We spend some time in lecture 3 discussing the proof using local coordinates that  the principal symbol of a (scalar) differential operator of order \(m\) is well defined as a function on the cotangent bundle (which is a homogeneous polynomial of order \(m\) in the \(\xi\)-variable).  As some found this hard to understand (probably for reasons of notation) I thought it might be helpful if I reviewed some of the ideas here in a little more detail. Continue reading →