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Mathematicsby A. D. Aleksandrov
Synopses & ReviewsPublisher Comments:Hailed by The New York Times Book Review as "nothing less than a major contribution to the scientific culture of this world," this major survey features the work of 18 outstanding mathematicians. Primary subjects include analytic geometry, algebra, ordinary and partial differential equations, the calculus of variations, functions of a complex variable, prime numbers, and theories of probability and functions. Other topics include linear and nonEuclidean geometry, topology, functional analysis, more. 1963 edition. Book News Annotation:This edition reprints in one volume the second edition of this title, which was published in three volumes by The MIT Press in 1969. The original edition was published in 1964, translated from the Russian. Eighteen Russian mathematicians survey the scope of math, from elementary to the advanced levels, writing to educate a lay audience—those with "secondary school mathematics"—who are motivated to know more. Discussion includes both the origins and the development of analytic geometry, algebra, ordinary differential equations, partial differential equations, curve and surface theories, prime numbers, probability, functions of a complex variable, linear algebra, nonEuclidean geometry, topology, functional analysis, and groups and other algebraic systems.
Annotation c. Book News, Inc., Portland, OR (booknews.com) Synopsis:Major survey offers comprehensive, coherent discussions of analytic geometry, algebra, differential equations, calculus of variations, functions of a complex variable, prime numbers, linear and nonEuclidean geometry, topology, functional analysis, more. 1963 edition. Synopsis:This major survey features the work of 18 outstanding mathematicians. Primary subjects include analytic geometry, algebra, ordinary and partial differential equations, the calculus of variations, functions of a complex variable, prime numbers, and theories of probability and functions. Other topics include linear and nonEuclidean geometry, topology, functional analysis, more. 1963 edition. About the AuthorThe 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 doityourself 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 Table of ContentsVolume 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 seconddegree equations 5. Descartes' method of solving third and fourthdegree algebraic equations 6. Newton's general theory of diameters 7. Ellipse, hyperbola, and parabola 8. The reduction of the general seconddegree 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. Initialvalue and boundaryvalue 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 highspeed electronic machines 3. Technical principles of the various units of a highspeed 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. NonEuclidean 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. Manydimensional 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. ndimensional space 2. Hilbert space (Infinitedimensional 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. 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