Cylindrical Coordinates

Cylindrical coordinates are simply another way to give a point an address: how far out from a central pole it is, which compass direction, and how high up. They're named after the cylinder because the simplest equation, "stay a fixed distance from the pole," draws one. The formulas $x = r\cos\theta$, $y = r\sin\theta$, $z = z$ look like a set of three, but the first two are polar coordinates from Calculus II, with $r$ (distance from the $z$-axis) and $\theta$ (angle around it), and the third does nothing at all. That's the whole system; it's less new than it looks.

Why bother with another coordinate system? Because the right coordinates can make a problem easy. Describing a soup can with $x$ and $y$ means writing $x^2 + y^2 \le 4$, while in cylindrical coordinates it's just $r \le 2$. When something is round around a vertical axis, $r$ and $\theta$ match its shape. This will matter a lot when we get to triple integrals.

Simple shapes:

• $r = c$: a cylinder of radius $c$ around the $z$-axis. Every point is the same distance from the axis, at any angle and any height.
• $\theta = c$: a half-plane hinged on the $z$-axis, like a door that's been opened to angle $c$.
• $z = c$: a horizontal plane.
• $z = r^2$: the paraboloid $z = x^2 + y^2$, since $r^2 = x^2 + y^2$. And $z = r$ is a cone: the height grows evenly with the distance out.

Converting works exactly like polar coordinates, with $z$ along for the ride.

A common mistake. It's tempting to think $r$ is the distance from the origin, since that's what it meant in the plane. In 3D it isn't: $r$ is the distance from the $z$-axis. A point high up the $z$-axis has $r = 0$ even though it's far from the origin.

Pipes, wires, drill bits and rotating machinery are naturally cylindrical; heat flow in a pipe wall and the magnetic field around a wire are written in these coordinates.