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 theory are given in Chapter 8, using both invariant and index notation.
Throughout the text supplementary topics are noted that may be downloaded from the internet from http://www.cds.caltech.edu/~marsden. 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.
Philosophy. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings (such as applications to fluid dynamics), the study of infinite-dimensional manifolds can be hard to motivate. Chapter 8 gives an intro- duction to these applications. Some readers may wish to skip the infinite-dimensional case altogether. To aid in this, we have separated some of the technical points peculiar to the infinite-dimensional case into sup- plements, either directly in the text or on-line. Our own research interests lean toward physical applications, and the choice of topics is partly shaped by what has been useful to us over the years.
We have tried to be as sympathetic to our readers as possible by providing ample examples, exercises, and applications. When a computation in coordinates is easiest, we give it and do not hide things behind com- plicated invariant notation. On the other hand, index-free notation sometimes provides valuable geometric and computational insight so we have tried to simultaneously convey this flavor.