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The Eulerian time derivative measures a continuum field's rate of change at a fixed Eulerian coordinate. For a velocity field $\mathbf{u}$, the material derivative instead follows a fluid element and satisfies $Df/Dt={\partial }_{t}f+\mathbf{u}\cdot \nabla f$. A nonuniform steady flow has zero Eulerian time derivative of velocity but can have nonzero material acceleration.

For a scalar fluid property $f(\mathbf{x},t)$, the Eulerian time derivative ${\partial }_{t}f$ measures change at a fixed Eulerian coordinate. The material derivative, also called the Lagrangian time derivative, measures change while following one fluid element. If its Lagrangian trajectory is $\mathbf{X}(\mathbf{a},t)$, where $\mathbf{a}$ is its fixed Lagrangian coordinate, then ${\partial }_t\mathbf{X}=\mathbf{u}(\mathbf{X},t)$.

The chain rule therefore givesThe last term accounts for entering a region where the property differs. For example, a spatially varying but time-independent velocity field has zero Eulerian time derivative, while a moving fluid element can have nonzero material acceleration. The same formula applies componentwise to a vector field.