Named after Karl Gustav Jacobi (1804-1851), a German mathematician who did important work on elliptic functions, partial differential equations, and mechanics. The "Jacobian" first appeared in an 1815 paper of the French mathematician Augustin Louis Cauchy (1789-1857), but Jacobi did write a detailed memoir about it in 1841.
Fr.: déterminant jacobien
Fr.: matrice jacobienne