Loomis L.H., Sternberg S. Advanced Calculus

Revised Edition. — Jones & Bartlett Publishers, 1989. — 592 p. — ISBN: 9780867201222, 0867201223.As Spivak's "Calculus on Manifolds", this book is labeled with a very modest title. It should be something as "All you wanted to know about analysis on manifolds but were afraid to ask". This book is a must-reading for the analyst. It covers everything from the most basic vector space concepts up to the fundamental theorems of classical mechanics, running through multivariate calculus, exterior calculus, integration of forms, and many topics more, always keeping a very modern and rigorous style. The undergraduate may find it a little difficult, but the effort is worth it. For the graduate student and the working mathematician it is an almost-daily reference.This book is based on an honors course in advanced calculus that we gave in the 1960's. 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 anyone 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.