". . . Nothing less than a major contribution to the scientific culture of this world." — The New York Times Book Review
This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated developement. As Professor Morris Kline of New York University noted, "This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."
Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It furthur examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and nonEuclidian geometry, topology, functional analysis, and groups and other algebraic systems.
Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.
"This is a masterful English translation of a stupendous and formidable mathematical masterpiece . . ." — Social Science
Author(s): A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev
Publisher: Dover Publications
Year: 1999
Language: English
Pages: 1120
Tags: Geometry & Topology;Algebraic Geometry;Analytic Geometry;Differential Geometry;Non-Euclidean Geometries;Topology;Mathematics;Science & Math;Algebra;Abstract;Elementary;Intermediate;Linear;Pure Mathematics;Mathematics;Science & Math;Calculus;Pure Mathematics;Mathematics;Science & Math
VOLUME ONE ... 2
PART I ... 11
CHAPTER I A GENERAL VIEW OF MATHEMATICS A. D. Aleksandrov ... 12
§ 1. The Characteristic Features of Mathematics ... 12
§ 2. Arithmetic ... 18
§ 3. Geometry ... 30
§ 4. Arithmetic and Geometry ... 35
§ 5. The Age of Elementary Mathematics ... 46
§ 6. Mathematics of Variable Magnitudes ... 54
§ 7. Contemporary Mathematics ... 66
Suggested Reading ... 75
CHAPTER II ANALYSIS M. A. Lavrent’ev and S. M. Nikol’ski? ... 76
§ 1. Introduction ... 76
§ 2. Function ... 84
§ 3. Limits ... 91
§ 4. Continuous Functions ... 99
§ 5. Derivative ... 103
§ 6. Rules for Differentiation ... 112
§ 7. Maximum and Minimum; Investigation of the Graphs of Functions ... 119
§ 8. Increment and Differential of a Function ... 128
§ 9. Taylor’s Formula ... 134
§ 10. Integral ... 139
§ 11. Indefinite Integrals; the Technique of Integration ... 148
§ 12. Functions of Several Variables ... 153
§ 13. Generalizations of the Concept of Integral ... 169
§ 14. Series ... 177
Suggested Reading ... 191
PART 2 ... 192
CHAPTER III ANALYTIC GEOMETRY B. N. Delone ... 194
§ 1. Introduction ... 194
§ 2. Descartes’ Two Fundamental Concepts ... 195
§ 3. Elementary Problems ... 197
§ 4. Discussion of Curves Represented by First-and Second-Degree Equations ... 199
§ 5. Descartes’ Method of Solving Thirdand Fourth-Degree Algebraic Equations ... 201
§ 6. Newton’s General Theory of Diameters ... 204
§ 7. Ellipse, Hyperbola, and Parabola ... 206
§ 8. The Reduction of the General Second-Degree Equation to Canonical Form ... 218
§ 9. The Representation of Forces, Velocities, and Accelerations by Triples of Numbers; Theory of Vectors ... 224
§ 10. Analytic Geometry in Space; Equations of a Surface in Space and Equations of a Curve ... 229
§ 11. A fine and Orthogonal Transformations ... 238
§ 12. Theory of Invariants ... 249
§ 13. Projective Geometry ... 253
§ 14. Lorentz Transformations Conclusion ... 260
Suggested Reading ... 270
CHAPTER IV ALGEBRA: THEORY OF ALGEBRAIC EQUATIONS B. N. Delone ... 272
§ 1. Introduction ... 272
§ 2. Algebraic Solution of an Equation ... 276
§ 3. The Fundamental Theorem of Algebra ... 291
§ 4. Investigation of the Distribution of the Roots of a Polynomial on the Complex Plane ... 303
§ 5. Approximate Calculation of Roots ... 313
