Le jeudi 5 janvier 2017 10:35:27 UTC+1, Dima Pasechnik a écrit :
>
>
> Thanks for your suggestion; however, I am not sure if this could fully
>> work: some computations require to take derivatives, i.e. to evaluate
>> d/dx (sqrt(-z)), where z is the rational function of (x,y) discussed
>> above. Could this work in our framework?
>>
>
> d/dx (-z)^{1/2}=-1/(2(-z)^{1/2})dz/dx=-(dz/dx)/(2w), and so you can carry
> one staying within rational functions in your variables original variables
> and w.
>
>
Yes indeed, but then what about dealing with a second volume element
sqrt(-z') (arising from another metric): we cannot work both in the
polynomial polynomial ring modulo the ideal generated by w^2+z and that
modulo the ideal generated by w'^2 + z', can we?
However, let me take the opportunity of this discussion to mention that all
the manifold stuff, as implemented in src/sage/manifolds, depends very
weakly on the Symbolic Ring: all the coordinate calculus is encapsulated in
the abstract base class CoordFunction
<http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/coord_func.html#sage.manifolds.coord_func.CoordFunction>,
which, at the moment, has a single concrete derived class: CoordFunctionSymb
<http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/coord_func_symb.html#sage.manifolds.coord_func_symb.CoordFunctionSymb>.
Only the latter involves the Symbolic Ring. In the future, we may conceive
other concrete derived classes, based on fraction fields of polynomial
rings, as Samuel and you suggest, or sympy, or Giac, or even numerical
methods.
Best wishes,
Eric.
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