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| Courant R., Robbins H. — What Is Mathematics?: An Elementary Approach to Ideas and Methods |
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| Ïðåäìåòíûé óêàçàòåëü |
"Infinitely small" 433—436 518—523
Absolute value 57
Acceleration 425
Addition, of complex numbers 90
Addition, of natural numbers 1—3
Addition, of rational numbers 53
Addition, of real numbers 70
Addition, of sets 110
Adjunction of irrationals 132
Alexander polynomial 503 505
Algebra, boolean 114
Algebra, fundamental theorem of 101—103 269—271
Algebra, of number fields 117—140
Algebra, of sets 108—116 494—495
Algebraic equations 101—103 269—271
Algebraic numbers 103—104
Algorithm and formula for primes 487
Algorithm and Steiner's problem 508—512
Algorithm, definition of 44
Algorithm, Euclidean 42—51 494 519
Algorithmic complexity theory 509—512
Analysis, nonstandard 518—523
Analytic geometry 72—77 191—196 488—494
Analytic geometry, of n dimensions 228—230
Angle of complex number 94
Antecedent point of mapping 141
Appollonius' problem 117 125—127 161—162
Archimedes' trisection of the angle 138 519—520
Area 399—401 464—465
Arithmetic, fundamental theorem of 23 46—48
Arithmetic, laws of 1—4
Arithmetical mean 361—365
Arithmetical progressions 12—13 487
Arithmetical progressions, primes in 26—27
Associative laws, for natural numbers 2
Associative laws, for rational numbers 54
Associative laws, for sets 110
Asymptotes of hyperbola 76
Asymptotically equal 29
Axes of conics 75—76
Axes of coordinates 73
Axiomatics 214—217
Axioms 214—217 494 519
Bicontinuous (= continuous in both directions) 241
Binomial series 475—476
Binomial theorem 16—18
Bisection of segment with compass alone 145
Biunique correspondence 78
Bolzano's theorem 312—313
Bolzano's theorem, applications of 317—321 506
Boolean algebra 114
Boundary conditions in extremum problems 376—379 506—507
Bounded sequence 295
Brachistochrone problem 379—381 383—384
Brianchon's theorem 190 191 209—212
Calculus 398—486 502—510 523
Calculus of variations 379—385
Calculus, Fundamental Theorem of 436—439
Cantor's middle thirds set 248—249 501
Cantor's theory of infinite sets 77—86 494
cardinal number 83—86 493
Cartesian coordinates 72—74
Center of circle, compass construction of 146
Characteristic, Euler's 236—240 258—259 262
Circle, equation of 74
Classification (topological) of surfaces 256—264 502—503
Coaxial planes 176
Collinear points 170
Combinatorial geometry 230—234
Commutative laws, for natural numbers 2
Commutative laws, for rational numbers 51
Commutative laws, for sets 110
Compact sets 316
Compass constructions 145—146 147—151
Complement of a set 111
Complete quadrilateral 170—180
Complex conjugate 93
complex numbers 88—103
Complex numbers, absolute value of 93
Complex numbers, angle of 94
Complex numbers, modulus of 93
Complex numbers, operations with 90—91
Complex numbers, trigonometric representation of 95
Complex variable, theory of functions of a 478—479
Composite numbers 22
Compound functions 281—283
Compound interest 457
Concurrent lines 170
Congruence of geometrical figures 166
Congruences (arithmetical) 31—40
Conics 198—212
Conics, equations of 74—77
Conics, line 207
Conics, metric definition of 199 494 496
Conics, point 204
Conics, projective definition of 204
Conjugate, complex 93
Conjugate, harmonic 175—176
Connectivity 243—244
Constant 273
Constructions, geometrical 117—164
Constructions, Mascheroni 147—151
Constructions, of number fields 120—127
Constructions, of rational quantities 120—122
Constructions, of regular polygons 122—125
Constructions, of square roots 122
Constructions, with compass alone 145—146 147—152
Constructions, with straightedge alone 151—152 196—198
Constructions, with various instruments 140—164
Constructive numbers and number fields 127—134
Constructive numbers and number fields, definition 132—133
Constructive proof 86
Continued fractions 49—51 301—303
Continuity of function of one variable 283—280 310—312 327—328 423
Continuity of function of several variables 288
Continuous variable 274
Continuum Hypothesis 88 493—494
Continuum of real numbers 68
Continuum of real numbers, denumerability 79—80
Contour lines 286—287
Convergence, of sequences 294
Convergence, of series 472 478—479
Coordinates, general 192
Coordinates, homogeneous 193—196
Coordinates, rectangular (Cartesian) 73
Coplanar lines 176
Correspondence, for sets 78
Correspondence, projective 178 204
Counting 78
Cross-cap 261 262
Cross-ratio 172—180 185
Curvature, mean 386—387
Curve, elliptic 491—493
Curve, equation of 74—77
Curve, length of 466—469
Cut (in real number system) 71—72
Cycloids 152—155 380—381
Cyclotomic equation 99—100
Damped oscillations 459
De Moivre's theorem 96 98—100
Decagon, construction of regular 122—123
Decimal fractions 61—63
Dedekind cut 71—72
Deformation 242
Delta ( ) 402
Density of rational numbers 58
