合成関数の2次偏導関数

$z = f (x, y)$ で $x = φ (t), y = ψ (t)$ ならば

$\frac{d^{2} z}{d t^{2}} = f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{x y} \frac{d x}{d t} \frac{d y}{d t}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$ 　　　⇒導出

$z = f (x, y)$ で $x = φ (u, v), y = ψ (u, v)$ ならば

$\frac{\partial^{2} z}{\partial u^{2}} = f_{x x} {(\frac{\partial x}{\partial u})}^{2} + 2 f_{x y} \frac{\partial x}{\partial u} \frac{\partial y}{\partial u}$ $+ f_{y y} {(\frac{\partial y}{\partial u})}^{2}$ $+ f_{x} \frac{\partial^{2} x}{\partial u^{2}} + f_{y} \frac{\partial^{2} y}{\partial u^{2}}$ 　　　　　⇒導出
$\frac{\partial^{2} z}{\partial v^{2}} = f_{x x} {(\frac{\partial x}{\partial v})}^{2} + 2 f_{x y} \frac{\partial x}{\partial v} \frac{\partial y}{\partial v}$ $+ f_{y y} {(\frac{\partial y}{\partial v})}^{2}$ $+ f_{x} \frac{\partial^{2} x}{\partial v^{2}} + f_{y} \frac{\partial^{2} y}{\partial v^{2}}$ 　　　　　⇒導出
$\frac{\partial^{2} z}{\partial u \partial v} = \frac{\partial^{2} z}{\partial v \partial u}$ $= f_{x x} \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} + f_{x y} (\frac{\partial x}{\partial u} \frac{\partial y}{\partial v} + \frac{\partial x}{\partial v} \frac{\partial y}{\partial u})$ $+ f_{y y} \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} + f_{x} \frac{\partial^{2} x}{\partial u \partial v} + f_{y} \frac{\partial^{2} y}{\partial u \partial v}$ 　　　　　⇒導出

学生スタッフ作成
最終更新日： 2023年1月21日