Jacobian Matrix | Sahithyan's Notes

For $F : \mathbb{R}^n \to \mathbb{R}^n$ with coordinate functions $f_1, f_2, \ldots, f_n$ , the Jacobian matrix $J(\vec{x})$ is the $n \times n$ matrix of first partial derivatives

J(\vec{x}) = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\[6pt] \dfrac{\partial f_2}{\partial x_1} & \dfrac{\partial f_2}{\partial x_2} & \cdots & \dfrac{\partial f_2}{\partial x_n} \\[6pt] \vdots & \vdots & \ddots & \vdots \\[6pt] \dfrac{\partial f_n}{\partial x_1} & \dfrac{\partial f_n}{\partial x_2} & \cdots & \dfrac{\partial f_n}{\partial x_n} \end{pmatrix}

Row $i$ of $J(\vec{x})$ is the gradient of $f_i$ .

Example

\begin{aligned} f_1(\vec{x}) &= 3x_1 - \cos(x_2 x_3) - \tfrac{1}{2} \\ f_2(\vec{x}) &= x_1^2 - 81(x_2 + 0.1)^2 + \sin x_3 + 1.06 \\ f_3(\vec{x}) &= e^{-x_1 x_2} + 20x_3 + \tfrac{10\pi - 3}{3} \end{aligned}

the partial derivatives give

J(\vec{x}) = \begin{pmatrix} 3 & x_3 \sin(x_2 x_3) & x_2 \sin(x_2 x_3) \\ 2x_1 & -162(x_2 + 0.1) & \cos x_3 \\ -x_2 e^{-x_1 x_2} & -x_1 e^{-x_1 x_2} & 20 \end{pmatrix}