jacobian

Jacobian refers to the Jacobian matrix and its determinant, concepts in multivariable calculus named after Carl Gustav Jacob Jacobi. For a differentiable map f: R^n -> R^m, the Jacobian J_f(x) is the m-by-n matrix of first-order partial derivatives with entries ∂f_i/∂x_j. If m = n, the determinant det J_f(x) is the Jacobian determinant and encodes the local linear approximation of f and how volumes are scaled near x.

A central application is the change of variables in multiple integrals. If y = f(x) is a diffeomorphism with invertible Jacobian, then integrals transform by the factor |det J_f(x)|. Concretely, ∫ g(y) dy = ∫ g(f(x)) |det J_f(x)| dx, with the domain adjusted by the inverse mapping and orientation determined by the sign of the determinant.

The Inverse Function Theorem states that if det J_f(a) ≠ 0, then f is locally invertible and its inverse is differentiable; equivalently, f is a local diffeomorphism near a. This relies on the Jacobian being non-singular at the point.

In probability and statistics, the Jacobian appears in transforming densities under nonlinear changes of variables. If Y = f(X) with invertible differentiable f, the density transforms as f_Y(y) = f_X(f^{-1}(y)) |det J_{f^{-1}}(y)|.

Numerical analysis uses Jacobians in solving systems of nonlinear equations (for example, Newton’s method for vector-valued F(x) = 0) by iteratively updating using the Jacobian.

In general, the Jacobian J_f is defined for any f: R^n -> R^m as an m-by-n matrix. A determinant exists only when m = n; in that case the determinant measures local volume scaling and orientation under f. The Jacobian is named after Jacobi.