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Elwin Bruno Christoffel


Elwin Bruno Christoffel

Elwin Bruno Christoffel
Born (1829-11-10)November 10, 1829
Montjoie, Prussia
Died March 15, 1900(1900-03-15) (aged 70)
Strasbourg, German Empire
Nationality German
Fields Mathematics
Institutions University of Strasbourg
Alma mater University of Berlin
Doctoral advisors Martin Ohm
Ernst Kummer
Heinrich Gustav Magnus
Doctoral students Rikitaro Fujisawa
Ludwig Maurer
Joseph Wellstein
Paul Epstein
Known for Christoffel symbols
Riemann–Christoffel tensor
Schwarz–Christoffel mapping
Influences Gustav Dirichlet
Influenced Gregorio Ricci-Curbastro
Tullio Levi-Civita

Elwin Bruno Christoffel (German: ; November 10, 1829 – March 15, 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provide the mathematical basis for general relativity.


  • Life 1
  • Work 2
    • Differential geometry 2.1
    • Complex analysis 2.2
    • Numerical analysis 2.3
    • Other research 2.4
  • Honours 3
  • Selected publications 4
  • Notes 5
  • References 6


Christoffel was born on 10 November 1829 in Montjoie (now Monschau) in Prussia in a family of cloth merchants. He was initially educated at home in languages and mathematics, then attended the Jesuit Gymnasium and the Friedrich-Wilhelms Gymnasium in Cologne. In 1850 he went to the University of Berlin, where he studied mathematics with Gustav Dirichlet (which had a strong influence over him)[1] among others, as well as attending courses in physics and chemistry. He received his doctorate in Berlin in 1856 for a thesis on the motion of electricity in homogeous bodies written under the supervision of Martin Ohm, Ernst Kummer and Heinrich Gustav Magnus.[2]

After receiving his doctorate, Christoffel returned to Montjoie where he has spent the following three years in isolation form the academic community. However, he continued to study mathematics (especially mathematical physics) from books by Bernhard Riemann, Dirichlet and Augustin-Louis Cauchy. He also continued his research, publishing two papers in differential geometry.[2]

In 1859 Christoffel returned to Berlin, earning his Prussian Academy of Sciences and of the Istituto Lombardo in Milan. In 1869 Christoffel returned to Berlin as a professor at the Gewerbeakademie (now part of the Technical University of Berlin), with Hermann Schwarz succeeding him in Zurich. However, strong competition from the close proximity to the University of Berlin meant that the Gewerbeakademie could not attract enough students to sustain advanced mathematical courses and Christoffel left Berlin again after three years.[2]

In 1872 Christoffel became a professor at the Alsace-Loraine in the Franco-Prussian War. Christoffel, together with his colleague Theodor Reye, has built a reputable mathematics department at Strassbourg. He continued to publish research and had several doctoral students: Otto Pauls, Victor Doerr, Rikitaro Fujisawa, Ludwig Maurer, Joseph Wellstein and Paul Epstein. Christoffel retired from the University of Strasbourg in 1894, being succeeded by Heinrich Weber.[2] After retirement he continued to work and publish, with the last treatise finished just before his death and published posthumously.[1]

Christoffel died on 15 March 1900 in Strassbourg. He was never married and has left no family behind.[2]


Differential geometry

Christoffel is mainly remembered for his seminal contributions to differential geometry. In a famous 1869 paper on the equivalence problem for differential forms in n variables, published in Crelle's Journal,[3] he introduced the fundamental technic later called covariant differentiation and used it to define the Riemann–Christoffel tensor (the most common method used to express the curvature of Riemannian manifolds). In the same paper he introduced the Christoffel symbols \Gamma_{ij,k} and \Gamma^{k}_{ij} which express the components of the Levi-Civita connection with respect to a system of local coordinates. Christoffel's ideas were generalized and greatly developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, who turned them into the concept of tensors and the absolute differential calculus. The absolute differential calculus, later named tensor calculus, forms the mathematical basis of the general theory of relativity.[2]

Complex analysis

Christoffel contributed to complex analysis, where the Schwarz–Christoffel mapping is the first nontrivial constructive application of the Riemann mapping theorem. The Schwarz–Christoffel mapping has many applications to the theory of elliptic functions and to areas of physics.[2] In the field of elliptic functions he also published results concerning abelian integrals and theta functions.

Numerical analysis

Christoffel generalized the Gaussian quadrature method for integration and, in connection to this, he also introduced the Christoffel–Darboux formula for Legendre polynomials[4] (he later also published the formula for general orthogonal polynomials).

Other research

Christoffel also worked on potential theory and the theory of differential equations, however much of his research in these areas went unnoticed. He published two papers on the propagation of discontinuities in the solutions of partial differential equations which represent pioneering work in the theory of shock waves. He also studied physics and published research in optics, however his contributions here quickly loosed their utility with the abandonment of the concept of the luminiferous aether.[2]


Christoffel was elected as a corresponding member of several academies:

Christoffel was also awarded two distinctions for his activity by the Kingdom of Prussia:

Selected publications

  • Christoffel, E. B. (1858). "Über die Gaußische Quadratur und eine Verallgemeinerung derselben.". Journal für Reine und Angewandte Mathematik (in German) 55: 61–82.  
  • Christoffel, E.B. (1869). "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades". Journal für Reine und Angewandte Mathematik (de Gruyter) 70. Retrieved 6 October 2015. 


  1. ^ a b Windelband, Wilhelm (1901). "Zum Gedächtniss Elwin Bruno Christoffel’s" ( 
  2. ^ a b c d e f g h Butzer, Paul L. (1981). "An Outline of the Life and Work of E. B. Christoffel (1829–1900)" ( 
  3. ^ Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Jour. für die reine und angewandte Mathematik, B. 70: 46–70 
  4. ^ Christoffel, E. B. (1858), "Über die Gaußische Quadratur und eine Verallgemeinerung derselben.", Journal für Reine und Angewandte Mathematik (in German) 55: 61–82,  


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