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.

These concepts are named after the mathematician Carl Gustav Jacob Jacobi. The term “Jacobian” is normally pronounced /dʒəˈkoʊbiən/, but sometimes also /jəˈkoʊbiən/.

http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant