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 > <javascript:>> wrote: > > kfp wrote: > > ... > >> Finally, what's your opinion to the following matter? > >> > >> Extending DeRhamComplex with some functions like scalar product and > Hodge > >> dual with respect to some metric g, what would be preferable: to > include a > >> (set-able) default g into the domain or to give it as a parameter? > >> > >> For instance, in the first case one had to write > >> > >> setMetric([...]), dot(a,b), hodgeStar(a) > >> > >> or othewise > >> > >> dot(g,a,b), hodgeStar(g,a) ? > >> > > > > IMO parameter is better. If there is strong desire to avoid > > giving parameter on each use, then we can create extra domain > > having g as domain parameter. Then all forms from this domain > > will use the same 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. > > Bill Page. >
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. 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. 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). BTW Hodge duality has a lot of generalizations (see e.g. http://ncatlab.org/nlab/show/Hodge+star+operator). I don't like the idea of a new domain while this one can be exteneded 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. Moreover, the unloved Expression(Integer) domain has the advantage that the DERHAM objects can be evaluated (if wished). I will prepare some examples for clarity which might convince you :) Kurt -- 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.
