Talk:Symmetry of second derivatives

More on counterexample
To convince a skeptical reader, at least one counterexample should be included in the following paragraph, or the paragraph should be deleted.

In most "real-life" circumstances the Hessian matrix is symmetric, although there are a great number of functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the second derivatives at a particular point. Schwarz' theorem gives a sufficient condition on f for this to occur.

--HopingToBeUseful (talk) 16:36, 3 May 2016 (UTC)

The link (1) is broken
131.111.16.20 (talk) 16:20, 18 May 2015 (UTC)

Counterexample
I think it would help, for the counterexample, to check whether it is locally integrable, at (0,0). Charles Matthews 22:39, 14 January 2006 (UTC)

Name
I'm very curious, as I know this property as "Young's Theorem," but this is not mentioned by name here. (dphrag 07:19, 14 June 2006 (UTC))
 * Yeah, I also heard it is called that way. I mentioned this in the article. Oleg Alexandrov (talk) 01:01, 15 June 2006 (UTC)

Copied out from article
[Could someone write up such an example here or in its own article, and add that to the list of mathematical examples?]

I removed it. --M1ss1ontom a rs2k4 (T 21:48, 22 June 2006 (UTC)

Merger
Definitely agree that this should be merged with Clairaut's theorem. --Macrakis 01:33, 1 June 2007 (UTC)

Agree. -- Hongooi 17:51, 15 June 2007 (UTC)

Counterexample:
How is the given function a counterexample? The article asserts that "Then the mixed partial derivatives of f exist, and are continuous everywhere except at (0,0)".

However, (0,0) is not in the domain of the original function. -- Heath 24.127.115.128 15:26, 30 August 2007 (UTC)
 * I tried to make that clearer. Oleg Alexandrov (talk) 04:21, 31 August 2007 (UTC)

Counterexample Again
I worked through the second partial derivatives of the counterexample and they seem to be equal; i.e., the "counterexample" isn't a counterexample. I got $$\frac{\partial^2f}{\partial x \,\partial y} = \frac{\partial^2f}{\partial y \,\partial x} = \frac{(x+y)(x-y)(x^4+10x^2y^2+y^4)}{(x^2+y^2)^3}$$ and my TI-89 confirmed it. Comments? Jonah 02:03, 20 October 2007 (UTC)


 * The counterexample is correct:


 * let $$ f_{xy}(0,0) \stackrel{df}{=} \lim_{h \to 0} \frac{f_x(0,h)-f_x(0,0)}{h} \mbox{ and } f_{yx}(0,0) \stackrel{df}{=} \lim_{h \to 0} \frac{f_y(h,0)-f_y(0,0)}{h} $$


 * You may also want to check, that $$ f_x(0,h)=-h\ f_x(0,0)=0\ f_y(h,0)=h\ f_x(0,0)=0.$$(it results from basic limits).
 * Now: $$ f_{xy}(0,0) = \lim_{h \to 0} \frac{-h-0}{h}=-1 \mbox{ and } f_{yx}(0,0) = \lim_{h \to 0} \frac{h-0}{h}=1

$$
 * Mkalinowski (talk) 20:56, 17 March 2008 (UTC)

Counterexample
I find it a bit confusing at first to read about a counterexample directly after a theorem (if there is a counterexample, it's not a theorem). However, the counterexample relates to the naive idea about symmetry of the Hessian where continuity is disrespected. Maybe the paragraph could be reformulated like: "If the condition of continuity of the second derivates is dropped, counterexamples with unsymmetric Hessian are possible". I think that would make it clearer that this is -- of course -- no counterexample to the preceding theorem, but only to the more naive idea. 134.169.77.186 (talk) 08:36, 8 April 2009 (UTC) (ezander)

Image
Alongside the 3d plot of the counterexample function, I think a 3d plot of its second derivative might be helpful. The piecewise function is continuous, and the first derivatives just about are as well (perhaps having at worst only removable singularities) but the second derivative is plainly discontinuous. It limits to different values (ranging between +/-1) according to the direction from which the origin is approached. (This seems to make it easier to understand why the actual value at that point, or even whether it is undefined, should depend on subtleties of how the derivatives are defined, and particularly on the direction in which the limits are calculated.) Cesiumfrog (talk) 02:38, 12 March 2013 (UTC)

Clairaut's constant
What does the "Clairaut's constant" section have to do with the topic of this page? Either this should be explained, or it should be moved elsewhere. McKay (talk) 03:53, 28 April 2009 (UTC)

Always Equal?
Just want to add my two cents - I think the language should be changed so that the theorem is correct, perhaps by adding a smoothness assumption, or by assuming that the "total derivative" / "Jacobian" exists — Preceding unsigned comment added by 199.38.133.55 (talk) 20:25, 7 March 2012 (UTC)

total derivatives
Should there be a note about the non-symmetry of total derivatives?

For example, in physics relating acceleration to the velocity as a function of distance:
 * $$\ddot x\equiv\frac{d\dot x}{dt}=\frac{dx}{dt}\frac{d\dot x}{dx}=\dot x\frac{d\dot x}{dx}

=\dot x D_x D_t x \neq \dot x D_t D_x x=0$$ Cesiumfrog (talk) 01:55, 11 March 2013 (UTC)

Naming and history
This topic has so many names, perhaps there needs to be a history section to explain it?
 * Clairaut-Schwarz's theorem
 * Clairaut's theorem (presumably published in french someplace 1713-1765 and one of several of Clairaut's theorems).
 * Schwarz's theorem (or Schwarz integrability condition?) (maybe published where it had more influence on germanic language authors 1843-1921)
 * Euler’s Theorem for mixed derivatives. Apparently, "Euler first published the Mixed Derivative Theorem in 1734, in a series of papers he wrote on hydrodynamics."
 * Young's theorem (W.H. Young ~1910? Or is it one coauthored with another Young?)

Funnily enough, there seems to be a uni math course that deals directly with the historical sublety here. Also. Seems they each may have proved something very slightly different. Cesiumfrog (talk) 22:47, 12 March 2013 (UTC)

two comments
might be useful to mark the comments that refer to totally obsolete texts?

the following is trivial but perhaps interesting let F(x,y)=int dxdy f(x,y), then d/dy(dF/dx) is the limit of f along y and d/dx(dF/dy) that along x because provides a construction of a pathology (by the way, I would have employed this word instead of counterexample avoiding the discussion above). I leave to someone more skilled than me to consider if this allows interesting conclusions on the size of the pathological domains.

in the section sufficiency of twice-differentiability, 'all partial derivatives' is 'all FIRST partial derivatives?'

pietro2A00:1620:C0:64:21C:61FF:FE03:A4C (talk) 13:12, 28 March 2014 (UTC)

Thomas Higgins, History section - false accusation of WP:COPYVIO
The Scripta Mathematica article of Higgins was inaccessible on wikipedia until I investigated carefully. With some effort I was able to retrieve a transcript of it on the Wayback Machine. The article was short and easy to summarise and paraphrase. Summarising and parpa=harsing is what we do on wikipedia. has claimed this was a copy-paste, but that is completely incorrect as a small child could verify. and Gumshoe2 already planned on User talk:D.Lazard how they would attempt to revert any edits I had made to this article. There plans have been summarised on User talk:Salix alba. It is a routine matter to chec the actual text and my paraphrase. D.Lazard did not check this properly; but we can easily check that with User:Diannaa, User:Moonriddengirl or user:Doug Weller (I should not really be pinging them). Mathsci (talk) 18:07, 19 August 2020 (UTC)
 * Just to clarify, the entirety of my and D.Lazard's comments are on D.Lazard's talk page, together with D.Lazard's two recent reversions on this page. The context is that I've had some exchanges with Mathsci which I found to be unpleasant. This is why I wanted to know if there was an indirect way to flag this article for attention which I felt it needed. Gumshoe2 (talk) 18:20, 19 August 2020 (UTC)
 * has copied Higgins's and Mathsci's version on my talk page. Everybody can see that the phrasings are very similar, with most sentences differing simply by replacing some words by equivalent ones. This is blatant plagiarism, and this is forbidden by the policy WP:COPYVIO, which asserts This is the case here, there is definitively a "substantial linguistic similarity in creative language and sentence structure". D.Lazard (talk) 19:31, 19 August 2020 (UTC)
 * By the way, even if there were no copyvio problem, this part of the hystory section should be deleted, as duplicating the first paragraph, also based on Higgin's work, and introducing pedagogical advices that do not belong here. D.Lazard (talk) 19:36, 19 August 2020 (UTC)


 * has not repsonded at all to the comments here. I actually have the text and my version next to each other so we can compare the text with the original.

(1) Let's just tae the first paragraph. My phrase, "The topic can be separated into be divided into two distinct line of attack" does not appear. My phrase, "Most advanced calculus texts contain sufficient conditions and proof for the equality of second mixed partial derivatives. Hence this is something that should interest those involved in teaching and learning that part of analysis," cannot be found anywhere. There is an original phrase, "Accordingly, a brief account of the mathematical history associated with this bit of analysis should not be without interest to the serious student or teacher," which is preceded by formulas.

(2) The next paragraph mentions Schwarz and Lindelõf. The paragraph doesn't have the sentence structure. Lindelõf is described as a Finnish mathematician.

(3) In the next paragraph I wrote, "Cantor outlined the historical status of second mixed derivatives before 1800". The original has "Indeed, in his comprehensive survey of mathematics prior to the nineteen century Moritz Cantor accords to ... " This is completly rewritten. There follow lists of unsuccessful proofs to establish the equality of mixed derivatives. Material on Euler was surpressed. A brief summary for Bernoulli. I write, "Nicolas Bernoulli had tacitly assumed the property without any formal proof". The original says, "tacitly assumed that they were equal, although he gave no formal proof in support of his assumption". Mathemticians and years are given for unsuccessful attempts. A long series of the original sentences are replaced by one sente by me, "All of the proposed proofs had been criticized, particularly when subtle points on limiting procedures arose. It was as a result of a detailed study of the deficiencies that Lindelöf could explicitly exhibit a counter-example, thus ending the stage of "primitive" investigations." The original final statement has, "this paper marked the close of the "primitive" period of investigation" which again is a paraphrase.

