Analysis on Manifolds

This book is intended as a text for a course in analysis, at the senior or first-year graduate level. The reader should have completed a one-term course in analysis that included a study of metric spaces and of functions of a single variable. The first chapter is devoted to reviewing the basic results from linear algebra and analysis. The main part of the book falls into two parts. The first, consisting of Chapters 2 through 4, covers derivatives, the inverse funtion theorem, the Riemann integral, and the change of variables theorem for multiple integrals. The second part of the book introduces manifolds and differential forms, providing the framework for proofs of the n-dimensional version of Stokes' theorem and of the poincare lemma. The final chapter discusses abstract manifolds. At the end of each section is a set of exercises. Some are computational and some are theoretical.