JacobiMatrix

The Jacobi matrix, or Jacobian matrix, of a differentiable vector-valued function f: R^n -> R^m at a point a in R^n, is the m×n matrix J_f(a) with entries J_{ij} = ∂f_i/∂x_j evaluated at a. It provides the best linear approximation to f near a, so f(a + h) ≈ f(a) + J_f(a) h for small h.

When m = n, the determinant det J_f(a) is called the Jacobian determinant. A nonzero Jacobian determinant at a point implies, by the inverse function theorem, that f is locally invertible near that point and that the local change of variables has a well-defined orientation and scale given by the determinant. If m ≠ n, the Jacobian is rectangular, and its rank describes local behavior: full rank indicates that the differential is as large as possible, while lower rank signals constraints on the local mapping.

The chain rule for Jacobians states that if g: R^m -> R^p and f: R^n -> R^m are differentiable, then J_{g∘f}(a) = J_g(f(a)) · J_f(a). For a scalar-valued function f: R^n -> R, the Jacobian is the row vector of partial derivatives, which is the transpose of the gradient: J_f(a) = ∇f(a)^T.

Applications of the Jacobian include change of variables in multiple integrals, linearization and stability analysis of dynamical systems near fixed points, and numerical methods such as Newton–Raphson that rely on the Jacobian to approximate solutions. In differential geometry, the Jacobian represents the differential of a map between manifolds. Computation is via symbolic differentiation or numerical methods such as finite differences or automatic differentiation.