Reasons Not to Use r, Theta as Variables in Cylindrical Coordinates
Notation of Cylindrical Coordinates
The cylindrical coordinate system represents points in 3D space as shown in $(\rho, \phi, z)$. Here,
- $\rho$: the magnitude of the vector projection of the position vector $\mathbf{r}$ onto the $xy$-plane
- $\phi$: the angle between the projected vector and the $x$ axis
- $z$: the magnitude of the vector projection of the position vector $\mathbf{r}$ onto the $z$ axis
However, the notation $(r,\theta, z)$ is also often used for cylindrical coordinates. This is probably because it simplifies the thought that cylindrical coordinates are an extension of polar coordinates $(r,\theta)$ with the addition of height $z$. Some might say it’s fine as long as the use of the notation conveys the idea of cylindrical coordinates, but on the contrary, using it this way undermines the consistency in the meaning of each symbol. In spatial coordinate systems, the variable $r$ signifies the linear distance from the origin to the said coordinate. Hence, the position vector is denoted as
$$ \mathbf{r}=r \hat{\mathbf{r}} $$
Comparison with Polar and Spherical Coordinates
Polar coordinates $(r, \theta)$ and spherical coordinates $(r, \theta, \phi)$, as shown in the figure, indeed represent $r$ as the linear distance between the origin and the coordinate. However, when using the notation $(r, \theta, z)$ for cylindrical coordinates, $r$ no longer represents the distance from the origin.
The first variable in the cylindrical coordinate system is the length of the vector obtained by projecting the position vector $\mathbf{r}$ onto the $xy-$ plane. Therefore, it should be denoted as $\rho$ or $s$ (anything but $r$). Likewise, $\theta$ is the angle between the position vector $\mathbf{r}$ and the coordinate axis. Moreover, the second variable is not the angle between the position vector and the coordinate axis but the angle between the projected vector onto the $xy-$ plane and the coordinate axis. Thus, $\theta$ should be denoted as $\phi$. For these reasons, cylindrical coordinates should be represented as $(\rho, \phi, z)$ to maintain consistency in the meaning of each symbol even when the coordinate system is changed. Even in the case where spherical coordinates are denoted as $(r, \phi, \theta)$, the meanings of $\phi$ and $\theta$ themselves likely remain unchanged, only the order of notation might have been switched.