Assume $m$ number of different multivariate functions as  $f_1(x),f_2(x),\dots, f_m(x)$, where $x=\begin{bmatrix} x_1\\ \vdots\\x_n \end{bmatrix}$. If we collect all the $f_i(x)$ functions in a column vector $F(x)$ as

$$F(x)=\begin{bmatrix} f_1(x)\\ \vdots\\f_m(x) \end{bmatrix},$$

$F(x)$ is itself a $m$-vector function of an $n$-vector variable $x$. For such a vector multivariate function, the $m\times n$ matrix of partial derivatives $\partial F_{i}/ \partial x_{j}$, denoted as $J_x$, is called  Jacobian matrix of $F$ defined as:

$$J_x=\partial_{x^{T}} F =\left[\begin{array}{ccc}\partial F_{1} / \partial x_{1} & \ldots & \partial F_{1}/ \partial x_{n} \\\partial F_{2} / \partial x_{1} & \ldots & \partial F_{2}/ \partial x_{n} \\ \vdots & & \vdots \\ \partial F_{m} / \partial x_{1} & \ldots & \partial F_{m}/ \partial x_{n}\end{array}\right].$$

For example, let $x=\begin{bmatrix} x_1\\ x_2\\x_3 \end{bmatrix}$ and  $F(x)=\begin{bmatrix} x_1+x_2+x_3 \\ x_1^2+2x_2 \\ x_1^2+x_2^3+\cos(x_3)\end{bmatrix}$. Then the Jacobian matrix of $F(x)$ is derived as:

$$J_x=\partial_{x^{T}} F =\left[\begin{array}{ccc}  1& 1& 1 \\ 2x_1&2 &0  \\ 2x_1 & 3x_2^2& -\sin(x_3) \end{array}\right].$$