Expansions of tensor densities

Expansions of tensor densities PhL 2.25.12 I added this enhancement into tensor doc in various places: Section 7 just after the vector expansions are stated Appendix E for the general expansion stuff Yes, it does affect the curlB stuff in Section 12 (but only cosmetically). I have a new Sec 12 ready. I never thought of this before today, and it might be causing trouble in my curl B section 9. I am writing this today after replacing |J| with J in Appendix D on tensor densities. Theorem: If V is a vector density of weight W, then an expansion of V looks like this V = Σn (JWV'n)en = JW ΣnV'nen where the extra factor JW suddenly appears. Proof: Expand with some unknown coefficients ( x-space has g = 1) V = αn en Using en Em = δn,m one finds that αn = En V = (En)k Vk = Rnk Vk // = 1 so AB = abAaBb = AkBk But since V has weight W we know that V'n = J-W Rnk Vk => Rnk Vk = JW V'n => α = JW V'n QED One might ask about all expansions of all kinds of objects if they are densities. I think the general rule is to take the regular expansion and make this replacement V'n → JW V'n and this would be true for any tensor density, not just a vector density. This might be worth a comment somewhere where I do these expansions. I think this is the rule: Rule: In any expansion, take the thing that is the tensor component in x' space, perhaps A'ijk..., and replace it by JW A'ijk.... Example: One has a tensor transforming this way A'ijk... = R R R R R Aijk If it has weight W, then this becomes A'ijk... = J-W R R R R R Aijk or JW A'ijk... = R R R R R Aijk