(4) There is then a discussion on Schwarz and Lindelõf. I wrote, "Six years after Lindelöf, Schwarz published the first satisfactory proof, thus starting the next stage of investigations." The original is, "Six years later H. A. Schwarz gave the first satisfactory proof, thus inaugurating the second period of investigation." Those lining sentences are close, but words have been changed. My sentences, "Mathematicians tried to relax some of the assumptions of Schwarz. After an unsuccessful attempt by Thomae in 1875, the Italian mathematician Dini made an improved on Schwarz by introducing the more general Dini-Schwarz conditions" are completely different from the original. The sentences, "Following another fruitless effort by Harnack in 1881, Jordan in 1882 was able to make headway." are completely different from the original. Made phrases "fruitless" and "was able to make headway" were originally "unsuccessful effort" and "made the next advance". Then there are lists of mathematicians and years. About Jordan my sentences write, "Assuming less than Dini, he published in 1883 the proof that can now be found in most text books" and "Some of these expositions were perfect, some not, but essentially apart from changing some points of view in a minor way, Jordan's proof was adopted." The originals are quite different, "Postulating a set of conditions somewhat less restrictive than those of Dini, he put forward the proof found in the majority of texts published since that year" and "Of these, some were perfect, some imperfect, but none were essentially new: most of them differed from previously published proofs only in minor points of analysis," parapahrase not a copy-paste.

(5) For the sentence on Hobson only the technical term "successive differentiation" occurs in my sentences and the original. For Young, I wrote, "independently found less restrictions than the Dini-Schwarz conditions". In the original it's, "Several years later, W. H. Young, without using Hobson's generalization, also obtained a set of conditions less restrictive than the Dini-Schwarz conditions." I quote the "fundamental theorem" of Young, which is direct quotation of his theorem from 1909. Finally the statement, "Finally, in 1918 Carathéodory gave an original and unique contribution in this context using Lebesgue integration" has in the original the year, the name Carathéodory, the word "unique" and the technical term Lebenesge integration.

So that seems like a paraphrase-summary, not omitting any facts. As I have written, the main technical experts are Monnriddengorl (Mdennis WMF), Diannaa and Doug Weller. There might be others. But this is the usual kind of paraphrase-summary. I have done these from French to English (for Auguste Pavie) and from German to English (John Butt's commentary on Canonic Variations on "Vom Himmel hoch da komm' ich her". I adssume that you will continue edit-warring. In that case you ahve to deal with a former arbitratpor. Mathsci (talk) 19:43, 19 August 2020 (UTC)
 * has not given any proper explanation of his actions. Above there is a fairly careful description of how my paraphrase-summary differs from the original. D.Lazard has not verified the two texts, which are very different: sentence structure, choice of words, etc. On User talk:D.Lazard, jointly discussed with D.Lazard how all of my edits on this article could be reverted. Gumshoe2 later described my paraphrase-summary in a dismmissive way as a student essay. He claimed that the material had been copy-pasted. But here on this talk page is concrete proof that that was not the case. (I have tried to use as little of the original material as possible.)


 * Anyway I hope that this can be checked with a trusted administator (perhaps a former arbitrator) so that these deletions can be restored. D.Lazard discussed gaming WP:3RR this morning on his user talk page and that is exactly what has happened. Mathsci (talk) 20:43, 19 August 2020 (UTC)
 * To be clear, my comment "I believe this would be quite bad in a student essay; as far as I can see, Wikipedia:Plagiarism doesn't explain standards for this kind of extensive copy-paste-modify" meant that in certain contexts, such as in a student-written essay in a course, the passages I quoted would likely be unambiguously considered as plagiarism, but that I didn't know the standards in the context of wikipedia. So I've made no plagiarism accusations, just waiting now for others to comment. Gumshoe2 (talk) 22:05, 19 August 2020 (UTC)
 * I have asked Doug Weller, a former arbitrator, for help here. I've also mentioned to him your current activities on User talk:D.Lazard. It's worth mentioning that it is extremely easy to rewrite the History section from scratch. Mathsci (talk) 22:57, 19 August 2020 (UTC)
 * Excellent! Gumshoe2 (talk) 23:06, 19 August 2020 (UTC)
 * I communicated by email with Doug Weller. I got a very kind and helpful response. Meanwhile I have completely rewritten the History section, merging the first paragraph. I removed one or two phrases from the first paragraph that were copy-pasted from the 1940 Scripta Mathematica article of Thomas Higgins. Mathsci (talk) 09:43, 20 August 2020 (UTC)

