As the impact of the American and French Revolutions was felt across Europe, a social atmosphere arose that encouraged ground breaking work in mathematics. Karl Gustav Jacob Jacobi, who attracted early attention from luminaries such as Adrien-Marie Legendre and Karl Gauss, appeared alongside and sometimes worked with a handful of innovative contemporaries like William Hamilton, Augustin-Louis Cauchy, Peter Dirichlet, and Niels Abel. Jacobi's own contributions range across older subjects in math such as number theoryand newer fields like analysis. Two mathematical terms he devised now bear Jacobi's name. He electrified the teaching profession with an unprecedented practice of opening up his theoretical notes to his students, thereby inventing the research seminar now common in universities. While the political instabilities of post-revolutionary times sometimes threatened Jacobi's livelihood, his sheer genius was always enough to attract a new protector to sponsor his continued work. Jacobi's mathematics had an immediate effect on the classical mechanics of Isaac Newton, Pierre Laplace and Joseph Lagrange, and later on the quantum mechanics and relativity theories of the 20th century.
Jacobi, whose first name is sometimes spelled "Carl," was born in Potsdam, Prussia (Germany) into a wealthy and well-educated Jewish family on December 10, 1804. He was one of two boys who would both gain a measure of fame during their lifetimes. Karl's older brother, Moritz, would later be celebrated--and even further on dismissed--as the founder of an experimental concept in electricity known as "galvanoplastics." The other children in the family were Eduard and Therese. All that is generally known of their mother is her own family name, Lehmann, and that her brother tutored Karl until he was 12. His father, Simon Jacobi, was a banker in Berlin.
Young Jacobi was a prodigy. After being home schooled in the classics and in mathematics, he entered Potsdam Gymnasium in 1816. He promptly rose to the top of his class within a few months, proving he was ready for university training. He was still only 12 years old. Early entry to higher education was forbidden by the authorities, so the Gymnasium kept him until the legal age of sixteen. The rector of the school described him as "a universal mind," expressing high hopes for his future.
The University of Berlin had to make way for Jacobi, who rebelled against his teacher, Heinrich Bauer, and preferred to read Leonhard Euler and Lagrange on his own. Jacobi graduated within a year with top marks for classical languages, history, and mathematics. As a graduate student Jacobi majored in philology, since he had no peer at Berlin in mathematics. Instead, he continued to read on his own and correspond with Gauss at Göttingen. Jacobi qualified for a teaching position at age 19, which he took without pay the next year, after completing his doctorate on partial fractions. Jacobi achieved all this as a Jew, but for perhaps more than one reason he converted to Christianity at age 20.
During his student days Jacobi continued to write to his uncle and former teacher, and corresponded with other mathematicians, a habit he continued throughout his career. To one who complained of the physical costs of intellectual labor, Jacobi replied, "Only cabbages have no nerves." How do they benefit, he asked rhetorically, from such well-being? This riposte shows Jacobi's lack of concern for his own health, which may well have contributed to his early death from a combination of illnesses. Some of his vacations were forced, whenever his schedule threatened to wear him down.
Jacobi came into his own as a lecturer. After six months at Berlin he transferred to the University of Königsberg on the recommendation of Legendre, who was impressed with Jacobi's additions to his own pioneering work in elliptic integrals. Jacobi combined new mathematical ventures and classwork to show his new inventions as they took shape. More importantly, he presented himself as one who knew little and desired to know more, inspiring his students to forge ahead according to his example. You do not have to "meet" all subjects in math, he argued by analogy, before "marrying" one.
Jacobi exclusively studied (at least for a while) the work of Legendre on elliptic integrals. Independently of Abel in Norway, Jacobi fully developed the new area of elliptic functions and introduced the notation used today for these functions. He investigated hyperelliptic integrals, making important discoveries about the generalizations of elliptic functions that were later to become known as Abelian functions. Because of his work in mathmatical physics, the ellipsoids of equilibrium for rotating liquid masses are known as Jacobi ellipsoids. Jacobi's work in functions inspired Cauchy and also Joseph Liouville. Other interests included determinants, especially those used in relation to partial differential equations and called the Jacobian determinants. Jacobi published three papers on determinants in 1841. For the first time he gave an algorithmic definition of the determinant that applied to cases when the entries were either numbers or functions. These papers helped to make the idea of a determinant more widely known. Partial differential equations came into play in dynamics, a subject which interested Jacobi. Here, he parlayed the findings of Hamilton into results later applied to quantum mechanics. To classical mechanics he contributed work on the three-body problem and other dynamical problems. In number theory, Jacobi proved an assertion of Pierre de Fermat's regarding the expression of integers as sums of squares.
Simon Jacobi died in 1832, and the family was able to live on his bequest for another eight years. In 1840, however, financial troubles forced them into bankruptcy. These difficulties likely contributed to Jacobi's subsequent collapse from overwork. Jacobi relinquished his chair of Ordinary Professor of Mathematics at Königsberg in 1842. He thereafter subsisted on a pension granted by Frederick William IV from the Prussian government, staying in Italy for a time due to ill health. For personal reasons, Jacobi ran for local office in Berlin as a liberal in 1848, leading to a temporary suspension of his Prussian funding, a great sorrow to Jacobi's wife and seven children. However, he was soon back in favor. He taught intermittently at Berlin meanwhile, but died February 18, 1851. Suffering from diabetes, he developed a fatal case of smallpox, contracted after a bout of influenza. Jacobi was memorialized by his friend and colleague Lejeune Dirichlet in 1852 with a special lecture given at the Berlin Academy of Sciences. In it, Dirichlet called Jacobi the greatest Academy member since Lagrange.
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