Who first proved the interchangeability of partial derivatives? I never see any reference in textbooks. This is not a trivial result.
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7$\begingroup$ For analytic functions it is trivial. The problem is that the modern definition of function was only given in 19th century. Previously most mathematicians thought of "functions" as analytic functions. $\endgroup$– Alexandre EremenkoCommented Jan 13, 2018 at 19:50
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$\begingroup$ @AlexandreEremenko What would be the contrast between the "modern definition of function" and "analytic functions"? $\endgroup$– NatCommented Jan 16, 2018 at 2:38
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$\begingroup$ clairaut? schwarz? young? $\endgroup$– BCLCCommented Nov 29, 2021 at 16:15
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2 Answers
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It looks as if it was Euler who first proved it. See A note on the history of mixed partial derivatives, by Thomas James Higgins (Scripta Mathematica 7 (1940), pp. 59–62).
answered Jan 13, 2018 at 13:31
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1$\begingroup$ Nice paper. One might add that (according to Euler himself) the first to put it in anything like the notation in the title was Fontaine (1738, p. 26). Clairaut (1742, footnote p. 294) discusses the independence of Euler’s, Fontaine’s, and his own proof. $\endgroup$ Commented Jan 13, 2018 at 15:30
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1$\begingroup$ José Carlos Santos link is dead? anyway what about clairaut? schwarz? young? $\endgroup$– BCLCCommented Nov 29, 2021 at 16:16
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2$\begingroup$ @BCLC: I just saw your comment while answering the MSE question Arbitrary Mixed Partial Derivatives. I notice that the URL for José Carlos Santos's answer is for a Math Forum archived sci.math post of mine, and those links no longer work (they might now redirect to behind a paywall, I don't know, as I'm not a NCTM member -- they purchased Math Forum a few years back, and then discontinued it for money reasons). Fortunately, that post is also in the google sci.math archive. $\endgroup$ Commented Jun 4, 2022 at 17:40
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This question could also be asked as: Who first found an example for the not-interchangeability of partial derivatives?
It was H.A. Schwarz who proved the theorem: If a function $f:\mathbb{R}^n \rightarrow\mathbb{R}^n$ is $m$ times differentiable and continuous, then the $m$th mixed derivatives are independent of the order.
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2$\begingroup$ Ok, I'm confused: what's meant here by a function which is differentiable but not continuous? $\endgroup$ Commented Jan 15, 2018 at 12:39
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1$\begingroup$ For an example see for instance W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin 2015, p. 246 or de.wikipedia.org/wiki/Satz_von_Schwarz $\endgroup$ Commented Jan 15, 2018 at 21:36
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$\begingroup$ I think you should mention notation $f \in C^m$, not for the sake of notation per se, but to help the person who asked the question. $\endgroup$ Commented Jan 20, 2018 at 6:21
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$\begingroup$ I am unaware of that paper by Schwarz. I only know the one where he proved the interchangeability of the partial derivatives for a function of two variables: archive.org/stream/gesammeltemathem02schwuoft#page/274/mode/2up $\endgroup$ Commented Jan 26, 2018 at 23:39