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"This book contains essential material that every graduate student must know. Written with Serge Lang´s inimitable wit and clarity, the volume introduces the reader to manifolds, differential forms, Darboux´s theorem, Frobenius, and all the central features of the foundations of differential geometry. Lang lays the basis for further study in geometric analysis, and provides a solid resource in the techniques of differential topology. The book will have a key position on my shelf.Steven Krantz, Washington University in St. Louis"This is an elementary, finite dimensional version of the author´s classic monograph, Introduction to Differentiable Manifolds (1962), which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginner´s entry into geometry, topology, and global analysis. The exposition is unencumbered by unnecessary formalism, notational or otherwise, which is a pitfall few writers of introductory texts of the subject manage to avoid. The author´s hallmark characteristics of directness, conciseness, and structural clarity are everywhere in evidence. A nice touch is the inclusion of more advanced topics at the end of the book, including the computation of the top cohomology group of a manifold, a generalized divergence theorem of Gauss, and an elementary residue theorem of several complex variables. If getting to the main point of an argument or having the key ideas of a subject laid bare is important to you, then you would find the reading of this book a satisfying experience."Hung-Hsi Wu, University of California, Berkeley Author is well-known and established book author (all Serge Lang books are now published by Springer); Presents a brief introduction to the subject; All manifolds are assumed finite dimensional in order not to frighten some readers; Complete proofs are given; Use of manifolds cuts across disciplines and includes physics, engineering and economics This book contains essential material that every graduate student must know. Written with Serge Lang´s inimitable wit and clarity, the volume introduces the reader to manifolds, differential forms, Darboux´s theorem, Frobenius, and all the central features of the foundations of differential geometry. Lang lays the basis for further study in geometric analysis, and provides a solid resource in the techniques of
differential topology. The book will have a key position on my shelf.
-Steven Krantz, Washington University in St. Louis
This is an elementary, finite dimensional version of the author´s classic monograph, Introduction to Differentiable Manifolds (1962), which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginner´s entry into geometry, topology, and global
analysis. The exposition is unencumbered by unnecessary formalism, notational or otherwise, which is a pitfall few writers of introductory texts of the subject manage to avoid. The author´s hallmark characteristics of directness, conciseness, and structural clarity are everywhere in evidence. A nice touch is the inclusion of more advanced topics at the end of the book, including the computation of the top cohomology group of a manifolds, a generalized divergence theorem of Gauss, and an elementary residue
theorem of several complex variables. If getting to the main point of an argument or having the key ideas of a subject laid bare is important to you, then you would find the reading of this book a satisfying experience.

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Foreword Acknowledgments Differential Calculus Manifolds Vector Bundles Vector Fields and Differential Equations Operations on Vector Fields and Differential Forms The Theorem of Frobenius Metrics Integration of Differential Forms Stokes´ Theorem Applications of Stokes´ Theorem

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