This new version is excellent. I have not compared it with Higgins version, but, if some sentences remain a close paraphrasing of Higgin's text, this is much less important, as the new version is limited to historical facts, and Higgin's comments have been all removed. D.Lazard (talk) 11:51, 20 August 2020 (UTC)
 * There was not much difference, just some merging to avoid duplication. The first paragraph had the sentence "Clairaut assumed all definite integrals could be differentiated under the integral sign". That was a copy-paste so was removed. The material thst \i preciously summarised from Higgins was just historical fact. Nothing added. Are you are suggesting that part of the summary was pure invention? Mathsci (talk) 13:45, 20 August 2020 (UTC)
 * When I write something, this means exactly what I write. Do not try to extract fallacious interpretations as you did in the last sentence of your previous post. D.Lazard (talk) 14:18, 20 August 2020 (UTC)

Thank you Mathsci for the good edit. I've just made some small changes. The most notable is that Higgins' article states that Euler's article was published in 1740, but his bibliography marks it as 1735. Clairaut's is marked as 1740; I suspect the body of his article made a typo. I've also removed the "period of uncertainty" phrasing, as it isn't clear from the source that there was any sense of uncertainty at the time. Based on my understanding of math history (I have no expertise), the modern understanding of rigor in analysis has its genesis sometime around the 1830s, so I don't think a sense of uncertainty can automatically be inferred from context. I've also changed "Jordan's proof" to "earlier proofs", as that is what is suggested by the article. Gumshoe2 (talk) 22:33, 20 August 2020 (UTC)
 * Mathsci, even if you think it is correct to mark it as 1740 instead of 1735, I hardly think it is appropriate to revert my entire edit. Gumshoe2 (talk) 23:34, 20 August 2020 (UTC)
 * Higgins writes 1740 and that seems to be correct; he refers to 1734/1735 in the reference, but that is not where the "theorem" appears. In "The Early Mathematics of Leonard Euler, Vol. 1", the paper is listed as 1740. It's listed in the footnote as
 * Comm. Acad. Sci. Imp. Petropol. 7 (1734/1735) 1740, 174-189, 180-183; Opera Omnia, 1.22, 34-56.


 * So 1740 seems correct. I will continue to make a few tweaks. Mathsci (talk) 00:13, 21 August 2020 (UTC)
 * I've clarified with a reference. The work was done in 1734 and published in 1740. Gumshoe2 (talk) 00:17, 21 August 2020 (UTC)
 * You appear to be following my edits at the moment. Why are you doing that? Mathsci (talk) 02:20, 21 August 2020 (UTC)
 * I think it's good at the moment, I don't plan to make further edits to that section. Gumshoe2 (talk) 04:02, 21 August 2020 (UTC)

Fubini's theorem
Why is there a proof of Fubini's theorem on this page? This seems to very clearly be not the place for it Gumshoe2 (talk) 04:06, 21 August 2020 (UTC)


 * I see that Mathsci has made several edits to the section, without addressing this question. If there has to be a proof of Fubini's theorem for continuous functions, then the present discussion is a really absurd overcomplication, just replaying more sophisticated measure-theoretic concepts in the case of continuous functions. But it could be proved in a couple of lines, very easily accessible to undergraduates: a continuous function can be uniformly approximated above and below by functions which take on finitely many values, and Fubini's theorem is trivial for functions which take on finitely many values. Fubini's theorem for continuous functions follows by the squeeze theorem. In light of this, I think the current version is clearly inappropriate Gumshoe2 (talk) 06:38, 22 August 2020 (UTC)
 * I've edited it out. Please do not revert to the previous version without addressing the above message Gumshoe2 (talk) 07:05, 22 August 2020 (UTC)


 * The revert that you made, planned for a long time in advance in discussions with User talk:D.Lazard, has not been made in good faith. Your discussions show that you were acting as a WP:TAGTEAM with D.Lazard: that is deprecated. As explained on User talk:Doug Weller and User talk:Salix alba, quotations ot diffs by you show that you have been waiting for the right moment to revert large amount of my edits for quite a while. You used the word "vendetta" this morning and that was probably not an accident. By concentrating on me in this way (see your recent edit history), you appear to have been WP:HOUNDING me. I continue editing in a humdrum way, anodyne and neutral, only to find that you want to create mayhem. That is disruptive whatever the cause.


 * The topic of this section in the article is fairly clear. The material is extremely elementary and could possibly be summarised in a briefer manner (Dieudonné's labelling of lemmas is not helpful). There are a few variants of the proof and they are all described in the citations. The main point is that all of the proofs have to be rigorous, otherwise why would there even be an article like this? One of the references is to American Mathematical Monthly. There's also a very recent 2020 book by Sheldon Axler, added by me just yesterday with page numbers. It's completely unlikely that you've checked these. It's also clear that your approach to this article has not been scholarly: no careful discussions. On Salix alba's talk page this morning, however, you grumbled about matters (worrying why administrators couldn't do more about Mathsci), referred to vendettas and then went about your task of reverting content with a WP:BATTLEGROUND attitude. I had been editing the article to improve the layout. The main problem is that you deliberately followed me here: that was the hounding aspect.


