埃尔温·布鲁诺·克里斯托费尔Elwin Bruno Christoffel

埃尔温·布鲁诺·克里斯托费尔以其在数学分析方面的工作而闻名，在这一领域他是约翰·彼得·古斯塔夫·勒热纳·狄利克雷和波恩哈德·黎曼的追随者。

克里斯托费尔的父母都来自从事布料贸易的家庭。他在蒙茹瓦（1918年更名为蒙绍）上小学，但随后有几年在家接受语言、数学和古典学的辅导。他从1844年到1849年上中学。起初他在科隆的耶稣会文理中学学习，但后来转到同一城市的弗里德里希-威廉文理中学，至少在那里完成了最后三年的中学教育。1849年，他以优异成绩获得了毕业证书。

克里斯托费尔从1850年起在柏林大学学习，师从卡尔·威廉·博尔夏特、费迪南·艾森斯坦、费迪南德·约阿希姆斯塔尔、雅各布·施泰纳和约翰·彼得·古斯塔夫·勒热纳·狄利克雷。对他影响最大的是约翰·彼得·古斯塔夫·勒热纳·狄利克雷，而克里斯托费尔被正确地认为是约翰·彼得·古斯塔夫·勒热纳·狄利克雷的学生。

在近卫炮兵旅服役一年后，他回到柏林攻读博士学位，并于1856年获得学位，学位论文是关于均匀物体中电的运动。他的考官包括数学家和物理学家，恩斯特·爱德华·库默尔是数学考官之一。

此时克里斯托费尔在学术界之外度过了三年。他回到蒙茹瓦，他母亲在那里身体不好，但他广泛阅读了约翰·彼得·古斯塔夫·勒热纳·狄利克雷、波恩哈德·黎曼和奥古斯丁·路易·柯西的著作。有人提出，这段学术隔绝时期对他的个性以及他对数学的独立态度产生了重大影响。Butzer在[3]中指出，克里斯托费尔的传记作者们把他描述为

一个孤独的人，……害羞、多疑、不善交际、易怒且粗鲁。

把这些性格特点仅仅归因于这三年似乎不合理，但显然这三年对他影响很大，并且肯定促成了他成为一个非常独立的思想者。正是在这段时间里，他发表了他的头两篇论文。这两篇发表于1858年的论文涉及数值分析，特别是数值积分。他推广了卡尔·弗里德里希·高斯的求积方法，并将所涉及的多项式表示为行列式。这现在被称为克里斯托费尔定理。

1859年，克里斯托费尔参加了成为大学教师的资格考试，并被任命为柏林大学的讲师。1862年，他被任命为苏黎世理工学院的讲席，填补了理查德·戴德金去布伦瑞克后留下的空缺。苏黎世理工学院成立于七年前，他们提供的数学课程主要面向工程专业的学生。克里斯托费尔对理工学院的数学产生了巨大影响，在那里建立了一个数学和自然科学研究所。

1868年，克里斯托费尔获得了柏林商业学院（即现在的柏林工业大学）的讲席职位。这并非首次有人试图吸引克里斯托费尔转到这所大学，因为当时设立了一个新职位，校方希望由一位著名数学家来担任。在1868年向克里斯托费尔发出邀请后不久，又向他提供了另一个职位，即担任亚琛新成立的多科技术学院的创始院长。这所新大学，即如今享有盛誉的亚琛莱茵-威斯特法伦技术大学，对在亚琛附近出生和长大的克里斯托费尔来说，想必是一个颇具吸引力的选择。

然而，克里斯托费尔没有接受亚琛的职位：也许他已经承诺去柏林的商业学院，因为他确实离开苏黎世前往柏林，于1869年4月1日就任新职。这一举动对克里斯托费尔来说可能是个错误。他和同事Aronhold试图为商业学院吸引高质量的学生，但事实证明这很困难，因为附近有享有盛誉的柏林大学，还有卡尔·魏尔斯特拉斯、恩斯特·爱德华·库默尔和利奥波德·克罗内克。

在柏林商业学院任职三年后，克里斯托费尔获得了斯特拉斯堡大学的数学讲席。这是一所历史悠久、声名显赫的大学，但在普鲁士占领阿尔萨斯-洛林后，该校正在进行重大重组。从1872年受聘起，克里斯托费尔开始在那里建立一个新的数学研究所，大致沿袭了他10年前在苏黎世的做法。他在建立这个极为成功的系的过程中得到了同事Reye的协助。

克里斯托费尔一直担任这个讲席，直到1892年因健康不佳被迫退休。他在退休前不久的一次事故中摔断了手臂，这无疑是他决定退休的原因之一。1895年，海因里希·马丁·韦伯被任命为他在斯特拉斯堡的继任者。

