[GiNaC-list] Simplifying indexed expressions not
workingonmulti-indices scalar_product
Richard Haney
rfhaney at yahoo.com
Fri Jul 28 02:55:40 CEST 2006
--- Chris Dams <Chris.Dams at 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 at hotmail.com> wrote:
> [--- Chris Dams <Chris.Dams at 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
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