I wish there was a decent and reliable way to write mathematics in email!
[image: \star \alpha = \frac{1}{k!(n-k)!} \epsilon_{i_1,\dots,i_n}
\sqrt{|det(g)|} \alpha_{j_1,\dots,j_k} g^{i_1,j_1} \ldots g^{i_k,j_k}
e^{i_{k+1}} \wedge \ldots \wedge e^{i_n}]
does that work?
On 6 October 2014 14:52, Bill Page <bill.p...@newsynthesis.org> wrote:
>
> ⋆ α = 1/k!(n-k)! εi1,...,in √|det(g)| αj1,...,jk gi1,j1 ·s gik,jk
> eik+1 ∧ ·s ∧ ein
>
On 6 October 2014 14:52, Bill Page <bill.p...@newsynthesis.org> wrote:
> 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.
