symbolic derivatives should be able to have non-variable arguments

[casamayou]

Here is an example from the french sagebook to compute the laplacian in polar, that no longer works with version 4.6 of Sage

sage: x, y, r, t = var('x, y, r, t'); f = function('f', x, y)
sage: F = f(x = r*cos(t), y = r*sin(t))
sage: d = (diff(F,r,2) + diff(F,t,2)/r**2 + diff(F,r)/r)
sage: d.simplify_full() 
---------------------------------------------------------------
NotImplementedError           Traceback (most recent call last)

NotImplementedError: arguments must be distinct variables

With older versions, you got it :

sage: d.simplify_full()
D[0, 0](f)(r, t) + D[1, 1](f)(r, t)

[kcrisman]

I don't know that this is necessarily a regression so much as having made some choices about what is 'allowed' to be a symbolic derivative. We don't allow (for reasons that are unclear to me, because I still don't understand why we changed how we represent symbolic derivatives) arguments that are not just variables, unlike here where they are expressions.

It's the conversion to Maxima that is failing, by the way.

sage: d.simplify() 
---------------------------------------------------------------
NotImplementedError           Traceback (most recent call last)

NotImplementedError: arguments must be distinct variables

This even happens with trying to simplify (send to Maxima)

sage: diff(F,r).simplify()
<boom>

This is at least tangentially related to #6840 and #6756, though probably not directly.

[burcin]

AFAIK, #7377 improves the conversion of derivatives to maxima. It might fix this as well.

[kcrisman]

Replying to burcin:

AFAIK, #7377 improves the conversion of derivatives to maxima. It might fix this as well.

Nice idea! Unfortunately, it seems to have the same problem.

[nbruin]

Replying to kcrisman:

Replying to burcin:

AFAIK, #7377 improves the conversion of derivatives to maxima. It might fix this as well.

Nice idea! Unfortunately, it seems to have the same problem.

but see ​http://trac.sagemath.org/sage_trac/ticket/7377#comment:54 for some ideas on how to improve this. The concerns about "at" in maxima are ungrounded: It's the documentation that is wrong and this will be fixed in the new version. It would be quite straightforward to extend the support to general derivatives, including conversion to/from maxima (we'll have to see how much trouble the "at" expressions give further down the line, of course). Furthermore, the conversion is easier to do with #7377 but it's not required. You could already do this in the old setup.

[kcrisman]

Thanks for that comment reference. I think you are right that this could work out. You are also right that "Sage support for functional derivatives is only rudimentary." This largely changed when the D[0] notation was introduced.

[dsm]

This seems to work for me in 5.0:

sage: x, y, r, t = var('x, y, r, t')
sage: f = function('f', x, y)
sage: F = f(x = r*cos(t), y = r*sin(t))
sage: d = (diff(F,r,2) + diff(F,t,2)/r**2 + diff(F,r)/r)
sage: d.simplify() 
(sin(t)*D[1, 1](f)(r*cos(t), r*sin(t)) + cos(t)*D[0, 1](f)(r*cos(t), r*sin(t)))*sin(t) + (sin(t)*D[0, 1](f)(r*cos(t), r*sin(t)) + cos(t)*D[0, 0](f)(r*cos(t), r*sin(t)))*cos(t) + (sin(t)*D[1](f)(r*cos(t), r*sin(t)) + cos(t)*D[0](f)(r*cos(t), r*sin(t)))/r - ((r*sin(t)*D[0, 1](f)(r*cos(t), r*sin(t)) - r*cos(t)*D[1, 1](f)(r*cos(t), r*sin(t)))*r*cos(t) - (r*sin(t)*D[0, 0](f)(r*cos(t), r*sin(t)) - r*cos(t)*D[0, 1](f)(r*cos(t), r*sin(t)))*r*sin(t) + r*sin(t)*D[1](f)(r*cos(t), r*sin(t)) + r*cos(t)*D[0](f)(r*cos(t), r*sin(t)))/r^2
sage: d.simplify_full()
D[0, 0](f)(r*cos(t), r*sin(t)) + D[1, 1](f)(r*cos(t), r*sin(t))
sage: diff(F,r).simplify()
sin(t)*D[1](f)(r*cos(t), r*sin(t)) + cos(t)*D[0](f)(r*cos(t), r*sin(t))
sage: diff(F,r).simplify_full()
sin(t)*D[1](f)(r*cos(t), r*sin(t)) + cos(t)*D[0](f)(r*cos(t), r*sin(t))

Although I'm too lazy to bisect to find out when it started working again, it'd probably be a good idea to add a doctest to make sure that it stays that way. Not quite sure where to put it, though.

[dsm]

Hmm. On second thought, I may have spoken too quickly:

D[0, 0](f)(r*cos(t), r*sin(t)) + D[1, 1](f)(r*cos(t), r*sin(t))

and

D[0, 0](f)(r, t) + D[1, 1](f)(r, t)

are a little different. Not incompatible, because the functions are abstract, but still not the same. Do we want to recover the original representation?

[nbruin]

Replying to dsm:

D[0, 0](f)(r, t) + D[1, 1](f)(r, t)

Wow, Sage before 4.6 was just plain wrong. We'll excuse casamayou, since superficially it looks like what one would sloppily write in a calculus book (the coordinate transformation implicitly assumed).

The formula for d is the expression for the laplacian in polar coordinates, so it should be equal to

(diff(f,x,x)+diff(f,y,y)).subs(x=r*cos(t),y=r*sin(t))

which is what 5.0 verifies.

I'm pretty sure #12796 fixed this, since this kind of computation was the point of that ticket. I think this ticket can be closed. Add a doctest if you think the current tests are insufficient.