约翰·弗里德里希·普法夫Johann Friedrich Pfaff

约翰·弗里德里希·普法夫的父亲伯克哈德普法夫是符腾堡的首席财务顾问，而他的母亲是符腾堡国库一位成员的千金。这是一个有着为符腾堡政府担任公务员传统的家族。

普法夫普法夫是他父母七个儿子中的第二个，虽然也许是获得最大名声的一个，但他肯定不是唯一在科学上出类拔萃的。家中最小的普法夫Wilhelm Pfaff生于1774年，也成为了一名数学家，并在维尔茨堡和埃尔朗根担任讲席。第二小的克里斯托夫Heinrich Pfaff生于1773年，对化学、医学和药学感兴趣，他与伏打合作研究动物电。

斯图加特有一所学校，即高等卡尔学校，其办学目的是培养符腾堡政府官员的儿子，普法夫普法夫从九岁起就读于这所学校。这是一所相当缺乏启发性的学校，纪律严格但学术水平较差。普法夫在那里没有学到多少数学，尽管他在该校一直读到将近二十岁。当他在1785年秋天离开时，他已经完成了法律学业，这对一名公务员来说是一个合适的科目。

尽管他在学校缺乏数学训练，普法夫自学了数学，并开始研究莱昂哈德·欧拉的著作。符腾堡公爵鼓励他转向科学课题，他在哥廷根大学学习了两年，在那里亚伯拉罕·哥特黑尔弗·凯斯特纳教他数学，他还学习了物理学。1787年夏天，普法夫从哥廷根搬到柏林。在那里，他在J E Bode指导下学习天文学，普法夫写了他的第一篇论文，是关于天文学中的一个问题。

1788年春天，普法夫出发前往维也纳，但途中访问了许多大学，特别是哈勒、耶拿、赫尔姆施泰特、德累斯顿和布拉格。Klügel是赫尔姆施泰特的数学教授，他接受了哈勒的讲席，留下了赫尔姆施泰特的职位空缺。普法夫在哥廷根的物理学教授推荐他担任该讲席，普法夫在当选赫尔姆施泰特大学数学教授之际提交了一篇学位论文。那里的大学新教授提交就职学位论文是一个传统。

普法夫的就职学位论文题为Programma inaugurale in quo peculiarem differentialia investigandi rationem ex theoria functionum deducit Ⓣ（就职程序，其中特殊的微分研究导致函数论）。它研究了一些函数方程的使用，以计算对数和三角函数的微分以及二项式展开和布鲁克·泰勒公式。这在[4]中有详细研究。

从1788年受聘到1810年，普法夫在Helmstedt担任讲席。他的聘任得到了符腾堡公爵的批准，但这并非最好的职位，因为薪水很低。他在那里为增强数学实力做了出色的工作，并在教学上投入了大量精力，成功地增加了数学学生的数量。一位在Helmstedt学习的学生是卡尔·弗里德里希·高斯。在哥廷根学习后，卡尔·弗里德里希·高斯于1798年来到Helmstedt。他听了普法夫的讲座，甚至住在他的房子里。Wussing在[1]中写道：-

普法夫推荐了卡尔·弗里德里希·高斯的学位论文，并在必要时大力协助他；卡尔·弗里德里希·高斯始终对普法夫作为教师和为人保持着友好的记忆。

当卡尔·弗里德里希·高斯在Helmstedt跟随普法夫学习时，这所大学正面临关闭的威胁。普法夫努力阻止此事，并成功维持了几年。普法夫于1803年与Caroline Brand结婚，但不幸的是他们的第一个孩子夭折了。到1810年，普法夫保存Helmstedt大学的努力最终失败，大学关闭。教职员工可以选择迁往不同的大学，普法夫选择迁往哈勒。

他于1810年被任命为哈勒的数学讲席，1812年，在Klügel去世后，他还担任了大学天文台的主任。

普法夫在分析方面做了重要工作，研究偏微分方程、特殊函数和级数理论。他使用约瑟夫·拉格朗日给出的带余项形式发展了布鲁克·泰勒定理。1810年，他为解决卡尔·弗里德里希·高斯提出的关于可以在给定四边形内画出的最大面积椭圆的问题做出了贡献。

他关于普法夫形式的最重要工作发表于1815年，当时普法夫已年近五十，但其重要性直到1827年卡尔·古斯塔夫·雅各布·雅可比发表了一篇关于普法夫方法的论文才被认识到。未能认识到这项工作的重要性很奇怪，尤其是考虑到卡尔·弗里德里希·高斯在该工作发表后不久就写了非常积极的评论。在1815年的论文中，普法夫于5月11日提交给柏林科学院，他提出了将一阶偏微分方程转化为微分系统的方法。这种全微分方程理论无疑是普法夫最重要的贡献。Wussing在[1]中写道，普法夫的这项工作：-

……构成了偏微分方程积分基本理论的起点，通过卡尔·古斯塔夫·雅各布·雅可比、索菲斯·李等人的工作，已发展成为现代嘉当的极微分形式微积分。

他的其他重要著作包括Disquisitiones analyticae maxime ad calculum integralem et doctrinam serierum pertinentes Ⓣ（关于积分学和相关级数）（1797年），这是一部以莱昂哈德·欧拉风格撰写的入门著作，以及Observationes ad Euleri institutiones calculi integralis Ⓣ（莱昂哈德·欧拉关于积分学的观察）。

Johann Friedrich Pfaff's father, Burkhard Pfaff, was chief financial counsellor of Württemberg while his mother was the daughter of a member of the exchequer of Württemberg. It was a family with a tradition of working as civil servants for the government of Württemberg.

