Manifolds, Tensor Analysis, and Applications (Applied Mathematical Sciences, 75) 2nd Edition by Ralph Abraham (PDF)

Description

The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me­ chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book’s contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.

User’s Reviews

Editorial Reviews: From the Back Cover The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This text in its second edition is the definitive text in its field written by highly respected authors. It contains some typos and minor errors, all of them easisly corrected by the reader. In exchange for corrections and suggestionis the authors offer three new chapters destined to appear in a third edition. There are supplementary sections at the end of each chapter for readers interested in delving deeper into the material. The book culminates with a chapter on practical applications. I find the notation of the first edition to be clearer than that of the second. For personal reasons I find the typesetting of the second edition to be inferior to the first. Therefore, I read the first edition side-by-side with the second edition. The typesetting of the second edition should be no big deal for the typical reader. I just have a preference for the typesetting of the first edition. (I was once a scientific magazine editor, so I am sensitive to the choice of typefaces for a book.) To be more specific, the use of boldface and other notation makes mathematical symbols clearer in the first edition. Anyway, if you need a text on manifolds and tensor analysis this is THE book to get.

⭐This book has provided an insightful and complete resource. Having little background in differentiable manifolds or geometry, I found it accessible, yet general and complete enough for advanced purposes.

⭐Abraham-Marsden-Ratiu is an excellent, enjoyable presentation of infinite dimensional manifolds and related topics: tensors, differential forms, functional equations, integration in (finite dimensional) manifolds, dynamical systems, Lagrange multipliers and more. It starts from the very beginning, analysis in Banach manifolds, following closely Dieudonne’s famous chapter VIII, vol. I of Treatise on Analysis. With its “uncountable” examples and applications, it is an overwhelming improvement of Lang’s book, which has a formal, dry style, not to mention complete absence of applications and examples. I think this book should be read by every mathematician. Tip: look for the last available edition.

⭐The only thing I can say about this book is that it is an absolute masterpiece of clarity, rigour and beauty.

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