Polar Coordinates

Polar coordinates are a type of 2D coordinate system characterised by the use of angles and radii. They are useful when the problem features arcs or has a circular symmetry.

Definition §

The polar coordinate system features is written as:

$$(r,\theta )$$

• $r$ : The radius, must be a non-zero number. $r\ge 0$
• $\theta$: The angle starting from the $x$-axis, going anti-clockwise. $\theta \in [0,2\pi )$

Conversions §

Conversion With Cartesian §

Every point in Cartesian coordinates $(x,y)$ can be converted to polar coordinates via the equation:

$$x=r\cos (\theta )y=r\sin (\theta )$$

And vice vera:

$$r=\Vert (x,y)\Vert =\sqrt{x^2+y^2}\theta =\{\begin{array}{ll}\arctan (\frac{y}{x}),& x>0,y>0(\text{Q1})\\ \pi +\arctan (\frac{y}{x}),& x<0,y>0(\text{Q2})\\ \pi -\arctan (\frac{y}{x}),& x>0,y<0(\text{Q3})\\ 2\pi -\arctan (\frac{y}{x}),& x<0,y<0(\text{Q4})\end{array}$$

Key Features §

• If $r=c$ (radius is constant), then we have a circle of radius $r$
• If $\theta =c$ (angle is constant), then we have a ray that starts from the origin

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Jacobian §

Cartesian Jacobian §

The Jacobian when converting from cartesian to polar is given by:

$$\begin{array}{rl}\det (\mathbf{J})& =\det \left[\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right]=\det \left[\begin{array}{cc}\cos (\theta )& -r\sin (\theta )\\ \sin (\theta )& r\cos (\theta )\end{array}\right]\\ & =r{\cos }^2(\theta )+r{\sin }^2(\theta )\\ & =r\end{array}$$

Hence, an integral change of variables is given by:

$${\iint }_{D}f(x,y)\text{d}x\text{d}y={\iint }_{D^{\ast }}f(r\cos (\theta ),r\sin (\theta ))\cdot r\text{d}r\text{d}\theta$$