Polar Coordinate

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.

Exchange between Polar coordinate and Cartesian system

Given a point $(r,\theta)$ in polar coordinates, its Cartesian equivalent is $(r\cos(\theta), r\sin(\theta))$. Conversely, a Cartesian coordinate $(x,y)$ translates to the polar coordinate $(\sqrt{x^2+y^2},\tan^{-1}(y/x))$ in polar form.

Integration in Polar Coordinates

Let $R$ be a region. Then, we have

\[\int\int_Rf(x,y)dxdy = \int\int_Rf(r\cos(\theta),r\sin(\theta))\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|drd\theta\]

where the Jacobian is

\[\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|=\det\left(\begin{matrix}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial\theta}\\\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial\theta}\end{matrix}\right)=\det\left(\begin{matrix}\cos\theta & -r\sin\theta\\\\ \sin\theta & r\cos\theta\end{matrix}\right)=r.\]

Note that the region $R$ on the right hand side is expressed in terms of the polar coordinate.

Example

\[\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\sqrt{x^2+y^2}dydx\]

This integrate might appear difficult to integrate at the first glance. However, after change to the polar coordinate, we get

\[\int_0^1\int_0^{2\pi}r^2drd\theta\]

which is very easy to integrate. Note that $R=\{(x,y)

-1\leq x\leq 1,\quad -\sqrt{1-x^2}\leq y\leq \sqrt{1-x^2}\}$ and $\{(r,\theta)

0\leq r\leq 1,\quad 0\leq \theta\leq 2\pi\}$ represent the same region.