Tensor Analysis - Lecture 08

This lecture marks the beginning of a more concrete turn to the choice of topics. While what I had spoken about earlier had been more about general features of vectors and tensors, from this one we focus basically on the specific ones that we need in physical applications. In this particular lecture I quickly review some important 3-vectors that we meet in physics. This brings us to the important case of the cross product of two vectors. I show that the Levi-Civita symbol is a numerically invariant third rank tensor under rotations - and this allows me to prove that the cross product is, actually, a vector. I also discuss why the Levi-Civita symbol is actually a tensor density for more general transformations - and us this to show that under general orthogonal transformations the cross product of two vectors is actually a pseudo-vector. I then go on to discuss the issue of the covariance under rotations of the laws of electrodynamics and in the process justify why the nabla or del operator deserved to be called a vector operator. This lecture touches on the mathematical aspect of vectors and pseudovectors. For a more physical discussion, you may want to watch the lecture at https://youtu.be/MPVMBTfyLOI