The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, :
where  is the covariant derivative of the tensor, and  is the flow velocity. Generally the convective derivative of the field , the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field , or as involving the streamline directional derivative of the field , leading to the same result. 
Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent of the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative , instead for only the spatial term . The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.