Manifolds, Tensor Analysis, And Applications

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 mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus.

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Mathematical scientists and advanced students of such fields will be familiar with the general style and quality of Abraham/Marsden work from previous publications. The present substantial volume, intended to be accessible to beginning graduate students, and to serve both as a text and as a reference work (the 1st edition appeared in 1983, and an expanded 3rd edition is planned), maintains that familiar high standard. The authors (joined here by T. Ratiu) strive to achieve sophistication with a minimum of formal fuss, and to maintain effective contact with principal physical applications (the subject of their final chapter, but present in spirit throughout the volume). The eight chapters treat topology, Banach spaces and differential calculus, manifolds and vector bundles, vector fields and dynamical systems, tensors, differential forms, integration on manifolds, applications, and are punctuated by exercises, examples and frequent asides. Splendid. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)