Coordinate Systems

coordinate_systems

## Cylindrical
![Imgur](https://i.imgur.com/hMRNeAw.png)

Essentially extends polar to include a $z$-component.
$(r,\theta,z)$

$$
r = x\cos\theta \\\
r = y\sin\theta \\\
z = z
$$

$$
r^2 = x^2 + y^2 \\\
\theta = \tan^{-1}\left(\frac{y}{x}\right)
$$

## Spherical
Uses a radius $\rho$ and two angles $\theta$ and $\phi$.

$$
\rho = \sqrt{x^2 + y^2 + z^2} \\\
x = \rho\sin\phi\cos\theta \\\
y = \rho\sin\phi\sin\theta
$$

$$
\sin\phi = \frac{r}{\rho} \implies r = \rho\sin\phi \\\
\cos\phi = \frac{z}{\rho} \\\
\tan\phi = \frac{r}{z} \\\
r^2 = x^2 + y^2
$$

![Imgur](https://i.imgur.com/1wJafWq.png)
![Imgur](https://i.imgur.com/6jCV3XL.png)