# Karl Gustav Jacob Jacobi

# Karl Gustav Jacob Jacobi

**1804-1851**

**German Mathematician**

Karl Jacobi made his most notable contributions to mathematics in the area of elliptic functions. His book *Concerning the Structure and Properties of Determinants* was an important work in that branch of mathematics, and his work on partial differential equations proved important in the formulation of quantum mechanics.

Jacobi was born into a relatively prosperous family in Potsdam, Germany, in 1804. His father was a banker, assuring Jacobi a good education as a child and, later, at the University of Berlin. He completed his Ph.D. at Berlin in 1825, then taught mathematics at the University of Königsberg from 1826 until 1844.

Jacobi's main area of interest was in the branch of mathematics that dealt with elliptic functions. These functions were first studied in the mid-seventeenth century when mathematicians began investigating ways to determine the length of an arc of arbitrary length and position in an ellipse. Since the curvature of an ellipse varies along its circumference, this can be a difficult problem. Such curves that vary with two levels of periodicity are called "doubly periodic" functions. Elliptic functions are important in dealing with problems in physics that examine the shape of bars under stress, the effects of stress on rods and bars, and other, similar problems that are frequently seen in engineering fields today. The problem of dealing with elliptic functions was addressed with varying degrees of success by such mathematicians as John Wallis (1616-1703), Isaac Newton (1642-1727), and Jakob Bernoulli (1654-1705).

In addition to his work on elliptic functions, Jacobi carried out very significant research into partial differential equations and their application to problems in dynamics (problems involving moving bodies). Some of this research was carried out in collaboration with the Irish mathematician William Hamilton (1805-1865), resulting in the Hamilton-Jacobi equation, which was to play a great role in the formulation of quantum mechanics in the early twentieth century.

Jacobi's other significant contribution to mathematics was in the area of determinants. A determinant is the result of a series of mathematical operations performed on a matrix. In this case, the matrix would be set up to help solve a set of equations called linear equations, and each line in the matrix would represent the numerical coefficients in a single mathematical equation. Jacobi was able to show that if functions with the same number of variables are related to each other, then the Jacobian determinant is equal to zero. Any non-zero value shows that these functions are not related.

In addition to his mathematical skills, Jacobi had a reputation as an excellent professor. Perhaps his most important innovation was the introduction of the seminar style of teaching, in which the students play an active role in both teaching and learning. Jacobi used seminars extensively to present the most advanced topics in mathematics to his students in a less formal setting. In fact, it was not uncommon for other respected mathematicians to attend Jacobi's seminars in order to learn better what their counterparts elsewhere were doing.

Jacobi once commented on the significance of mathematics: "It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world."

**P. ANDREW KARAM**

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**Karl Gustav Jacob Jacobi**