Édouard Goursat

Édouard Goursat

Edouard Goursat
Born 21 May 1858
Lanzac, Lot
Died 25 November 1936 (aged 78)
Paris
Nationality French
Fields Mathematics
Institutions University of Paris
Alma mater École Normale Supérieure
Doctoral advisor Jean Gaston Darboux
Doctoral students Georges Darmois
Dumitru Ionescu
Known for Goursat tetrahedron, Goursat's theorem

Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century. It set a standard for the high-level teaching of mathematical analysis, especially complex analysis. This text was reviewed by William Fogg Osgood for the Bulletin of the American Mathematical Society.[1][2] This led to its translation in English by Earle Raymond Hedrick published by Ginn and Company. Goursat also published texts on partial differential equations and hypergeometric series.

Life

Edouard Goursat was born in Lanzac, Lot. He was a graduate of the École Normale Supérieure, where he later taught and developed his Cours. At that time the topological foundations of complex analysis were still not clarified, with the Jordan curve theorem considered a challenge to mathematical rigour (as it would remain until L. E. J. Brouwer took in hand the approach from combinatorial topology). Goursat’s work was considered by his contemporaries, including G. H. Hardy, to be exemplary in facing up to the difficulties inherent in stating the fundamental Cauchy integral theorem properly. For that reason it is sometimes called the Cauchy–Goursat theorem.

Work

Goursat was the first to note that the generalized Stokes theorem can be written in the simple form

\int_S \omega = \int_T d \omega

where \omega is a p-form in n-space and S is the p-dimensional boundary of the (p + 1)-dimensional region T. Goursat also used differential forms to state the Poincaré lemma and its converse, namely, that if \omega is a p-form, then d\omega=0 if and only if there is a (p − 1)-form \eta with d \eta=\omega. However Goursat did not notice that the "only if" part of the result depends on the domain of \omega and is not true in general. E. Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development of the De Rham cohomology of a differential manifold.

Books by Edouard Goursat

See also

References

  1. ↑ Osgood, W. F. (1903). "Review: Cours d'analyse mathématique. Tome I.". Bull. Amer. Math. Soc. 9 (10): 547–555. doi:10.1090/s0002-9904-1903-01028-3.
  2. ↑ Osgood, W. F. (1908). "Review: Cours d'analyse mathématique. Tome II.". Bull. Amer. Math. Soc. 15 (3): 120–126. doi:10.1090/s0002-9904-1908-01704-x.
  3. ↑ 3.0 3.1 3.2 Lovett, Edgar Odell (1898). "Review: Goursat's Partial Differential Equations". Bull. Amer. Math. Soc. 4 (9): 452–487.
  4. ↑ Szegő, G. (1938). "Review: Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent by É. Goursat". Bull. Amer. Math. Soc. 44 (1, Part 1): 16. doi:10.1090/s0002-9904-1938-06652-9.
  5. ↑ Dresden, Arnold (1924). "Review: Leçons sur le problème de Pfaff". Bull. Amer. Math. Soc. 30 (7): 359–362. doi:10.1090/s0002-9904-1924-03903-2.
  6. ↑ Osgood, W. F. (1896). "Review: Théorie des fonctions algébriques et de leurs intégrales, by P. Appell and É. Goursat". Bull. Amer. Math. Soc. 2 (10): 317–327. doi:10.1090/s0002-9904-1896-00353-0.

External links