合成関数の2次偏導関数

$3$\displaystyle{ z=f $ x,y $ }$ で $3$\displaystyle{x={\varphi} $t$ ,y={\psi} $t$ }$ ならば

$3$\displaystyle{ \frac{\hspace{2} {d}^{2}z \hspace{2}}{\hspace{2} d{t}^{2} \hspace{2}}=\hspace{1}{f}_{ xx }{\left({\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}}\right)}^{2}+ 2\hspace{1}{f}_{ xy }\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}} }$ $3$\displaystyle{ + \hspace{1}{f}_{ yy }{\left({\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}}\right)}^{2} }$ $3$\displaystyle{ + \hspace{1}{f}_{x}\frac{\hspace{2} {d}^{2}x \hspace{2}}{\hspace{2} d{t}^{2} \hspace{2}}+ \hspace{1}{f}_{y}\frac{\hspace{2} {d}^{2}y \hspace{2}}{\hspace{2} d{t}^{2} \hspace{2}} }$ 　　　⇒導出

$3$\displaystyle{z=f $ x,y $ }$ で $3$\displaystyle{x={\varphi} $ u,v $ ,y={\psi} $ u,v $ }$ ならば

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}=\hspace{1}{f}_{ xx }{ \left({ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }\right) }^{2}+2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }$ $3$\displaystyle{ +\hspace{1}{f}_{ yy }{ \left({ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }\right) }^{2} }$ $3$\displaystyle{ + \hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}+ \hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}} }$ 　　　　　⇒導出
$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{v}^{2} \hspace{2}}=\hspace{1}{f}_{ xx }{\left({\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}}\right)}^{2}+ 2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} }$ $3$\displaystyle{ + \hspace{1}{f}_{ yy }{\left({\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}}\right)}^{2} }$ $3$\displaystyle{ + \hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{v}^{2} \hspace{2}}+ \hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{v}^{2} \hspace{2}} }$ 　　　　　⇒導出
$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}u{\partial}v \hspace{2}}=\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}v{\partial}u \hspace{2}} }$ $3$\displaystyle{ =\hspace{1}{f}_{ xx }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}+\hspace{1}{f}_{ xy }\left({ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}+\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }\right) }$ $3$\displaystyle{ +\hspace{1}{f}_{ yy }\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}u{\partial}v \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}u{\partial}v \hspace{2}} }$ 　　　　　⇒導出

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最終更新日： 2025年4月25日