We review the Eulerian continuity and Euler momentum equations, clarify the relationship between Eulerian and Lagrangian descriptions via the material derivative, and briefly connect these fundamentals to the physics of hydraulic jumps as transitions between supercritical and subcritical flow.

We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative \(\mathrm{d} \mathbf{v} / \mathrm{d} t=\partial_t \mathbf{v}+(\mathbf{v} \cdot \nabla) \mathbf{v}\), and explain how Eulerian field descriptions relate to Lagrangian particlebased descriptions via \(\rho(\mathbf{x}(t), t)\).