关于经典场论中两类互补的方向导数与流场描述
On Two Complementary Types of Directional Derivative and Flow Field Specification in the Classical Field Theory
摘要：
讨论了任意张量流场方向导数的一般定义及其在物理学中的应用。结果表明，拉格朗日和欧拉描述作为现代理论流体力学中采用的流场规范的互补类型，意味着两种互补类型的方向导数作为相应的数学构造。其中之一是欧拉实质性的衍生物只在初始柯西问题的背景下有用，另一个，这里称为局部方向导数，只出现在所谓的最终柯西问题的背景下。拉格朗日和欧拉规范之间的选择被证明是等价的选择与欧几里德或闵可夫斯基度量的任何流场域的时空，分别。本文在一般四维可微流形的商模变换的框架内，给出了数学上的一些考虑。以定理的形式给出了局部方向导数的解析表达式。虽然这是对理想单分量（标量）流场的考虑，但它可以很容易地推广到任何张量场。局部方向导数概念的经典场论的一些影响也进行了探讨。
英文摘要：
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.
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