chapter  3
74 Pages

Tensor Analysis

WithJohn G. Papastavridis

So far we have seen tensor algebra, that is, the rules of combination of tensor components at one and the same point of a manifold In this chapter we shall examine tensor analysis, that is, how tensors vary from point to point of that manifold, or the application of differential calculus to tensors — better, tensor fields. More specifically, and since, as shown below, ordinary and partial derivatives of tensor components do not transform as tensors under general admissible coordinate transformations (CT), we will formulate special invariant (or absolute, or covariant) derivatives of these components that do transform as tensors. As Weyl (1922, p. 58) puts it, “…Tensor analysis tells us how, by differentiating with respect to the space co-ordinates, a new tensor can be derived from the old one in a manner entirely independent of the co-ordinate system.” Thus, tensor analysis is the ideal tool for the mathematical description of the states of a spatially and temporally extended system via form-invariant differential equations, consisting entirely of such covariant derivatives.*