Who first proved that $f_{xy} = f_{yx}$?

Question

Who first proved the interchangeability of partial derivatives? I never see any reference in textbooks. This is not a trivial result.

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. — Alexandre Eremenko, Jan 13 '18 at 19:50
@AlexandreEremenko What would be the contrast between the "modern definition of function" and "analytic functions"? — Nat, Jan 16 '18 at 02:38

score 9 · Accepted Answer · edited Sep 30 '22 at 18:14

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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).

edited Sep 30 '22 at 18:14

Glorfindel

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answered Jan 13 '18 at 13:31

José Carlos Santos

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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. – Francois Ziegler Jan 13 '18 at 15:30
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One might also add that Lindelöf in 1867 criticized all proofs that had been given so far and that the first rigorous proof was subsequently presented by Schwarz in 1873. – Jan Peter Schäfermeyer Jan 26 '18 at 23:42
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José Carlos Santos link is dead? anyway what about clairaut? schwarz? young? – BCLC Nov 29 '21 at 16:16
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@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. – Dave L Renfro Jun 04 '22 at 17:40

score 5 · Answer 2 · answered Jan 13 '18 at 16:07

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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.

answered Jan 13 '18 at 16:07

Franz Kurz

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Ok, I'm confused: what's meant here by a function which is differentiable but not continuous? – Carl Witthoft Jan 15 '18 at 12:39
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For an example see for instance W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin 2015, p. 246 or https://de.wikipedia.org/wiki/Satz_von_Schwarz – Franz Kurz Jan 15 '18 at 21:36
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. – Jean Marie Becker Jan 20 '18 at 06:21
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: https://archive.org/stream/gesammeltemathem02schwuoft#page/274/mode/2up – Jan Peter Schäfermeyer Jan 26 '18 at 23:39

Who first proved that $f_{xy} = f_{yx}$?

2 Answers2