克里斯托费尔在斯特拉斯堡期间指导了六名博士生。其中至少有四人后来成为大学数学教授，包括Paul 保罗·爱泼斯坦。

克里斯托费尔发表了关于函数论（包括共形映射）、几何学和张量分析、波恩哈德·黎曼的o函数、不变量理论、正交多项式以及连分数、微分方程和potential theory、光和冲击波的论文。

克里斯托费尔早期的一些工作是关于将由多边形围成的单连通区域共形映射到圆上的。这项关于共形映射的工作在1868年至1870年间发表于四篇论文中。其中第一篇论文是在克里斯托费尔于苏黎世期间写的，其余三篇关于克里斯托费尔-赫尔曼·阿曼杜斯·施瓦茨公式的论文是在他于柏林商业学院期间写的。

在1865年至1871年间，克里斯托费尔发表了四篇关于位势理论的重要论文，其中三篇涉及约翰·彼得·古斯塔夫·勒热纳·狄利克雷问题。1877年，克里斯托费尔发表了一篇关于平面波在具有表面间断的介质中传播的论文。这是对激波理论的早期贡献，并延续了波恩哈德·黎曼此前关于一维气体流动的工作。

克里斯托费尔对不变量理论感兴趣。他写了六篇关于这个主题的论文。他写了重要论文，对C G 格雷戈里奥·里奇-库尔巴斯托罗和Tullio 图利奥·列维-齐维塔的张量微积分的发展做出了贡献。他引入的克里斯托费尔符号[ij,k]是张量分析研究的基础。由菲利克斯·克莱因命名的克里斯托费尔约化定理解决了两个二次微分形式的局部等价问题。在[3]中Butzer写道：-

克里斯托费尔在解决等价问题中采用的程序就是Gregorio 格雷戈里奥·里奇-库尔巴斯托罗后来所称的协变微分，克里斯托费尔也使用后一概念来定义基本的波恩哈德·黎曼-克里斯托费尔曲率张量。……这种方法的重要性以及克里斯托费尔至少隐含地引入的两个概念，只有在考虑它所产生的影响时才能评判。

确实，这种影响显而易见，因为这使得格雷戈里奥·里奇-库尔巴斯托罗和图利奥·列维-齐维塔能够发展出一种无坐标微分微积分，阿尔伯特·爱因斯坦在马塞尔·格罗斯曼的帮助下将其转变为广义相对论的张量分析数学基础。

克里斯托费尔写了35篇论文，但这并不代表他数学工作的全部范围。事实上，与当时许多其他人一样，他的许多原创研究被放入他的讲座课程中，只有通过那个来源才为人所知。Timerding描述了克里斯托费尔的教学，这一描述被引用于[3]中：-

克里斯托费尔是有史以来最优雅的教师之一。他的讲座准备得一丝不苟，细致入微……他的讲授清晰明了，具有极高的美学完美性……讲座的核心是复变函数论课程，以波恩哈德·黎曼这个鼓舞人心的名字著称。克里斯托费尔独立发展了波恩哈德·黎曼的函数论，特别是在超椭圆函数领域，但没有发表他的研究，只是按照卡尔·魏尔斯特拉斯的模式在讲座中呈现。

给数学家排名非常困难。如何将只在一个领域工作的人与在多个领域做出贡献的人进行比较？又如何将研究微分方程的人与几何学家进行比较？尽管存在明显的困难和细微的意见分歧，但在这样的排名上仍有如此多的共识，这仍然令人惊讶。在[5]中，Butzer和Fehér试图将克里斯托费尔纳入这样的排名中：-

很难将微分几何学家与函数论专家进行比较，或者将研究常微分方程和偏微分方程的人与数值分析学家进行比较。克里斯托费尔不仅对所有这些领域都有贡献，而且他的兴趣还扩展到正交多项式和连分数，以及他的工作对张量分析基础、大地测量科学、激波理论、光的色散的应用。尽管如此，人们普遍认为，至少在欧洲德语国家，波恩哈德·黎曼是19世纪最好的数学家，仅次于卡尔·弗里德里希·高斯，领先于卡尔·魏尔斯特拉斯。在我们看来，克里斯托费尔的老师约翰·彼得·古斯塔夫·勒热纳·狄利克雷属于下一个最重要的数学家群体，该群体包括（按出生时间顺序）卡尔·古斯塔夫·雅各布·雅可比、恩斯特·爱德华·库默尔、利奥波德·克罗内克、理查德·戴德金、格奥尔格·康托尔和菲利克斯·克莱因。克里斯托费尔本人应被置于紧随其后的第二群体中。这第二群体，可能与前一群体的部分重叠，将包括诸如奥古斯特·费迪南德·莫比乌斯、卡尔·冯·施陶特、尤里乌斯·普吕克、爱德华·海涅、Du Bois-Reymond、卡尔·诺伊曼、鲁道夫·利普希茨、拉扎勒斯·福克斯、赫尔曼·阿曼杜斯·施瓦茨、阿道夫·赫维兹和赫尔曼·闵可夫斯基等显赫的名字。

