Manifolds, Tensor Analysis, and Applications

by Ralph H. Abraham, Jerrold E. Marsden, Tudor Ratiu

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.

Genres: Mathematics

667 pages, Hardcover

First published August 1, 1983

About the author

Dr. Ralph Herman Abraham was a mathematician who specialized in the development of dynamical systems theory. He was a Professor of Mathematics at the University of California, Santa Cruz. He also consulted on chaos theory and its applications in fields such as medical physiology, ecology, mathematical economics, and psychotherapy.

Abraham was also interested in alternative ways of expressing mathematics, for example visually or aurally. He has staged performances in which mathematics, visual arts and music are combined into one presentation. He also developed an interest in "Hip" activities in Santa Cruz in the 1960s and he wrote several books and one website on the topic. He credited his use of psychedelics for inspiring this interest.

Reviews

[Noah · Rating 4 out of 5]

Indispensable for my research. Excellent exposition. Lots of errors though.