Suggested Reading ... 320
CHAPTER V ORDINARY DIFFERENTIAL EQUATIONS I. G. Petrovski? ... 322
§ 1. Introduction ... 322
§ 2. Linear Differential Equations with Constant Coefficients ... 334
§ 3. Some General Remarks on the Formation and Solution of Differential Equations ... 341
§ 4. Geometric Interpretation of the Problem of Integrating Differential Equations; Generalization of the Problem ... 343
§ 5. Existence and Uniqueness of the Solution of a Differential Equation; Approximate Solution of Equations ... 346
§ 6. Singular Points ... 354
§ 7. Qualitative Theory of Ordinary Differential Equations ... 359
Suggested Reading ... 367
VOLUME TWO ... 371
PART 3 ... 380
CHAPTER VI PARTIAL DIFFERENTIAL EQUATIONS S. L. Sobolev and O. A. Ladyzenskaja ... 381
§ 1. Introduction ... 381
§ 2. The Simplest Equations of Mathematical Physics ... 383
§ 3. Initial-Value and Boundary-Value Problems; Uniqueness of a Solution ... 393
§ 4. The Propagation of Waves ... 403
§ 5. Methods of Constructing Solutions ... 405
§ 6. Generalized Solutions ... 426
Suggested Reading ... 432
CHAPTER VII CURVES AND SURFACES A. D. Aleksandrov ... 435
§ 1. Topics and Methods in the Theory ... 435
§ 2. Theory of Curves ... 439
§ 3. Basic Concepts in the Theory of Surfaces ... 454
§ 4. Intrinsic Geometry and Deformation of Surfaces ... 469
§ 5. New Developments in the Theory of Curves and Surfaces ... 486
Suggested Reading ... 495
CHAPTER VIII THE CALCULUS OF VARIATIONS V. I. Krylov ... 497
§ 1. Introduction ... 497
§ 2. The Differential Equations of the Calculus of Variations ... 502
§ 3. Methods of Approximate Solution of Problems in the Calculus of Variations ... 513
Suggested Reading ... 516
CHAPTER IX FUNCTIONS OF A COMPLEX VARIABLE M. V. Keldyš ... 517
§ 1. Complex Numbers and Functions of a Complex Variable ... 517
§ 2. The Connection Between Functions of a Complex Variable and the Problems of Mathematical Physics ... 531
§ 3. The Connection of Functions of a Complex Variable with Geometry ... 541
§ 4. The Line Integral; Cauchy’s Formula and Its Corollaries ... 552
§ 5. Uniqueness Properties and Analytic Continuation ... 565
§ 6. Conclusion ... 572
Suggested Reading ... 573
PART 4 ... 575
CHAPTER X PRIME NUMBERS K. K. Mardzanisvili and A. B. Postnikov ... 577
§ 1. The Study of the Theory of Numbers ... 577
§ 2. The Investigation of Problems Concerning Prime Numbers ... 582
§ 3. Cebyšev’s Method ... 589
§ 4. Vinogradov’s Method ... 595
§ 5. Decomposition of Integers into the Sum of Two Squares; Complex Integers ... 603
Suggested Reading ... 606
CHAPTER XI THE THEORY OF PROBABILITY A. N. Kolmogorov ... 607
§ 1. The Laws of Probability ... 607
§ 2. The Axioms and Basic Formulas of the Elementary Theory of Probability ... 609
§ 3. The Law of Large Numbers and Limit Theorems ... 616
§ 4. Further Remarks on the Basic Concepts of the Theory of Probability ... 625
§ 5. Deterministic and Random Processes ... 633
§ 6. Random Processes of Markov Type ... 638
Suggested Reading ... 642
CHAPTER XII APPROXIMATIONS OF FUNCTIONS S. M. Nikol? ski? ... 643
§ 1. Introduction ... 643
§ 2. Interpolation Polynomials ... 647
§ 3. Approximation of Definite Integrals ... 654
§ 4. The ?ebyšev(Chebyshev) Concept of Best Uniform Approximation ... 660
§ 5. The ?ebyšev(Chebyshev) Polynomials Deviating Least from Zero ... 663
§ 6. The Theorem of Weierstrass; the Best Approximation to a Function as Related to Its Properties of Differentiability ... 666
