ヤコビアン（Jacobian）

■2変数関数の場合

$3$\displaystyle{x={\varphi} $ u,v $ }$ ， $3$\displaystyle{y={\psi} $ u,v $ }$ とし， $3$\displaystyle{{\varphi},{\psi}}$ は $3$\displaystyle{u,v}$ で偏微分可能であるとする．

$3$\displaystyle{x,y}$ の全微分は

$3$\displaystyle{dx=\frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}du+\frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}dv}$

$3$\displaystyle{dy=\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}du+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}dv}$

となる．行列を用いて表すと

$3$\displaystyle{ $ \begin{array} dx & \vspace{6}\\ dy & \vspace{6}\\ \end{array} $ = $ \begin{array} \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \end{array} $ $ \begin{array} du & \vspace{6}\\ dv & \vspace{6}\\ \end{array} $ }$

となる．

行列

$3$\displaystyle{ $ \begin{array} \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \end{array} $ }$

の行列式

$3$\displaystyle{ \| \begin{array} \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\varphi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \vspace{6}\\ \end{array} \| }$

をヤコビアン（Jacobian）といい， $3$\displaystyle{\frac{\hspace{2} {\partial} $ {\varphi},{\psi} $ \hspace{2}}{\hspace{2} {\partial} $ u,v $ \hspace{2}}}$ で表す．

■3変数関数の場合

$3$\displaystyle{x={\phi} $ u,v,w $ \text{\hspace{1} },\text{\hspace{1} }y={\psi} $ u,v,w $ \text{\hspace{1} },\text{\hspace{1} }z={\omega} $ u,v,w $ \text{\hspace{1} }}$ とし， $3$\displaystyle{{\phi}\text{\hspace{1} },\text{\hspace{1} }{\psi}\text{\hspace{1} },\text{\hspace{1} }{\omega}\text{\hspace{1} }}$ は $3$\displaystyle{u\text{\hspace{1} },\text{\hspace{1} }v\text{\hspace{1} },\text{\hspace{1} }w\text{\hspace{1} }}$ で偏微分可能とする．

$3$\displaystyle{x\text{\hspace{1} },\text{\hspace{1} }y\text{\hspace{1} },\text{\hspace{1} }z\text{\hspace{1} }}$ の全微分は

$3$\displaystyle{dx=\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}du+\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}dv+\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}}dw}$

$3$\displaystyle{dy=\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}du+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}dv+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}}dw}$

$3$\displaystyle{dz=\frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}du+\frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}dv+\frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}}dw}$

となる．これを行列を用いて表すと，

$3$\displaystyle{ $\begin{array}{lll}dx& \vspace{6}\\ dy& \vspace{6}\\ dz& \vspace{6}\\ \end{array}$ = $ \begin{array} \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \end{array} $ $ \begin{array} du & \vspace{6}\\ dv & \vspace{6}\\ dw & \vspace{6}\\ \end{array} $ }$

行列

$3$\displaystyle{ $ \begin{array} \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \end{array} $ }$

の行列式

$3$\displaystyle{ \| \begin{array} \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} & \frac{\hspace{2} {\partial}{\omega} \hspace{2}}{\hspace{2} {\partial}w \hspace{2}} & \vspace{6}\\ \end{array} \| }$

をヤコビアン（Jacobian）といい， $3$\displaystyle{\frac{\hspace{2} {\partial} $ {\phi},{\psi},{\omega} $ \hspace{2}}{\hspace{2} {\partial} $ u,v,w $ \hspace{2}}\text{\hspace{1} }}$ で表す．

■一般に

n変数関数の場合，行列を用いて表すと，

$3$\displaystyle{ $ \begin{array} \hspace{1}{f}_{1} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{f}_{n} & \vspace{6}\\ \end{array} $ = $ \begin{array} \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \vdots &\ddots &\vdots & \vspace{6}\\ \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \end{array} $ $ \begin{array} d\hspace{1}{x}_{1} & \vspace{6}\\ \vdots & \vspace{6}\\ d\hspace{1}{x}_{n} & \vspace{6}\\ \end{array} $ }$

行列

$3$\displaystyle{ $ \begin{array} \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \vdots &\ddots &\vdots & \vspace{6}\\ \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \end{array} $ }$

の行列式

$3$\displaystyle{ \| \begin{array} \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{1} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \vdots &\ddots &\vdots & \vspace{6}\\ \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{1} \hspace{2}} &\cdots & \frac{\hspace{2} {\partial}\hspace{1}{f}_{n} \hspace{2}}{\hspace{2} {\partial}\hspace{1}{x}_{n} \hspace{2}} & \vspace{6}\\ \end{array} \| }$

をヤコビアン（Jacobian）といい， $3$\displaystyle{\frac{\hspace{2} {\partial} $ \hspace{1}{f}_{1}\cdots ,\hspace{1}{f}_{n} $ \hspace{2}}{\hspace{2} {\partial} $ \hspace{1}{x}_{1}\cdots ,\hspace{1}{x}_{n} $ \hspace{2}}\text{\hspace{1} }}$ で表す．

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最終更新日： 2016年12月14日