 * The treatment for Fubini's theorem and Clairaut's theorem was clear. Dieudonné proved that any continuous function on a rectangle can be approximated by simple functions. He checked that certain functionals are bounded on simple functions. Then he checked the passage to the limit for continuous functions. The other versions prior to my edits contained no detailed account of how Fubini's theorem implies to prove Clairout's theorem. Fubini's theorem is often proved non-rigorously. Proofs using Lebesgue integration and measures are not elementary. As I mentioned on Salix alba's user talk page, Lebesgue integration theory is either a graduate topic (à la M. H. Stone, von Neumann); or can be treated in an ad hoc way, as done for undergraduates, using a variety of approaches, measures on σ-algebras, the Daniell integral, Tonelli's quasi-continuous functions, etc, etc.


 * I will probably contact Newyorkbrad privately about your conduct, depending on how things pan out. If you continue reverting, I will report you on the vandalism noticeboard. Mathsci (talk) 10:34, 22 August 2020 (UTC)
 * I have checked Gumshoe2's version. It is very poorly written, almost like a high school essay. Gumshooe2 has copied several of my references. Then he's placed them at the end of the paragraph. Wow! But there's hardly any resemblance between the sources and Gumshoe2's essay. References to Dieudonné have been removed, even though Gumshoe2 is aware of the source. The treatment of Fubuni's theorem for continuous functions on a rectangle is not at all well explained. At some stage it has to be proved that integration in the $x$ or $y$ parameter results in continuous functions. No proof, no explanation, no references. Omitting material like that just makes a mockery of providing a rigorous proof (the whole point of the article). Gumshoe2 has concentrated on Clairaut's theorem. The theorem occupies 5 lines that can be read in Robert Marshall's notes from the University of Washington and were reproduced by me. Even Marshall's punchline has been mangled. Gumshoe2 has assumed that he writes infinitely better than me. But is it not a better idea to stop following me, to stop playing games of one-upmanship. There are plenty of articles out there ... Mathsci (talk) 12:06, 22 August 2020 (UTC)
 * As there is a page Fubini's theorem, if Wikipedia must have one of several proofs of this theorem, they must be there. Otherwise this would be confusing for readers. Therefore, the proof of Fubini's theorem must be removed from here, and this is independent from the editor or the editors who have added it. As you are the only editor that consider that a proof of Fubini's theorem belongs to the article Symmetry of second derivatives, I'll restore version that you have reverted without discussing here the issue (discussing the proof is not discussing whether thee proof belongs to this article). D.Lazard (talk) 14:19, 22 August 2020 (UTC)
 * Mathsci, it is clear that I removed the discussion of Fubinis theorem from the page, not that I gave an incomplete discussion of it. Please focus on the actual questions at hand: 1) why must there be a proof of Fubinis theorem here; 2) if there is going to be a proof of it here, why not the elementary one I sketched in my previous message? Dieudonne Theorem (for instance) is important for more general situations, not the one at hand.


 * Also, if you contact an administrator, which I strongly encourage, please do so publicly. Gumshoe2 (talk) 15:10, 22 August 2020 (UTC)


 * In this section the notation seems to switch between using $$f(x,y)$$ and $$F(x,y)$$ which should it be?
 * Its not clear to me why we need to have a summary of the proof of the theorem. WP:SUMMARY style suggests that proof of theorems should go in the article being summarised and Wikipedia talk:WikiProject Mathematics/Proofs questions the need for proofs at all. I do not find the argument that this is what the sources have done to be compelling. The sources are generally academic textbooks or monographs, hence secondary sources. We are writing an encyclopedia, a tertiary source, with different aims. Too much technical detail detracts from the flow of the article, conflicting with the aim of WP:TECHNICAL. --Salix alba (talk): 15:19, 22 August 2020 (UTC)


 * (ec) D.Lazard: So far you have not provided any context for Fubini's theorem for continuous functions on a rectangle. I produced a self-contained account mostly drawn from Dieudonné's Treatise on Analysis, Vol.2. It also be can found in other places, but so far neither you nor Gumshoe2 have given us clear statements or citations. That's not a very helpful for producing a wikipedia article. My proof of Clairaut's theorem follows Robert Marshall, who supplied details about Fubini's theorem for continuous functions using Riemann approximating sums. I don't that Gumshoe2 and you, essentially acting as if you're the same editor (is that called tag teaming?), provided that degree of detail.


