Gaston Darboux

How to use this course

Gaston Darboux appears in almost every branch of nineteenth-century mathematics that touches geometry. Analysts know the Darboux integral and Darboux’s theorem on derivatives. Geometers know the Darboux frame and the Darboux vector. Symplectic geometers know that every symplectic manifold looks locally the same, by Darboux’s theorem. Specialists in integrable systems use Darboux transformations and Darboux integrability. Behind these names is one coherent programme: to treat the geometry of curves and surfaces, in the tradition of Gaspard Monge, with the full power of the analysis of his day.

Darboux was also a great teacher and administrator. He trained a generation of French mathematicians, among them Émile Picard, Édouard Goursat, Émile Borel and Élie Cartan. He ran the Paris Faculty of Science while the Sorbonne was rebuilt, and he served as perpetual secretary of the Academy of Sciences for the last seventeen years of his life.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order of his work: orthogonal systems and cyclides (1864–1873), real analysis (1875), differential equations (1870–1882), and the great treatise on surfaces (1887–1896).

Timeline

Nîmes and Montpellier, 1842–1861

Jean-Gaston Darboux was born in Nîmes, in Languedoc, in August 1842. His birth certificate records the early hours of 14 August, but all his life he gave the 13th, including on his Legion of Honour file. His father, François Darboux, kept a haberdashery shop; his mother was Alix Gourdoux. The family was Protestant, and Gaston first attended a Protestant school before the lycée in Nîmes. A younger brother, Louis, born in 1844, also became a mathematics teacher and spent almost his whole career at the lycée in Nîmes.

François Darboux died in December 1849, when Gaston was seven. Alix Darboux took over the shop and made sure that both boys received the best education the town could offer. Gaston entered the lycée in 1853 and left it in July 1859 with the baccalaureate.

That October he moved to Montpellier to join the class of special mathematics taught by Charles Berger, which prepared students for the competitive entrance examinations of the grandes écoles. He sat the examination of the École polytechnique after one year, largely to please Berger, and was declared eligible, but he chose to repeat the year. In 1861 he came first in the entrance lists of both the École polytechnique and the École normale supérieure. The Polytechnique offered a career in engineering or the army, the École normale a career in teaching and research. Pasteur, then administrator of the École normale, and the minister of education both pressed Darboux and his teacher not to choose the Polytechnique. Darboux chose the École normale.

The École normale and orthogonal surfaces, 1861–1866

At the École normale Darboux followed the courses of the school and of the Faculty of Science, and Pasteur arranged for him to attend Joseph Bertrand’s lectures at the Collège de France. He heard Charles Briot and Victor Puiseux, and won the respect of Chasles, Charles Hermite, Joseph Serret and Claude Bouquet. He read the classics of geometry, Monge and Carl Friedrich Gauss above all, and was drawn to the geometry of surfaces.

He took the licence in mathematical sciences and the licence in physical sciences in the summer of 1863. While still a student he found a new family of surfaces cutting each other at right angles. His first results on orthogonal systems were presented to the Academy of Sciences on 1 August 1864. On the same day Théodore Moutard announced the same system independently. On 20 September 1864 he was placed first in the mathematics agrégation.

Pasteur thought so highly of him that he created a new post for him at the school, that of mathematics préparateur, so that he could stay in Paris and continue research. Pasteur’s note in the school register predicted that Darboux would become one of the most eminent mathematicians of his time. On 14 July 1866 Darboux defended his doctoral thesis, Sur les surfaces orthogonales, before Chasles, Serret and Bouquet. The Mathematics Genealogy Project lists Chasles as his advisor. Lesson 04 presents the subject of the thesis.

Teacher, editor, rising geometer, 1866–1880

For fifteen years Darboux combined heavy teaching with research. In 1866–67 he replaced Bertrand at the Collège de France in the course on mathematical physics. In 1867 he became substitute for Bouquet in the special mathematics class of the Lycée Louis-le-Grand, the school where Évariste Galois had studied, and in 1868 he took the post in his own right. A former pupil recalled that he trained the critical sense and reasoning of his students, not their memory. From 1872 he taught at the École normale, and from 1873 to 1878 he replaced Liouville in rational mechanics at the Sorbonne, where he built a course that combined geometric and analytic methods.

