Includes bibliographical references and index.

Part I: Mathematical bases -- Part II: Dynamical systems -- Part III: Flows of ideal fluids -- Part IV: Geometry of integrable systems -- Appendixes: -- A. Topological space and mappings -- B. Exterior forms, products and differentials -- C. Lie groups and rotation groups -- D.A curve and a surface in Rp3s -- E. Curvature transformation -- F. Function spaces Lp, H p8s and orthogonal decomposition -- G. Derivation of KdV equation for a shallow water wave -- H. Two-cocycle, central extensions and bott cocycle -- I. Additional comment on the gauge theory of "."7.3 -- J. Frobenius integration theorem and Pfaffian system -- K. Orthogonal coordinate net and lines of curvature. "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, and the approach is very fundamental. The underlying concepts are based on differential geometry and theory of Lie groups in the mathematical aspect, and on transformation symmetries and gauge theory in the physical aspect. A great deal of effort has been directed toward making the description elementary, clear and concise, so that beginners will have easy access to the topics."--Jacket.