 * That is somehow the point: it's important to give a satisfactory account that makes it quite clear why Fubini's thoerem for continuous functions is important. In most expositions, mathematicians have understood that Fubini's theorem and the symmetry of mixed derivatives are equivalent. It will take some time to work out further reference for this easy Fubini theorem for continuous functions. I don't want to feel hurried. There is no rush. Mathsci (talk) 15:39, 22 August 2020 (UTC)
 * This is under construction so I suggest D.Lazard stops edit-warring on this article. Please see the text. As far as I'm maware, neither D.Lazard nor Gumshoe2 have written careful acounts of Fubini's theorem for continuous functions on a rectangle. In the source that is explained in detail by Robert Marshall and in the Treatise of Dieudonné. Mathsci (talk) 15:42, 22 August 2020 (UTC)
 * I’m not sure how to respond to this, as Marshall’s note clearly follows the sketched proof of Fubini which I gave above. And as I said, Dieudonne’s treatment is important for a more general scenario which is irrelevant here. I am also deeply suspicious that Fubini and second derivative symmetry could possibly be equivalent; do you have any kind of reference for that claim? Gumshoe2 (talk) 15:50, 22 August 2020 (UTC)
 * (ec) Neither Gumshoe2 nor D.Lazard has explained the content gaps in the section about Fubini's theorem for continuous functions. From the content that is not covered by Fubini's theorem, which could be about measurable functions. If I look at the article it's completely vague. Bourbaki has integration theory for locally compact spaces. Dieudonné, as a member of Bourbaki, covers the same material and that is more or less what I've used in the case of a rectangle. The point is not the 5 pages tagged on to Ropert Mashall's account of the proof of Clairaut's theorem. It is about worrying about the context. You cannot simply write "Fubini's theorem." Here it is about integration over compact intervals: tensor products C(I) ⊗ C(J) and their states or probability measures. But D.Lazard and GUmshoe2 should really concentrate in the content which is absent from the section. D.Lazard has also ignored the "under construction." That is not permitted. Mathsci (talk) 16:08, 22 August 2020 (UTC)
 * I have no issue with explicitly stating the form of Fubini’s theorem being made use of. Hope this clarifies. Gumshoe2 (talk) 16:15, 22 August 2020 (UTC)

"This section's factual accuracy is disputed."
Several hours ago Mathsci added this tag to the Fubini section of the page, saying "too many accuracies - see talk page". I assume he meant "inaccuracies". However, none of the above discussion has involved accusations of incorrect statements or factual inaccuracies. Perhaps can clarify what he means here. Gumshoe2 (talk) 00:35, 24 August 2020 (UTC)


 * These are standard templates. There are anyway quite a number of errors: some are serious—conceptual concerns that need attention to repair, without using my previous material or at least having some clear statements with WP:RS (see below); and some are careless student errors.


 * The actual passage only involved rewriting the sections on Clairaut's theorem, 10 lines or so. Originally that was just a brief paragraph, written entirely by me: it was adapted from the notes of Donald Marshall, an analyst from the University of Washington. His notes were entitled, "Theorems of Fubini and Clairaut." In the paragraph it was made crystal clear that two applications of integrations by parts were required. But Gumshoe2 wrote three. And he made that error twice. Schoolboy errors. While reverting in his disruptive edits, D.Lazard simply copied the errors.


 * Stylistically there are several problems. The section should either start with some assumption on C2 (more natural because coordinate-free); or assume, in a 19C manner, the weaker hypothesis of Marshall that $f$ is differentiable and the mixed partial derivative $f_{xy}$ and $f_{yx}$ exist and are continuous. (No assumptions on the existence or continuity of $f_{xx}$ and $f_{yy}$ are made.) But as I've written, only this brief paragraph was rewritten. The notes of Marshall was very good; my summary-adaptation was unsurpricingly similar.


 * The second sentence would be improved by, "Replacing $f$ by $−f$ is becessary, it may be supposed that $f_{yx} − f_{xy}$ is stricly positive at some point". Amazingly that is what Marshall (and I) wrote. Yet instead, Gumshoe2's presentation is needlessly long-winded. His argument involves running through the same argument twice. The presentation was slapdash.


 * Gumshoe2 and, by proxy, D.Lazard have made a wikilink to Fubini's theorem, assuming incorrectly that there is a version of it that applies to continuous functions on a rectangle. But Fubini's theorem on wikipedia is only treated for measurable functions. The section makes no statements at all about Riemann integration of continuous functions in two variables. Donald Marshall's notes provided a self-contained account of Fubini's theorem as iterated integrals of continuous functions. The notes are labelled "Math 136" so it was probably an optional handout on Clairaut's theorem. Marshall used Riemann approximating sums and uniform continuity.


 * Relying on the English classic A Course of Pure Mathematics of G. H. Hardy, another English classic The Functions of Functions (1932) by E.C. Titchmarsh gives the proofs of Fubini's theorems on §1.52 (uniform continuity) and §1.81. It is short. Tichmarsh does not refer to Fubini's theorem, just "repeated integrals."


 * Other approaches, such as the one in the 1973 GTM of Beals or Peter Walker's "Examples and Theorems in Analysis," involve partial differentiation or are even equivalent to equality of mixed derivatives.


 * Another standard proof, in the French tradition, is to use a simplified version of the Daniell integral for spaces of continuous functions, à la Bourbaki, Dieudonné or Loomis. It is an approach using functional analysis and a very simple version of the Stone-Weierstrass theorem. Bourbaki first shows that the algebraic tensor product $C(I) ⊗ C(J)$ is uniformly dense in $C(I × J)$ (partition of unity); then they check that the iterated integrals have the required properties and that the integrals / iterated integrals (operators / functionals) are norm-bounded; finally, passing to the limit of the algebraic tensor product, they deduce Fubini's theorem for continuous functions. Tensor product of function algebras are discussed in Bourbaki, but in this case it reduces to finite expressions $∑ g_{i}(x) h_{i}(y)$, what I have called simple functions. (An exercise in Dieudonné's Treatise of Analysis, Tome II, shows that for $I, J$ = $[0,1]$, the function |$x − y$| is not a simple function.)


