Jacobianmatrix

The Jacobian matrix of a vector-valued function f: R^n → R^m at a point x is the m-by-n matrix J_f(x) = [∂f_i/∂x_j]. It provides the best linear approximation to f near x: f(x + h) ≈ f(x) + J_f(x) h. The differential df_x is the linear map h ↦ J_f(x) h.

Notation and relationship to gradient: J_f(x) or Df(x) are common. When m = 1, the Jacobian is a 1×n row vector of partial derivatives, which is often identified with the gradient of f; the gradient is the transpose of that row.

If m = n, J_f(x) is square and det J_f(x) is the Jacobian determinant. A nonzero determinant implies local invertibility by the inverse function theorem, and det J_f(x) measures local volume change under f in change-of-variables formulas.

Example: Let f: R^2 → R^2 with f(x, y) = (u, v) where u = x^2 − y and v = sin x + y. Then J_f(x, y) = [ [∂u/∂x, ∂u/∂y], [∂v/∂x, ∂v/∂y] ] = [ [2x, −1], [cos x, 1] ].

Applications include solving nonlinear systems with Newton's method, analysis of dynamical systems, sensitivity analysis, optimization, and automatic differentiation. In numerical work, the Jacobian is often evaluated symbolically or approximated; its factorization informs iterative updates and stability.