This 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering.The book covers both classical surface theory and the modern theory of connections and curvature, and includes a chapter on applications to theoretical physics.The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed.The powerful and concise calculus of differential forms is used throughout.Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds.There are nearly 200 exercises, making the book ideal for both classroom use and self-study.