Jacobian matrix and determinant

Jacobian matrix and determinant

From Wikipedia, the free encyclopedia

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:

J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n}  \end{bmatrix}.

This matrix is also denoted by J_F(x_1,\ldots,x_n) and \frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)}. 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\left(\nabla y_i\right). 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/.



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