§ 7. Fourier Series ... 669
§ 8. Approximation in the Sense of the Mean Square ... 676
Suggested Reading ... 680
CHAPTER XIII APPROXIMATION METHODS AND COMPUTING TECHNIQUES V. I. Krylov ... 681
§ 1. Approximation and Numerical Methods ... 681
§ 2. The Simplest Auxiliary Means of Computation ... 697
Suggested Reading ... 707
CHAPTER XIV ELECTRONIC COMPUTING MACHINES S. A. Lebedev and L. V. Kantorovi? ... 709
§ 1. Purposes and Basic Principles of the Operation of Electronic Computers ... 709
§ 2. Programming and Coding for High-Speed Electronic Machines ... 714
§ 3. Technical Principles of the Various Units of a High-Speed Computing Machine ... 728
§ 4. Prospects for the Development and Use of Electronic Computing Machines ... 743
Suggested Reading ... 752
INDEX OF NAMES ... 753
VOLUME THREE ... 756
PART 5 ... 765
CHAPTER XV THEORY OF FUNCTIONS OF A REAL VARIABLE S. B. Ste?kin ... 766
§ 1. Introduction ... 766
§ 2. Sets ... 768
§ 3. Real Numbers ... 775
§ 4. Point Sets ... 781
§ 5. Measure of Sets ... 788
§ 6. The Lebesgue Integral ... 793
Suggested Reading ... 799
CHAPTER XVI LINEAR ALGEBRA D. K. Faddeev ... 800
§ 1. The Scope of Linear Algebra and Its Apparatus ... 800
§ 2. Linear Spaces ... 811
§ 3. Systems of Linear Equations ... 824
§ 4. Linear Transformations ... 837
§ 5. Quadratic Forms ... 847
§ 6. Functions of Matrices and Some of Their Applications ... 854
Suggested Reading ... 858
CHAPTER XVII NON-EUCLIDEAN GEOMETRY A. D. Aleksandrov ... 860
§ 1. History of Euclid’s Postulate ... 860
§ 2. The Solution of Loba?evski? ... 864
§ 3. Loba?evski? Geometry ... 868
§ 4. The Real Meaning of Loba?evski? Geometry ... 877
§ 5. The Axioms of Geometry; Their Verification in the Present Case ... 885
§ 6. Separation of Independent Geometric Theories from Euclidean Geometry ... 892
§ 7. Many-Dimensional Spaces ... 899
§ 8. Generalization of the Scope of Geometry ... 914
§ 9. Riemannian Geometry ... 927
§ 10. Abstract Geometry and the Real Space ... 941
Suggested Reading ... 952
PART 6 ... 954
CHAPTER XVIII TOPOLOGY P. S. Aleksandrov ... 956
§ 1. The Object of Topology ... 956
§ 2. Surfaces ... 960
§ 4. The Combinatorial Method ... 967
§ 5. Vector Fields ... 975
§ 6. The Development of Topology ... 981
§ 7. Metric and Topological Spaces ... 984
Suggested Reading ... 987
CHAPTER XIX FUNCTIONAL ANALYSIS I. M. Gel? fand ... 990
§ 1. n-Dimensional Space ... 990
§ 2. Hilbert Space (Infinite-Dimensional Space) ... 995
§ 3. Expansion by Orthogonal Systems of Functions ... 1000
§ 4. Integral Equations ... 1008
§ 5. Linear Operators and Further Developments of Functional Analysis ... 1015
Suggested Reading ... 1024
CHAPTER XX GROUPS AND OTHER ALGEBRAIC SYSTEMS A. I. Mal? cev ... 1026
§ 1. Introduction ... 1026
§ 2. Symmetry and Transformations ... 1027
§ 3. Groups of Transformations ... 1036
§ 4. Fedorov Groups (Crystallographic Groups) ... 1048
§ 5. Galois Groups ... 1056
§ 6. Fundamental Concepts of the General Theory of Groups ... 1060
§ 7. Continuous Groups ... 1068
§ 8. Fundamental Groups ... 1071
§ 9. Representations and Characters of Groups ... 1087
§ 10. The General Theory of Groups ... 1082
§ 11. Hypercomplex Numbers ... 1083
§ 12. Associative Algebras ... 1093
§ 13. Lie Algebras ... 1102
§ 14. Rings ... 1105
§ 15. Lattices ... 1110
§ 16. Other Algebraic Systems ... 1112
Suggested Reading ... 1114
INDEX OF NAMES ... 1116
