Tensor Analysis : Lecture 06

In this lecture I focus on the "meaning" of a tensor, especially the way in which this is used in Newtonian physics. We have already seen that it is perhaps best to see a tensor as a machine that "eats" covectors and vectors and churns out numbers. We can also use the idea of filling up some, but not all slots of a tensor - and thus creating new kind of machines. In particular, a very useful avatar of a second rank tensor is one where we can think of it as a linear map from one vector to another. This brings us to some very common tensors in physics - like the moment of inertia tensor and the permittivity tensor. I describe how the former denotes the linear relationship between the angular velocity and the angular momentum of a rigid body, and the latter that between the electric field and the electric displacement vector. I also describe how the transformation law for a rank 2 tensor under rotations, which becomes a similarity transformation for the corresponding matrix version, leads to the concept of principal axes. I also briefly talk about a few tensors that are used in continuum mechanics, perhaps better known as "elasticity".