Calculus, Vol II - T.M. Apostol

Название: Calculus, Vol II
Автор: T.M. Apostol
Категория: Математика
Тип: Книга
Дата: 30.12.2008 16:13:13
Скачано: 103
Описание: This book is a continuation of the author's Calculus, Volume I, Second Edition. The present volume has been written with the same underlying philosophy that prevailed in the first. Sound training in technique is combined with a strong theoretical development. Every effort has been made to convey the spirit of modern mathematics without undue emphasis on formalization. As in Volume I, historical remarks are included to give the student a sense of participation in the evolution of ideas. The second volume is divided into three parts, entitled Linear Analysis, Nonlinear Analysis, and Special Topics. The last two chapters of Volume I have been repeated as the first two chapters of Volume П so that all the material on linear algebra will be complete in one volume. Part 1 contains an introduction to linear algebra, including linear transformations, matrices, determinants, eigenvalues, and quadratic forms. Applications are given to analysis, in particular to the study of linear differential equations. Systems of differential equations are treated with the help of matrix calculus. Existence and uniqueness theorems are proved by Picard's method of successive approximations, which is also cast in the language of contraction operators. Part 2 discusses the calculus of functions of several variables. Differential calculus is unified and simplified with the aid of linear algebra. It includes chain rules for scalar and vector fields, and applications to partial differential equations and externum problems. Integral calculus includes line integrals, multiple integrals, and surface integrals, with applications to vector analysis. Here the treatment is along more or less classical lines and does not include a formal development of differential forms. The special topics treated in Part 3 are Probability and Numerical Analysis. The material on probability is divided into two chapters, one dealing with finite or countably infinite sample spaces; the other with uncountable sample spaces, random variables, and distribution functions. The use of the calculus is illustrated in the study of both one- and two-dimensional random variables. The last chapter contains an introduction to numerical analysis, the chief emphasis being on different kinds of polynomial approximation. Here again the ideas are unified by the notation and terminology of linear algebra. The book concludes with a treatment of approximate integration formulas, such as Simpson's rule, and a discussion of Euler's summation formula.
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