微积分/极坐标微分

微分学

我们有以下公式

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

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

为了找到极坐标曲线 $r(\theta )$ 在任何给定点的切线的笛卡尔斜率，曲线首先被表示为一组参数方程。

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

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

对两个等式关于 $\theta$ 进行微分产生

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

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

用第二个等式除以第一个等式产生曲线在点 $(r,r(\theta ))$ 处的切线的笛卡尔斜率

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