History of mixed partial derivatives equality

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Dave L. Renfro

Jun 9, 2007, 4:37:02 PM6/9/07
While looking for something else this morning I came
across a paper that I thought it would be useful
to have archived in sci.math. Just before the paper
(below) I've given URLs to some related posts.

I have not tried to verify any of the references nor
have I included anything beyond what was in the paper.
I've made two changes beyond the expected ASCII format
changes to correct two errors I noticed while typing
the paper: (1) In the 3'rd paragraph, "Georg Cantor"
was in the original when it should have been "Moritz
Cantor" (the bibliography correctly lists the person
as "Cantor, M."; (2) in the original, a comma was
inadvertently omitted after the year in the
bibliography entry [24].

Dave L. Renfro



second partial derivatives commute (Clairaut's Thm.)

Continuity in each variable vs. joint continuity


Thomas James Higgins, "A note on the history of mixed partial
derivatives" [*], Scripta Mathematica 7 (1940), 59-62.

[*] Adapted from the author's M.A. thesis, done at Cornell
University under the guidance of Prof. W. A. Hurwitz.

This note is devoted to a brief résumé of the efforts, successful
and otherwise, that have been made to prove the equality of the
mixed second partial derivatives, (partial^2 u)/(partial x partial y)
and (partial^2 u)/(partial y partial x) of the function u = f(x,y).
Since a set of sufficient conditions and proof for the same are
generally given in most advanced calculus texts, the student of
analysis is usually familiar with at least one proof. 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.

Proofs of the equality of the mixed partial derivatives can be
assigned to one or the other of two sharply defined groups.
One comprises the proofs published before 1867, in which year
the Finnish mathematician L. Lindelöf, spurred to criticism
by the publication of several widely disseminated texts
containing imperfect proofs, showed that not every function
f(x,y) has equal mixed second derivatives. The second is
composed of the proofs published since 1873. In that year
the German analyst H. A. Schwarz proved that if a function
f(x,y) satisfies a certain set of sufficient conditions it
has equal mixed second derivatives. Schwarz was the first
to publish a rigorous proof; yet for more than a century
and a half previously mathematicians of note had offered

Indeed, in his comprehensive survey of mathematics prior to
the nineteen century Moritz Cantor [1] accords to L. Euler [2]
the distinction of being the first to offer a proof -- in a
paper published in 1740. Although Euler was the first to
advance a proof, he was not, however, the first to suspect
that the mixed derivatives might be equal. In 1901
G. Eneström [3], writing in the _Bibliotheca_Mathematica,
pointed out that Nicolas Bernoulli I [4] had, as early
as 1721, tacitly assumed that they were equal, although
he gave no formal proof in support of his assumption.

Simultaneously with Euler, A. C. Clairaut [5] advanced
a proof, after which no further ones appeared until the
end of the century. Then, in succession, J. L. Lagrange [6]
in 1797, A. L. Cauchy [7] in 1823, P. H. Blanchet [8] in
1841, J. M. Duhamel [9] in 1856, C. Sturm [10] in 1857,
O. Schlömilch [11] in 1862, and J. L. Bertrand [12] in
1864 published proofs. None of these were without fault:
Euler did not discriminate between the differential
and the increment of the dependent variable; Clairaut
assumed all definite integrals could be differentiated
under the integral sign; Lagrange inferred that the terms
of any infinite series could be rearranged without changing
its value; Cauchy asserted that the limit of the derivative
of a function of one variable was equal to the derivative
of the limit of the function; Duhamel predicated that the
order of a repeated limit could be arbitrarily reversed;
similarly, the others faltered on some subtle point of
analysis. Finally, in 1867 Lindelöf [13] published a paper,
criticizing in detail those proofs with which he was
familiar, and demonstrating by explicit example that
not every function f(x,y) has equal mixed derivatives.
It may be said that the publication of this paper marked
the close of the "primitive" period of investigation.

Six years later H. A. Schwarz [14] gave the first satisfactory
proof, this inaugurating the second period of investigation.
A valid proof having been given, his successors bent their
efforts towards formulating proofs based on less restrictive
hypotheses than those of Schwarz. After an abortive attempt
by J. Thomae [15] in 1875, the Italian mathematician U. Dini
[16] gave in 1877 the set of conditions familiarly known as
the Dini-Schwarz conditions. Following an unsuccessful effort
by A. Harnack [17] in 1881, C. Jordan [18] in 1882 made the
next advance. 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. Subsequent
to this popular proof appeared others by H. Laurent [19] in
1885, G. Peano [20] [21] [*] in 1889 and 1893, J. Edwards [22]
in 1892, P. Haag [23] in 1893, J. K. Whittemore [24] in 1898,
G. Vivanti [25] in 1899, and J. Pierpont [26] in 1905. 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.

[*] Most authors attribute Peano's proof to Schwarz. The
former's conditions, however, are less restrictive
than the latter's.

Not until 1907, when E. W. Hobson turned his attention to
this problem, were further advances made.

Hobson [27], by formulating a new generalization of successive
differentiation, was able to obtain a set of conditions less
restrictive than the Dini-Schwarz conditions. Several years
later, W. H. Young [28], without using Hobson's generalization,
also obtained a set of conditions less restrictive than the
Dini-Schwarz conditions. Immediately following publication of
the paper containing this work appeared another [29] in which
Young gave a theorem, termed by him the fundamental theorem
of the theory of differentials of two variables -- "if
(partial u)/(partial x) and (partial u)/(partial y) each
have differentials of the first order, the function u = f(x,y)
possesses a differential of the second order." In the course
of proving this theorem, he showed that if the function f(x,y)
satisfied the conditions stated in the quoted theorem it had
equal mixed second derivatives.

