Jacobian matrix

Concept

$$[J(x_k){]}_{ij}=\left[\frac{\partial f(x_i)}{\partial x_j}\right]$$

Example

$$\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)=\left(\begin{array}{c}{e}^{x_1{x}_2}\\ e^{x_1}+e^{x_2}\end{array}\right)$$ $$J=\left(\begin{array}{cc}\frac{\partial f_1}{x_1}& \frac{\partial f_1}{x_2}\\ \frac{\partial f_2}{x_1}& \frac{\partial f_2}{x_2}\end{array}\right)=\left(\begin{array}{cc}{x}_2{e}^{x_1{x}_2}& x_1{e}^{x_1{x}_2}\\ e^{x_1}& e^{x_2}\end{array}\right)$$