Tensor Formalism

Without any exaggeration, we can say that in physics tensor formalism is needed and used as much as linear algebra itself. For example, it is impossible to understand electrodynamics, relativity, and many aspects of classical mechanics without tensors. Therefore, there is no doubt that a better understanding of tensor formalism leads to a better understanding of physics. Engineers and physicists first came across tensors in terms of indices to describe certain states of solids. Later, mathematicians found out that these objects correspond to a very precise and exciting mathematical structure. It turned out that this structure is a generalization of linear algebra. This chapter will discuss tensor formalism, also known as multilinear algebra, in a basis-independent way. But of course, as we know from linear algebra, we cannot do without basis-dependent representations of tensors. From Sect.  3.5 and Chap.  8 we already know what tensors are, and we know at least one possibility to arrive at tensor spaces. Before, this was obtained by explicitly utilizing bases of vector spaces. Now, we would like to achieve this differently, which will allow us to further expand and consolidate the theory of tensors.