This book presents modern vector analysis and carefully describes the classical notation and understanding of the theory. It covers all of the classical vector analysis in Euclidean space, as well as on manifolds, and goes on to introduce de Rham Cohomology, Hodge theory, elementary differential geometry, and basic duality. The material is accessible to readers and students with only calculus and linear algebra as prerequisites. A large number of illustrations, exercises, and tests with answers make this book an invaluable self-study source.

* Differentiable manifolds * Tangent vector space * Differential forms * Orientability * Integration on manifolds * Open manifolds * The intuitive meaning of Stoke''s theorem * The hat product and the definition of Cartan''s derivative * Stoke''s theorem * Classical vector analysis * De Rham cohomology * Differential forms on Riemannian manifolds * Calculating in coordinates * Answers * References * Index