Advanced Calculus (4th Edition)

Book Preface

This edition differs from the previous one in two major aspects and in a number of mmor ones:

The chapter on ordinary differential equations ( chapter 8) has been omitted and a chapter on conformal mapping ( chapter 9) has been added, using mainly material from the first edition ( the topic was not included in the second and third editions).  Omission of the chapter on ordinary differential equations required adding to chapter 10, on partial differential equations, a brief discussion of ordinary linear differential equations.

Chapter 1, on linear algebra, has been shortened, especially in the treatment of Gaussian elimination. In chapter 2, the treatment of the implicit and inverse function theorems has been made more complete and the discussion of functional dependence has been improved. In chapter 3, sections have been added providing a fuller treatment of tensors. In chapters 2 and 3, some classical geometry of curves and surfaces is now provided, mainly in problems. In chapter 5, there is now a concise introduction to differential forms.

Throughout, problems have been added; in a number of cases, these signifi­cantly extend the coverage of the theory. For example, problems 11-15 following section 7-13 provide many additional complete orthogonal systems.

As with the previous editions, the purpose of this book is to provide sufficient material for a course in advanced calculus up to one year in length. It is hoped that the great variety of topics covered will also make this work useful as a reference book.

The background assumed is that usually obtained in the freshman-sophomore calculus sequence. Linear algebra is not assumed known but is developed in the first chapter.  Subjects discussed include all the topics usually found in texts on advanced calculus.

However, there is more than the usual emphasis on applications and on physical motivation. Vectors are introduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations.

Numerical methods are touched upon at various points, both because of their practical value and because of the insights they give into the theory.

A sound level of rigor is maintained throughout. Definitions are clearly labeled as such and all important results are formulated as theorems. A few of the finer points of real variable theory are treated at the ends of chapters 2, 4, and 6.

A large number of problems (with answers) are distributed throughout the text. These include simple exercises as well as complex ones planned to stimulate critical reading. Some points of the theory are relegated to the problems, with hints given where appropriate.

Generous references to the literature are given, and each chapter concludes with a list of books for supplementary reading.