The Bulletin

In 1870 Darboux founded the Bulletin des sciences mathématiques et astronomiques, published by Gauthier-Villars. It printed reviews and abstracts of the mathematical literature of all countries, along with original papers, and kept French readers informed of German, Italian and British work. Contemporaries called it Darboux’s journal. He was still contributing to it in the last year of his life.

Sophus Lie, 1870

In 1870 the Norwegian Sophus Lie and Felix Klein were in Paris, where they met Camille Jordan and Darboux. When war broke out between France and Prussia, Klein left. Lie stayed, then set out on foot toward Italy and was arrested near Fontainebleau as a suspected German spy, his notes on line complexes taken for coded messages. He spent about a month in prison until Darboux obtained his release. The two men remained in contact, and Lie’s ideas on contact transformations and differential equations run parallel to Darboux’s own (Lessons 07 and 09).

Marriage and research

In July 1872 Darboux married Célina Carbonnier, from Beauvais. The 1870s were his most productive research decade: the theory of cyclides (1873), the memoir on discontinuous functions (1875), the prize memoir on singular solutions of first-order equations (1876), and the memoir on algebraic first-order equations (1878). In 1878 he became Chasles’s substitute in the chair of higher geometry at the Sorbonne, created for Chasles in 1846, and in 1880 he succeeded him. He held the chair for the rest of his life. His first doctoral students, Picard (1877) and Goursat (1881), were already at work.

Triply orthogonal systems

For the general reader. Space can be filled with three families of surfaces that cross each other at right angles everywhere, the way the walls, floors and ceilings of a building do. Planes are the obvious example; spheres, cones and planes through an axis are another. Such systems give curvilinear coordinates in which physics problems separate. Darboux’s thesis was about finding and classifying them, and he returned to the subject all his life.

Orthogonal coordinates

A triply orthogonal system is a triple of one-parameter families of surfaces \(u_1=\text{const}\), \(u_2=\text{const}\), \(u_3=\text{const}\) in a region of \(\mathbb R^3\), such that surfaces of different families meet at right angles. Equivalently, the position vector \(\mathbf r(u_1,u_2,u_3)\) satisfies \(\mathbf r_i\cdot\mathbf r_j=0\) for \(i\ne j\), where \(\mathbf r_i=\partial\mathbf r/\partial u_i\). The metric is then diagonal,

with Lamé coefficients \(H_i=|\mathbf r_i|\). Gabriel Lamé had used such coordinates for the heat equation in the 1830s and had found the conditions on the \(H_i\) for the metric to be flat.

Proof. Differentiate \(\mathbf r_1\cdot\mathbf r_2=0\) with respect to \(u_3\), and similarly for the other pairs:

Three linear equations in the three quantities \(\mathbf r_{12}\cdot\mathbf r_3\), \(\mathbf r_{13}\cdot\mathbf r_2\), \(\mathbf r_{23}\cdot\mathbf r_1\) force all of them to vanish. On a surface \(u_3=\text{const}\), the unit normal is \(\mathbf n=\mathbf r_3/H_3\), so the second fundamental form has \(M=\mathbf r_{12}\cdot\mathbf n=0\). The first fundamental form has \(F=\mathbf r_1\cdot\mathbf r_2=0\). When \(F=M=0\) the coordinate directions are principal, so the coordinate curves are lines of curvature.

Confocal quadrics

Fix \(a>b>c\). Through every point \((x,y,z)\) with \(xyz\ne0\) pass exactly three surfaces of the family

one for each root \(\lambda\) of the cubic obtained by clearing denominators: an ellipsoid (\(\lambda<c\)), a hyperboloid of one sheet (\(c<\lambda<b\)), and a hyperboloid of two sheets (\(b<\lambda<a\)). The three surfaces are mutually orthogonal, and the roots \(\lambda_1,\lambda_2,\lambda_3\) are the elliptic coordinates of Carl Jacobi and Lamé. By Dupin’s theorem, the lines of curvature of the ellipsoid are its intersections with the confocal hyperboloids, a result Monge had found by direct computation.

