Talk:Symbol of a differential operator

Is there anyone else who doesn't understand this? It's coming up in my chemistry course and I just can't fathom my notes and it doesn't seem to be in any textbooks and rarely online. I think we need an expert to edit this...

Incidentally, I don't know how to edit this into the page, but this term is also known as the 'principal part of the derivative', 'principal part' or 'dominant part of the derivative'. So redirect from there? - Nessa Ancalimë ♥ (talk) 16:51, 8 November 2009 (UTC)

Principal and dominant part refer only to the principal symbol, not the symbol itself, although there is considerable abuse of terminology in the literature conflating the symbol with the principal symbol. The symbol of a differential operator is what the article says: take all of the partial derivatives and replace them with new variables. The principal symbol is then just the part that has highest degree in the new variables. 71.182.244.158 (talk) 21:38, 8 November 2009 (UTC)

The space in which principal symbols live is WRONG in the article: It is a map of S^k(TX) ⊗ E → F (and NOT of S^k(T*X) ⊗ E → F!), thus giving a polynomial on the cotangent space! Elements of the tangent and the cotangent space do not transform in the same way, so it is important to distinguish them here!