In this chapter, the differential mathematical formulation for fluid dynamics is presented. The only assumptions and restrictions on what is presented are the continuous hypotheses and Newtonian fluids. First, the Reynolds–Leibniz transport theorem is presented in Sect. 2.2, from which balances for mass, linear momentum, total energy, thermic energy, kinetic energy and vorticity were developed and presented in Sect. 2.3. It is an elegant and mathematically formal derivation. It starts from the integral formulation and leads to the differential formulation with the help of the localization theorem. A list of exercises is presented at the end of this chapter.