A Quick Introduction to Tensors

Abstract

This chapter introduces the notion of a tensor as a multilinear map and explores its implications through the examples of the Levi–Civita tensor and a generic second rank tensor. This discussion sheds light on the Levi–Civita symbol and also answers 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, 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.