Jean-Gaston Darboux FAS MIF FRS

FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and letters, judged to be "eminently distinguished in their subject". This soci ...

(14 August 1842 – 23 February 1917) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Life

According t
his birth certificate
he was born in

Nîmes Nîmes ( , ; oc, Nimes ; Latin: ''Nemausus'') is the prefecture of the Gard department in the Occitanie region of Southern France. Located between the Mediterranean Sea and Cévennes, the commune of Nîmes has an estimated population of ...

France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...

on 14 August 1842, at 1 am. However, probably due to the midnight birth, Darboux himself usually reported his own birthday as 13 August, ''e.g.'' i
his filled form for Légion d'Honneur
His parents were François Darboux, businessman of mercery, and Alix Gourdoux. The father died when Gaston was 7. His mother undertook the mercery business with great courage, and insisted that her children receive good education. Gaston had a younger brother, Louis, who taught mathematics at the Lycée Nîmes for almost his entire life. He studied at the Nîmes Lycée and the

Montpellier Montpellier (, , ; oc, Montpelhièr ) is a city in southern France near the Mediterranean Sea. One of the largest urban centres in the region of Occitania, Montpellier is the prefecture of the department of Hérault. In 2018, 290,053 people l ...

Lycée before being accepted as the top qualifier at the

École normale supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, S ...

in 1861, and received his

PhD PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * '' Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. al ...

there in 1866. His thesis, written under the direction of

Michel Chasles Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...

, was titled ''Sur les surfaces orthogonales''. During his studies at the ENS, he also took lectures in

Sorbonne University Sorbonne University (french: Sorbonne Université; la Sorbonne: 'the Sorbonne') is a public research university located in Paris, France. The institution's legacy reaches back to 1257 when Sorbonne College was established by Robert de Sor ...

and

Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris n ...

Darboux as a student of the Ecole Normale

In 1870, he co-founded the journal
Bulletin des sciences mathématiques et astronomiques
', called "Darboux's Journal" by his contemporary mathematicians. The editorial board was also formed by the mathematicians

Paul Émile Appell :''M. P. Appell is the same person: it stands for Monsieur Paul Appell''. Paul Émile Appell (27 September 1855, in Strasbourg – 24 October 1930, in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials ...

Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...

and

Amedeo Guillet Baron Amedeo Guillet (February 7, 1909 – June 16, 2010) was an officer of the Italian Army and an Italian Diplomat. Dying at the age of 101, he was one of the last men to have commanded cavalry in war. He was nicknamed ''Devil Commander'' a ...

, with Darboux in the role of President. The publishing house was the Henry Gauthier-Villars et Cle Éditeurs, located in Paris. In 1872, Darboux married the Beauvaisian

milliner Hat-making or millinery is the design, manufacture and sale of hats and other headwear. A person engaged in this trade is called a milliner or hatter. Historically, milliners, typically women shopkeepers, produced or imported an inventory of ...

Amélie ''Célina'' Carbonnier (1848-1911), daughter of Charles Louis Carbonnier, tailor, and Marie Victorine Anastase Hènocq. He and Célina had two children, Jean-''Gaston'' (1870-1921), who was born at the time of the Siege of Paris and later became a marine zoologist at the Faculty of Science in Marseille, and Anaïs Berthe ''Lucie'' (1873-1970). He participated in the foundation of the

École normale supérieure de jeunes filles The ''École normale supérieure de jeunes filles'' (also, ''École normale supérieure de Sèvres'') was a French institute of higher education, in Sèvres, now a commune in the suburbs of Paris. The school educated girls only, especially as tea ...

in 1880, an institute that aimed at training female educators and ran parallel to the

on rue d'Ulm. Its first director was

Julie Favre Julie Velten Favre (November 15, 1833 – ), sometimes called Madame Jules Favre, was a French philosopher and educator. She is known for her work educating young women and for advancing a moral philosophy that advocated living a virtuous life, r ...

. In 1884, Darboux was elected to the

Académie des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...

. Darboux made several important contributions to

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

(see, for example, linear PDEs). He was a

biographer Biographers are authors who write an account of another person's life, while autobiographers are authors who write their own biography. Biographers Countries of working life: Ab=Arabia, AG=Ancient Greece, Al=Australia, Am=Armenian, AR=Ancient Rome ...

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...

and he edited the Selected Works of

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...

. Among his students were

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...

Édouard Goursat 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 se ...

Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at ...

Gheorghe Țițeica Gheorghe Țițeica (; 4 October 1873 – 5 February 1939) publishing as George or Georges Tzitzéica) was a Romanian mathematician who made important contributions in geometry. He is recognized as the founder of the Romanian school of differential ...

and Stanisław Zaremba. In 1900, he was appointed the Academy's permanent secretary of its Mathematics section. In 1902, he was elected to the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

and the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

; in 1916, he received the

Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...

from the Society. In 1908, he was a plenary speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rena ...

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus ( legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

. He continued to contribute to the French ''Bulletin des sciences mathématiques'', even after 1916.

Named in his honour

There are many things named after him: * Darboux basis * Darboux chart *

Darboux cubic In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equ ...

Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundamental ...

* Darboux equation *

Darboux frame In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...

Darboux integral In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a functi ...

* Darboux net invariants * Darboux or Goursat problem * Darboux transformation *

Darboux vector In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it.. It is also called angular momentum vecto ...

* Darboux's problem *

Darboux's theorem Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

, related to the

intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...

* Darboux's formula * Christoffel–Darboux identity *

Christoffel–Darboux formula In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states that : \sum_^n \frac = \frac \frac where ''f'j''(''x'') is the ''j''th term of a set of orthogonal polyn ...

* Euler–Darboux equation * Darboux–Froda's theorem *

Euler–Poisson–Darboux equation In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation : u_+\frac=0. This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave e ...

* Laplace–Darboux transformations

Work

Papers and essays (incomplete list)

* 1870
Sur les équations aux dérivées partielles du second ordre
''Annales scientifiques de l’É.N.S.'' 1re série, tome 7 (1870), p. 163-173 * 1871
Sur la représentation des surfaces algébriques
''Bulletin des sciences mathématiques et astronomiques'', tome 2 (1871), p. 155-158 * 1872
Mémoire sur les surfaces cyclides
''Annales scientifiques de l’É.N.S.'' 2e série, tome 1 (1872), p. 273-292 * 1872
Sur les relations entre les groupes de points, de cercles et de sphères dans le plan et dans l’espace
''Annales scientifiques de l’É.N.S''. 2e série, tome 1 (1872), p. 323-392 * 1872
Sur un théorème relatif à la continuité des fonctions
''Bulletin des sciences mathématiques et astronomiques'', tome 3 (1872), p. 307-313 * 1875
Mémoire sur les fonctions discontinues
''Annales scientifiques de l’É.N.S. 2e série'', tome 4 (1875), p. 57-112. In it were introduced the

(based on the limit of upper and lower integrals) and

in analysis. * 1875
Sur la composition des forces en statique
''Bulletin des sciences mathématiques et astronomiques'', tome 9 (1875), p. 281-288 * 1890
Sur le déplacement d’une figure invariable
''Annales scientifiques de l’É.N.S. 3e série'', tome 7 (1890), p. 323-326

Books

1873
Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires.
Gauthier-Villars. Darboux's contribution to the

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...

appears in the four-volume collection of studies he published between 1887 and 1896; see links below for access to these texts. 1887–96. ''Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal''. Gauthier-Villars: *
Vol. 1.
' The

is introduced in Section 4 of this volume. *
Vol. 2.
' *
Vol. 3.
' *
Vol. 4.
' 1898. ''Leçons sur les systèmes orthogonaux et les coordonnées curvilignes''. Tome I. Gauthier-Villars.

Notes

References

* * *

External links

* * * {{DEFAULTSORT:Darboux, Gaston 1842 births 1917 deaths 19th-century French mathematicians 20th-century French mathematicians French geometers Differential geometers Members of the French Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences École Normale Supérieure alumni People from Nîmes Lycée Louis-le-Grand teachers Members of the Royal Society of Sciences in Uppsala