Darboux’s problem

Given one family of surfaces \(u(x,y,z)=\text{const}\), can it be completed to a triply orthogonal system? In general it cannot. Darboux showed that the function \(u\) must satisfy a partial differential equation of the third order, and Arthur Cayley wrote this equation out in 1872. Darboux also argued that triply orthogonal systems depend on three arbitrary functions of two variables, a count that Élie Cartan later justified with his theory of exterior differential systems. He then constructed whole classes of examples, including systems of cyclides (Lesson 05). His Leçons sur les systèmes orthogonaux (1898; enlarged 1910) is the standard nineteenth-century account, and the subject came back to life a century later when orthogonal systems were linked to integrable equations of hydrodynamic type.

Cyclides

For the general reader. A torus is covered by circles in two ways, the meridians and the parallels. Charles Dupin found a wider class of surfaces whose lines of curvature are all circles, now called Dupin cyclides. Darboux placed them inside a still larger family of quartic surfaces, the cyclides, and studied them with a geometry in which spheres, planes and points are treated alike. Cyclides are now used in computer-aided design and architecture because they can be built from circular arcs.

The equation

Write \(S=x^2+y^2+z^2\). A (Darboux) cyclide is a surface with an equation

where \(\lambda\) is a constant, \(\ell\) is a polynomial of degree at most 1 and \(q\) a polynomial of degree at most 2. When \(\lambda\ne0\) the surface is a quartic that contains the imaginary circle at infinity \(S=0\) of the plane at infinity as a double curve; when \(\lambda=0\) and \(\ell\ne0\) it is a cubic, and when \(\lambda=0\) and \(\ell=0\) it is a quadric. Darboux developed the theory in a memoir of 1870 and in his book of 1873, Sur une classe remarquable de courbes et de surfaces algébriques, using coordinates built from five mutually orthogonal spheres, the pentaspherical coordinates.

Exercise 5.1 carries out the proof. The proposition explains the central role of spheres in the theory: inversion maps spheres and planes to spheres and planes, circles and lines to circles and lines, and preserves angles. Every result about cyclides can be moved around by inversions to its simplest form.

Dupin cyclides

The key fact behind (iii) is Liouville’s theorem of 1850: every conformal map between open subsets of \(\mathbb R^3\) is the restriction of a Möbius transformation. Conformal maps preserve lines of curvature, so the class of Dupin cyclides is closed under inversion. General Darboux cyclides carry several families of circles that are not lines of curvature; modern work by Helmut Pottmann, Ling Shi and Mikhail Skopenkov (2012) shows that a Darboux cyclide can contain up to six real families of circles, which makes them attractive for architecture.

Discontinuous functions and the Darboux integral

For the general reader. In 1875 Darboux wrote a long memoir on functions that jump, oscillate and misbehave. He gave the cleanest account yet of Bernhard Riemann’s integral, squeezing the area under a graph between rectangles from above and below. He also proved a surprising fact: a derivative, even a discontinuous one, can never jump. It must pass through every intermediate value.

Upper and lower sums

Let \(f:[a,b]\to\mathbb R\) be bounded and \(P:\ a=x_0<x_1<\cdots<x_n=b\) a partition. With \(M_i=\sup_{[x_{i-1},x_i]}f\) and \(m_i=\inf_{[x_{i-1},x_i]}f\), define

Refining a partition can only lower \(U\) and raise \(L\), so \(L(f,P)\le U(f,Q)\) for any two partitions. The upper integral is \(\inf_P U(f,P)\) and the lower integral is \(\sup_P L(f,P)\).

Proof idea for (ii). Every Riemann sum \(\sum f(\xi_i)(x_i-x_{i-1})\) lies between \(L(f,P)\) and \(U(f,P)\), and by choosing the tags \(\xi_i\) near the suprema or infima, Riemann sums can be brought within any \(\varepsilon\) of \(U\) or \(L\). So Riemann sums converge exactly when \(U\) and \(L\) have the same limit, and part (i) identifies these limits with the upper and lower integrals. Part (i) is Darboux’s main technical lemma: inserting one point into a partition changes \(U\) by at most \(2\sup|f|\) times the mesh.

