This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures. Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
具體描述
讀後感
给一本数学书写书评，这似乎是一件很不知天高地厚的事情。然则这本书很有趣，不只是在它本身，还在于和它关联着的许多东西。我想把这些东西写下来，于是这大概就是书评吧。 这本书是好书。在学完数学分析以及点集拓扑以后，学校开的下一门课叫微分几何。用书是http://book.d...
評分给一本数学书写书评，这似乎是一件很不知天高地厚的事情。然则这本书很有趣，不只是在它本身，还在于和它关联着的许多东西。我想把这些东西写下来，于是这大概就是书评吧。 这本书是好书。在学完数学分析以及点集拓扑以后，学校开的下一门课叫微分几何。用书是http://book.d...
評分给一本数学书写书评，这似乎是一件很不知天高地厚的事情。然则这本书很有趣，不只是在它本身，还在于和它关联着的许多东西。我想把这些东西写下来，于是这大概就是书评吧。 这本书是好书。在学完数学分析以及点集拓扑以后，学校开的下一门课叫微分几何。用书是http://book.d...
評分给一本数学书写书评，这似乎是一件很不知天高地厚的事情。然则这本书很有趣，不只是在它本身，还在于和它关联着的许多东西。我想把这些东西写下来，于是这大概就是书评吧。 这本书是好书。在学完数学分析以及点集拓扑以后，学校开的下一门课叫微分几何。用书是http://book.d...
評分给一本数学书写书评，这似乎是一件很不知天高地厚的事情。然则这本书很有趣，不只是在它本身，还在于和它关联着的许多东西。我想把这些东西写下来，于是这大概就是书评吧。 这本书是好书。在学完数学分析以及点集拓扑以后，学校开的下一门课叫微分几何。用书是http://book.d...
用戶評價
這是最後一本瞭！！
评分一周橫掃五百麵（好多定理直接跳過證明，題當然也是沒做）。這書好在極其嚴謹，幾何的東西很多感覺就那麼迴事，但想嚴密描述齣來就得花一番功夫，更何況嚴密證明。這本書做到瞭這些，定義和定理非常細緻精確。不可避免的缺點就是太長瞭——沒辦法啊。這本跟Spivak“純幾何”的微分幾何教材很好地互補。
评分當年不太會看書，所以看得很痛苦_(:з」∠)_
评分For physics students who study GR, manifold, differential geometry，強推Sean Carrol 寫的書。寫的比數學的定義intrinsic非常多。簡單到根本不可能看不懂。https://douc.cc/4jbB7Y
评分前16章，清晰完整
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