The next contribution was that of C. Carathéodory [30] in 1918.
His proof is unique in that it is the only one based on the
use of Lebesgue integration. In 1928 E. J. Townsend [31] also
published a rather unusual proof, his being based on point-set
theory. The most recent proofs are those of H. Y. Chen [32],
published in 1936.


[1] Cantor, M., "Geschichte der Mathematik", Leipzig, 1898,
v. 3, p. 734.

[2] Euler, L., "De infinitis curvis eiusdem generis",
Commentarii Academiae Scientiarum Imperialis
Petropolitanae, v. 7 (1734-35), p. 177-178.

[3] Eneström, G., "Note, Bibliotheca Mathematica", (3), v. 2
(1901), p. 443.

[4] Bernoulli, I. N., "Exercitatio geometrica de trajectoriis
orthogonalibus", Actorum Eruditorum Lipsiae Supplementa,
v. 7 (1721), p. 310-311.

[5] Clairaut, A. C., "Sur l'integration ou la construction des
equations différentielles du premier ordre", Memoires de
l'Académie Royale des Sciences, v. 2 (1740), p. 420-421.

[6] Lagrange, J. L., "Théorie des fonctions analytiques",
Journal de L'École Polytechnique, v. III, 9 (1797), p. 91-93.

[7] Cauchy, A. L., "Résumé des leçons donnés à l'École Royale
Polytechnique sur le calcul infinitésimal", Paris, 1823,
v. 1, p. 42-50.

[8] Blanchet, P. H., "Extrait d'une lettre addressèe à M. Liouville",
Journal de Mathématiques pures et appliquées, v. 6 (1841), p.

[9] Duhamel, J. M., "Eléments de calcul infinitésimal", Paris, 1856,
v. 1, p. 266-267.

[10] Sturm, C., "Cours d'analyse de l'École Polytechnique", Paris,
1858, v. 1.

[11] Schlömilch, O., "Compendium der höheren analysis", Braunschweig,
3rd ed., 1868, v. 1, p. 69-72.

[12] Bertrand, J. L., "Traité de calcul différentiel et de calcul
intégral", Paris, 1864, p. 156-158.

[13] Lindelöf, L., "Remarques sur les différentes manières d'établir
la formule (d^2 z)/(dx dy) = (d^2 z)/(dy dx)", Acta Societatis
Scientiarum Fennicae, v. 8, part 1 (1867), p. 205-213.

[14] Schwarz, H. A., "Communication", Archives des Sciences Physiques
et Naturelles, v. 48 (1873), p. 38-44.

[15] Thomae, J., "Einleitung in die Theorie der bestimmten Integrale",
Halle, 1875, p. 22.

[16] Dini, U., "Lezioni di analisi infinitesimale", Pisa, 1877-78,
p. 122.

[17] Harnack, A., "Die elemente der Differential- und
Leipzig, 1881, p. 98-99.

[18] Jordan, C., "Cours d'analyse", Paris, 1882, v. 1, p. 31-32.

[19] Laurent, H., "Traité d'analyse", Paris, 1885, v. 1, p. 138-139.

[20] Peano, G., "Sur l'interversion des dérivations partielles",
Mathesis, v. 10 (1889), p. 153-154.

[21] Peano, G., "Lezioni di analisi infinitesimale", Turin, 1893,
v. 2, p. 150-151.

[22] Edwards, J., "An Elementary Treatise on the Differential
Calculus", London, 1892, p. 122-123.

[23] Haag, P., "Cours de calcul différentiel et intégral", Paris,
1893, p. 46-47.

[24] Whittemore, J. K., "A Proof of the Theorem (partial^2 u)/
(partial x partial y) = (partial^2 u)/(partial y partial x)",
Bulletin of the American Mathematical Society, v. 4 (1897-98),
p. 389-390.

[25] Vivanti, G., "Corso di calcolo infinitesimale", Messina, 1899,
p. 209-210.

[26] Pierpont, J., "Lectures on the Theory of Functions of Real
Variables", Boston, 1905, v. 1, p. 265-267.

[27] Hobson, E. W., "On Partial Differential Coefficients and
on Repeated Limits in General", Proceedings of the London
Mathematical Society, (2), v. 5 (1907), p. 225-236.

[28] Young, W. H., "On the Conditions for the Reversibility of
the Order of Partial Differentiation", Proceedings Royale
Society of Edinburgh, v. 29 (1908-09), p. 136-164.

[29] Young, W. H., "On Differentials", Proceedings of the London
Mathematical Society, (2), v. 7 (1909), p. 157-180.

[30] Carathéodory, C., "Vorlesungen über reele Funktionen", Leipzig,
1918, p. 650-651.

[31] Townsend, E. J., "Functions of Real Variables", New York,
1928, p. 189-195.

[32] Chen, H. Y., "A Theorem on the Difference Quotient", Tohoku
Mathematical Journal, v. 42, part 1 (June 1936), p. 86-89.



Mar 2, 2020, 6:30:47 AM3/2/20
This is really useful. Thank you for your post.
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