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-
mechanics, 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-
introduction 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-
supplements, 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-
complicated 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.
Prerequisites and Links. The prerequisites required are solid undergraduate courses in linear algebra
and advanced calculus along with the usual mathematical maturity. At various points in the text contacts are
made with other subjects. This provides a good way for students to link this material with other courses. For
example, Chapter 1 links with point-set topology, parts of Chapters 2 and 7 are connected with functional
analysis, Section 4.3 relates to ordinary differential equations and dynamical systems, Chapter 3 and Section
7.5 are linked to differential topology and algebraic topology, and Chapter 8 on applications is connected
with applied mathematics, physics, and engineering.