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 |