Derivatives cannot jump

Proof. Suppose \(f'(a)<c<f'(b)\) and put \(g(x)=f(x)-cx\). Then \(g'(a)<0<g'(b)\). The continuous function \(g\) has a minimum on \([a,b]\). Since \(g'(a)<0\), \(g\) takes values below \(g(a)\) just to the right of \(a\); since \(g'(b)>0\), it takes values below \(g(b)\) just to the left of \(b\). So the minimum is at an interior point \(\xi\), where \(g'(\xi)=0\), that is \(f'(\xi)=c\). No continuity of \(f'\) is assumed.

The same memoir gave examples of continuous functions without derivatives, such as sums of the form \(\sum_n \sin\big((n+1)!\,x\big)/n!\), found independently of Karl Weierstrass, whose example was presented in Berlin in 1872 and first printed by Paul du Bois-Reymond in 1875. Darboux’s memoir helped make such functions an accepted object of study in France.

Second-order equations and the Laplace cascade

For the general reader. Many important partial differential equations in two variables have two families of characteristic curves. Pierre-Simon Laplace found a way to solve some linear ones by factoring the equation into two first-order pieces, and a way to transform the equation step by step until it factors. Darboux turned this into a systematic method and extended it to nonlinear equations in his memoir of 1870. Equations that can be solved this way are now called Darboux integrable.

Laplace invariants

Consider the linear hyperbolic equation in characteristic coordinates

with \(a,b,c\) functions of \(x,y\). A direct expansion gives two factorizations up to a remainder:

with the Laplace invariants \(h=a_x+ab-c\) and \(k=b_y+ab-c\). They are unchanged by the gauge transformation \(u\mapsto e^{\varphi}u\), which changes \(a,b,c\).

Iterating gives the Laplace cascade \(\dots,h_{-1},h,h_1,h_2,\dots\), infinite in both directions. If some \(h_n\) vanishes, the equation is solved in closed form with arbitrary functions of \(x\) and \(y\). Darboux’s account in volume 2 of his Leçons sur la théorie générale des surfaces made the cascade a standard tool in surface theory, where conjugate nets on surfaces satisfy equations of exactly this form.

Darboux’s method for nonlinear equations

In 1870 Darboux attacked general second-order equations \(F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0\). He looked for functions of \(x,y,u\) and finitely many derivatives that are constant along one family of characteristics. When there are enough of these characteristic invariants in each family, the equation reduces to ordinary differential equations and its general solution can be written down. The Liouville equation is the classic case.

Invariant algebraic curves

For the general reader. A system of two differential equations whose right-hand sides are polynomials moves points along curves in the plane. Some of these trajectories may lie on algebraic curves. Darboux discovered in 1878 that if there are enough such curves, they can be multiplied together, with suitable exponents, to give a quantity that never changes along the motion. This simple counting argument is the basis of a modern theory of integrability for polynomial systems.

Cofactors

Let \(P,Q\in\mathbb C[x,y]\) with \(m=\max(\deg P,\deg Q)\), and let \(X=P\,\partial_x+Q\,\partial_y\) be the associated vector field, the derivative along solutions of \(\dot x=P,\ \dot y=Q\). A polynomial \(f\) defines an invariant algebraic curve \(f=0\) if

for some polynomial \(K\), the cofactor. Comparing degrees shows \(\deg K\le m-1\). The equation says that \(f\) vanishes along a whole trajectory once it vanishes at one point.

Proof. The cofactors lie in the space of polynomials of degree at most \(m-1\) in two variables, which has dimension \(\tfrac12m(m+1)\). So \(q>\tfrac12m(m+1)\) cofactors are linearly dependent: \(\sum\lambda_iK_i=0\) with some \(\lambda_i\ne0\). Since \(X\) is a derivation, \(XH=H\sum_i\lambda_i\frac{Xf_i}{f_i}=H\sum_i\lambda_iK_i=0\).

