[Maxima-discuss] Fwd: Integrate Differential Form Over Manifold "Easily"?

[Jeffrey R. <air...@gm...>]

---------- Forwarded message ---------
From: Jeffrey Rolland <air...@gm...>
Date: Sun, Jun 6, 2021 at 12:04 PM
Subject: Re: [Maxima-discuss] Integrate Differential Form Over Manifold
"Easily"?
To: Dimiter Prodanov <dim...@gm...>


Here are two more examples where the chain homotopy defined by the integral
fails (One is supposed to have arctan(y/x) for its antiderivative on any
star-shaped region [or contractible subset] and the other is supposed to
have ln|x^2+y^2| on the whole plane less the origin)

On Sun, Jun 6, 2021 at 11:04 AM Jeffrey Rolland <air...@gm...>
wrote:

> A couple of things:
>
> 1) I think you may be talking about the antiderivative script; the
> integral over k-manifold script works jsut fine (See attached test01.wxmx).
>
> 2) I'm not sure how well any of the antiderivative scripts work with
> improper integrals (See attached test02, test03, and test04). Test02 and
> test 03 do not have the t-integral for the coefficient converge, but test04
> does. As you point out, the test02 has additional problems, as, "even if"
> the t-integral converged, the script would return a constant for the
> antiderivative, not ln|y| possibly times a constant.
>
> I think when the form has a pole of some sort in [0,1] and the integral is
> improper, special care may need to be taken. But, then, I think one of the
> hypotheses for the theorem on which the script is based is that the form is
> smooth. I'll actually defer to someone with more knowledge of chain
> homotopy operators for forms with discontinuities at this point.
>
> On Sun, Jun 6, 2021 at 4:36 AM Dimiter Prodanov <dim...@gm...>
> wrote:
>
>> How can you integrate the from
>>
>> dy/y
>>
>> using your method?
>> The substitution y -> t *y does not work.
>>
>> best regards,
>>
>> Dimiter
>>
>> On Fri, Jun 4, 2021 at 12:45 PM Francisco Carbajo <
>> fra...@up...> wrote:
>>
>>> Here is the wxmaxima file that solves the proposed example using the
>>> cartan package.
>>>
>>> Regards
>>> El 4/6/21 a las 10:55, Jeffrey Rolland escribió:
>>>
>>> Is there a way to integrate a differential k-form "easily" over a
>>> k-manifold, say using the dii_form package? Here is an example of
>>> integrating over a 3-torus in Mathematica, illustrating Stokes's Theorem.
>>> This would be the last piece to replace Mathematica with wxMaxima for me.
>>> (You need to have the DifferentialForms.m file in your Mathematica path --
>>> your HOME directory should work -- for the notebook to function properly.)
>>>
>>> --
>>> Jeffrey Rolland
>>> <air...@gm... <jro...@gm...>>
>>>
>>> "The weed of crime bears bitter fruit; crime does NOT pay! The Shadow
>>> knows!"
>>>  - The Shadow, _The Shadow_ (1994)
>>>
>>>  -----BEGIN GEEK CODE BLOCK-----
>>> Version: 3.1
>>> GM d-- s:+ a+ C++>$ UL+>$
>>> P? L+++>+++++$ E--- W+++>$ N+++>+++$ o? K--? !w--- !O---- !M- !V-- PS++
>>> PE- Y? PGP+++ t+++ 5? X+ R+>$ tv++ !b DI+++>+++++ !D G+ e++++$ h+ r--
>>> y++
>>> ------END GEEK CODE BLOCK------
>>>
>>>
>>> _______________________________________________
>>> Maxima-discuss mailing lis...@li...://lists.sourceforge.net/lists/listinfo/maxima-discuss
>>>
>>> _______________________________________________
>>> Maxima-discuss mailing list
>>> Max...@li...
>>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss
>>>
>>
>
> --
> Jeffrey Rolland
> <air...@gm... <jro...@gm...>>
>
> "The weed of crime bears bitter fruit; crime does NOT pay! The Shadow
> knows!"
>  - The Shadow, _The Shadow_ (1994)
>
>  -----BEGIN GEEK CODE BLOCK-----
> Version: 3.1
> GM d-- s:+ a+ C++>$ UL+>$
> P? L+++>+++++$ E--- W+++>$ N+++>+++$ o? K--? !w--- !O---- !M- !V-- PS++
> PE- Y? PGP+++ t+++ 5? X+ R+>$ tv++ !b DI+++>+++++ !D G+ e++++$ h+ r--
> y++
> ------END GEEK CODE BLOCK------
>


-- 
Jeffrey Rolland
<air...@gm... <jro...@gm...>>

"The weed of crime bears bitter fruit; crime does NOT pay! The Shadow
knows!"
 - The Shadow, _The Shadow_ (1994)

 -----BEGIN GEEK CODE BLOCK-----
Version: 3.1
GM d-- s:+ a+ C++>$ UL+>$
P? L+++>+++++$ E--- W+++>$ N+++>+++$ o? K--? !w--- !O---- !M- !V-- PS++
PE- Y? PGP+++ t+++ 5? X+ R+>$ tv++ !b DI+++>+++++ !D G+ e++++$ h+ r--
y++
------END GEEK CODE BLOCK------


-- 
Jeffrey Rolland
<air...@gm... <jro...@gm...>>

"The weed of crime bears bitter fruit; crime does NOT pay! The Shadow
knows!"
 - The Shadow, _The Shadow_ (1994)

 -----BEGIN GEEK CODE BLOCK-----
Version: 3.1
GM d-- s:+ a+ C++>$ UL+>$
P? L+++>+++++$ E--- W+++>$ N+++>+++$ o? K--? !w--- !O---- !M- !V-- PS++
PE- Y? PGP+++ t+++ 5? X+ R+>$ tv++ !b DI+++>+++++ !D G+ e++++$ h+ r--
y++
------END GEEK CODE BLOCK------