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Courant R., Robbins H. Ч What Is Mathematics?: An Elementary Approach to Ideas and Methods
Courant R., Robbins H. Ч What Is Mathematics?: An Elementary Approach to Ideas and Methods

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Ќазвание: What Is Mathematics?: An Elementary Approach to Ideas and Methods

јвторы: Courant R., Robbins H.


Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Brought up to date with a new chapter by Ian Stewart, this second edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

»здание: 2nd

√од издани€: 1996

 оличество страниц: 566

ƒобавлена в каталог: 12.06.2007

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
"Infinitely small"      433Ч436 518Ч523
Absolute value      57
Acceleration      425
Addition, of complex numbers      90
Addition, of natural numbers      1Ч3
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      101Ч103 269Ч271
Algebra, of number fields      117Ч140
Algebra, of sets      108Ч116 494Ч495
Algebraic equations      101Ч103 269Ч271
Algebraic numbers      103Ч104
Algorithm and formula for primes      487
Algorithm and Steiner's problem      508Ч512
Algorithm, definition of      44
Algorithm, Euclidean      42Ч51 494 519
Algorithmic complexity theory      509Ч512
Analysis, nonstandard      518Ч523
Analytic geometry      72Ч77 191Ч196 488Ч494
Analytic geometry, of n dimensions      228Ч230
Angle of complex number      94
Antecedent point of mapping      141
Appollonius' problem      117 125Ч127 161Ч162
Archimedes' trisection of the angle      138 519Ч520
Area      399Ч401 464Ч465
Arithmetic, fundamental theorem of      23 46Ч48
Arithmetic, laws of      1Ч4
Arithmetical mean      361Ч365
Arithmetical progressions      12Ч13 487
Arithmetical progressions, primes in      26Ч27
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      75Ч76
Axes of coordinates      73
Axiomatics      214Ч217
Axioms      214Ч217 494 519
Bicontinuous (= continuous in both directions)      241
Binomial series      475Ч476
Binomial theorem      16Ч18
Bisection of segment with compass alone      145
Biunique correspondence      78
Bolzano's theorem      312Ч313
Bolzano's theorem, applications of      317Ч321 506
Boolean algebra      114
Boundary conditions in extremum problems      376Ч379 506Ч507
Bounded sequence      295
Brachistochrone problem      379Ч381 383Ч384
Brianchon's theorem      190 191 209Ч212
Calculus      398Ч486 502Ч510 523
Calculus of variations      379Ч385
Calculus, Fundamental Theorem of      436Ч439
Cantor's middle thirds set      248Ч249 501
Cantor's theory of infinite sets      77Ч86 494
cardinal number      83Ч86 493
Cartesian coordinates      72Ч74
Center of circle, compass construction of      146
Characteristic, Euler's      236Ч240 258Ч259 262
Circle, equation of      74
Classification (topological) of surfaces      256Ч264 502Ч503
Coaxial planes      176
Collinear points      170
Combinatorial geometry      230Ч234
Commutative laws, for natural numbers      2
Commutative laws, for rational numbers      51
Commutative laws, for sets      110
Compact sets      316
Compass constructions      145Ч146 147Ч151
Complement of a set      111
Complete quadrilateral      170Ч180
Complex conjugate      93
complex numbers      88Ч103
Complex numbers, absolute value of      93
Complex numbers, angle of      94
Complex numbers, modulus of      93
Complex numbers, operations with      90Ч91
Complex numbers, trigonometric representation of      95
Complex variable, theory of functions of a      478Ч479
Composite numbers      22
Compound functions      281Ч283
Compound interest      457
Concurrent lines      170
Congruence of geometrical figures      166
Congruences (arithmetical)      31Ч40
Conics      198Ч212
Conics, equations of      74Ч77
Conics, line      207
Conics, metric definition of      199 494 496
Conics, point      204
Conics, projective definition of      204
Conjugate, complex      93
Conjugate, harmonic      175Ч176
Connectivity      243Ч244
Constant      273
Constructions, geometrical      117Ч164
Constructions, Mascheroni      147Ч151
Constructions, of number fields      120Ч127
Constructions, of rational quantities      120Ч122
Constructions, of regular polygons      122Ч125
Constructions, of square roots      122
Constructions, with compass alone      145Ч146 147Ч152
Constructions, with straightedge alone      151Ч152 196Ч198
Constructions, with various instruments      140Ч164
Constructive numbers and number fields      127Ч134
Constructive numbers and number fields, definition      132Ч133
Constructive proof      86
Continued fractions      49Ч51 301Ч303
Continuity of function of one variable      283Ч280 310Ч312 327Ч328 423
Continuity of function of several variables      288
Continuous variable      274
Continuum Hypothesis      88 493Ч494
Continuum of real numbers      68
Continuum of real numbers, denumerability      79Ч80
Contour lines      286Ч287
Convergence, of sequences      294
Convergence, of series      472 478Ч479
Coordinates, general      192
Coordinates, homogeneous      193Ч196
Coordinates, rectangular (Cartesian)      73
Coplanar lines      176
Correspondence, for sets      78
Correspondence, projective      178 204
Counting      78
Cross-cap      261 262
Cross-ratio      172Ч180 185
Curvature, mean      386Ч387
Curve, elliptic      491Ч493
Curve, equation of      74Ч77
Curve, length of      466Ч469
Cut (in real number system)      71Ч72
Cycloids      152Ч155 380Ч381
Cyclotomic equation      99Ч100
Damped oscillations      459