Jean-Pierre Jouanolou proved in 1979 that with at least \(\tfrac12m(m+1)+2\) invariant curves, the system has a rational first integral. Darboux’s memoir also treated the singular solutions and the geometry of these equations, and it was a starting point for Poincaré’s qualitative theory of differential equations and for the open problem of bounding the degree of invariant curves, the Poincaré problem.

The Pfaff problem and Darboux’s theorem

For the general reader. An expression like \(P\,dx+Q\,dy+R\,dz\) describes an infinitesimal quantity, such as heat or work, that need not come from any function. Johann Friedrich Pfaff asked in 1814–15 how simply such an expression can be written after a change of variables. Darboux answered completely in 1882. His answer implies that the phase spaces of classical mechanics all look the same locally: there are no local invariants in symplectic geometry, unlike in Riemannian geometry, where curvature is one.

Normal forms for 1-forms

Let \(\alpha\) be a smooth 1-form on an \(n\)-manifold. Its class at a point is determined by which of the forms \((d\alpha)^p\) and \(\alpha\wedge(d\alpha)^p\) vanish there.

Darboux proved this by induction on the number of variables, building on Pfaff and on work of Jacobi and Clebsch. Georg Frobenius reached equivalent results in 1877 in a different language, and the normal form is often called the Pfaff–Darboux theorem. When \(n=2p+1\) and \(\alpha\wedge(d\alpha)^p\ne0\), case (ii) says that every contact form is locally \(dz+\sum x_i\,dy_i\), the form underlying Lie’s contact transformations.

Proof by Moser’s method

Jürgen Moser’s argument of 1965 gives a short modern proof. Work near a point \(0\) in \(\mathbb R^{2n}\). By linear algebra, choose coordinates in which \(\omega_0=\sum dp_i\wedge dq_i\) agrees with \(\omega_1=\omega\) at 0. Put \(\omega_t=\omega_0+t(\omega_1-\omega_0)\); for small neighbourhoods each \(\omega_t\) is nondegenerate. By the Poincaré lemma, \(\omega_1-\omega_0=d\beta\) with \(\beta(0)=0\). Define the time-dependent vector field \(v_t\) by \(\iota_{v_t}\omega_t=-\beta\). Its flow \(\phi_t\) fixes 0, and Cartan’s formula gives

So \(\phi_1^*\omega_1=\omega_0\): the map \(\phi_1\) carries the standard form to \(\omega\). Alan Weinstein extended the argument in 1971 to neighbourhoods of submanifolds.

The Leçons: moving frames and transformations

For the general reader. Darboux’s four-volume Leçons sur la théorie générale des surfaces (1887–1896) gathered a century of surface theory into one treatise. Its central device is a frame of three perpendicular unit vectors that moves along a curve or surface, and whose rotation records the geometry. Élie Cartan turned this device into the method of moving frames. The Leçons also contain transformations that make new surfaces, and new solutions of equations, out of old ones.

The Darboux frame

Let \(\gamma(s)\) be a curve parametrized by arc length on an oriented surface with unit normal \(\mathbf n\). Set \(\mathbf T=\gamma'\) and \(\mathbf g=\mathbf n\times\mathbf T\). Then \((\mathbf T,\mathbf g,\mathbf n)\) is an orthonormal frame adapted to both curve and surface, and

The geodesic curvature \(\kappa_g\) vanishes along geodesics, the normal curvature \(\kappa_n\) vanishes along asymptotic curves, and the geodesic torsion \(\tau_g\) vanishes along lines of curvature. The matrix is skew-symmetric because the frame stays orthonormal. Every rotating orthonormal frame has an angular velocity vector \(\boldsymbol\omega\) with \(\mathbf e'=\boldsymbol\omega\times\mathbf e\) for each frame vector.

Kinematics runs through the Leçons. The first volume begins with moving frames and the motion of rigid bodies, and the fourth treats one surface rolling on another, with the paths traced by points and lines carried by the rolling surface. Luther Pfahler Eisenhart praised Darboux’s geometric proofs of the rolling theorems as pure and simple. In between lie the deformation of surfaces, minimal surfaces, conjugate nets and the Laplace cascade (Lesson 07), and geodesics.

