Jacobian matrix and determinant
![A nonlinear map f : R2 → R2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.](/Images/godic/202501/21/Jacobian_determinant_and_distortion.svg0457.png")
In vector calculus, the Jacobian matrix (/dʒᵻˈkoʊbiən/, /jᵻˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.