 * The 1932 approach of Titchmarsh is probably the shortest. It is perfect as a WP:RS and in addition is extremely readable. Mathsci (talk) 14:28, 24 August 2020 (UTC)
 * There seems to be only one statement of Mathsci which alleges a factual inaccuracy, which is addressed first:
 * "In the paragraph it was made crystal clear that two applications of integrations by parts were required. But Gumshoe2 wrote three. And he made that error twice. Schoolboy errors."
 * I indeed count three:
 * $$\begin{align}

\int_c^d\frac{\partial^2f}{\partial y\partial x}\,dy&=\frac{\partial f}{\partial x}(x,d)-\frac{\partial f}{\partial x}(x,c)&\int_a^b \frac{\partial f}{\partial x}(x,d)\,dx&=f(b,d)-f(a,d)&\int_a^b\frac{\partial f}{\partial x}(x,c)\,dx&=f(b,c)-f(a,c) \end{align}$$
 * I believe this is the simplest and most direct way to count.
 * "Gumshoe2 and, by proxy, D.Lazard have made a wikilink to Fubini's theorem, assuming incorrectly that there is a version of it that applies to continuous functions on a rectangle. But Fubini's theorem on wikipedia is only treated for measurable functions."
 * This is a strange comment, as every continuous function is measurable in the present settinng. As I said in the previous discussion, I have no issue with giving an explicit statement of Fubini's theorem for continuous functions in this section. I believe it would be highly appropriate, for expository and accessibility reasons, to edit the Fubini's theorem page to include a section with a proof of this simpler special case, as it can be done rather economically. However, as a question of factual inaccuracies on this page, this is not so relevant.
 * "The second sentence would be improved by, "Replacing f by −f is becessary, it may be supposed that fyx − fxy is stricly positive at some point". Amazingly that is what Marshall (and I) wrote. Yet instead, Gumshoe2's presentation is needlessly long-winded. His argument involves running through the same argument twice. The presentation was slapdash."
 * This long-windedness refers to the sentence "The same reasoning shows that $∂^{2}f⁄∂x∂y$ cannot be lesser than $∂^{2}f⁄∂y∂x$ at any point." It seems to me that in an expository sense this is just about exactly as long-winded as "Replacing f by −f is becessary, it may be supposed that $f_{yx} − f_{xy}$ is stricly positive at some point." The difference is just a matter of personal choice.
 * "Stylistically there are several problems. The section should either start with some assumption on C2 (more natural because coordinate-free); or assume, in a 19C manner, the weaker hypothesis of Marshall that f is differentiable and the mixed partial derivative fxy and fyx exist and are continuous."
 * Another strange comment, as the paragraph presently starts with "Let $f$ be a function on an open set, whose second partial derivatives are continuous." This is the definition of $C^{2}$. Gumshoe2 (talk) 16:50, 24 August 2020 (UTC)
 * I have checked Gumshoe2's version, and fixed some details. The proof is now completely correct. It seems that Mathsci's main concern is that the article Fubini's theorem does not state the theorem in the standard case of continuous functions, and even not in the case of Riemann integral functions. So, I have tagged Fubini's theorem, as it is there that an elementary proof must be added if one is needed. D.Lazard (talk) 17:52, 24 August 2020 (UTC)
 * Like I said, for expository reasons I believe it is marginally better to count as three uses of fundamental theorem of calculus. But I understand that some would count the latter two of the three as the same. So I have no issue at all with counting as two, especially if it is both your and Mathsci's opinion.


 * Just to be absolutely clear: I fully agree with D.Lazard that any proof of Fubini's theorem belongs on the page Fubini's theorem. (I do believe that a proof of Fubini's theorem for continuous functions would be an excellent thing to include on that page, as even a complete version can be done rather briefly. I suggest that further discussion of this happen on that article's talk page.) I'm not opposed to including a one-sentence statement of Fubini's theorem on this page, although it doesn't seem necessary to me, given some small modification of the Fubini's theorem page. Gumshoe2 (talk) 18:10, 24 August 2020 (UTC)


 * Using Fubini's theorem for this section is misguided. Any readers would have no idaa what was going on. This is supposedly a rigorous account that is comprehensible to high school student. Looking at Titchmarsh and Hardy, the elementary theory of Riemann sums is completely separate from measure theory and Lebesgue integration. Titchmarsh covers Lebesgue theory much later. It's also very well-known that Fubini's theorem is used as a tool for proving the Plancherel theorem for locally compact Abelian groups. This is well-trodden territory. Mathsci (talk) 19:23, 24 August 2020 (UTC)
 * I have no idea why you bring up the Plancherel theorem. Anyway, it seems that we have all agreed that the page Fubini's theorem should be improved to have a section accessible to Riemann integration methods, possibly including a proof. I think the material in section 1.81 of Titchmarsh, which you have referenced in the article, is perfectly suited for such a section. I suggest that further discussion of this happen on the Fubini's theorem talk page. Gumshoe2 (talk) 19:47, 24 August 2020 (UTC)

