The purpose of this text is to present an integrated treatment of a number of those topics in mathematics which can be made to depend only upon a sound course in elementary calculus, and which are of common importance in many fields of application. An attempt is made to deal with the various topics in such a way that a student who may not proceed into the more profound areas of mathematics may still obtain an intelligent working knowledge of a substantial number of useful mathematical methods, together with an appropriate awareness of the foundations, interrelations, and limitations of these methods. At the same time, it is hoped that a student who is to progress, say, into a rigorous course in mathematical analysis will be provided, in addition, with increased incentive and motivation in that direction, as, for example, when he is confronted by the phrase HIt can be shown” within the derivation of a useful concrete result, or when he has led to a sense that a certain new concept is a fertile one and is deserving of being expanded and made more precise.
The book is a revision of Advanced Calculus for Engineers, published in 1949, incorporating not only a number of minor changes for the purpose of increased clarity or precision but also some added textual material, as well as a very substantial number of additional problems. The first four chapters are concerned chiefly with ordinary differential equations, including analytical, operational, and numerical methods of solution, and with special functions generated as solutions of such equations. In particular, the material of the first chapter can be considered as either a systematic review or an initial introduction to the elementary concepts and techniques. associated with linear equations and with special
solvable types of nonlinear equations, which are needed in subsequent chapters. The fifth chapter deals with boundary-value problems governed by ordinary differential equations, with the associated characteristic functions, and with series and integral representations of arbitrary functions in terms of these functions. Chapter 6 develops the useful ideas and tools of vector analysis; Chapter 7 provides brief introductions to some special topics in higher-dimensional calculus which are rather frequently needed in applications. The treatment here occasionally consists essentially of indicating the plausibility and practical significance of a result and stating conditions under which its validity is rigorously established in listed references.
|Title: Advanced Calculus for Applications|
Author: Francis B. Hildebrand
Publisher: Pearson; 2 edition (31 March 1976)