For example, for a macroscopic scalar field  and a macroscopic vector field  the definition becomes:
In the scalar case  is simply the gradient of a scalar, while  is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of  as a function of ).  
In particular for a scalar field in a three-dimensional Cartesian coordinate system , the components of the velocity  are , and the convective term is then: