The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).
Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F(x, t). Now one might ask about the total rate of change of F experienced by a specific flow parcel. This can be computed as
where ∇ denotes the nabla operator with respect to x, and the operator u⋅∇ is to be applied to each component of F. This tells us that the total rate of change of the function F as the fluid parcels moves through a flow field described by its Eulerian specification u is equal to the sum of the local rate of change and the convective rate of change of F. This is a consequence of the chain rule since we are differentiating the function F(X(x, t), t) with respect to t.
Conservation laws for a unit mass have a Lagrangian form, which together with mass conservation produce Eulerian conservation; on the contrary, when fluid particles can exchange a quantity (like energy or momentum), only Eulerian conservation laws exist.