如果也将数学物理学家考虑在内，那么Butzer和Fehér认为克里斯托费尔将不得不与乔治·格林、威廉·哈密顿、詹姆斯·约瑟夫·西尔维斯特、赫尔曼·冯·亥姆霍兹、阿瑟·凯莱、古斯塔夫·基尔霍夫、詹姆斯·克拉克·麦克斯韦、贝尔特拉米、索菲斯·李、路德维希·玻尔兹曼、儒勒·昂利·庞加莱和埃里克·伊瓦尔·弗雷德霍姆进行比较。我[EFR]必须说，我发现Butzer和Fehér认为其中一些数学家是数学物理学家，这令人惊讶。

Elwin Christoffel was noted for his work in mathematical analysis, in which he was a follower of Dirichlet and Riemann.

Christoffel's parents both came from families who were in the cloth trade. He attended an elementary school in Montjoie (which was renamed Monschau in 1918) but then spent a number of years being tutored at home in languages, mathematics and classics. He attended secondary schools from 1844 until 1849. At first he studied at the Jesuit Gymnasium in Cologne but moved to the Friedrich-Wilhelms Gymnasium in the same town for at least the three final years of his school education. He was awarded the final school certificate with a distinction in 1849.

Christoffel studied at the University of Berlin from 1850 where he was taught by Borchardt, Eisenstein, Joachimsthal, Steiner and Dirichlet. It was Dirichlet who had the greatest influence on him and Christoffel is rightly thought of as a student of Dirichlet's.

After one year of military service in the Guards Artillery Brigade, he returned to Berlin to study for his doctorate which he was awarded in 1856 with a dissertation on the motion of electricity in homogeneous bodies. His examiners included mathematicians and physicists, Kummer being one of the mathematics examiners.

At this point Christoffel spent three years outside the academic world. He returned to Montjoie where his mother was in poor health but read widely from the works of Dirichlet, Riemann and Cauchy. It has been suggested that this period of academic isolation had a major effect on his personality and on his independent approach towards mathematics. Butzer, in [3], remarks that Christoffel's biographers have described him as

a lonely man, ... shy, distrustful, unsociable, irritable and brusque.

It would seem unreasonable to attribute these aspects of his character solely to these three years, yet clearly these years had a large influence on him and certainly contributed to his being an very independent thinker. It was during this time that he published his first two papers. These papers, which appeared in 1858, are on numerical analysis, in particular numerical integration. He generalised Gauss's method of quadrature and expressed the polynomials which are involved as a determinant. This is now called Christoffel's theorem.

In 1859 Christoffel took the qualifying examination to become a university teacher and was appointed a lecturer at the University of Berlin. In 1862 he was appointed to the chair at the Polytechnicum in Zürich, filling the post left vacant when Dedekind went to Brunswick. The Polytechnic School in Zürich had been set up seven years before and the mathematics courses which they offered were mainly aimed at engineering students. Christoffel was to have a huge influence on mathematics at the Polytechnicum, setting up an institute for mathematics and the natural sciences there.

In 1868 Christoffel was offered a chair at the Gewerbsakademie in Berlin which is now the University of Technology of Berlin. This was not the first time an attempt had been made to interest Christoffel in moving to this university since a new position had been set up and the university authorities wanted an eminent mathematician to fill he post. Shortly after the 1868 offer to Christoffel another position was offered to him, namely to become a founding director of the new Polytechnicum at Aachen. This new university, now the prestigious Rheinisch- Westfälische Technische Hochschule at Aachen, must have been an attractive idea to Christoffel who was born and brought up close to Aachen.

Christoffel, however, did not accept the Aachen position: perhaps he was already committed to the Gewerbsakademie in Berlin for he certainly left Zürich for Berlin to take up his new post on 1 April 1869. This move may have been a mistake for Christoffel. He and his colleague Aronhold tried to attract high quality students to the Gewerbsakademie but this proved difficult with the highly prestigious University of Berlin with Weierstrass, Kummer and Kronecker close by.

After three years at the Gewerbsakademie in Berlin, Christoffel was offered the chair of mathematics at the University of Strasbourg. This was a university with a long and distinguished past, but the university was undergoing a major reorganisation following the Prussian capture of Alsace- Lorraine. From his appointment in 1872 Christoffel began to built up a new Institute for mathematics there much along the lines which he had followed in Zürich 10 years before. He was assisted in his efforts to build this new highly successful department by his colleague Reye.

