# 雅可比矩阵

（重定向自雅可比行列式

## 雅可比矩阵

${\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}}$

${\displaystyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}}$

${\displaystyle Df}$ ${\displaystyle \mathrm {D} \mathbf {f} }$ ${\displaystyle \mathbf {J} _{\mathbf {f} }(x_{1},\ldots ,x_{n})}$  或者 ${\displaystyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}}$

${\displaystyle f(x)\approx f(p)+\mathbf {J} _{\mathbf {f} }(p)\cdot (x-p)}$

${\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {f} (\mathbf {p} )+\mathbf {J} _{\mathbf {f} }(\mathbf {p} )(\mathbf {x} -\mathbf {p} )+o(\|\mathbf {x} -\mathbf {p} \|)}$

## 例子

### 例一

{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi ;\\y&=r\sin \theta \sin \varphi ;\\z&=r\cos \theta \end{aligned}}}

${\displaystyle \mathbf {J} _{\mathbf {F} }(r,\theta ,\varphi )={\begin{bmatrix}{\dfrac {\partial x}{\partial r}}&{\dfrac {\partial x}{\partial \theta }}&{\dfrac {\partial x}{\partial \varphi }}\\[1em]{\dfrac {\partial y}{\partial r}}&{\dfrac {\partial y}{\partial \theta }}&{\dfrac {\partial y}{\partial \varphi }}\\[1em]{\dfrac {\partial z}{\partial r}}&{\dfrac {\partial z}{\partial \theta }}&{\dfrac {\partial z}{\partial \varphi }}\end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \varphi &r\cos \theta \cos \varphi &-r\sin \theta \sin \varphi \\\sin \theta \sin \varphi &r\cos \theta \sin \varphi &r\sin \theta \cos \varphi \\\cos \theta &-r\sin \theta &0\end{bmatrix}}}$

### 例二

F : ℝ3 → ℝ4，其各分量為

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

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

## 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

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