Material derivative

Remarks

If $X(a, t)$ is the Lagrangian path of the parcel labelled $a$, then the chain rule gives $\frac{d}{dt} f(X(a, t), t) = \frac{Df}{Dt}(X(a, t), t)$. Example: for $f(x, t) = x_1$ and $\mathbf{u} = (2, 0, 0)$, $\frac{Df}{Dt} = 0 + 2 \cdot 1 = 2$. Non-example: the partial derivative $\partial f / \partial t = 0$ in this example is not the material derivative, because it omits the transport term $\mathbf{u} \cdot \nabla f$. The acceleration of a fluid parcel is $\frac{D\mathbf{u}}{Dt} = \partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u}$. Other names are convective derivative, substantial derivative and Lagrangian derivative.