Eulerian coordinate ($\mathbf{x}$)

An Eulerian coordinate labels a fixed spatial location at which a continuum field is evaluated. A probe fixed in a pipe records $\mathbf{v}({\mathbf{x}}_0,t)$ as different fluid particles pass it. This description is convenient when the apparatus or boundary is fixed and the identities of individual particles are unimportant.

It differs from a Lagrangian coordinate, which labels one material particle. If the velocity field varies with position, that particle experiences changes both because the field evolves and because it moves into a different region. Its material derivative is therefore $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. In a steady narrowing pipe, ${\partial }_t\mathbf{v}=0$ at every Eulerian coordinate, yet a travelling particle can accelerate through the convective term. Steady observations do not imply zero particle acceleration.