Christoffel was to hold this chair until he was forced to retire due to ill health in 1892. He broke his arm in an accident shortly before he retired and this was certainly one of the reasons he decided to retire. Heinrich Weber was appointed to succeed him at Strasbourg in 1895.

Christoffel supervised six doctoral students while at Strasbourg. At least four of these were to become university professors of mathematics, including Paul Epstein.

Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis, Riemann's o-function, the theory of invariants, orthogonal polynomials and continued fractions, differential equations and potential theory, light, and shock waves.

Some of Christoffel's early work was on conformal mappings of a simply connected region bounded by polygons onto a circle. This work on conformal mappings was published in four papers between 1868 and 1870. The first of these papers was written while Christoffel was at Zürich, the remaining three papers on the Christoffel-Schwarz formula were written while he was at the Gewerbsakademie in Berlin.

Between 1865 and 1871 Christoffel published four important papers on potential theory, three of them dealing with the Dirichlet problem. In 1877 Christoffel published a paper on the propagation of plane waves in media with a surface discontinuity. This was an early contribution to the theory of shock waves and followed earlier work on one dimensional gas flows by Riemann.

Christoffel was interested in the theory of invariants. He wrote six papers on this topic. He wrote important papers which contributed to the development of the tensor calculus of C G Ricci-Curbastro and Tullio Levi-Civita. The Christoffel symbols [ij,k] which he introduced are fundamental in the study of tensor analysis. The Christoffel reduction theorem, so named by Klein, solves the local equivalence problem for two quadratic differential forms. In [3] Butzer writes:-

The procedure Christoffel employed in his solution of the equivalence problem is what Gregorio Ricci-Curbastro later called covariant differentiation, Christoffel also used the latter concept to define the basic Riemann-Christoffel curvature tensor. ... The importance of this approach and the two concepts Christoffel introduced, at least implicitly, can only be judged when one considers the influence it has had.

Indeed this influence is clearly seen since this allowed Ricci-Curbastro and Levi-Civita to develop a coordinate free differential calculus which Einstein, with the help of Grossmann, turned into the tensor analysis mathematical foundation of general relativity.

Christoffel wrote 35 papers but this does not represent the full extent of his mathematical work. In fact, in common with many others at that time, much of his original research was put into his lecture courses and only through that source was it known. Timerding described Christoffel's teaching, this description is quoted in [3]:-

Christoffel was one of the most polished teachers ever to occupy a chair. His lectures were meticulously prepared, to the smallest detail ... His delivery was lucid and of the greatest aesthetic perfection ... The core of the lectures was the course on complex function theory, distinguished by the inspirational name of Riemann. Christoffel had developed Riemann's function theory independently, particularly in the area of ultraelliptic functions, but did not publish his research, presenting them only in his lectures, after the model of Weierstrass.

It is very difficult to rank mathematicians. How does one compare someone who worked solely in one area with another who contributed to many areas? Again how does one compare someone who worked on differential equations with a geometer? Despite the obvious difficulties, and minor differences of opinion, it is still surprising how much agreement there is on such a ranking. In [5] Butzer and Fehér attempt to fit Christoffel into such a ranking:-

It is difficult to compare a differential geometer with a function theorist, or those working on ordinary and partial differential equations with numerical analysts. Christoffel not only contributed to all these fields, but his interests extended to orthogonal polynomials and continued fractions, and the applications of his work to the foundations of tensor analysis, to geodetical science, to the theory of shock waves, to the dispersion of light. Nevertheless, it is widely recognised, at least in the German speaking countries of Europe, that Riemann was the best mathematician of the 19th century, behind Gauss and ahead of Weierstrass. In our opinion Christoffel's teacher Dirichlet, belongs to the next most important group of mathematicians which includes (in chronological order of birth) Jacobi, Kummer, Kronecker, Dedekind, Cantor and Klein. Christoffel himself should be placed in a second group following these. This second group, which may partly overlap with the former, would include such illustrious names as Möbius, von Staudt, Plücker, Heine, Du Bois-Reymond, Carl Neumann, Lipschitz, Fuchs, Schwarz, Hurwitz and Minkowski.

If mathematical physicists are also taken into account then Butzer and Fehér believe that Christoffel would have to be compared with Green, Hamilton, Sylvester, Helmholtz, Cayley, Kirchhoff, Maxwell, Beltrami, Lie, Boltzmann, Poincaré and Fredholm. I [EFR] must say that I find it surprising that some of these mathematicians are considered by Butzer and Fehér to be mathematical physicists.