In the Eulerian specification of a field, the field is represented as a function of position x and time t. For example, the flow velocity is represented by a function 
On the other hand, in the Lagrangian specification, individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector field x. (Often, x is chosen to be the position of the center of mass of the parcels at some initial time t. It is chosen in this particular manner to account for the possible changes of the shape over time. Therefore the center of mass is a good parameterization of the flow velocity u of the parcel.) In the Lagrangian description, the flow is described by a function
giving the position of the particle labeled x at time t.
The two specifications are related as follows:
because both sides describe the velocity of the particle labeled x at time t.
Within a chosen coordinate system, x and x are referred to as the Lagrangian coordinates and Eulerian coordinates of the flow respectively.