Parametric Equation

$$x=f(t)y=g(t)$$

• parameter $t$
• parametric curve $(x,y)$
  • initial point/ terminal point

derivative

$$\begin{gathered} \frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt} \\ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\left(\frac{dx}{dt}\ne 0\right) \end{gathered}$$

arc length

$$\begin{gathered} L={\int }_{\alpha }^{\beta }ds \\ ={\int }_{\alpha }^{\beta }\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt \end{gathered}$$

polar coordinates $(r,\theta )$

• pole
• polar axis

transformation between Cartesian coordinates and polar coordinates

$$\begin{gathered} x=r\cos \theta y=r\sin \theta \\ {r}^2=x^2+y^2\tan \theta =\frac{y}{x} \end{gathered}$$

polar curve

graph of polar equation

$$F(r,\theta )=0$$

area

$$A={\int }_a^b\frac{1}{2}{r}^{2}d\theta$$

arc length in polar coordinates

$$L={\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta$$

Last edited: 2024-09-13