Johann Friedrich was the second of his parents seven sons and, although perhaps the one to attain the greatest fame, he was certainly not the only one to excel in science. The youngest of the family, Johann Wilhelm Pfaff who was born in 1774, also became a mathematician and held chairs in Würtzburg and Erlangen. The second youngest, Christoph Heinrich Pfaff was born in 1773 and, with interests in chemistry, medicine and pharmacy, he worked with Volta on electricity in animals.

There was a school in Stuttgart, the Hohe Karlsschule, which was run to train sons of government officials of Württemberg and Johann Friedrich attended this school from the age of nine. It was a rather uninspiring school, strong on discipline but less good academically. Pfaff did not learn much in the way of mathematics there despite attending the school until he was nearly twenty. When he left in the autumn of 1785 he had completed his studies in law, a fitting subject for a civil servant.

Despite a lack of training in mathematics at his school, Pfaff had studied mathematics on his own and began to study the works of Euler. He was encouraged to move toward scientific topics by the Duke of Württemberg, and he spent two years studying at the University of Göttingen where he was taught mathematics by Kästner and he also studied physics. From Göttingen, Pfaff moved to Berlin in the summer of 1787. There he studied astronomy under J E Bode, and Pfaff wrote his first paper which was on a problem in astronomy.

In the spring of 1788 Pfaff set off on a journey to Vienna but he visited many universities on the way, in particular Halle, Jena, Helmstedt, Dresden, and Prague. Klügel was professor of mathematics at Helmstedt and he accepted a chair at Halle leaving the position at Helmstedt vacant. Pfaff's physics professor at Göttingen recommended him for the chair, and Pfaff submitted a dissertation on the occasion of his election as professor of mathematics at the University of Helmstedt. It was a tradition that new professors at the university there submitted an inaugural dissertation.

Pfaff's inaugural dissertation was titled Programma inaugurale in quo peculiarem differentialia investigandi rationem ex theoria functionum deducit Ⓣ. It investigates the use of some functional equations in order to calculate the differentials of logarithmic and trigonometrical functions as well as the binomial expansion and Taylor formula. This is studied in detail in [4].

From his appointment in 1788 until 1810 Pfaff held the chair at Helmstedt. His appointment was approved by the Duke of Württemberg but it was not the best of positions, since it was poorly paid. He did good work building the strength of mathematics there and he put much effort into teaching and was successful in increasing the number of students of mathematics. One student who studied at Helmstedt was Gauss. After studying at Göttingen, Gauss came to Helmstedt in 1798. He attended Pfaff's lectures and even lived in his house. Wussing writes in [1]:-

Pfaff recommended Gauss's doctoral dissertation and, when necessary, greatly assisted him; Gauss always retained a friendly memory of Pfaff both as a teacher and as a man.

By the time Gauss studied with Pfaff at Helmstedt, the university was under threat of closure. Pfaff fought hard to prevent this and for a few years he was successful. Pfaff married in 1803 to Caroline Brand but sadly their first child died as an infant. By 1810 Pfaff's attempts to preserve the University of Helmstedt finally failed with the closure of the university. The staff were given a number of different choices as to which university they might move to, and Pfaff chose to move to Halle.

He was appointed to the chair of mathematics at Halle in 1810 and in 1812, on the death of Klügel, he took on the directorship of the University Observatory too.

Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series. He developed Taylor's Theorem using the form with remainder as given by Lagrange. In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral.

His most important work on Pfaffian forms was published in 1815 when Pfaff was nearly fifty years old but its importance was not recognised until 1827 when Jacobi published a paper on Pfaff's method. This failure to recognise the importance of the work is strange, particularly given the very positive review which Gauss wrote of the work shortly after it was published. In the 1815 paper, which Pfaff submitted to the Berlin Academy on 11 May, he presented a transformation of a first-order partial differential equation into a differential system. This theory of equations in total differentials is undoubtedly Pfaff's most significant contribution. Wussing writes in [1] that this work by Pfaff:-

... constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Lie, and others, has developed into a modern Cartan calculus of extreme differential forms.

Among his other important works are Disquisitiones analyticae maxime ad calculum integralem et doctrinam serierum pertinentes Ⓣ (1797), an introductory work written in the style of Euler, and Observationes ad Euleri institutiones calculi integralis Ⓣ.