Definition:Jacobian

Definition
Let $U$ be an open subset of $\R^n$.

Let $f = (f_1,\ldots,f_m)^T: U \to \R^m$ be a vector valued function, differentiable at $x = (x_1,\ldots,x_n) \in U$.

The Jacobian of $f$ at $x$ is defined to be the matrix of partial derivatives:
 * $\displaystyle J_f =

\begin{pmatrix} \frac{\partial f_1}{\partial x_1}(x) & \cdots & \frac{\partial f_1}{\partial x_n}(x) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(x) & \cdots & \frac{\partial f_m}{\partial x_n}(x) \end{pmatrix}$