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:

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