Cylindrical Coordinates

Summary

Cylindrical coordinates $(r, θ, z)$ extend polar coordinates by a vertical $z$ . They suit problems with axial symmetry about the $z$ -axis. The volume element is $r d r d θ d z$ .

Prerequisites

Polar Coordinates

Definition

$r = \sqrt{x^{2} + y^{2}} \geq 0$ : distance from the $z$ -axis.
$θ$ : angle in the $x y$ -plane from the positive $x$ -axis, typically $[0, 2 π)$ or $(- π, π]$ .
$z$ : same as Cartesian $z$ .

Cartesian relations:

x = r \cos θ, y = r \sin θ, z = z .

Inverse Formulas

r = \sqrt{x^{2} + y^{2}}, θ = atan 2 (y, x), z = z .

Use $atan 2 (y, x)$ (two-argument arctangent), not $\arctan (y / x)$ alone, so the quadrant is correct when $x \leq 0$ .

Formula

Volume element:

d V = r d r d θ d z .

Gradient in orthonormal cylindrical basis (for reference):

\nabla f = \frac{\partial f}{\partial r} e_{r} + \frac{1}{r} \frac{\partial f}{\partial θ} e_{θ} + \frac{\partial f}{\partial z} e_{z} (r > 0) .

Conditions / Assumptions

$r = 0$ is the entire $z$ -axis; $θ$ is undefined there.
Jacobian factor $r$ is required for integrals.

Worked Example

Cartesian point $(2 \sqrt{3}, 6, 4)$ :

r = \sqrt{12 + 36} = \sqrt{48} = 4 \sqrt{3}, θ = atan 2 (6, 2 \sqrt{3}) = π / 3, z = 4.

So $(r, θ, z) = (4 \sqrt{3}, π / 3, 4)$ .

Common Mistakes

Using $θ = \arctan (y / x)$ without quadrant correction.
Omitting the factor $r$ in triple integrals.
Allowing negative $r$ without adjusting $θ$ by $π$ .

Connections

Polar Coordinates, Spherical Coordinates, Triple Integrals

References

Cylindrical coordinates appear in OpenStax Calculus Volume 3.^[1]

OpenStax, Calculus Volume 3, Section 5.5, https://openstax.org/details/books/calculus-volume-3 ↩︎