Jacobi

Jacobi is a surname of German origin, derived from the given name Jacob. The name is associated with several figures in science, most notably Carl Gustav Jacob Jacobi (1804–1851), a German mathematician whose work spanned analysis, number theory, and dynamics.

Because of Jacobi's influence, numerous mathematical constructs carry the name. The Jacobi method, or Jacobi iteration, is an algorithm for solving systems of linear equations by successive approximations. Jacobi polynomials are a family of classical orthogonal polynomials. Jacobi elliptic functions, including sn, cn, and dn, generalize trigonometric functions to elliptic curves. The Jacobi symbol is used in number theory to generalize the Legendre symbol in quadratic reciprocity. The Jacobi identity is a fundamental relation in the theory of Lie algebras that constrains the Lie bracket.

Other uses include the Jacobian, a matrix of first-order partial derivatives, from which the Jacobian determinant arises in multivariable calculus and coordinate transformations. The Jacobi constant, arising in the circular restricted three-body problem, is a conserved quantity in celestial mechanics. Jacobi fields describe infinitesimal variations of geodesics in differential geometry, and Jacobi coordinates are a system of relative coordinates in few-body problems.

The surname Jacobi also appears in various regions and disciplines, reflecting the broad influence of Jacobi's work across mathematics and physics.