Jacobianmátrix

Jacobianmátrix, commonly referred to in English as the Jacobian matrix, is the matrix of all first-order partial derivatives of a vector-valued function. For a differentiable map f: R^n -> R^m with components f1,...,fm, the Jacobian J_f(x) is the m-by-n matrix whose entry in row i and column j is ∂f_i/∂x_j, evaluated at the point x. The Jacobian represents the best linear approximation to f near x: Df_x(v) ≈ J_f(x) v for small v.

It plays a central role in multivariable calculus. When m = n, the determinant det J_f(x) indicates local invertibility at x by the inverse function theorem: if det J_f(x) ≠ 0, f is locally a diffeomorphism around x. In integration, the absolute value of det J_f(x) is used in change of variables to transform volume elements: ∫ g(y) dy = ∫ g(f(x)) |det J_f(x)| dx.

Chain rule: for g ∘ f, J_{g∘f}(x) = J_g(f(x)) J_f(x). The rank of J_f gives differential rank; singular points occur where rank is less than min(m,n).

Example: For f(x,y) = (x^2 − y, sin x + y^2), J_f = [ [2x, −1], [cos x, 2y] ].

Jacobian matrices are used in dynamical systems to linearize systems near equilibria; in differential geometry to study coordinate changes; and in applications such as computer graphics and robotics to model how motion and transformations affect coordinates. The term derives from the 19th‑century mathematician Carl Jacobi; in Hungarian mathematical terminology the matrix is referred to as the Jacobianmátrix.