In Chapter 3 we identified velocity \(\underline{V}(\underline{X},t)\) , density \(\rho (\underline{X},t)\) , temperature \(T(\underline{X},t)\) and pressure \(p(\underline{X},t)\) as the dependent variables of the Eulerian description of a flow field. To have the ability to compute and predict the values of these dependent variables, we must know their governing equations. These governing equations must emerge from the fundamental laws and axioms of classical mechanics, namely the law of conservation of mass, the Euler’s axioms, Newton’s third law of motion, and the laws of thermodynamics. Furthermore, we may also need an appropriate thermodynamic state equation for the medium relating to pressure, temperature, and density of the fluid. When the particle itself is an independent variable (the Lagrangian description), the application of these laws and axioms is straightforward because all the fundamental laws and axioms in their original forms are indeed applicable to independently identified particles/systems of particles. However, in the Eulerian description, wherein the particle is not an independent variable and instead the spatial location vector ( \(\underline{X}\) ) is an independent variable, we need some more advanced concepts to implement these fundamental laws/axioms in order to extract the desired governing equations of the dependent variables of our interest.