Chap. II - Integrals. Densities. Derivatives

Summary

The subject-matter of the previous chapter is called tensor algebra. It is characterized by the fact that only relations between invariants, vectors or tensors referring to the same point of the continuum are contemplated. From the point of view taken here, algebraic relations between vectors and tensors referring to different points are meaningless.

Remember, however, that we based the notion of tensors on that of vectors, and the latter on the notion of the gradient, and there is hardly any simple and natural alternative to this procedure. Now in forming the gradient we actually had to compare the values of an invariant at different points, and at the same time we made the first step at introducing analysis into our continuum. In this and the following chapters we shall have to extend it. Analysis will involve derivatives and integrals. We shall have to study both from the point of view of general invariance. However, this does not mean to look out only for invariants, but also for entities with tensorial character, because, as we have seen, an equation between them (or in other words a system of equations saying that a tensor vanishes) is conserved on transformation. We begin with space-time-integrals. That leads to a certain extension of the notion of tensors, viz. to tensor densities.

We had emphasized that there is no point in adding (or, more generally, in forming linear aggregates of) tensors or vectors referring to different points. This would have no simple meaning.