Jacobians

Jacobians refer mainly to two related concepts in mathematics: the Jacobian matrix of a multivariable function, and the Jacobian variety associated with an algebraic curve. In multivariable calculus, let f: R^n → R^m be differentiable. The Jacobian matrix J_f(x) is the m×n matrix whose entries are the partial derivatives (∂f_i/∂x_j)(x). It represents the best linear approximation to f at x, i.e., df_x(v) = J_f(x) v. If m = n and det J_f(x0) ≠ 0, then f is locally invertible near x0 by the inverse function theorem, and the derivative of the inverse is (J_f(x0))^{-1}. The Jacobian determinant det J_f is used in the change of variables formula for integrals: ∫ g(y) dy = ∫ g(f(x)) |det J_f(x)| dx over a suitable domain, and it describes how volumes scale under f. The chain rule also expresses the Jacobian of a composition as J_{g∘f}(x) = J_g(f(x)) J_f(x).

In algebraic geometry, the Jacobian of a smooth projective curve C of genus g is a principally polarized abelian variety of dimension g, denoted Jac(C). It parameterizes degree-zero line bundles on C and arises via the Abel–Jacobi map from divisors to line bundles. Jacobians play a central role in the study of curves, with applications such as the Torelli theorem, which reconstructs a curve from its Jacobian and polarization.

The term Jacobian is named after the 19th-century German mathematician Carl Gustav Jacob Jacobi.