# 雅可比矩陣

## 雅可比矩陣

${\displaystyle {\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\end{bmatrix}}.}$

${\displaystyle J_{F}(x_{1},\ldots ,x_{n})}$ ，或者${\displaystyle {\frac {\partial (y_{1},\ldots ,y_{m})}{\partial (x_{1},\ldots ,x_{n})}}.}$

${\displaystyle F(x)\approx F(p)+J_{F}(p)\cdot (x-p)}$

### 例子

${\displaystyle x_{1}=r\sin \theta \cos \phi \,}$
${\displaystyle x_{2}=r\sin \theta \sin \phi \,}$
${\displaystyle x_{3}=r\cos \theta \,}$

${\displaystyle J_{F}(r,\theta ,\phi )={\begin{bmatrix}{\frac {\partial x_{1}}{\partial r}}&{\frac {\partial x_{1}}{\partial \theta }}&{\frac {\partial x_{1}}{\partial \phi }}\\[3pt]{\frac {\partial x_{2}}{\partial r}}&{\frac {\partial x_{2}}{\partial \theta }}&{\frac {\partial x_{2}}{\partial \phi }}\\[3pt]{\frac {\partial x_{3}}{\partial r}}&{\frac {\partial x_{3}}{\partial \theta }}&{\frac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \phi &r\cos \theta \cos \phi &-r\sin \theta \sin \phi \\\sin \theta \sin \phi &r\cos \theta \sin \phi &r\sin \theta \cos \phi \\\cos \theta &-r\sin \theta &0\end{bmatrix}}.}$

${\displaystyle \mathbb {R} ^{4}}$${\displaystyle F}$函數:

${\displaystyle y_{1}=x_{1}\,}$
${\displaystyle y_{2}=5x_{3}\,}$
${\displaystyle y_{3}=4x_{2}^{2}-2x_{3}\,}$
${\displaystyle y_{4}=x_{3}\sin x_{1}\,}$

${\displaystyle J_{F}(x_{1},x_{2},x_{3})={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{1}}{\partial x_{3}}}\\[3pt]{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{3}}}\\[3pt]{\frac {\partial y_{3}}{\partial x_{1}}}&{\frac {\partial y_{3}}{\partial x_{2}}}&{\frac {\partial y_{3}}{\partial x_{3}}}\\[3pt]{\frac {\partial y_{4}}{\partial x_{1}}}&{\frac {\partial y_{4}}{\partial x_{2}}}&{\frac {\partial y_{4}}{\partial x_{3}}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&5\\0&8x_{2}&-2\\x_{3}\cos x_{1}&0&\sin x_{1}\end{bmatrix}}.}$

## 雅可比行列式

### 例子

${\displaystyle y_{1}=5x_{2}\,}$
${\displaystyle y_{2}=4x_{1}^{2}-2\sin(x_{2}x_{3})\,}$
${\displaystyle y_{3}=x_{2}x_{3}\,}$

${\displaystyle {\begin{vmatrix}0&5&0\\8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}=-40x_{1}x_{2}.}$

## 逆矩陣

${\displaystyle J_{F^{-1}}\circ f=J_{F}^{-1}}$