Jacobian matrix

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, and let $a \in U$. Suppose that each partial derivative $\partial f_i / \partial x_j$ exists at $a$, for $1 \le i \le m$ and $1 \le j \le n$. The Jacobian matrix of $f$ at $a$ is the $m \times n$ matrix $$J_f(a) = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1}(a) & \cdots & \dfrac{\partial f_1}{\partial x_n}(a) \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1}(a) & \cdots & \dfrac{\partial f_m}{\partial x_n}(a) \end{pmatrix},$$ whose entry in row $i$ and column $j$ is $\partial f_i / \partial x_j (a)$. Other notations are $Df(a)$, $f'(a)$ and $\dfrac{\partial (f_1, \dots, f_m)}{\partial (x_1, \dots, x_n)}$.