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
". . . 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 nonEuclidian 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
Inhalt
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. Postni…
