Introduction to Differential Manifolds 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. This book contains essential material that every graduate student must know. Written with Serge Langs inimitable wit and clarity the volume introduces the reader to manifolds differential forms Darbouxs 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 authors classic monograph Introduction to Differentiable Manifolds (1962) which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginners 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 authors 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.