Volume 1. Part 1 |

Chapter 1. A general view of mathematics (A.D. Aleksandrov) |

1. The characteristic features of mathematics |

2. Arithmetic |

3. Geometry |

4. Arithmetic and geometry |

5. The age of elementary mathematics |

6. Mathematics of variable magnitudes |

7. Contemporary mathematics |

Suggested reading |

Chapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii) |

1. Introduction |

2. Function |

3. Limits |

4. Continuous functions |

5. Derivative |

6. Rules for differentiation |

7. Maximum and minimum; investigation of the graphs of functions |

8. Increment and differential of a function |

9. Taylor's formula |

10. Integral |

11. Indefinite integrals; the technique of integration |

12. Functions of several variables |

13. Generalizations of the concept of integral |

14. Series |

Suggested reading |

Part 2. |

Chapter 3. Analytic Geometry (B. N. Delone) |

1. Introduction |

2. Descartes' two fundamental concepts |

3. Elementary problems |

4. Discussion of curves represented by first- and second-degree equations |

5. Descartes' method of solving third- and fourth-degree algebraic equations |

6. Newton's general theory of diameters |

7. Ellipse, hyperbola, and parabola |

8. The reduction of the general second-degree equation to canonical form |

9. The representation of forces, velocities, and accelerations by triples of numbers; theory of vectors |

10. Analytic geometry in space; equations of a surface in space and equations of a curve |

11. Affine and orthogonal transformations |

12. Theory of invariants |

13. Projective geometry |

14. Lorentz transformations |

Conclusions; Suggested reading |

Chapter 4. Algebra: Theory of algebraic equations (B. N. Delone) |

1. Introduction |

2. Algebraic solution of an equation |

3. The fundamental theorem of algebra |

4. Investigation of the distribution of the roots of a polynomial on the complex plane |

5. Approximate calculation of roots |

Suggested reading |

Chapter 5. Ordinary differential equations (I. G. Petrovskii) |

1. Introduction |

2. Linear differential equations with constant coefficients |

3. Some general remarks on the formation and solution of differential equations |

4. Geometric interpretation of the problem of integrating differential equations; generalization of the problem |

5. Existence and uniqueness of the solution of a differential equation; approximate solution of equations |

6. Singular points |

7. Qualitative theory of ordinary differential equations |

Suggested re |

Volume 2 Part 3 |

Chapter 6. Partial differential equations (S. L. Sobolev and O. A. Ladyzenskaja) |

1. Introduction |

2. The simplest equations of mathematical physics |

3. Initial-value and boundary-value problems; uniqueness of a solution |

4. The propagation of waves |

5. Methods of constructing solutions |

6. Generalized solutions |

Suggested reading |

Chapter 7. Curves and surfaces (A. D. Aleksandrov) |

1. Topics and methods in the theory of curves and surfaces |

2. The theory of curves |

3. Basic concepts in the theory of surfaces |

4. Intrinsic geometry and deformation of surfaces |

5. New Developments in the theory of curves and surfaces |

Suggested reading |

Chapter 8. The calculus of variations (V. I. Krylov) |

1. Introduction |

2. The differential equations of the calculus of variations |

3. Methods of approximate solution of problems in the calculus of variations |

Suggested reading |

Chapter 9. Functions of a complex variable (M. V. Keldys) |

1. Complex numbers and functions of a complex variable |

2. The connection between functions of a complex variable and the problems of mathematical physics |

3. The connection of functions of a complex variable with geometry |

4. The line integral; Cauchy's formula and its corollaries |

5. Uniqueness properties and analytic continuation |

6. Conclusion |

Suggested reading |

Part 4. |

Chapter 10. Prime numbers (K. K. Mardzanisvili and A. B. Postnikov) |

1. The study of the theory of numbers |

2. The investigation of problems concerning prime numbers |

3. Chebyshev's method |

4. Vinogradov's method |

5. Decomposition of integers into the sum of two squares; complex integers |

Suggested reading |

Chapter 11. The theory of probability (A. N. Kolmogorov) |

1. The laws of probability |

2. The axioms and basic formulas of the elementary theory of probability |

3. The law of large numbers and limit theorems |

4. Further remarks on the basic concepts of the theory of probability |

5. Deterministic and random processes |

6. Random processes of Markov type |

Suggested reading |

Chapter 12. Approximations of functions (S. M. Nikol'skii) |

1. Introduction |

2. Interpolation polynomials |

3. Approximation of definite integrals |

4. The Chebyshev concept of best uniform approximation |

5. The Chebyshev polynomials deviating least from zero |

6. The theorem of Weierstrass; the best approximation to a function as related to its properties of differentiability |

7. Fourier series |

8. Approximation in the sense of the mean square |

Suggested reading |

Chapter 13. Approximation methods and computing techniques (V. I. Kr |

1. Approximation and numerical methods |

2. The simplest auxiliary means of computation |

Suggested reading |

Chapter 14. Electronic computing machines (S. A. Lebedev and L. V. Kantorovich) |

1. Purposes and basic principles of the operation of electronic computers |

2. Programming and coding for high-speed electronic machines |

3. Technical principles of the various units of a high-speed computing machine |

4. Prospects for the development and use of electronic computing machines |

Suggested reading |

Volume 3. Part 5. |

Chapter 15. Theory of functions of a real variable (S. B. Stechkin) |

1. Introduction |

2. Sets |

3. Real Numbers |

4. Point sets |

5. Measure of sets |

6. The Lebesque integral |

Suggested reading |

Chapter 16. Linear algebra (D. K. Faddeev) |

1. The scope of linear algebra and its apparatus |

2. Linear spaces |

3. Systems of linear equations |

4. Linear transformations |

5. Quadratic forms |

6. Functions of matrices and some of their applications |

Suggested reading |

Chapter 17. Non-Euclidean geometry (A. D. Aleksandrov) |

1. History of Euclid's postulate |

2. The solution of Lobachevskii |

3. Lobachevskii geometry |

4. The real meaning of Lobachevskii geometry |

5. The axioms of geometry; their verification in the present case |

6. Separation of independent geometric theories from Euclidean geometry |

7. Many-dimensional spaces |

8. Generalization of the scope of geometry |

9. Riemannian geometry |

10. Abstract geometry and the real space |

Suggested reading |

Part 6. |

Chapter 18. Topology (P. S. Aleksandrov) |

1. The object of topology |

2. Surfaces |

3. Manifolds |

4. The combinatorial method |

5. Vector fields |

6. The development of topology |

7. Metric and topological space |

Suggested reading |

Chapter 19. Functional analysis (I. M. Gelfand) |

1. n-dimensional space |

2. Hilbert space (Infinite-dimensional space)< |

4. Integral equations |

5. Linear operators and further developments of functional analysis |

Suggested reading |

Chapter 20. Groups and other algebraic systems (A. I. Malcev) |

1. Introduction |

2. Symmetry and transformations |

3. Groups of transformations |

4. Fedorov groups (crystallographic groups) |

5. Galois groups |

6. Fundamental concepts of the general theory of groups |

7. Continuous groups |

8. Fundamental groups |

9. Representations and characters of g |

10. The general theory of groups |

11. Hypercomplex numbers |

12. Associative algebras |

13. Lie algebras |

14. Rings |

15. Lattices |

16. Other algebraic systems |

Suggested reading |

Index |