Learning Higher Mathematics: Part I: The Method of Coordinates Part II: Analysis of the Infinitely Small

ISBN-13: 9783540123514 / Angielski / Miękka / 1983 / 308 str.

L. S. Pontrjagin; E. Hewitt

Opis

Lev Semenovic Pontrjagin (1908) is one of the outstanding figures in 20th century mathematics. In a long career he has made fundamental con tributions to many branches of mathematics, both pure and applied. He has received every honor that a grateful government can bestow. Though in no way constrained to do so, he has through the years taught mathematics courses at Moscow State University. In the year 1975 he set himself the task of writing a series of books on secondary school and beginning university mathematics. In his own words, "I wished to set forth the foundations of higher mathematics in a form that would have been accessible to myself as a lad, but making use of all my experience as a scientist and a teacher, ac cumulated over many years. " The present volume is a translation of the first two out of four moderately sized volumes on this theme planned by Pro fessor Pontrjagin. The book begins at the beginning of modern mathematics, analytic ge ometry in the plane and 3-dimensional space. Refinements about limits and the nature of real numbers come only later. Many concrete examples are given; these may take the place of formal exercises, which the book does not provide. The book continues with careful treatment of differentiation and integration, of limits, of expansions of elementary functions in power se ries."

Spis treści

I The Method of Coordinates.- to Part I.- I. Coordinates in the Plane.- § 1. Rectangular Cartesian Coordinates and Vectors in the Plane.- §2. Polar Coordinates.- §3. Geometric Representation of Complex Numbers.- Supplement to Chapter I.- 1. Coordinates in Space (25). 2. Vectors in Space (26).- II. Coordinates and Lines in the Plane.- § 4. Graphs of Functions and Functions.- §5. Ellipses, Hyperbolas, and Parabolas.- § 6. Parametric Representation of Curves.- §7. Closed Curves.- § 8. Polynomials in a Complex Variable.- Supplement to Chapter II.- 1. Functions of Two Variables and their Graphs in Space (63)..- 2. Functions of Three Variables and the Surfaces that Correspond to them (64). 3. Surfaces of Revolution (66). 4. Equations of Planes (68). 5. Surfaces of the First and Second Orders (69).- III. Analytic Geometry in the Plane.- § 9. Transformations of Cartesian Coordinates in the Plane.- § 10. Curves of the First and Second Orders.- §11. Conic Sections.- Supplement to Chapter III.- 1. Transformation of Coordinates (96). 2. Classification of Surfaces of Orders 1 and 2 (99). 3. Conic Sections Revisited (102).- II Analysis of the Infinitely Small.- to Part II.- IV. Series.- § 12. Convergent Sequences of Numbers.- §13. Infinitely Small Quantities.- §14. Cauchy’s Convergence Criterion.- §15. Applications of Cauchy’s Convergence Criterion.- §16. Convergent Series.- §17. Absolutely Convergent Series.- §18. The Function exp(z).- §19. The Elementary Transcendental Functions.- §20. Power Series.- V. The Differential Calculus.- §21. The Derivative.- §22. Computing Derivatives.- §23. The Indefinite Integral.- § 24. Computation of Some Indefinite Integrals.- §25. The Definite Integral.- §26. Taylor Series.- VI. The Integral Calculus.- §27. The Definite Integral as an Area.- §28. The Definite Integral as the Limit of a Sequence of Finite Sums.- §29. Area and Curve Length.- § 30. The Length of a Curve Given in Parametric Form.- VII. Analytic Functions.- §31. Integration of Functions of a Complex Variable.- § 32. Cauchy’s Theorem.- §33. Taylor Series and Laurent Series.- §34. Residues.- §35. Finding Inverse Functions.- §36. Entire Functions and Singular Points.