A Quick Introduction to Tensors

Abstract

The Introduction gives a brief overview of the modern component-free definition of tensors as multilinear maps, and then uses this definition to answer many of the questions students often have when seeing tensors for the first time. In particular, we discuss the meaning of components and the origin of the tensor transformation law (which is taken as the definition of a tensor in the old-fashioned formulation), as well as the difference between a second rank tensor and a matrix. We also demonstrate how second rank tensors are related to linear operators. We then make these considerations concrete by applying them to the moment of inertia tensor from classical mechanics. The discussion is neither totally complete nor precise but is meant to introduce the main ideas quickly, to give the reader a sense of where the next two chapters are heading.