Jacobi's research advice to students was three words: invert, always invert
Carl Jacobi told students hunting for a problem to take a known result and turn it upside down. He had done exactly that with elliptic integrals, flipping them into elliptic functions and opening a field physicists still use for periodic motion. Several ideas in modern mathematics carry his name.
Jacobi was born in Potsdam in 1804, the second of four children of a Jewish banker, and began life as Jacques Simon before his name was Germanised. An uncle taught him classics and basic mathematics at home, and he was so far ahead at the Potsdam Gymnasium that he was moved into the senior year within months. Because the university would not take anyone under 16, he waited there until 1821, spending the time on everything from Greek to an attempt at solving the quintic equation by radicals.
At Berlin University he was torn between philology, where the classicist August Böckh admired him, and mathematics, whose classes he found too basic; he taught himself from Euler, Lagrange and Laplace instead. In 1823 he chose mathematics. He earned his doctorate in 1825, converted to Christianity as he qualified to teach at university, and at 21 was lecturing on curves and surfaces. From 1829 he held the mathematics chair at Königsberg until 1842.
His 1829 treatise on elliptic functions and their link to theta functions was his landmark, and he was the first to turn elliptic functions on number theory, proving Fermat's two-square and Lagrange's four-square theorems by that route. He proved that a single-valued function can have at most two periods. The Jacobian determinant, built from partial derivatives and used when changing variables in multiple integrals, is his, and in 1841 he revived Legendre's curly partial-derivative sign, which became standard. Physicists meet him in Hamilton–Jacobi theory and the Jacobi identity.
Overwork broke his health in 1843. After a few months recovering in Italy he lived in Berlin on a royal pension, which was cut off when he stood unsuccessfully for parliament for a liberal club during the 1848 revolution, then restored after Alexander von Humboldt intervened. Smallpox killed him in 1851. A lunar crater bears his name.
Source: Carl Gustav Jacob Jacobi
