Cylindrical coordinates

Definition

The cylindrical coordinate map is $C \colon [0, \infty) \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}^3$, $$C(r, \theta, z) = (r \cos \theta, \ r \sin \theta, \ z).$$ A triple $(r, \theta, z)$ with $C(r, \theta, z) = (x, y, z)$ is a triple of cylindrical coordinates of $(x, y, z)$. Here $r = \sqrt{x^2 + y^2}$ is the distance to the $z$-axis, $\theta$ is the azimuth, and $z$ is the height.

Remarks

The pair $(r, \theta)$ is a pair of polar coordinates of $(x, y)$. So, away from the $z$-axis, the coordinates are unique if $r > 0$ and $\theta \in [0, 2\pi)$. The Jacobian determinant of $C$ is $r$, so the volume element is $dV = r \, dr \, d\theta \, dz$. Example: the solid cylinder $x^2 + y^2 \le a^2$, $0 \le z \le h$ has volume $\int_0^h \int_0^{2\pi} \int_0^a r \, dr \, d\theta \, dz = \pi a^2 h$. Some texts write $\rho$ or $s$ for $r$, and $\varphi$ for $\theta$.

Explanations

L1 human

Cylindrical coordinates are polar coordinates in the floor plan, together with the height. Surfaces $r =$ constant are cylinders around the $z$-axis, which gives the coordinates their name.