How to write jacobian symbol in Latex ? The jacobian matrix of vector-valued function f is the matrix of all its first-order partial derivatives.

Write Latex jacobian symbol

You can use \mathbb function:

$$\mathbb{J}$$
\[\mathbb{J}\]

Definition of Jacobian matrix in Latex

Suppose $\mathbf{f}:\mathbb{R}^n \rightarrow \mathbb{R}^m$, defined for all $\mathbf{x}=(x_1,x_2,\dots,x_n)$ by

$$
\begin{aligned}
\mathbf{f}(\mathbf{x})&=\mathbf{f}(x_1,x_2,\dots,x_n)\\
&=(f_1(x_1,x_2,\dots,x_n),,\dots,f_m(x_1,x_2,\dots,x_n))\\
&=(f_1(\mathbf{x}),,\dots,f_m(\mathbf{x}))
\end{aligned}
$$
\[\begin{aligned} \mathbf{f}(\mathbf{x})&=\mathbf{f}(x_1,x_2,\dots,x_n)\\ &=(f_1(x_1,x_2,\dots,x_n),,\dots,f_m(x_1,x_2,\dots,x_n))\\ &=(f_1(\mathbf{x}),,\dots,f_m(\mathbf{x})) \end{aligned}\] 
$$
\mathbb{J}=\left[\begin{array}{ccc}
\dfrac{\partial \mathbf{f}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial \mathbf{f}(\mathbf{x})}{\partial x_{n}}
\end{array}\right]=\left[\begin{array}{c}
\nabla^{T} f_{1}(\mathbf{x}) \\
\vdots \\
\nabla^{T} f_{m}(\mathbf{x})
\end{array}\right]=\left[\begin{array}{ccc}
\dfrac{\partial f_{1}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial f_{1}(\mathbf{x})}{\partial x_{n}} \\
\vdots & \ddots & \vdots \\
\dfrac{\partial f_{m}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial f_{m}(\mathbf{x})}{\partial x_{n}}
\end{array}\right]
$$
\[\mathbb{J}=\left[\begin{array}{ccc} \dfrac{\partial \mathbf{f}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial \mathbf{f}(\mathbf{x})}{\partial x_{n}} \end{array}\right]=\left[\begin{array}{c} \nabla^{T} f_{1}(\mathbf{x}) \\ \vdots \\ \nabla^{T} f_{m}(\mathbf{x}) \end{array}\right]=\left[\begin{array}{ccc} \dfrac{\partial f_{1}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial f_{1}(\mathbf{x})}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_{m}(\mathbf{x})}{\partial x_{1}} & \cdots & \dfrac{\partial f_{m}(\mathbf{x})}{\partial x_{n}} \end{array}\right]\]

where: $\nabla^{T} f_{i}$ is the transpose of the gradient of the $i$-th component.

Then we have:

$$\mathbb{J}_{i,j}=\dfrac{\partial f_{i}(\mathbf{x})}{\partial x_{j}}$$
\[\mathbb{J}_{i,j}=\dfrac{\partial f_{i}(\mathbf{x})}{\partial x_{j}}\]