On Two Complementary Types of Directional Derivative and Flow Field Specification in the Classical Field Theory

We discuss a general definition of directional derivative of any tensor flow field and its practical applications in physics. It is shown that both Lagrangian and Eulerian descriptions as complementary types of flow field specifications adopted in modern theoretical hydrodynamics, imply two complementary types of directional derivatives as corresponding mathematical constructions. One of them is the Euler substantive derivative useful only in the contexts of initial Cauchy problem and the other, called here as the local directional derivative, arises only in the context of so-called final Cauchy problem. The choice between Lagrangian and Eulerian specifications is demonstrated to be equivalent to the choice between space-time with Euclidean or Minkowski metric for any flow field domain, respectively. Mathematical consideration is developed within the framework of diffeomorfic transformations for general 4-dimensional differentiable manifold. The analytical expression for local directional derivative is formulated in form of a theorem. Although the consideration is developed for ideal one-component (scalar) flow field, it can be easily generalized to any tensor field. Some implications of the local directional derivative concept for the classical theory of fields are also explored.