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This textbook gives a modern treatment of incompressible and compressible fluid flows and it is intended as a companion to my Textbook of Dynamics. It is suitable for undergraduate students of mathematics in universities and colleges of technology and it should also satisfy the needs of those general science degree students who are offering mathematics as a principal subject in the final part of their course. The kinematics of a moving fluid is considered in Chapter 2. The basic ideas of fluid velocity streamlines vortex motion rotational and irrotational flows are first introduced and distinctions are drawn between local and particle rates of change. The chapter ends with the general analysis of fluid motion.