--- Chris Dams <Chris.Dams@mi.infn.it> wrote: [...]
On Thu, 27 Jul 2006, Alejandro Limache wrote: [...]
1) By definition an inner product must be a scalar so all tensor indices should get contracted.
Yes, obviously.
Hmmm..., seems to me that in general you would also like to have scalar products that may be dependent on other variables, including integers. --- Alejandro Limache <alejandrolimache@hotmail.com> wrote:
[--- Chris Dams <Chris.Dams@mi.infn.it> wrote:]
(1) What if the indices are of different dimension? [[...]] Maybe user-defined inner products should only be used if all dimensions of indices involved are the same. [[...]] [...] one might have different dimensions. [...] [...] there are times where you can get products M.k.i.j*A.i.j of multidimensional arrays M,A where for example i and j have dimension of space (3) and k the dimension of the number of nodes or elements. [...]
(M.k.i.j*A.i.j){k=1...N} here seems to be such an example of an *indexed scalar*. I also vaguely seem to recall seeing some treatments in general relativity where some indices of indexed quantities run over "space" dimensions only, whereas other indices run over all four space-time dimensions. My memory is hazy here, so I suppose I could be a bit mistaken on this. There is also some mumbo-jumbo concerning what indexed entities constitute a "tensor" -- something about the indices relating to coordinates and the indexed entities conforming to certain rules of transformation -- so that, I suppose, the tensors constitute a kind of equivalence class with such an indexed entity being a representative of that class. So it seems that one would like to be able to distinguish indices in various convenient ways; some indices would relate to the "tensor" properties and some would (perhaps) be extraneous to the concept of a tensor -- perhaps relevant to some other type of algebraic concept. And then there are some who like to work with "coordinate-free" representations of tensors. I'm a bit of a newbie to GiNaC; so I don't know yet how relevant all of this is to GiNaC. Richard Haney __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com