雅可比矩阵

雅可比矩阵

$\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} \end{bmatrix}.$

$J_F(x_1,\ldots,x_n)$ ，或者 $\frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)}.$

$F(\mathbf{x}) \approx F(\mathbf{p}) + J_F(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p})$

例子

$x_1 = r \sin\theta \cos\phi \,$
$x_2 = r \sin\theta \sin\phi \,$
$x_3 = r \cos\theta \,$

$J_F(r,\theta,\phi) =\begin{bmatrix} \frac{\partial x_1}{\partial r} & \frac{\partial x_1}{\partial \theta} & \frac{\partial x_1}{\partial \phi} \\[3pt] \frac{\partial x_2}{\partial r} & \frac{\partial x_2}{\partial \theta} & \frac{\partial x_2}{\partial \phi} \\[3pt] \frac{\partial x_3}{\partial r} & \frac{\partial x_3}{\partial \theta} & \frac{\partial x_3}{\partial \phi} \\ \end{bmatrix}=\begin{bmatrix} \sin\theta \cos\phi & r \cos\theta \cos\phi & -r \sin\theta \sin\phi \\ \sin\theta \sin\phi & r \cos\theta \sin\phi & r \sin\theta \cos\phi \\ \cos\theta & -r \sin\theta & 0 \end{bmatrix}.$

R4的f函数:

$y_1 = x_1 \,$
$y_2 = 5x_3 \,$
$y_3 = 4x_2^2 - 2x_3 \,$
$y_4 = x_3 \sin(x_1) \,$

$J_F(x_1,x_2,x_3) =\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \frac{\partial y_1}{\partial x_3} \\[3pt] \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \frac{\partial y_2}{\partial x_3} \\[3pt] \frac{\partial y_3}{\partial x_1} & \frac{\partial y_3}{\partial x_2} & \frac{\partial y_3}{\partial x_3} \\[3pt] \frac{\partial y_4}{\partial x_1} & \frac{\partial y_4}{\partial x_2} & \frac{\partial y_4}{\partial x_3} \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix}.$

雅可比行列式

例子

$y_1 = 5x_2 \,$
$y_2 = 4x_1^2 - 2 \sin (x_2x_3) \,$
$y_3 = x_2 x_3 \,$

$\begin{vmatrix} 0 & 5 & 0 \\ 8x_1 & -2x_3\cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix}=-8x_1\cdot\begin{vmatrix} 5 & 0\\ x_3&x_2\end{vmatrix}=-40x_1 x_2.$