On 05/11/2010 04:18 PM, jvkersch wrote:
Dear mailing list,
I was wondering if there is any interest in a Sage package to compute
with differential forms. Right now, I have the rudiments of a
differential form class with the obvious operations (wedge product,
exterior differential, Hodge star, ...) but I want to take this
somewhat further: my goal is to be able to deal with Cartan's
structure equation, exterior differential systems, point symmetries
for ODEs, ... These are all very algorithmic procedures, so they
would be relatively easy to implement.
I am interested. I'm particularly interested in applications to
teaching multivariable calculus.
Secondly, a more detailed question: while my class works well from a
Python point of view, I don't really know how to incorporate it into
the Sage hierarchy. More specifically: the set of all differential
forms is a module over the ring of functions on R^n -- how would I
incorporate this algebraic structure in Sage? Is there a parent
representing a function ring on R^n?
Since the result of "f = function(...); f.parent()" is "Symbolic
Ring", and I want to be able to coerce functions into 0-forms, should
I start with a module over the Symbolic Ring, or is another ring more
appropriate?
Maybe a module over callable symbolic expressions would be more
appropriate starting point? I've been doing some work on these in the
past week. With the patch up at
http://trac.sagemath.org/sage_trac/ticket/8947 (needs review) and the
patch http://trac.sagemath.org/sage_trac/ticket/8866 (already merged in
4.4.2.alpha0), we have:
sage: f(x,y)=[3*x,e^x,2*x*y]
sage: f
(x, y) |--> (3*x, e^x, 2*x*y)
sage: f.parent()
Vector space of dimension 3 over Callable function ring with arguments
(x, y)
This sort of gives us functions over modules (sort of a function from
SR^2 --> SR^3 in this example). However, Sage doesn't treat f as a
function as much as it treats f as a vector of functions (though you can
still call f, take its derivative, etc.).
Thanks,
Jason
--
Jason Grout
--
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