4 - CARTESIAN VECTOR AND TENSOR FIELDS

Abstract

For a given Euclidean manifold, a tensor field is uniquely determined when its kind, its rank, and its representative component functions are specified in any one specified coordinate system. In particular, a rectangular Cartesian coordinate system can be chosen for specifying the component functions of any general tensor field of given kind and rank. But the same component functions also determine uniquely a Cartesian tensor field of the same rank. There is, therefore, a one-to-one correspondence between general tensor fields of specified kind and rank, and Cartesian tensor fields of the same rank. For a Euclidean manifold whose metric tensor field is independent of time, Cartesian tensors and general tensors represent two different sets of tools, either of which can be used in applications. For most purposes, Cartesian tensors are simpler to use and have been widely used in the literature on elasticity, hydrodynamics, and continuum mechanics.