The Darboux transformation

In a note of 1882 Darboux observed that a second-order linear equation can be transformed into another of the same type using one known solution. In modern notation, for the operator \(L=-\partial_x^2+u(x)\):

In operator language, \(L-\lambda_0=A^*A\) with \(A=\partial_x-\sigma\), and the new operator is \(\tilde L-\lambda_0=AA^*\). This is the factorization method of supersymmetric quantum mechanics. Iterated Darboux transformations generate the multi-soliton solutions of the Korteweg–de Vries equation, and Darboux transformations for systems of equations are now a basic tool of the theory of integrable systems.

Dean and perpetual secretary, 1880–1912

From the 1880s Darboux’s energy turned more and more to teaching and institutions. His course in higher geometry at the Sorbonne became the source of the Leçons, and he gathered a school around it. The Mathematics Genealogy Project lists twelve doctoral students, among them Picard (1877), Goursat (1881), Gabriel Koenigs (1882), Stanisław Zaremba (1889), Borel (1893), Cartan (1894), Émile Cotton (1899) and the Romanian geometer Gheorghe Țițeica (1899). He was among the founders of the École normale supérieure for young women at Sèvres, which trained women teachers for the new girls’ secondary schools.

He was elected to the Academy of Sciences on 3 March 1884, in the seat left by Puiseux. With Joseph Serret he edited the Œuvres of Joseph-Louis Lagrange, completing the edition after Serret’s death, and from 1888 to 1890 he edited the works of Fourier.

Dean of the Faculty of Science

In November 1889 the Paris Faculty of Science chose him as dean. He was reappointed term after term and served until 1903. These were the years in which the Sorbonne was rebuilt and the separate faculties were joined into a new University of Paris in 1896. Darboux oversaw the faculty’s expansion into laboratories across Paris and marine stations at Roscoff, Banyuls and Wimereux.

Perpetual secretary

On 21 May 1900 he succeeded Bertrand as perpetual secretary of the Academy of Sciences for the mathematical sciences. The post made him the official voice of French science. He wrote many historical éloges of deceased members, collected in 1912 as Éloges académiques et discours. He served on the Bureau des longitudes, the International Committee for Weights and Measures and the International Association of Academies, and represented France at Lord Kelvin’s funeral in Westminster Abbey in December 1907.

In 1904, at the request of the Court of Cassation, he joined Poincaré and Paul Appell in examining the pseudo-mathematical analysis of the handwriting evidence used against Alfred Dreyfus. Their report demolished it. At the International Congress of Mathematicians in Rome in 1908 he gave a plenary lecture on the origins, methods and problems of infinitesimal geometry, and when Poincaré fell ill he read Poincaré’s address on the future of mathematics.

Last years, death, legacy, 1912–1917

In 1912 French mathematicians celebrated Darboux’s jubilee, and Poincaré reviewed his work in analysis, differential equations and geometry. Poincaré died that July, and on 15 December 1913 Darboux read his historical éloge of Poincaré to the Academy, one of the first full accounts of Poincaré’s life. Darboux succeeded him as president of the council of the astronomical observatories. He also presided over the council of the Institut Pasteur, honouring his first patron.

Honours came from abroad. He was an honorary member of the London Mathematical Society from 1878, a foreign member of the Royal Society from 1902, and a member of more than a hundred academies and learned societies. The Berlin Academy awarded him the Steiner Prize in 1910, and in 1916, in the middle of the war, the Royal Society gave him its Sylvester Medal. He rose to the rank of grand officer of the Legion of Honour.

His wife Célina died in 1911. Darboux died in Paris on 23 February 1917, aged 74, and was buried in the Montparnasse cemetery. Picard succeeded him as perpetual secretary. A street in the 18th arrondissement of Paris bears his name.

