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
The Inverse Function Theorem states that if det J_f(a) ≠ 0, then f is locally invertible and its
In probability and statistics, the Jacobian appears in transforming densities under nonlinear changes of variables. If
Numerical analysis uses Jacobians in solving systems of nonlinear equations (for example, Newton’s method for vector-valued
In general, the Jacobian J_f is defined for any f: R^n -> R^m as an m-by-n matrix. A