De Moivre's theorem      96 98Ч100
Decagon, construction of regular      122Ч123
Decimal fractions      61Ч63
Dedekind cut      71Ч72
Deformation      242
Delta ($\Delta$)      402
Density of rational numbers      58
Denumerability of rational numbers      79Ч80
Dependent variable      275
Derivative      414Ч433
Derivative, second      426 435
Desargues's theorem      170Ч172 187Ч188
Differential equations      453Ч461
Differential quotient      434
Differentials      433Ч436 518
Differentiation      417 423 427Ч433 462Ч464
DIMENSION      248Ч251 499Ч501
Diophantine equations      50Ч51 487 491
Diriehlet's principle      368
Discontinuities of functions      261Ч280
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 134Ч135 146 147
Duality, principle of, in algebra of sets      112
Duality, principle of, in geometry      191 193Ч196 209 217
Duodecimal system      6
Dyadic system      8
Dynamics, Newtonian      460Ч461 506
e, as base of natural logarithms      445
e, as limit      448Ч450
e, Euler's number      297Ч299
e, expressions for      298 302 303
e, irrationality of      298Ч299
Eccentricity (of conics)      75
Ecole Polytechnique      167
Ellipse, equation of      75
Ellipse, tangent properties of      333Ч334
Elliptic curves, theory of      491Ч493
Elliptic geometry      224Ч227 489Ч490
Elliptic points      226
Empirical induction      10
Empty set      18 494
Epicycloid      155
Equation, cyclotomic      99Ч100
Equation, Diophantine      50Ч55 487 491
Equation, multiplicity of roots of      102
Equation, of a curve      74Ч77
Equation, of circle      74
Equation, of ellipse      75 404
Equation, of hyperbola      75Ч76 494
Equation, of straight line      75 490Ч492
Equation, quadratic      91Ч92 302
Equation, roots of      101
Equations of motion      460Ч461
Equivalence of sets      78
Eratosthenes, sieve of      25 489
Ergodic motion      353Ч854
Erlanger Program      167
Euclidean algorithm      42Ч51 494 519
Euler's characteristic      236Ч240 258Ч259 262 496Ч497
Euler's phi-function      48 49
Excluded middle, law of      86
Exhaustion, method of      400
Existence proofs      86 366Ч373
Existence, mathematical      88
Experimental solution of minimum problems      385Ч397
Exponential function      440Ч447 449Ч450
Exponential function, differential equation of      454Ч467
Exponential function, order of magnitude of      469Ч470
Extension field      129
Extraction of square root, geometrical      122
Extrema and inequalities      361Ч366
Extreme distances to given curve      336Ч338
Extremum problems      329Ч397
Extremum problems in elementary geometry      330Ч338
Extremum problems with boundary conditions      376Ч379 506Ч507
Extremum problems, general principle in      338Ч341
Factorial n      17
Factorization, unique      23 46Ч48
Factors, prime      23
Fermat numbers      25 119
Fermat's last theorem      40Ч42 491Ч493
Fermat's principle      381Ч383
Fermat's theorem      37Ч38 49
Field      56
Fields, algebra of number      117Ч140
Fields, geometric construction of      120Ч127
Five color theorem      261Ч267 495Ч499
Fixed point theorem      251Ч255
Focus of conic      75
Formalism      88 215
Foundations of mathematics      87Ч88
Four color problem      246Ч248 496Ч499
Fractals      499Ч501
Fractions, continued      49Ч51 301Ч303
Fractions, decimal      61Ч63
Frey's elliptic curves      491Ч492 493
Functions (and limits)      272Ч328
Functions of a complex variable      478Ч479
Functions of several variables      286Ч288
Functions, compound      282Ч283
Functions, continuity of      283Ч286 288 310Ч312 327Ч328
Functions, convex      500
Functions, definition of      274
Functions, graphs of      278
Functions, inverse      278Ч281
Functions, monotone      280
Functions, primitive      438
Fundamental theorem of algebra      101Ч103 269Ч271
Fundamental Theorem of Arithmetic      23 46Ч48
Fundamental theorem of the calculus      436Ч439
Generalization, principle of      56
Genus of surface      256Ч258 262
Geodesic      226
Geodesics on a sphere      384Ч385
Geometric measure theory      518
Geometrical constructions, theory of      117Ч164
Geometrical mean      361Ч365
Geometrical progression      13Ч14
Geometrical series      65Ч66
Geometrical transformations      140Ч141 165Ч167
Geometry, analytic      72Ч77 191Ч196 488Ч494
Geometry, axioms in      214Ч217
Geometry, combinatorial      230Ч234
Geometry, elliptic      224Ч227 489Ч490
Geometry, extremum problems in elementary      330Ч338
Geometry, hyperbolic      218Ч224
Geometry, inversion      140Ч146 158Ч164
Geometry, n-dimensional      227Ч234
Geometry, non-Euclidean      218Ч227
Geometry, projective      165Ч214
Geometry, Riemannian      224Ч227 489Ч490
Geometry, synthetic      165
Geometry, theory of constructions in      117Ч164 196Ч198
Geometry, topological      235Ч272 502Ч503
Goldbach's theorem      30Ч231 488Ч490
Graph of a function      279
Greatest common divisor      413Ч445
Greek problems, three famous      117 134Ч140
Group      168
Growth, law of      457
Harmonic conjugate      175Ч176
Harmonic cross-ratio      175Ч176
Harmonic series      479Ч480
Hart's inventor      157Ч158
Hausdorff dimension      499Ч501
Heptagon, impossibility of constructing regular      138Ч139
Heron's theorem      330Ч332
Hexagon, construction of regular      123
Homfly polynomial      505
Homogeneous coordinates      193Ч296
Hyperbola, equations of      75Ч76
Hyperbola, tangent properties of      334Ч336
Hyperbolic functions      503Ч504
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