Ironically, it seems that Mathsci's new rewrite has introduced some errors.
 * The middle expressions in both formulas involving the fundamental theorem of calculus are now written incorrectly
 * The order of integration in the first expressions are backwards, so that (in the first formula) one is integrating a x-derivative with respect to y, which the fundamental theorem of calculus does not immediately apply to.
 * In the statement of Fubini's theorem, the RHS should use "dx dy" instead of "dy dx", in order to clearly reflect the order of integration.
 * The use of "It follows that" in the sentence presenting Fubini's theorem is misleading, since the preceding sentence applies to verify the meaningfulness of both sides of Fubini's theorem. However, it is not so relevant for the proof. This directly matches Titchmarsh's exposition (as well as my brief sketch of a proof of Fubini's theorem above, which is the same as Titchmarsh's), which is the reference Mathsci has chosen.
 * The sentence "Since $∂_{y}∂_{x} f(x,y)$ is continuous, the second iterated integral can be performed by first integrating over $y$ and then afterwards over $x$" is confusing, as it comes after the corresponding formula, which was previously remarked to be "similar" to the first formula; now after the fact it is clarified that Fubini's theorem (which I think should be explicitly invoked to here) has been implicitly inserted into the "similar" formula, making it not so directly similar.

I don't want to make spend time making edits which I suspect would just be reverted, so I hope mathsci will fix these himself. My only small issue with Mathsci's presentation format is that I think it is a little unfortunate, for purely expository reasons, that the sentence presenting Fubini's theorem is used once with $F$ taken to be $∂_{y}∂_{x} f$ and once with it taken to be $∂_{x}∂_{y} f − ∂_{y}∂_{x} f$. This cause a small confusion for some readers. Just making an observation, I don't mind if it's not changed.

One more observation: Mathsci's edits have introduced separate discussion of "Schwarz theorem" and "Clairaut theorem," the distinction between which is somewhat unclear, especially since at certain places in the article they are said to be synonymous. This is reflected again by his new title for the section. I suggest this be fixed. Gumshoe2 (talk) 19:52, 24 August 2020 (UTC)


 * Titchmarsh has his own notation. Corrections are always made cumulatively. On WP, for Fubini's theorem in measure theory we write


 * $$\int_X \int_Y f(x,y)\, d\mu(y) \,d\lambda(x) = \int_Y \int_X f(x,y) \, d\lambda(x) \, d\mu(y) = \int_{X\times Y} f(x,y)\, d\nu(x,y),$$


 * where $$\nu = \lambda \times \mu$$. Mathsci (talk) 20:29, 24 August 2020 (UTC)
 * I don't see how this relates to anything I said, sorry. But I see you have started making some of the corrections, thank you for that. Gumshoe2 (talk) 20:38, 24 August 2020 (UTC)

History and conditions
Looking at the history section there seems to be various conditions used. Some questions:


 * 1) In  Clairaut first attempted proof what were the conditions
 * 2) What condition was used by Schwarz
 * 3) Dini strengthen the conditions, how?
 * 4) What were the Hobson and Young conditions?

Later down in the Sufficiency of twice-differentiability section we have the condition that
 * that all partial derivatives are themselves differentiable.

but it does not say who's result this is. The section also notes a strengthening of the theorem due to Peno but this does not seem to be mentioned in the history.

Is there a way we can be a bit more precise about who did what when? --Salix alba (talk): 05:33, 26 August 2020 (UTC)
 * The historical account of Higgins can be found by clicking on the wayback link in the article. I don't think there's much more detail for most of the questions you've asked. Rudin's book on Principles of analysis gives various conditions and proofs. There is also the 1907/1921 book of Hobson. Dini, Schwarz, Young and Peano appear in the index. And the counterexample of Peano comes from Hobson and Apostol. Hope that helps. Mathsci (talk) 07:30, 26 August 2020 (UTC)
 * See also this. Mathsci (talk) 07:39, 26 August 2020 (UTC)
 * I strongly agree with the issues raised by Salix Alba. The historical account given in the article is a good starting point, but some further information would improve it a lot. At the moment, it seems that the best statement and proof are (as usual) those by Giuseppe Peano. His statement has a stronger content (it gives the existence of the other mixed partial derivative for free!), and the proof is even simpler than the one reported in the article. Further results by Hobson or Young may be even better. In any case, I think this wiki article should pinpoint the best statement and the best proof --and maybe give the right credit to the authors. pm a  18:06, 19 October 2021 (UTC)

sufficiency of twice-differentiability
this is equivalent to the existence of the second (frechet) derivative, right? in this form its called dieudonne theorem, maybe this should be mentioned. Peter Grabs (talk) 19:39, 26 May 2024 (UTC)