On 5 October 2014 23:18, Kurt Pagani <nil...@gmail.com> wrote: > > On Monday, 6 October 2014 04:29:54 UTC+2, Bill Page wrote: >> >> On 5 October 2014 20:59, Waldek Hebisch <heb...@math.uni.wroc.pl> wrote: >> > kfp wrote: >> >> >> >> Extending DeRhamComplex with some functions like scalar product and >> >> Hodge dual with respect to some metric g, >> ... >> It seems to me that the concept of De Rham Complex (Cohomology of >> differential forms) does not depend in any way on the notion of >> "metric" and as I understand it, Hodge duality requires the >> coefficients to form a field. Therefore I think it would be much >> better to define a new domain. Depending on just how general one >> wanted to be, you could start by defining an InnerProductSpace(F,G,V) >> as a finite VectorSpace(F) with a bilinear form G (metric) and basis V >> (independent variables), then consider the >> ExteriorAlgebra(InnerProductSpace(F,G,V ) which inherits the exterior >> derivative from DerhamComplex(F,V). And alternate name for >> ExteriorAlgebra might be GrassmannAlgebra. This sort of domain would >> have a lot of applications in physics. >> > > At first sight you are right, however, the name DeRhamComplex is certainly > a bit misleading for what the domain is capable of. I also expected some tools > to compute cohomology groups and the like. Actually it implements a bundle > of Grassmann algebras over some coordinate patch, just what one needs to > have for a useful tool. Of course the notion is entirely independent of any > metric/connection - but of limited computational use.
In my opinion in FriCAS/Axiom a lot depends on the name and to some extent also your imagination. Names provide the "semantic content" and imagination fills in the gap between what FriCAS can provide and what you want to do. So if you argue that DeRhamComplex is misleading then the first thing I would consider is a name change. Perhaps you are suggesting that it should simply be called GrassmannAlgebra? I do agree that the available documentation does not make it obvious that what the original authors had in mind was really cohomology. > Fact is, we have at each point a (graded) algebra hence a vector space (over > whatever field), the CoefRing needs merely have 'IntegralDomain' to allow > reasonable computation of various objects. Are you sure you don't need CoefRing to be a Field? > I suppose you don't like the terms 'scalar product' and 'metric' in this > connection? Truly, one should speak of bilinear forms and pseudo > Riemannian manifolds (e.g. Minkowki space). 'scalar product' and 'metric' seem fine to me, but just not in the context of the De Rham complex. 'bilinear form' also seems fine instead of metric but I am not sure why we need to say anything about manifolds. > BTW Hodge duality has a lot of generalizations (see e.g. > http://ncatlab.org/nlab/show/Hodge+star+operator). I think this is a reasonable though rather terse treatment. (Wikipedia is probably better.) When I mentioned Field above what I was concerned about was the "Component formulas" such as given in the link above: ⋆ α = 1/k!(n-k)! εi1,...,in √|det(g)| αj1,...,jk gi1,j1 ·s gik,jk eik+1 ∧ ·s ∧ ein > I don't like the idea of a new domain while this one can be extended with > economic means. When using the 'metric' g with the functions instead of > embedding it into the domain then I see no problems at all. I explained the reason why I don't like the idea of passing a metric to operations exported by DeRhamComplex. The problem is not a technical one. It is certainly possible to do what you suggest. For me it is rather more a conceptual design issue: The best way to describe the mathematical content of a given subject. For me this is a critically important aspect which differentiates FriCAS/Axiom from most other computer algebra systems. > Moreover, the unloved Expression(Integer) domain has the advantage that > the DERHAM objects can be evaluated (if wished). Actually I do rather love, or at least admire Expression Integer. Notice that Expression Integer is a Field. At present DERHAM does not require CoefRing to be a Field, so operations such as Hodge dual as defined above would not be possible. > I will prepare some examples for clarity which might convince you :) > Yes, thanks. I am interested in continuing this discussion. Bill Page. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To post to this group, send email to fricas-devel@googlegroups.com. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
