Manifolds, Tensor Analysis, and Applications (Applied Mathematical Sciences (75))

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. 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. MTA_Ch1_1-7-01.pdf......Page 1 MTA_Ch2_1-9-01.pdf......Page 37 MTA_Ch3_1-21-01.pdf......Page 127 MTA_Ch4_1-30-01.pdf......Page 210 MTA_Ch5_2-11-01.pdf......Page 291 MTA_Ch6_2-16-01.pdf......Page 336 MTA_Ch7_2-26-01.pdf......Page 398 MTA_Ch8_3-7-01.pdf......Page 481 ms_Ch9_2-1-00.pdf......Page 552 The purpose of this chapter is to introduce just enough topology for later requirements. R. Abraham, J.e. Marsden, T. Ratiu. Includes Index. Bibliography: P. [631]-641.