On 5 October 2014 20:59, Waldek Hebisch <[email protected]> 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. -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