Denumerability of rational numbers 79—80
Dependent variable 275
Derivative 414—433
Derivative, second 426 435
| Desargues's theorem 170—172 187—188
Differential equations 453—461
Differential quotient 434
Differentials 433—436 518
Differentiation 417 423 427—433 462—464
DIMENSION 248—251 499—501
Diophantine equations 50—51 487 491
Diriehlet's principle 368
Discontinuities of functions 261—280
Discontinuous functions as limits of continuous functions 325
Distance 74 316
Distributive laws, for natural numbers 2
Distributive laws, for rational numbers 54
Distributive laws, for sets 110
Divergence, of sequences 204
Divergence, of series 472
Division by zero excluded 56 90
Domain of a variable 273
Doubling of the cube 117 134—135 146 147
Duality, principle of, in algebra of sets 112
Duality, principle of, in geometry 191 193—196 209 217
Duodecimal system 6
Dyadic system 8
Dynamics, Newtonian 460—461 506
e, as base of natural logarithms 445
e, as limit 448—450
e, Euler's number 297—299
e, expressions for 298 302 303
e, irrationality of 298—299
Eccentricity (of conics) 75
Ecole Polytechnique 167
Ellipse, equation of 75
Ellipse, tangent properties of 333—334
Elliptic curves, theory of 491—493
Elliptic geometry 224—227 489—490
Elliptic points 226
Empirical induction 10
Empty set 18 494
Epicycloid 155
Equation, cyclotomic 99—100
Equation, Diophantine 50—55 487 491
Equation, multiplicity of roots of 102
Equation, of a curve 74—77
Equation, of circle 74
Equation, of ellipse 75 404
Equation, of hyperbola 75—76 494
Equation, of straight line 75 490—492
Equation, quadratic 91—92 302
Equation, roots of 101
Equations of motion 460—461
Equivalence of sets 78
Eratosthenes, sieve of 25 489
Ergodic motion 353—854
Erlanger Program 167
Euclidean algorithm 42—51 494 519
Euler's characteristic 236—240 258—259 262 496—497
Euler's phi-function 48 49
Excluded middle, law of 86
Exhaustion, method of 400
Existence proofs 86 366—373
Existence, mathematical 88
Experimental solution of minimum problems 385—397
Exponential function 440—447 449—450
Exponential function, differential equation of 454—467
Exponential function, order of magnitude of 469—470
Extension field 129
Extraction of square root, geometrical 122
Extrema and inequalities 361—366
Extreme distances to given curve 336—338
Extremum problems 329—397
Extremum problems in elementary geometry 330—338
Extremum problems with boundary conditions 376—379 506—507
Extremum problems, general principle in 338—341
Factorial n 17
Factorization, unique 23 46—48
Factors, prime 23
Fermat numbers 25 119
Fermat's last theorem 40—42 491—493
Fermat's principle 381—383
Fermat's theorem 37—38 49
Field 56
Fields, algebra of number 117—140
Fields, geometric construction of 120—127
Five color theorem 261—267 495—499
Fixed point theorem 251—255
Focus of conic 75
Formalism 88 215
Foundations of mathematics 87—88
Four color problem 246—248 496—499
Fractals 499—501
Fractions, continued 49—51 301—303
Fractions, decimal 61—63
Frey's elliptic curves 491—492 493
Functions (and limits) 272—328
Functions of a complex variable 478—479
Functions of several variables 286—288
Functions, compound 282—283
Functions, continuity of 283—286 288 310—312 327—328
Functions, convex 500
Functions, definition of 274
Functions, graphs of 278
Functions, inverse 278—281
Functions, monotone 280
Functions, primitive 438
Fundamental theorem of algebra 101—103 269—271
Fundamental Theorem of Arithmetic 23 46—48
Fundamental theorem of the calculus 436—439
Generalization, principle of 56
Genus of surface 256—258 262
Geodesic 226
Geodesics on a sphere 384—385
Geometric measure theory 518
Geometrical constructions, theory of 117—164
Geometrical mean 361—365
Geometrical progression 13—14
Geometrical series 65—66
Geometrical transformations 140—141 165—167
Geometry, analytic 72—77 191—196 488—494
Geometry, axioms in 214—217
Geometry, combinatorial 230—234
Geometry, elliptic 224—227 489—490
Geometry, extremum problems in elementary 330—338
Geometry, hyperbolic 218—224
Geometry, inversion 140—146 158—164
Geometry, n-dimensional 227—234
Geometry, non-Euclidean 218—227
Geometry, projective 165—214
Geometry, Riemannian 224—227 489—490
Geometry, synthetic 165
Geometry, theory of constructions in 117—164 196—198
Geometry, topological 235—272 502—503
Goldbach's theorem 30—231 488—490
Graph of a function 279
Greatest common divisor 413—445
Greek problems, three famous 117 134—140
Group 168
Growth, law of 457
Harmonic conjugate 175—176
Harmonic cross-ratio 175—176
Harmonic series 479—480
Hart's inventor 157—158
Hausdorff dimension 499—501
Heptagon, impossibility of constructing regular 138—139
Heron's theorem 330—332
Hexagon, construction of regular 123
Homfly polynomial 505
Homogeneous coordinates 193—296
Hyperbola, equations of 75—76
Hyperbola, tangent properties of 334—336
Hyperbolic functions 503—504
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