Jacobian Matrix

Let $\boldsymbol{f}:X\mapsto \mathbb{R}^m$ be a function where $\boldsymbol{f}=(f_1,f_2,\dots,f_m)$ and $X\subset \mathbb{R}^n$ is open.

We define Jacobian Matrix of $\boldsymbol{f}$ is the $m\times n$ matrix given by

$$D\boldsymbol{f}=\Big[ \frac{\partial f_i}{\partial x_j} \Big]_{ij}$$