Mathematics: Its Content, Methods and Meaning

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