Elwin Bruno Christoffel | |
|---|---|
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| Born | (1829-11-10)10 November 1829 |
| Died | 15 March 1900(1900-03-15) (aged 70) |
| Alma mater | University of Berlin |
| Known for | Christoffel symbols Christoffel equation Christoffel–Darboux formula Riemann–Christoffel tensor Schwarz–Christoffel mapping |
| Scientific career | |
| Fields | Mathematics Physics |
| Institutions | University of Strasbourg |
| Doctoral advisors | Martin Ohm Ernst Kummer Heinrich Gustav Magnus |
| Doctoral students | Rikitaro Fujisawa Ludwig Maurer Paul Epstein |
Elwin Bruno Christoffel (German:[kʁɪˈstɔfl̩]; 10 November 1829 – 15 March 1900) was a Germanmathematician andphysicist. He introduced fundamental concepts ofdifferential geometry, opening the way for the development oftensor calculus, which would later provide the mathematical basis forgeneral relativity.
Christoffel was born on 10 November 1829 in Montjoie (nowMonschau) inPrussia in a family of cloth merchants. He was initially educated at home in languages and mathematics, then attended the Jesuit Gymnasium and the Friedrich-WilhelmsGymnasium inCologne. In 1850 he went to theUniversity of Berlin, where he studied mathematics withGustav 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 ofelectricity in homogeneous bodies written under the supervision ofMartin Ohm,Ernst Kummer andHeinrich Gustav Magnus.[2]
After receiving his doctorate, Christoffel returned to Montjoie where he spent the following three years in isolation from the academic community. However, he continued to study mathematics (especially mathematical physics) from books byBernhard Riemann, Dirichlet andAugustin-Louis Cauchy. He also continued his research, publishing two papers indifferential geometry.[2]
In 1859 Christoffel returned to Berlin, earning hishabilitation and becoming aPrivatdozent at the University of Berlin. In 1862 he was appointed to a chair at thePolytechnic School in Zürich left vacant byDedekind. He organised a new institute of mathematics at the young institution (it had been established only seven years earlier) that was highly appreciated. He also continued to publish research, and in 1868 he was elected a corresponding member of thePrussian Academy of Sciences and of theIstituto Lombardo in Milan. In 1869 Christoffel returned to Berlin as a professor at the Gewerbeakademie (now part ofTechnische Universität Berlin), withHermann Schwarz succeeding him in Zürich. 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 theUniversity of Strasbourg, a centuries-old institution that was being reorganized into a modern university after Prussia's annexation ofAlsace-Lorraine in theFranco-Prussian War. Christoffel, together with his colleagueTheodor Reye, built a reputable mathematics department at Strasbourg. He continued to publish research and had several doctoral students includingRikitaro Fujisawa,Ludwig Maurer andPaul Epstein. Christoffel retired from the University of Strasbourg in 1894, being succeeded byHeinrich 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 Strasbourg. He never married and left no family.[2]
Christoffel is mainly remembered for his seminal contributions todifferential geometry. In a famous 1869 paper on the equivalence problem fordifferential forms inn variables, published inCrelle's Journal,[3] he introduced the fundamental technique later calledcovariant differentiation and used it to define theRiemann–Christoffel tensor (the most common method used to express thecurvature ofRiemannian manifolds). In the same paper he introduced theChristoffel symbols and which express the components of theLevi-Civita connection with respect to a system of local coordinates. Christoffel's ideas were generalized and greatly developed byGregorio Ricci-Curbastro and his studentTullio Levi-Civita, who turned them into the concept oftensors and theabsolute differential calculus. The absolute differential calculus, later namedtensor calculus, forms the mathematical basis of thegeneral theory of relativity.[2]
Christoffel contributed tocomplex analysis, where theSchwarz–Christoffel mapping is the first nontrivial constructive application of theRiemann mapping theorem. The Schwarz–Christoffel mapping has many applications to the theory ofelliptic functions and to areas of physics.[2] In the field of elliptic functions he also published results concerningabelian integrals andtheta functions.
Christoffel generalized theGaussian quadrature method for integration and, in connection to this, he also introduced theChristoffel–Darboux formula forLegendre polynomials[4] (he later also published the formula for generalorthogonal polynomials).
Christoffel also worked onpotential theory and the theory ofdifferential 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 ofshock waves. He also studied physics and published research inoptics, however his contributions here quickly lost their utility with the abandonment of the concept of theluminiferous 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:
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