Advanced Calculus, Revised Edition

This book is based on an honors course in Advanced Calculus given at Department of Mathematics, Hardvard University. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis. These prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, reader should read Differential and Integral Calculus by R. Courant, Calculus by T. Apostol, Calculus by M. Spivak, and Pure Mathematics by G. Hardy. The reader should also have some experience with partial derivatives.

In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.

There are exercises of many different kinds spread throughout the book. Some are in the nature of routine applications. Others ask the reader to fill in or extend various proofs of results presented in the text. Sometimes whole topics, such as the Fourier transform or the residue calculus, are presented in exercise form. Due to the rather abstract nature of the textual material, the student is strongly advised to work out as many of the exercises as he possibly can.