Jacobimatrisen

Jacobimatrisen, commonly referred to as the Jacobian matrix in English, is a mathematical construct used to describe the best linear approximation of a vector-valued function at a given point. It captures how small input perturbations propagate to outputs, locally describing a transformation.

For a differentiable map f: R^n → R^m, the Jacobimatrisen at x is the m×n matrix J_f(x) with entries ∂f_i/∂x_j, arranged with i indexing outputs and j inputs. Each row corresponds to one component of f and each column to one input variable.

When m = n, the determinant det J_f(x) is the Jacobian determinant, measuring local volume scaling and orientation under the transformation. A nonzero determinant implies local invertibility by the inverse function theorem, and the absolute value gives how volumes are scaled to first order.

The Jacobimatrisen is also central to the chain rule: J_{g∘f}(x) = J_g(f(x)) · J_f(x). This property enables change of variables in integrals and the analysis of composite maps, including coordinate transformations in higher dimensions.

Example: f: R^2 → R^2, f(x,y) = (x^2, x y). Then J_f(x,y) = [[2x, 0], [y, x]]. At (1,2) this becomes [[2, 0], [2, 1]].

Applications span multivariable calculus, differential geometry, dynamical systems, optimization, robotics, computer graphics, and numerical analysis. Etymology: The matrix is named after Carl Gustav Jacob Jacobi; in Nordic languages the term Jacobimatrisen is commonly used to refer to this construct. See also Jacobian determinant, Jacobian matrix.