微積分/極座標微分

微分學

我們有以下公式

$r{\frac {\partial }{\partial r}}=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}$

${\frac {\partial }{\partial \theta }}=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}$

為了找到極座標曲線 $r(\theta )$ 在任何給定點的切線的笛卡爾斜率，曲線首先被表示為一組引數方程。

$x=r(\theta )\cos(\theta )$

$y=r(\theta )\sin(\theta )$

對兩個等式關於 $\theta$ 進行微分產生

${\frac {\partial x}{\partial \theta }}=r'(\theta )\cos(\theta )-r(\theta )\sin(\theta )$

${\frac {\partial y}{\partial \theta }}=r'(\theta )\sin(\theta )+r(\theta )\cos(\theta )$

用第二個等式除以第一個等式產生曲線在點 $(r,r(\theta ))$ 處的切線的笛卡爾斜率

${\frac {dy}{dx}}={\frac {r'(\theta )\sin(\theta )+r(\theta )\cos(\theta )}{r'(\theta )\cos(\theta )-r(\theta )\sin(\theta )}}$