Definition talk:Differential

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The previous version of the page utilized only concepts in a Calculus I curriculum. This new one is no longer accessible to students new to Calculus (and I don't know what a Frechet derivative is). --GFauxPas (talk) 18:53, 28 September 2012 (UTC)

Tried using Google? --prime mover (talk) 18:55, 28 September 2012 (UTC)
I did, and it utilizes concepts that are too advanced for me. The main point was the first part of my comment, that this page is no longer accessible to people new to Calculus. --GFauxPas (talk) 18:59, 28 September 2012 (UTC)
If by that you mean "by people who don't have the particular textbook that you had access to in college", I'm afraid it probably never was ... --prime mover (talk) 19:03, 28 September 2012 (UTC)
I don't think it's particular to my textbooks. The new definition utilizes open sets and bounded linear operators. --GFauxPas (talk) 19:10, 28 September 2012 (UTC)
Beg your pardon. I see what you mean now. It has indeed become seriously inaccessible. --prime mover (talk) 19:19, 28 September 2012 (UTC)
Agreed the Frechet derivative is post-undergraduate. It's the same thing as all the usual stuff in finite dimensions. I used it because Definition:Derivative only has one dimension, it wasn't really a good solution. --Linus44 (talk) 20:32, 28 September 2012 (UTC)
I'm an undergraduate. This page reminds me of Definition:Polynomial. I didn't know what it was talking about at first. But everything was linked and what was once cloudy very soon became clear.
I also want to say that on several pages (Definition:Integer and Definition:Sequence for example) you have an informal subsection for accessibility. If you can use this new definition here to prove results it should be preserved somehow. Just my 2 cents. --Jshflynn (talk) 20:50, 28 September 2012 (UTC)
It's hard to know where the line is between demanding the reader expects some opacity in the definitions, and just being more confusing than is necessary. Def::Polynomials is a good example; the definition is completely obscure, and only exists so we can say "all of that stuff we assumed we can do is fine", and from then on you can stop worrying. As you say, an informal introduction before the more pedantic stuff is a good solution

In fact, there is a more general problem: at the undergraduate calculus (for me) was all done from $\R^n \to \R^m$. That is, it was done for vector spaces, but only identifying them all with $\R^n$. Generally it'd be nicer to have the results formulated without fixing a basis like this, and the translation of results doesn't mean anything more than fixing an identification with $\R^n$, but it would probably be unhelpful when trying to learn the stuff.

So should everything be done in just $\R^n$? Or just vector spaces? It'd get pretty messy doing it for both, and having a load of pages saying after fixing a basis the results are the same.

My opinion: calculus should all be done in $\R^n$. Then anything that might be put under Category:Differential Geometry should be done in vector spaces, rather than repeating everything, and having a load of somewhat-trivial-but-still-kind-of-necessary-for-rigor linking pages. --Linus44 (talk) 21:55, 28 September 2012 (UTC)

My opinion: Calculus should have a page (or suite of pages) doing it all in $\R$. Then there should be another series of pages where the results are generalised to $\R^n$. Then you can go into general vector algebra and algebraic topology as a set of results from there.
The base line is: if you've got a result in a book which approaches it from any particular (limited) context, then use it. We do not subsume the simple results into more general all-encompassing results if by so doing you obscure the details.
Having said that, I am not a fan (although I admit having used it) of an "informal introduction" as it then looks like Shitipedia as it should be possible to make a straightforward definition that applies to the "everyday" case.
As for "demanding the reader expects some opacity in the definitions", I hope you mean "transparency"? We are not into opacity, I had enough of that when I was an undergraduate, when certain mathematicians seemed to think that it was important to make things as difficult to follow as possible, on the grounds that it weeded out the slower students, leaving only the elite able to continue. That is exactly what ProofWiki is not about. --prime mover (talk) 22:22, 28 September 2012 (UTC)
As for the pages on Polynomials, it's in the plan to rewrite them again to make them once more accessible. --prime mover (talk) 22:23, 28 September 2012 (UTC)
Seconded; this is in the same line as the 'characteristic function of a relation' discussion a few days ago. I recognise the stuff about obscuring information to sift out the talented; this has caused some of the most frustrating (and, OTOH, delighting once I finally understood) moments in my brief mathematical career. Clarity is everything; even when skipping through trivial parts, mention that they are trivial: as such, the reader can be confident that nothing was glossed over in constructing the proof. --Lord_Farin (talk) 22:35, 28 September 2012 (UTC)
I didn't mean in saying that calculus should be done in $\R^n$ that it shouldn't be done over $\R$ too. I meant that it should be done up to $\R^n$. My personal preference would be for having, say, the mean value theorem from $\R \to \R$, $\R^n \to \R^n$ or between banach spaces tanscluded into one page (or with the second two versions as corollaries); I think it would be easier to navigate the site if the three results were transcluded into one page, since all three theorems are called the same thing, and each version contains the previous one as a special case. But this is just a matter of taste, I'm not especially raising an objection to using separate pages and then disambiguating or something.
I think it is right to ask that the definitions be given is a manner such that they are as transparent as possible. I don't in any way advocate omitting or obscuring details to sift out the talented; partly because I think this gives a poor description of talent, and more personally because this too often sifts me out as well. But I also think that having definitions that are as transparent as possible, one still has to expect some opacity. I often find that the link between a mathematical definition and what it means is provided only by a long-winded and entirely un-rigorous explanation by someone who understands it. I don't think one can demand complete transparancy without engaging in a wikipedia-format prelude in these cases. In fact, a lot of the time I find that the wikipedia article doesn't shed any light on the matter either. --Linus44 (talk) 09:21, 29 September 2012 (UTC)
I'm more in favour of the transclusion/subpage technique to bundle these results. There is no problem with a long page IMO, and transclusion can be used to collect only the theorem statement, leaving the proofs to the "dedicated" (transcluded) page. For the intuition-like sections, I would like to see those at the bottom of the page. I prefer it when an object or notion is defined, and then its use for solving/describing the issues at hand be explained. An explanatory remark is necessary even if the purpose of a definition is made clear in advance (so as to ensure the reader that the notion behaves as it is claimed to do). If PW is intended to make every notion accessible to a sufficiently dedicated reader, then we need to provide that reader with some words pointing out the correct lines of thought. --Lord_Farin (talk) 09:45, 29 September 2012 (UTC)

