I think a table of contents would be useful. I use this book as a supplement to the "Princeton Companion to Mathematics", a companion companion as it were, um . . . companion squared? If we differentiate (companion)(companion) we get 2companion or "Companion Le Deaux".
VOLUME ONE, PART 1 CHAPTER I 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 II ANALYSIS M. A. Lavrent’ev and S. M. Nikol’ski?
§ 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 III 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 Thirdand 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. A fine and Orthogonal Transformations
§ 12. Theory of Invariants
§ 13. Projective Geometry
§ 14. Lorentz Transformations Conclusion
Suggested Reading
CHAPTER IV 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 V ORDINARY DIFFERENTIAL EQUATIONS I. G. Petrovski?
§ 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 TWO, PART 3 CHAPTER VI 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 VII CURVES AND SURFACES A. D. Aleksandrov
§ 1. Topics and Methods in the Theory
§ 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 VIII 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 IX FUNCTIONS OF A COMPLEX VARIABLE M. V. Keldyš
§ 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 X 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. ?ebyšev’s Method
§ 4. Vinogradov’s Method
§ 5. Decomposition of Integers into the Sum of Two Squares; Complex Integers
Suggested Reading
CHAPTER XI 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 XII APPROXIMATIONS OF FUNCTIONS S. M. Nikol? ski?
§ 1. Introduction
§ 2. Interpolation Polynomials
§ 3. Approximation of Definite Integrals
§ 4. The ?ebyšev(Chebyshev) Concept of Best Uniform Approximation
§ 5. The ?ebyšev(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 XIII APPROXIMATION METHODS AND COMPUTING TECHNIQUES V. I. Krylov
§ 1. Approximation and Numerical Methods
§ 2. The Simplest Auxiliary Means of Computation
Suggested Reading
CHAPTER XIV ELECTRONIC COMPUTING MACHINES S. A. Lebedev and L. V. Kantorovi?
§ 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 THREE, PART 5 CHAPTER XV THEORY OF FUNCTIONS OF A REAL VARIABLE S. B. Ste?kin
§ 1. Introduction
§ 2. Sets
§ 3. Real Numbers
§ 4. Point Sets
§ 5. Measure of Sets
§ 6. The Lebesgue Integral
Suggested Reading
CHAPTER XVI 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 XVII NON-EUCLIDEAN GEOMETRY A. D. Aleksandrov
§ 1. History of Euclid’s Postulate
§ 2. The Solution of Loba?evski?
§ 3. Loba?evski? Geometry
§ 4. The Real Meaning of Loba?evski? 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 XVIII 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 Spaces
Suggested Reading
CHAPTER XIX FUNCTIONAL ANALYSIS I. M. Gel? fand
§ 1. n-Dimensional Space
§ 2. Hilbert Space (Infinite-Dimensional Space)
§ 3. Expansion by Orthogonal Systems of Functions
§ 4. Integral Equations
§ 5. Linear Operators and Further Developments of Functional Analysis
Suggested Reading
CHAPTER XX GROUPS AND OTHER ALGEBRAIC SYSTEMS A. I. Mal? cev
§ 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
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From the Publisher
Now in One Volume! A Complete Mathematical Journey
- A Complete Mathematical Journey – Covers everything from analytic geometry, algebra, and differential equations to complex variables, topology, group theory, and more.
- Now in One Volume – Originally published in three parts, this edition brings the full work together in one accessible, affordable volume ideal for study or reference.
- Organized for Learning – Part I introduces core topics like algebra, analytic geometry, and ordinary differential equations. Part II expands into partial differential equations, calculus of variations, complex functions, and probability theory. Part III explores real analysis, linear algebra, non-Euclidean geometry, topology, and algebraic structures.
- Rich Visuals – Includes numerous diagrams and illustrations to clarify abstract ideas and support visual learners.
- Further Reading Included – Each chapter ends with suggested reading for deeper exploration.
About the author
Including Material On Both Elementary And Advanced Levels
"This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and INCLUDING MATERIAL ON BOTH ELEMENTARY AND ADVANCED LEVELS,....." [from the book of the back cover]
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