This is an introductory textbook on the geometrical theory of dynamical
systems, fluid flows, and certain integrable systems. The subjects are
interdisciplinary and extend from mathematics, mechanics and physics to
mechanical engineering. The approach is very fundamental and would be
traced back to the times of Poincar´e, Weyl and Birkhoff in the first half of
the 20th century. The theory gives geometrical and frame-independent characterizations of various dynamical systems and can be applied to chaotic
systems as well from the geometrical point of view. For integrable systems,
similar but different geometrical theory is presented.