All courses › Multivariable Calculus › Polar coordinates

Polar coordinates

In polar coordinates a point is described by its distance $r$ from the origin and angle $\theta$ . The area element gets an extra factor of $r$ , coming from the Jacobian of the change of variables.

dA = r\,dr\,d\theta

area element in polar coordinates

x = r\cos\theta,\ y = r\sin\theta

relation to Cartesian coordinates

Symbols

$r$	distance from the origin
$\theta$	angle	rad

Example

Area of a unit disk: $\displaystyle\int_0^{2\pi}\!\int_0^1 r\,dr\,d\theta = 2\pi\cdot\tfrac12 = \pi$ .

Don't forget the factor

r

in the area element - that is the most common mistake when switching to polar coordinates.

Practise multiple integrals for free →

← Double integral · Spherical coordinates →

Part of Multivariable Calculus: Multiple integrals.