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Édouard Goursat

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(Redirected from Edouard Goursat) French mathematician (1858–1936)

Édouard Goursat
Edouard Goursat
Born(1858-05-21)21 May 1858
Lanzac, Lot
Died25 November 1936(1936-11-25) (aged 78)
Paris
NationalityFrench
Alma materÉcole Normale Supérieure
Known forGoursat tetrahedron
Goursat theorem
Goursat's lemma
Inverse function theorem
Scientific career
FieldsMathematics
InstitutionsUniversity of Paris
Doctoral advisorJean Gaston Darboux
Doctoral studentsGeorges Darmois
Dumitru Ionescu [ro]

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. This led to its translation into 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, along with Möbius, Schläfli, Cayley, Riemann, Clifford and others, was one of the 19th century mathematicians who envisioned and explored a geometry of more than three dimensions.

He was the first to enumerate the finite groups generated by reflections in four-dimensional space, in 1889. The Goursat tetrahedra are the fundamental domains which generate, by repeated reflections of their faces, uniform polyhedra and their honeycombs which fill three-dimensional space. Goursat recognized that the honeycombs are four-dimensional Euclidean polytopes.

He derived a formula for the general displacement in four dimensions preserving the origin, which he recognized as a double rotation in two completely orthogonal planes.

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

∫ S ω = ∫ T d ω {\displaystyle \int _{S}\omega =\int _{T}d\omega }

where ω {\displaystyle \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 ω {\displaystyle \omega } is a p-form, then d ω = 0 {\displaystyle d\omega =0} if and only if there is a (p − 1)-form η {\displaystyle \eta } with d η = ω {\displaystyle d\eta =\omega } . However Goursat did not notice that the "only if" part of the result depends on the domain of ω {\displaystyle \omega } and is not true in general. Élie 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. Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
  4. Coxeter 1973, p. 209, §11.x.
  5. Coxeter 1973, p. 216, §12.1 Orthogonal transformations.
  6. ^ Lovett, Edgar Odell (1898). "Review: Goursat's Partial Differential Equations". Bull. Amer. Math. Soc. 4 (9): 452–487. doi:10.1090/S0002-9904-1898-00540-2.
  7. Szegő, G. (1938). "Review: Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent by É. Goursat" (PDF). Bull. Amer. Math. Soc. 44 (1, Part 1): 16–17. doi:10.1090/s0002-9904-1938-06652-9.
  8. 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.
  9. 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.

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