Legacy

• Differential geometry: the Leçons remained the reference work on classical surface theory for half a century. The Darboux frame and Darboux vector are standard, and Cartan’s method of moving frames grew directly from Darboux’s trièdre mobile.
• Analysis: the Darboux integral is how most students first meet the Riemann integral, and Darboux’s theorem on derivatives is a standard result of real analysis.
• Differential equations and integrable systems: Darboux integrability of hyperbolic equations, Darboux first integrals of polynomial vector fields, and Darboux transformations are active research areas. The Christoffel–Darboux formula is basic to the theory of orthogonal polynomials.
• Symplectic and contact geometry: Darboux’s theorem is the starting point of both subjects, and coordinates in normal form are called Darboux charts.
• Teaching and institutions: his students Picard, Goursat, Borel and Cartan wrote the treatises and held the chairs that defined French mathematics in the early twentieth century.

Connections

People linked to Gaston Darboux across this collection. A name without a link has no page here yet. See Gaston Darboux in the network of all 337 lives →

Teachers and advisors

• Michel Chasles 1866

Students

• Thomas Joannes Stieltjes
• Élie Cartan 1894
• Émile Borel 1893
• Émile Picard 1877
• Gabriel Koenigs 1882
• Gheorghe Țițeica 1899
• Stanisław Zaremba 1889
• Édouard Goursat 1881
• Émile Cotton 1899

Influenced by

• Bernhard Riemann
• Carl Friedrich Gauss
• Gaspard Monge
• Joseph Fourier
• Pierre-Simon Laplace
• Joseph Bertrand

Colleagues and collaborators

• Arthur Cayley
• Charles Hermite
• Georg Frobenius
• Gösta Mittag-Leffler
• Henri Poincaré
• Joseph Liouville 1873-1878
• Louis Pasteur

Friends

• Sophus Lie 1870

Also mentioned in this life

• Peter Gustav Lejeune Dirichlet
• Évariste Galois
• Carl Gustav Jacob Jacobi
• Camille Jordan
• William Thomson, Lord Kelvin
• Felix Klein
• Joseph-Louis Lagrange
• August Ferdinand Möbius
• Erwin Schrödinger
• Jakob Steiner
• Vito Volterra
• Karl Weierstrass

Fields

• differential geometry 15 lives
• real analysis 10 lives
• differential equations 13 lives

On CoursesHub

• Differential Geometry
• ODEs / PDEs / Green Functions · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jean Gaston Darboux, MacTutor History of Mathematics.
2. Gaston Darboux, Wikipedia; Gaston Darboux, Wikipédia (French).
3. D. J. Struik, Darboux, Jean-Gaston, Dictionary of Scientific Biography.
4. P. Appell, obituary of Gaston Darboux (English translation at MacTutor).
5. J. Larmor, obituary of Gaston Darboux, Nature 99 (1917).
6. É. Picard, La vie et l’œuvre de Gaston Darboux, Académie des sciences.
7. E. Lebon, Gaston Darboux: biographie, bibliographie analytique des écrits, Gauthier-Villars, 1910; 2nd ed. 1913.
8. L. P. Eisenhart, Darboux’s contribution to geometry, Bull. Amer. Math. Soc. 24 (1918), 227–237.
9. Gaston Darboux, Mathematics Genealogy Project.

Mathematics

1. G. Darboux, Sur les surfaces orthogonales, thesis, Paris, 1866.
2. G. Darboux, Sur une classe remarquable de courbes et de surfaces algébriques, Gauthier-Villars, 1873.
3. G. Darboux, Mémoire sur les fonctions discontinues, Ann. Sci. École Norm. Sup. (2) 4 (1875), 57–112.
4. G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. (2) 2 (1878).
5. G. Darboux, Sur le problème de Pfaff, Bull. Sci. Math. (2) 6 (1882), 14–36, 49–68.
6. G. Darboux, Leçons sur la théorie générale des surfaces, 4 vols., Gauthier-Villars, 1887–1896.
7. G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, 1898; 2nd ed. 1910.
8. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.
9. J.-P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Math. 708, Springer, 1979.
10. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, 1991.
11. H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles, Comput. Aided Geom. Design 29 (2012).
12. Darboux’s theorem, Darboux integral and Darboux transformation, Wikipedia.