TITLE: Symbol of pseudodiff operator
QUESTION [10 upvotes]: Hello,
I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator locally, then prove that it transforms well, hence becomes a "global" object. Is there any good way to define the the principal symbol without coordinate patches? Am I asking too much here :)?

REPLY [2 votes]: I think that Hormander  beautiful short paper

Pseudo-differential operators, Comm.
  Pure Appl. Math. 18 1965, 501–517

is always a good place to start.  In this paper  he gives a coordinate free  definition of a (scalar) pseudo-differential operator.  This paper did it for me.