Tensor Analysis - Lecture 07

In this lecture I discuss a basic question - why don tensors exist? In other words, why are there precisely some transformation laws and not others? I begin by showing that the only possible transformation law that can work single component object under rotations is that of a scalar (invariant). I then go on to stress the underlying principle - while the numbers describing an object can depend on the coordinate system, they can not depend on how you reached that coordinate system in the first place. This puts a severe restriction on allowed transformation rules, and I then go on to show how scalars, vectors, covectors, higher rank tensors are possible because of these restrictions. In a sense this lecture goes back to the "transforms according to" aspect of a tensor rather than the "multilinear map" aspect that has dominated the last couple of lectures - but treats it in a different way. Rather than ask how do the components of a tensor transform, what I ask here is "why do they transform in that way"?