A. D. Aleksandrov & A. N. Kolmogorov 
Mathematics 
Its Content, Methods and Meaning

‘. . . 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 non Euclidian 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

Table of Content

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 reading

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. Krylov)

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 groups

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

About the author

The Russian Equation Representative of the tremendous impact which Russian mathematicians have had on the Dover list since the Sputnik era is this outstanding book edited by A. D. Aleksandrov and others.

Critical Acclaim for Mathematics: Its Content, Methods and Meaning:’In effect, these volumes present a do-it-yourself course for the person who would like to know what the chief fields of modern mathematics are all about but who does not aspire to be a professional mathematician or a professional user of mathematics. The coverage is extremely wide, including such important areas as linear algebra, group theory, functional analysis, ordinary and partial differential equations, the theory of functions of real and complex variables, and related subjects. . . . What makes these volumes so readable as compared with usual mathematics textbooks is the emphasis here upon basic concepts and results rather than upon the intricate and wearying proofs that make such demands in conventional textbooks and courses. There are proofs in these volumes, but usually they are presented only for the most important results, and even then to emphasize key areas and to illustrate the kind of methodology employed. . . . It is hard to imagine that any intelligent American with a curious mind and some good recollection of his high school and college mathematics would not find many entrancing discoveries in the intellectual gold mine that is this work.’ — The New York Times Book Review

‘An excellent reference set for bright high school students and beginning college students . . . also of value to their teachers for lucid discussions and many good elementary examples in both familiar and unfamiliar branches. The intelligentsia of laymen who care to tackle more than today’s popular magazine articles on mathematics will find many rewarding introductions to subjects of current interest.’ — The Mathematics Teacher

‘Whether a physicist wishes to know what a Lie algebra is or how it is related to a Lie group, or an undergraduate would like to begin the study of homology, or a crystallographer is interested in Fedorov groups, or an engineer in probability, or any scientist in computing machines, he will find here a connected, lucid account.’ — Science