Jacobian矩陣

Jacobian矩陣是函式對向量求導，其結果是一階偏導陣列成的矩陣。假設:$F:R_n \to R_m$也就是一個n維歐式空間向m維歐式空間的一個對映。

舉例：

1. 由球座標系轉換到直角座標系，存在對映形式化表示如下：

$R \times [0, \pi] \times [0, 2\pi] \to R^3$

原始座標： $x_1 = rcos\theta sin\phi$ $x_2 = rsin\theta cos\phi$

$J_F(x_1, x_2, x_3)=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3}\\ \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3}\\ \dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3}\\ \dfrac{\partial y_4}{\partial x_1} & \dfrac{\partial y_4}{\partial x_2} & \dfrac{\partial y_4}{\partial x_3} \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & bx_2 & -2 \\ x_3cosx_1 & 0 & sinx_1 \end{bmatrix}$