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Jun 9, 2007, 4:37:02 PM6/9/07

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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.

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

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RELATED POSTS

second partial derivatives commute (Clairaut's Thm.)

http://groups.google.com/group/sci.math/msg/78a769e88081ec8b

http://groups.google.com/group/sci.math/msg/3b7eeedf51fc5b92

http://groups.google.com/group/sci.math/msg/cd695899296573b6

Continuity in each variable vs. joint continuity

http://groups.google.com/group/sci.math/msg/a1b3752adec7650e

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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

solutions.

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.

BIBLIOGRAPHY

[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.

65-68.

[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

Integralrechnung",

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.

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Mar 2, 2020, 6:30:47 AM3/2/20

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