Re: "They Use ∂ Differently in Math and Physics. Which is Better?"

Lovely discussion!  And the use of ∂ as boundary shows up beautifully in
Stokes' theorem on differential forms, where it looks just like rearranging
"d"s,

       ∫_C dω = ∫_{∂C} ω

of which

      ∫_a^b df(x) = f(b)-f(a)

is a special case.

Stephen



On Thu, Jun 20, 2024 at 10:33 AM David Carlisle <david.carlisle@nag.co.uk>
wrote:

> On 20/06/2024 15:21, Deyan Ginev wrote:
>
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> Hi Abbas, all,
>
> Interesting additions, thank you.
>
> For k-forms, since those are "differential forms", I wonder if the use
> already fits in the conventions described in the video.
>
> I see you are referring to overloading the notational use of "∂" with the
> boundary example in homology.
> I wasn't familiar with it until now, wikipedia has a nice overview here:
>
> https://en.wikipedia.org/wiki/Homology_(mathematics)#Construction_of_homology_groups
>
> Different "named concept" uses of ∂ would be relevant additions for the
> Intent Open list, especially in cases where - quoting you - "it would be
> nice to say boundary". That is useful for accessibility.
>
> The original youtube video is closer to the OpenMath questions, as it
> reveals how an operator with the appearance of full formalization -
> "partial derivative" - may represent (at least) two different formal
> definitions.
> In OpenMath terms, one could have cast the video exposition as two symbols
> "convention-m:partial-derivative" and "convention-p:partial-derivative".
>
> But I doubt generator tools can realistically infer these without some
> very purposeful additional help from authors - such as a brand new
> notational convention.
>
> Deyan
>
>
> On Thu, Jun 20, 2024 at 8:56 AM Abbas Jaffary <abbas.jaffary2@gmail.com>
> wrote:
>
>> Very interesting!  There is also the differential geometry use of
>> derivatives as one-forms (and k-forms), and the as boundary operator in
>> algebraic topology. In homology, for example, one could likely infer
>> "partial x" as the "boundary of x", though it would be nice to have it say
>> "boundary".
>>
>> I imagine there will be some crowdsourced effort to accommodate the most
>> popular use cases for fundamental symbols.  Not sure where OpenMath is with
>> this?
>>
>> On Wed, Jun 19, 2024 at 8:49 AM Deyan Ginev <deyan.ginev@gmail.com>
>> wrote:
>>
>>> Hello everyone,
>>>
>>> I stumbled on a very well-made exposition video, which covers a subtlety
>>> in the meaning of  "partial derivative" between mathematics and physics:
>>>
>>> https://youtu.be/QFHSHhpbo00
>>>
>>> Aside: This topic is not directly related to current conversations about
>>> derivative syntax for "intent".
>>>
>>> Instead, the presenter has some well-reasoned general discussion, a
>>> community proposal for adding yet-another notation, and showcases some of
>>> the problems that our generator tools also face when trying to infer
>>> Content MathML expressions from human-authored math syntax.
>>>
>>> Greetings,
>>> Deyan
>>>
>> use of ∂  for boundary (read as "boundary") is already flagged in an
> example in the draft open concept list
>
> https://w3c.github.io/mathml-docs/intent-open-concepts/#boundary1function
>
> this list of course is never complete or correct but it's a start....
>
> David
>
>
>
>
>
>
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