Polar Coordinates

Summary

Polar coordinates $(r,\theta )$ describe a plane point by distance from the origin and angle from the positive $x$ -axis. They simplify circles, spirals, and many double integrals.

Prerequisites

Prerequisites: trigonometry and the Cartesian plane.

Formula

Polar to Cartesian

Cartesian to polar

The two-argument arctangent fixes the quadrant; $\arctan (y/x)$ alone is insufficient when $x\le 0$ .

Conditions / Assumptions

• Common ranges: $r\ge 0$ and $\theta \in [0,2\pi )$ or $(-\pi ,\pi ]$ .
• Non-uniqueness: $(r,\theta )$ and $(r,\theta +2\pi k)$ represent the same ray; with signed radius, $(-r,\theta +\pi )$ is the same point as $(r,\theta )$ .

Worked Example

$(r,\theta )=(2,\pi /3)$ gives $x=2\cos (\pi /3)=1$ , $y=2\sin (\pi /3)=\sqrt{3}$ .

For area integrals, $dA=rdrd\theta$ (see Double Integrals in Polar Coordinates).

Common Mistakes

• Forgetting quadrant adjustments for $\theta$ .
• Using $dA=drd\theta$ without the Jacobian factor $r$ .

Connections

• Double integrals: Double Integrals in Polar Coordinates
• 3D extensions: Cylindrical Coordinates, Spherical Coordinates

References

Polar coordinates are reviewed in OpenStax Calculus Volume 2/3 as needed for integration.^[1]