Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector– or scalar-valued function with respect to another vector. Suppose F :Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1(x1,…,xn), …, ym(x1,…,xn). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix J of F, as follows:
This matrix is also denoted by and . If (x1,…,xn) are the usual orthogonal Cartesian coordinates, the i th row (i = 1, …, m) of this matrix corresponds to the gradient of the ith component function yi: . Note that some books define the Jacobian as the transpose of the matrix given above.
The Jacobian determinant (often simply called the Jacobian) is the determinant of the Jacobian matrix.