Editing Symmetry of second derivatives (section)

== History ==
The result on the equality of mixed partial derivatives under certain conditions has a long history. The list of unsuccessful proposed proofs started with [[Leonard Euler|Euler]]'s, published in 1740,{{sfn|Euler|1740}} although already in 1721 [[Nicolas Bernoulli|Bernoulli]] had implicitly assumed the result with no formal justification.{{sfn|Sandifer|2007|pages=[https://books.google.com/books?id=3-DyDwAAQBAJ&pg=PA142 142–147] |loc=footnote: Comm. Acad. Sci. Imp. Petropol. '''7''' (1734/1735) '''1740''', 174-189, 180-183; ''Opera Omnia'', 1.22, 34-56.}} [[Alexis Clairaut|Clairaut]] also published a proposed proof in 1740, with no other attempts until the end of the 18th century. Starting then, for a period of 70 years, a number of incomplete proofs were proposed. The proof of [[Lagrange]] (1797) was improved by [[Cauchy]] (1823), but assumed the existence and continuity of the partial derivatives <math>\tfrac{\partial^2 f}{\partial x^2}</math> and <math>\tfrac{\partial^2 f}{\partial y^2}</math>.{{sfn|Minguzzi|2015}} Other attempts were made by P. Blanchet (1841), [[Jean-Marie Duhamel|Duhamel]] (1856), [[Jacques Charles François Sturm|Sturm]] (1857), [[Schlömilch]] (1862), and [[Joseph Bertrand|Bertrand]] (1864). Finally in 1867 [[Lorenz Leonard Lindelöf|Lindelöf]] systematically analyzed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal.{{sfn|Lindelöf|1867}}{{sfn|Higgins|1940}}

Six years after that, [[H. A. Schwarz|Schwarz]] succeeded in giving the first rigorous proof.{{sfn|Schwarz|1873}} [[Ulisse Dini|Dini]] later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by [[Camille Jordan|Jordan]] in 1883 that is still the proof found in most textbooks. Minor variants of earlier proofs were published by [[Paul Matthieu Hermann Laurent|Laurent]] (1885), [[Peano]] (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), [[Giulio Vivanti|Vivanti]] (1899) and [[James Pierpont (mathematician)|Pierpont]] (1905). Further progress was made in 1907-1909 when [[E. W. Hobson]] and [[W. H. Young]] found proofs with weaker conditions than those of Schwarz and Dini. In 1918, [[Carathéodory]] gave a different proof based on the [[Lebesgue integral]].{{sfn|Higgins|1940}}