Introduction to Analysis of the Infinite: Book II

ISBN-13: 9780387971322 / Angielski / Twarda / 1989 / 504 str.

Leonhard Euler; Leonard Euler; John D. Blanton

Opis

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra, even of all the algebraic technique so far discovered, still there are topics whose con- sideration prepares a student for a deeper understanding. However, in the ordinary treatise on the elements of algebra, these topics are either completely omitted or are treated carelessly. For this reason, I am cer- tain that the material I have gathered in this book is quite sufficient to remedy that defect. I have striven to develop more adequately and clearly than is the usual case those things which are absolutely required for analysis. More- over, I have also unraveled quite a few knotty problems so that the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite. There are also many questions which are answered in this work by means of ordinary algebra, although they are usually discussed with the aid of analysis. In this way the interrelationship between the two methods becomes clear.

Spis treści

Book II.- I On Curves in General.- II On the Change of Coordinates.- III On the Classification of Algebraic Curves by Orders.- IV On the Special Properties of Lines of Any Order.- V On the Second Order Lines.- VI On the Subdivision of Second Order Lines into Genera.- VII On the Investigation of Branches Which Go to Infinity.- VIII Concerning Asymptotes.- IX On the Species of Third Order Lines.- X On the Principal Properties of Third Order Lines.- XI On Fourth Order Lines.- XII On the Investigation of the Configuration of Curves.- XIII On the Disposition of Curves.- XIV On the Curvature of a Curve.- XV Concerning Curves with One or Several Diameters.- XVI On Finding Curves from the Properites of the Ordinate.- XVII On Finding Curves from Other Properties.- XVIII On the Similarities and Affinities of Curves.- XIX On the Intersection of Curves.- XX On the Construction of Equations.- XXI On Transcendental Curves.- XXII On the Solution to Several Problems Pertaining to the Circle.- I On the Surfaces of Solids.- II On the Intersection of a Surface and an Arbitrary Plane.- III On Sections of Cylinders, Cones, and Spheres.- IV On the Change of Coordinates.- V On Second Order Surfaces.- VI On the Intersection of Two Surfaces.- Figures.

O autorze

Der Schweizer Mathematiker, Natur- und Technikwissenschaftler Leonhard Euler (1707-83) prägte mit seinen wegweisenden Resultaten zur Zahlentheorie, Geometrie, Reihenlehre und zur Theorie der Differentialgleichungen die Mathematik seiner Zeit. Euler gilt als bedeutender Vertreter einer mathematisch orientierten Naturwissenschaft, er lieferte u.a. fundamentale Ergebnisse zur Hydrodynamik und zur Mechanik. Zahlreiche Symbole und Bezeichnungen, die nach wie vor in verschiedenen Disziplinen Geltung haben, wurden von Euler eingeführt.