This book develops a fundamental understanding of geophysical fluid dynamics based on a mathematical description of the flows of inhomogeneous fluids. It covers these topics:1. development of the equations of motion for an inhomogeneous fluid  2. review of thermodynamics3. thermodynamic and kinetic energy equations4. equations of state for the atmosphere and the ocean, salt, and moisture effects5. concepts of potential temperature and potential density6. Boussinesq and quasi-geostrophic approximations7. conservation equations for vorticity, mechanical and thermal energy instability theories, internal waves, mixing, convection, double-diffusion, stratified turbulence, fronts, intrusions, gravity currentsGraduate students will be able to learn and apply the basic theory of geophysical fluid dynamics of inhomogeneous fluids on a rotating earth, including:1. derivation of the governing equations for a stratified fluid starting from basic principles of physics2. review of thermodynamics, equations of state, isothermal, adiabatic, isentropic changes3. scaling of the equations, Boussinesq approximation, applied to the ocean and the atmosphere4. examples of stratified flows at geophysical scales, steady and unsteady motions, inertia-gravity internal waves, quasi-geostrophic theory5. vorticity and energy conservation in stratified fluids6.boundary layer convection in stratified containers and basins