From Attraction Theory to Existence Proofs: The Evolution of Potential-Theoretic Methods in the Study of Boundary-Value Problems, 1860–1890

Thomas Archibald

Revue d'histoire des mathématiques (1996)

  • Volume: 2, Issue: 1, page 67-93
  • ISSN: 1262-022X

Abstract

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This paper examines developments in the study of boundary-value problems between about 1860 and 1890, in the context of the general evolution of this theory from the physical models in which the subject has its roots to a free-standing part of pure mathematics. The physically-motivated work of Carl Neumann and his method of the arithmetic mean appear as an initial phase in this development, one which employs physical models as an integral part of its reasoning and which concentrates on geometrical hypotheses concerning the regions under study. The alternating method of Hermann Amandus Schwarz, roughly contemporary to that of Neumann, exhibits more strongly the analytic influence of Weierstrass. Both methods form the essential background to Émile Picard’s method of successive approximations, developed by him following a reading of both men’s work. Picard’s work, analytically rigorous and remote from physical argument, marks both a transition of the subject matter from applied to pure mathematics, and the full comprehension and mastery of Weierstrassian methods in the French context.

How to cite

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Archibald, Thomas. "From Attraction Theory to Existence Proofs: The Evolution of Potential-Theoretic Methods in the Study of Boundary-Value Problems, 1860–1890." Revue d'histoire des mathématiques 2.1 (1996): 67-93. <http://eudml.org/doc/252026>.

@article{Archibald1996,
abstract = {This paper examines developments in the study of boundary-value problems between about 1860 and 1890, in the context of the general evolution of this theory from the physical models in which the subject has its roots to a free-standing part of pure mathematics. The physically-motivated work of Carl Neumann and his method of the arithmetic mean appear as an initial phase in this development, one which employs physical models as an integral part of its reasoning and which concentrates on geometrical hypotheses concerning the regions under study. The alternating method of Hermann Amandus Schwarz, roughly contemporary to that of Neumann, exhibits more strongly the analytic influence of Weierstrass. Both methods form the essential background to Émile Picard’s method of successive approximations, developed by him following a reading of both men’s work. Picard’s work, analytically rigorous and remote from physical argument, marks both a transition of the subject matter from applied to pure mathematics, and the full comprehension and mastery of Weierstrassian methods in the French context.},
author = {Archibald, Thomas},
journal = {Revue d'histoire des mathématiques},
keywords = {potential theory; boundary value problems; Carl Neumann; Jules Riemann; E. Picard},
language = {eng},
number = {1},
pages = {67-93},
publisher = {Société mathématique de France},
title = {From Attraction Theory to Existence Proofs: The Evolution of Potential-Theoretic Methods in the Study of Boundary-Value Problems, 1860–1890},
url = {http://eudml.org/doc/252026},
volume = {2},
year = {1996},
}

TY - JOUR
AU - Archibald, Thomas
TI - From Attraction Theory to Existence Proofs: The Evolution of Potential-Theoretic Methods in the Study of Boundary-Value Problems, 1860–1890
JO - Revue d'histoire des mathématiques
PY - 1996
PB - Société mathématique de France
VL - 2
IS - 1
SP - 67
EP - 93
AB - This paper examines developments in the study of boundary-value problems between about 1860 and 1890, in the context of the general evolution of this theory from the physical models in which the subject has its roots to a free-standing part of pure mathematics. The physically-motivated work of Carl Neumann and his method of the arithmetic mean appear as an initial phase in this development, one which employs physical models as an integral part of its reasoning and which concentrates on geometrical hypotheses concerning the regions under study. The alternating method of Hermann Amandus Schwarz, roughly contemporary to that of Neumann, exhibits more strongly the analytic influence of Weierstrass. Both methods form the essential background to Émile Picard’s method of successive approximations, developed by him following a reading of both men’s work. Picard’s work, analytically rigorous and remote from physical argument, marks both a transition of the subject matter from applied to pure mathematics, and the full comprehension and mastery of Weierstrassian methods in the French context.
LA - eng
KW - potential theory; boundary value problems; Carl Neumann; Jules Riemann; E. Picard
UR - http://eudml.org/doc/252026
ER -

References

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