Differential geometry textbook pdf7/28/2023 ![]() Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. This is a textbook on differential geometry well-suited to a variety of courses on this topic. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. ![]() Hints and solutions are provided to many of the exercises and problems. ![]() The requisite point-set topology is included in an appendix of twenty pages other appendices review facts from real analysis and linear algebra. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. The book is furnished with an index, extensive sets of exercises and many cross references, which are hyperlinked for the ebook users, to facilitate linking related concepts and sections. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. The book, which consists of 260 pages, is about differential geometry of space curves and surfaces.
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