Declaration of Intent

I am now mentally prepared to clarify my position as to why I feel uncomfortable with the Differential page as it stood, and where it stands now (work in progress still).

A definition page should be just that: a succinct, clear, unambiguous statement of what a concept means. This will probably be in terms of other notation which is already defined elsewhere, and it will be defined in a specific context (e.g. in this case, real functions, real valued functions, more general vector spaces).

In that definition, there may be a line or two of explanatory matter, e.g. to take as a simple example:

$\forall x, y \in S: x \circ y \in S$
That is, $\circ$ is closed in $S$. (Link omitted so as not to confuse the dedicated user of "What links here".)

Implications of the definition are discouraged, as once you allow this, it is difficult to know where to stop. While interesting details are interesting to the writer, anything which is not essential to the understanding of a concept can often serve to confuse the reader. (The implication being: if a statement is included in the definition, then that statement is necessary to the definition of the concept.) This carries through to notational variants and their derivations - a plurality of notational variants can just get in the way, so (a fairly recent development) such material is best placed in a separate "notation" section.

If there are immediate implications of the definition, then, as with all proof material, these need to be put into separate pages. Definition pages are for definitions, statements go on proof pages (even if the proof is no more than "follows directly from the definition". A link to that page (and possibly a brief description of its contents) goes in the "Also see" section.

Historical information is encouraged, and goes near the bottom of the page, along with any namedfor section. IMO this is one of the areas in which ProofWiki has the potential to shine - many sources do not give a historical perspective, but this is arguably a mistake as it give insight into why the train of intellectual development went in that particular direction.

I suspect that the above can be expanded, clarified and further tightened, and placed somewhere on the House Style page. I'm not scared of the house style instructions becoming larger - after all, have you seen the style guide at Wikipedia? --prime mover (talk) 12:30, 29 September 2012 (UTC)

Agreed --Linus44 (talk) 12:38, 29 September 2012 (UTC)
There, that's about finished now, except for the work on the Vector Spaces section. Thoughts? --prime mover (talk) 12:59, 29 September 2012 (UTC)
I think instead of vector spaces it should be done $\R^n \to \R^m$, so it's still calculus-style. Then the stuff on vector spaces should be yet another section, done on manifolds, or banach spaces, which trivially contain vector spaces as a special case. This differentiates (no pun intended) between differential geometry and calculus; to combine the two I think will likely be confusing, and isn't really necessary in the cases covered. If this is ok with everyone I'll modify this after I've done the definition of differentiability and the Jacobian and linked it with the derivative --Linus44 (talk) 13:18, 29 September 2012 (UTC)
Differential geometry and topology need to be separated from the "applied" context of $\R^n$ and matrices. Definitions should be allowed to have natural subdefinitions (like "the space of all continuous functions from $\R^n$ to $\R$ is denoted $C(\R^n)$.") which will be necessary or convenient in any relevant discussion of the stuff. Of course, these subdefs come after the main definition and generally serve only to introduce terminology. In the past, I've pleaded for an Examples section, and I take this opportunity to do so again. A definition will generally not "come to life" for the reader until a number of examples has been discussed. Other than that I wholeheartedly agree with PMs DoI for ProofWiki. --Lord_Farin (talk) 15:27, 29 September 2012 (UTC)
I have no problem with an "Examples" section, although whether they would be best written into the page, transcluded or linked to is something we haven't explored (my pref would be transclusion). If they go on different pages (which is best) then we need a separate category for examples, on the same level as definitions, proofs, axioms and subcategorised in the same way, so as to allow maximum use of standard templates for boilerplate.
We do have a few examples floating round, but the structure has not been properly defined and explored. No reason except that I haven't got round to doing it yet. --prime mover (talk) 16:47, 29 September 2012 (UTC)