On the invariant formulation of fluid mechanics
  It can be observed that the differential operators of fluid mechanics can be defined in terms of the complete derivative on the finite - dimensional affine space. It follows from the fact that all norms on the finite - dimensional vector space are equivalent and from the definition of the complete derivative on the normed affine spaces (see: L.Schwartz, Analyse Mathematique, Hermann, 1967). In particular, it is shown that the "substantial derivative" of the standard formulation is a directional derivative along the "non - relativistic four - velocity". 
S.Piekarski