Purpose

This book represents an extended version of my lecture notes for a one-semester course on differential geometry, aimed at students without knowledge of topology. Indeed the only prerequisites are a solid grasp of multivariate calculus and of linear algebra. The goal is to train advanced undergraduates and beginning graduate students in exterior calculus (including integration), covariant differentiation (including curvature calculations), and the identification and uses of submanifolds and vector bundles. It is hoped that this will serve both the minority who proceed to study advanced texts in differential geometry, and the majority who specialize in other subjects, including physics and engineering.

Summary of the Contents

Every generation since Newton has seen a richer and deeper presentation of the differential and integral calculus. The nineteenth century gave us vector calculus and tensor analysis, and the twentieth century has produced, among other things, the exterior calculus and the theory of connections on vector bundles. As the title implies, this book is based on the premise that differential forms provide a concise and efficient approach to many constructions in geometry and in calculus on manifolds.

Chapter 1 is algebraic; Chapters 2, 4, 8, and 9 are mostly about differential forms; Chapters 4, 9, and 10 are about connections; and Chapters 3, 5, 6, and 7 are about underlying structures such as manifolds and vector bundles. The reader is not mistaken if he detects a strong influence of Harley Flanders's delightful 1989 text.