 Material derivative

The material derivative measures change seen by a moving material parcel rather than by an observer fixed in space. For a scalar field $f(x,t)$ and a Lagrangian trajectory satisfying $\dot{X}(t)=u(X(t),t)$, the chain rule gives $df(X(t),t)/dt={\partial }_{t}f+u\cdot \nabla f$.

The material derivative follows a moving continuum parcel:

For steady $f(x)=\alpha x$ in a one-dimensional flow with constant velocity $U$, the fixed-position derivative vanishes but $Df/Dt=\alpha U$. The parcel experiences change because it crosses a spatial gradient. This advection term explains why a steady velocity field can still produce parcel acceleration. ^[ref]