Jacobimaatriks

Jacobimaatriks, also known as the Jacobian matrix in many contexts, is the matrix of all first-order partial derivatives of a vector-valued function. For a differentiable function f: R^n → R^m with components f_i(x_1,...,x_n), the Jacobimaatriks J_f(x) has entries J_f(x)_{i,j} = ∂f_i/∂x_j. It is an m-by-n matrix that describes the linear approximation of f at x and maps small changes Δx in the domain to changes Δf ≈ J_f(x) Δx in the codomain.

Example: If f: R^2 → R^2, f(x,y) = (x^2 + y, sin x + y^2), then J_f(x,y) = [[2x, 1], [cos x, 2y]].

Properties and applications: J_f depends on the point x. When m = n and det J_f(x) ≠ 0, the inverse function theorem guarantees a local inverse and a local diffeomorphism. The Jacobimaatriks determinant is called the Jacobian determinant. In dynamical systems, the Jacobimaatriks of the vector field at a fixed point informs stability via eigenvalues. In numerical methods, J_f is used in Newton-type methods for systems of equations, and in machine learning and automatic differentiation to propagate gradients.

Origin and terminology: The concept is named after Carl Gustav Jacob Jacobi, a 19th-century mathematician. The term Jacobimaatriks appears in some languages as a direct translation or adaptation of "Jacobian matrix."

See also: Jacobian determinant; inverse function theorem; sensitivity analysis.