normal to surface 419
Odd functions 252
Open interval 5
Ordered variables 43
Osculation, point of 183
Ostrogradskii — Hermite formula 510
Ovals, Cassini's 200
parabola 23 25 256 265
Paraboloid, hyperbolic 427
Partial derivatives of first order 402
Partial derivatives of higher orders 405
Partial fractions 506
Passage to limit under differentiation sign 382
Passage to limit under integral sign 381
Polar equation of curve 194
Polynomial(s) 332 479—489
Polynomial(s) with real coefficients 489
Polynomial(s), relatively prime 488
Polytropic curves 29
Poncelet's formula 284
Primitives 213 244
Radius of curvature 169
Rational fraction, expansion into partial fractions 506
Real numbers 3 80
Real numbers, operations on 83
Reduction formula for integrals 254
Relative error 120
Remainder of series 316 327
Remainder term, Taylor's formula 335
Revolution, surface of 271
Riemann integral 299
Rolle's theorem 147
Roots of complex numbers 455
Section of numerical set 80
Semi-open interval 5 6
Semicubical parabola 181
Separation of algebraic part from integral 509
Sequence of functions (infinite) 374—377
Series, absolutely convergent 328 359 372
Series, alternating 327
Series, convergent 315
Series, divergent 315
Series, double 369
Series, Maclaurin 337
Series, power 386
Series, Taylor 337
Series, trigonometric 374
Series, uniformly convergent 374
Set, numerical, bounded above, below 79 85
Single-valued functions 34
Singular points of curve 180
Sinusoidal quantity 470
Slope of straight line 18
Slope of tangent 103
spirals 196
Strict bounds of numerical set 85
Subnormal 185
Substitution by Euler's method 230 512
Substitution by rational-fractional method 511 512
Substitution by trigonometric method 516
Substitution, change of variables by 230
Subtangent 185
Sum of series 315
Tabular method of specifying function 11
Tangent 184
Tangent formula 281
Tangent method 502
Tangent plane 417
Tests for convergence of series, Abel's 384
Tests for convergence of series, alternating 327
Tests for convergence of series, Cauchy's 320
Tests for convergence of series, d'Alembert's 321
Tests for convergence of series, Gauss's 365
Tests for convergence of series, integral 324
Tests for convergence of series, Kummer's 363
Tests for convergence of series, Weierstrass's 384
Tests for convergence of series, with positive terms 319
Torus 278
Total differential 162 402
Total differential of function of function 404
Transcendental curves 184
Trifolium 260
Trigonometric functions 38 93 105 106 108 460
Trigonometric functions, expanded into power series 340
Trochoid 189
Undetermined coefficients, method of 232 507
Unicursal curves 516
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