合成関数の偏導関数

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

$\frac{d z}{d t} = f_{x} \frac{d x}{d t} + f_{y} \frac{d y}{d t}$ $\frac{d z}{d t} = f_{x} \frac{d x}{d t} + f_{y} \frac{d y}{d t}$ 　　　⇒導出
$z = f (x, y)$ $z = f (x, y)$ で $x = φ (u, v), y = ψ (u, v)$ $x = φ (u, v), y = ψ (u, v)$ ならば
$\frac{\partial z}{\partial u} = f_{x} \frac{\partial x}{\partial u} + f_{y} \frac{\partial y}{\partial u}$ $\frac{\partial z}{\partial u} = f_{x} \frac{\partial x}{\partial u} + f_{y} \frac{\partial y}{\partial u}$ 　　　　　⇒導出
$\frac{\partial z}{\partial v} = f_{x} \frac{\partial x}{\partial v} + f_{y} \frac{\partial y}{\partial v}$ $\frac{\partial z}{\partial v} = f_{x} \frac{\partial x}{\partial v} + f_{y} \frac{\partial y}{\partial v}$ 　　　　　⇒導出
$z = f (u)$ $z = f (u)$ ， $u = g (x, y)$ $u = g (x, y)$ ならば

$\frac{\partial z}{\partial x} = \frac{d z}{d u} \frac{\partial u}{\partial x}$ $\frac{\partial z}{\partial x} = \frac{d z}{d u} \frac{\partial u}{\partial x}$ 　　　　　⇒導出

$\frac{\partial z}{\partial y} = \frac{d z}{d u} \frac{\partial u}{\partial y}$ $\frac{\partial z}{\partial y} = \frac{d z}{d u} \frac{\partial u}{\partial y}$

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最終更新日： 2024年5月15日