Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German
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 ...
and
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...
. He introduced fundamental concepts of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, opening the way for the development of
tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
, which would later provide the mathematical basis for
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
Life
Christoffel was born on 10 November 1829 in Montjoie (now
Monschau
Monschau (; french: Montjoie, ; wa, Mondjoye) is a small resort town in the Eifel region of western Germany, located in the Aachen district of North Rhine-Westphalia.
Geography
The town is located in the hills of the North Eifel, within the ...
) in
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...
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
Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western States of Germany, state of North Rhine-Westphalia (NRW) and the List of cities in Germany by population, fourth-most populous city of Germany with 1.1 m ...
. In 1850 he went to the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
, where he studied mathematics with
Gustav Dirichlet (which had a strong influence over him)
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
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
in homogeneous bodies written under the supervision of
Martin Ohm
Martin Ohm (May 6, 1792 in Erlangen – April 1, 1872 in Berlin) was a German mathematician and a younger brother of physicist Georg Ohm.
Biography
He earned his doctorate in 1811 at Friedrich-Alexander-University, Erlangen-Nuremberg where his ...
,
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
and
Heinrich Gustav Magnus
Heinrich Gustav Magnus (; 2 May 1802 – 4 April 1870) was a notable German experimental scientist. His training was mostly in chemistry but his later research was mostly in physics. He spent the great bulk of his career at the University of Ber ...
.
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 by
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
, Dirichlet and
Augustin-Louis Cauchy. He also continued his research, publishing two papers in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
.
[
In 1859 Christoffel returned to Berlin, earning his ]habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
and becoming a Privatdozent
''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...
at the University of Berlin. In 1862 he was appointed to a chair at the Polytechnic School
Polytechnic School, often referred to simply as Poly, is a college preparatory private day school located in Pasadena, California with approximately 850 students enrolled in grades Kindergarten through 12.
The school is a former member of the ...
in Zürich left vacant by Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
. 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 the Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
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
The Technical University of Berlin (official name both in English and german: link=no, Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public research university located in Berlin, Germany. It was ...
), with Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life
Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, ...
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.[
In 1872 Christoffel became a professor at the ]University of Strasbourg
The University of Strasbourg (french: Université de Strasbourg, Unistra) is a public research university located in Strasbourg, Alsace, France, with over 52,000 students and 3,300 researchers.
The French university traces its history to the ea ...
, a centuries-old institution that was being reorganized into a modern university after Prussia's annexation of Alsace-Lorraine in the Franco-Prussian War. Christoffel, together with his colleague Theodor Reye
Karl Theodor Reye (born 20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) was a German mathematician. He contributed to geometry, particularly projective geometry and synthetic geometry. He is best known for his ...
, built a reputable mathematics department at Strasbourg. He continued to publish research and had several doctoral students including Rikitaro Fujisawa, Ludwig Maurer and Paul Epstein. Christoffel retired from the University of Strasbourg in 1894, being succeeded by Heinrich Weber.[ After retirement he continued to work and publish, with the last treatise finished just before his death and published posthumously.][
Christoffel died on 15 March 1900 in Strasbourg. He never married and left no family.][
]
Work
Differential geometry
Christoffel is mainly remembered for his seminal contributions to differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. In a famous 1869 paper on the equivalence problem for differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s in ''n'' variables, published in Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
, he introduced the fundamental technique later called covariant differentiation
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...
and used it to define the Riemann–Christoffel tensor (the most common method used to express the curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of Riemannian manifolds
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
). In the same paper he introduced the Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
and which express the components of the Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
with respect to a system of local coordinates. Christoffel's ideas were generalized and greatly developed by Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus.
With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the ...
and his student Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significa ...
, who turned them into the concept of tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s and the absolute differential calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
. The absolute differential calculus, later named tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
, forms the mathematical basis of the general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
.[
]
Complex analysis
Christoffel contributed to complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, where the Schwarz–Christoffel mapping In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard ...
is the first nontrivial constructive application of the Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...
. The Schwarz–Christoffel mapping has many applications to the theory of elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s and to areas of physics.[ In the field of elliptic functions he also published results concerning ]abelian integral In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
:\int_^z R(x,w) \, dx,
where R(x,w) is an arbitrary rational function of the two variables x and w, whi ...
s and theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
s.
Numerical analysis
Christoffel generalized the Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for mor ...
method for integration and, in connection to this, he also introduced the 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 ...
for Legendre polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
s (he later also published the formula for general orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
).
Other research
Christoffel also worked on potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
and the theory of differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, 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
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
. He also studied physics and published research in optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, however his contributions here quickly lost their utility with the abandonment of the concept of the luminiferous aether
Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
.[
]
Honours
Christoffel was elected as a corresponding member of several academies:
* Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
(1868)
* Istituto Lombardo (1868)
* Göttingen Academy of Sciences
Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911.
General information
The or ...
(1869)
Christoffel was also awarded two distinctions for his activity by the Kingdom of Prussia:
* Order of the Red Eagle
The Order of the Red Eagle (german: Roter Adlerorden) was an order of chivalry of the Kingdom of Prussia. It was awarded to both military personnel and civilians, to recognize valor in combat, excellence in military leadership, long and faithful se ...
3rd Class with bow (''Schleife'') (1893)
* Order of the Crown 2nd Class (1895)
Selected publications
*
*
* 2 volumes, edited by Ludwig Maurer with the assistance of Adolf Krazer and Georg Faber;Erster Band
Zweiter Band
(Service Commun de Documentation de l'Université Louis Pasteur, Strasbourg)
Notes
References
* P.L. Butzer & F. Feher (editors) ''EB Christoffel: the influence of his work on mathematics and the physical sciences'', Birkhäuser Verlag
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields:
* Springer continues to publish science (particu ...
, 1981 .
*
*
External links
{{DEFAULTSORT:Christoffel, Elwin Bruno
1829 births
1900 deaths
People from Monschau
Differential geometers
19th-century German mathematicians
ETH Zurich faculty
19th-century German physicists