# ヤコビアン（Jacobian）

## ■2変数関数の場合

$x=\phi \left(u,v\right)$$y=\psi \left(u,v\right)$ とし，$\phi ,\psi$$u,v$  で偏微分可能であるとする．

$x,y$  の全微分

$dx=\frac{\partial \phi }{\partial u}du+\frac{\partial \phi }{\partial v}dv$

$dy=\frac{\partial \psi }{\partial u}du+\frac{\partial \psi }{\partial v}dv$

となる．  行列を用いて表すと

$\left(\begin{array}{c}dx\\ dy\end{array}\right)=\left(\begin{array}{cc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}\end{array}\right)\left(\begin{array}{c}du\\ dv\end{array}\right)$

となる．

$\left(\begin{array}{cc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}\end{array}\right)$

$|\begin{array}{cc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}\end{array}|$

ヤコビアン（Jacobian）といい， $\frac{\partial \left(\phi ,\psi \right)}{\partial \left(u,v\right)}$  で表す．

## ■3変数関数の場合

$x=\phi \left(u,v,w\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}y=\psi \left(u,v,w\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}z=\omega \left(u,v,w\right)\text{\hspace{0.17em}}$ とし， $\phi \text{\hspace{0.17em}},\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}},\text{\hspace{0.17em}}\omega \text{\hspace{0.17em}}$$u\text{\hspace{0.17em}},\text{\hspace{0.17em}}v\text{\hspace{0.17em}},\text{\hspace{0.17em}}w\text{\hspace{0.17em}}$偏微分可能とする．

$x\text{\hspace{0.17em}},\text{\hspace{0.17em}}y\text{\hspace{0.17em}},\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$全微分

$dx=\frac{\partial \phi }{\partial u}du+\frac{\partial \phi }{\partial v}dv+\frac{\partial \phi }{\partial w}dw$

$dy=\frac{\partial \psi }{\partial u}du+\frac{\partial \psi }{\partial v}dv+\frac{\partial \psi }{\partial w}dw$

$dz=\frac{\partial \omega }{\partial u}du+\frac{\partial \omega }{\partial v}dv+\frac{\partial \omega }{\partial w}dw$

となる．  これを行列を用いて表すと，

$\left(\begin{array}{l}dx\\ dy\\ dz\end{array}\right)=\left(\begin{array}{ccc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}& \frac{\partial \phi }{\partial w}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}& \frac{\partial \psi }{\partial w}\\ \frac{\partial \omega }{\partial u}& \frac{\partial \omega }{\partial v}& \frac{\partial \omega }{\partial w}\end{array}\right)\left(\begin{array}{c}du\\ dv\\ dw\end{array}\right)$

$\left(\begin{array}{ccc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}& \frac{\partial \phi }{\partial w}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}& \frac{\partial \psi }{\partial w}\\ \frac{\partial \omega }{\partial u}& \frac{\partial \omega }{\partial v}& \frac{\partial \omega }{\partial w}\end{array}\right)$

$|\begin{array}{ccc}\frac{\partial \phi }{\partial u}& \frac{\partial \phi }{\partial v}& \frac{\partial \phi }{\partial w}\\ \frac{\partial \psi }{\partial u}& \frac{\partial \psi }{\partial v}& \frac{\partial \psi }{\partial w}\\ \frac{\partial \omega }{\partial u}& \frac{\partial \omega }{\partial v}& \frac{\partial \omega }{\partial w}\end{array}|$

ヤコビアン（Jacobian）といい， $\frac{\partial \left(\phi ,\psi ,\omega \right)}{\partial \left(u,v,w\right)}\text{\hspace{0.17em}}$ で表す．

## ■一般に

n変数関数の場合，行列を用いて表すと，

$\left(\begin{array}{c}{f}_{1}\\ ⋮\\ {f}_{n}\end{array}\right)=\left(\begin{array}{ccc}\frac{\partial {f}_{1}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{1}}{\partial {x}_{n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial {f}_{n}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{n}}{\partial {x}_{n}}\end{array}\right)\left(\begin{array}{c}d{x}_{1}\\ ⋮\\ d{x}_{n}\end{array}\right)$

$\left(\begin{array}{ccc}\frac{\partial {f}_{1}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{1}}{\partial {x}_{n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial {f}_{n}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{n}}{\partial {x}_{n}}\end{array}\right)$

$|\begin{array}{ccc}\frac{\partial {f}_{1}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{1}}{\partial {x}_{n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial {f}_{n}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{n}}{\partial {x}_{n}}\end{array}|$

ヤコビアン（Jacobian）といい， $\frac{\partial \left({f}_{1}\cdots ,{f}_{n}\right)}{\partial \left({x}_{1}\cdots ,{x}_{n}\right)}\text{\hspace{0.17em}}$ で表す．

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