user talk : jacobolus

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Again, welcome! --Lst27 (talk) 23:54, 15 Jan 2005 (UTC)

Dialogue and talks in Wikipedia

[... clipped; see 2024 archive and Talk:Function (mathematics) ... ]

Indeed, that's why I decided terminating my yearly visit to this article (over the past 10 years I have seen no progress), and coming back in 2025. As regards the codomain issue, since the 1980s I've been regularly asking mathematicians using codomains why they might want it. The answer always amounted to "tradition in certain fields" or "it's a convenience" (without a technical justification). Yet, most definitions for composition of functions with codomains are very restrictive. Of course, since the term codomain exists, it must be covered by an article meant to be encyclopedic, but a solid technical justification would be a genuine added value.

If you have the incentive to continue working on this article, you may find Rogaway's remarks very helpful [1], in particular his remark #18: 'Definitional choices that don't capture strong intuition are usually wrong, they may come back to haunt you". As for the of informal introduction, I recently found very high praise for the educational style of Michael Spivak's Calculus (now in its 4th edition, freely available on the web). Since the perspective of the article must be far wider than calculus (ideally, all of mathematics) the intuitive discussion on functions must also be wider. A suitable preamble to the formal definition might run as follows (after a few examples). (begin excerpt) In our examples, we have been writing f(x) for the "output value" of the function f for a given "input value" x. This immediately raises two questions: (a) for which values of x is f defined? and (b) if x is such a value, what, then, is the value of f(x)? (end excerpt). Answering those questions provides the justification for a simple formal definition to follow.

I hope this helps. Other work currently prevents me from providing more input or even having an occasional look at these pages. Boute (talk) 09:17, 3 March 2024 (UTC)

The most obvious purpose I can see is to avoid extra bookkeeping. It's easier to say e.g. a square matrix represents a function from $\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ and not worry up front about noting that the image might be some restricted subset in degenerate cases. Etc. –jacobolus (t) 09:46, 3 March 2024 (UTC)

Was reading through this talk page out of curiosity and might have something helpful to add. When talking about functions $\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}$ you don't want to exclude maps that only map to subspaces. Also the transpose map turns the codomain of a matrix $M$ into the domain of $M^{\top }$. I think the category theorists quite appreciate the codomain for these reasons and others. Anyway happy late new year I guess. Shoe Deceiver (talk) 21:36, 7 January 2025 (UTC)

Oh and also the preimage map of clearly depends on what the codomain is. Shoe Deceiver (talk) 21:48, 7 January 2025 (UTC)

Subtended angle intro

Hi,

Obviously it's disappointing to have clarified something for the worse, but I'm happy to go with your reverting of it.

What I'm wondering, though, is whether there's a way to introduce readers in a really clear way to the central idea first, rather than (as it seems to me) hit them straight away with several definitions in succession, all applied to different situations, and a large number of wikilinks? That's the problem I was trying to solve, really, and it's clear from the talk page that at least one visitor had trouble understanding the article when they visited in 2019, though I've not checked what it looked like back then. Musiconeologist (talk) 21:13, 5 January 2025 (UTC)

Afterthought: maybe a better diagram would go some way. Show one angle, with a a line segment, an circular arc and an arbitrary curve all subtending it (and having the same endpoints?). We can also say the line segment subtends both the arcs if we want, and everything subtends the angle. Musiconeologist (talk) 21:44, 5 January 2025 (UTC)

Moved to Talk:Subtended angle § Subtended angle intro

Fundamental theorem of calculus

• User:Dedhert.Jr/sandbox/1

Might be planning to implement it anyway. But do you think the introduction of differentiation and integral seems superfluous and somewhat unrelated before the fundamental theorem of calculus? I somehow managed to relate those two with the fundamental theorem in order to describe it mathematically. The second theorem's proof is the only problem I could not comprehend anymore. Dedhert.Jr (talk) 04:30, 6 January 2025 (UTC)

I'm not quite sure what the question is, or what you mean by "implement it". If you are asking whether it is worth giving a quick introduction about what derivatives and integrals are in the article about the fundamental theorem of calculus, I would say yes. Some readers who are not familiar with calculus might be curious about it. I think we should if possible give a (brief) explanation at the start which is accessible to e.g. high school students taking an algebra or trigonometry course who have not yet seen any calculus. But we also shouldn't belabor it, as that may be distracting for more expert readers. –jacobolus (t) 23:57, 7 January 2025 (UTC)

Reordered Formulas

Hi, I reordered the formulas on the page about stereographic projections because I think sums are nicer if they don't lead with a minus. It's also more consistent since, as it stands, the formula for the polar form has both $1+R^{2}$ and $R^{2}+1$ in the denominator. That said I won't contest. Shoe Deceiver (talk) 10:42, 7 January 2025 (UTC)

@Shoe Deceiver: There are a wide variety of different variants in use in different sources, but it's typical to put the 1 first. If it were up to me we'd use a north-pole centered stereographic projection, in which case the relevant quantities end up as ⁠${\tfrac {1}{2}}(1+x^{2}+y^{2})$⁠ and ⁠${\tfrac {1}{2}}(1-x^{2}-y^{2})$⁠. (If I ever manage to get the time and energy for a substantial rewrite I might put this one in place. It also has the beneficial property of not reversing the orientation of the sphere. I think it's easier to make sense of, it accords better with the conventional spherical coordinates which measure the polar angle from the north pole as 0, and it generalizes better to uses such as taking the tangent of half an angle or taking the stereographic projection of unit quaternions as a representation of rotations ["modified Rodrigues parameters"].) –jacobolus (t) 19:59, 7 January 2025 (UTC)