First Look at Tensors

In Sect. 3.5 , we discussed in very elementary terms an origin of tensors. Yet there are many ways to introduce tensors. Some of them are very abstract. We believe that the most direct way, which is also very appropriate for physics, is to use bases of V and \(V^*\) . This definition is, of course, basis-dependent, but as we have often seen thus far, this is not a disadvantage. In this chapter, we shall proceed on this path and leave a basis-independent definition for later (see Chap. 14 ). Thus far, we have gained considerable experience with bases. We already used the simplest possible tensors (e.g. scalars, vectors, and covectors) with their indices on various occasions. Therefore, it is very instructive to summarize first what we already know from linear algebra about the use of indices and tensors. This leads us to the following section.