Polar Coordinates

The polar coordinates $r$ and $\theta$ are defined by

$$x = r\cos\theta$$ (1)

$$y = r \sin \theta.$$ (2)

In terms of $x$ and $y$,

$$r = \sqrt{x^2 + y^2}$$ (3)

$$\theta = \tan^{-1}\left({y\over x}\right).$$ (4)

The Arc Length of a polar curve given by $r = r(\theta)$ is

$$s = \int^{\theta_2}_{\theta_1} \sqrt{r^2 +\left({dr\over d\theta}\right)^2}\, d\theta.$$ (5)

The Line Element is given by

$$ds^2 = r^2\,d\theta^2,$$ (6)

and the Area element by

$$dA = r\,dr\,d\theta.$$ (7)

The Area enclosed by a polar curve $r = r(\theta)$ is

$$A = {\textstyle{1\over 2}}\int^{\theta_2}_{\theta_1} r^2 \,d\theta.$$ (8)

The Slope of a polar function $r = r(\theta)$ at the point $(r,\theta)$ is given by

$$m = {r+\tan \theta {dr\over d\theta}\over -r \tan \theta +{dr\over d\theta}}.$$ (9)

The Angle between the tangent and radial line at the point $(r,\theta)$ is

$$\psi = \tan^{-1}\left({r\over {dr\over d\theta}}\right).$$ (10)

A polar curve is symmetric about the $x$-axis if replacing $\theta$ by $-\theta$ in its equation produces an equivalent equation, symmetric about the $y$-axis if replacing $\theta$ by $\pi -\theta$ in its equation produces an equivalent equation, and symmetric about the origin if replacing $r$ by $-r$ in its equation produces an equivalent equation.

In Cartesian coordinates, the Position Vector and its derivatives are

$${\bf r} = \sqrt{x^2+y^2}\,\hat{\bf r}$$ (11)

$$\dot {\bf r} = \dot{\hat {\bf r}}\sqrt{x^2+y^2}+\hat {\bf r}\,(x^2+y^2)^{-1/2}(x\dot x+y\dot y)$$ (12)

$$\hat {\bf r} = {x\hat {\bf x}+y\hat {\bf y}\over \sqrt{x^2+y^2}}$$ (13)

$$\dot{\hat {\bf r}} = {\dot x\hat {\bf x}+\dot y\hat {\bf y}\over\sqrt{x^2+y^2}}-{\textstyle{1\over 2}}(x^2+y^2)^{-3/2}(2)(x\dot x+y\dot y)(x\hat{\bf x}+y\hat{\bf y})$$

$$= {(x\dot y-y\dot x)(x\hat{\bf y}-y\hat{\bf x})\over(x^2+y^2)^{3/2}}.$$ (14)

In polar coordinates, the Unit Vectors and their derivatives are

$${\bf r} \equiv \left[\begin{array}{c}r\cos\theta\\ r\sin\theta\end{array}\right]$$ (15)

$$\hat {\bf r} \equiv {{d{\bf r}\over dr}\over \left\vert{d{\bf r}\over dr}\right\vert} = \left[\begin{array}{c}\cos\theta\\ \sin\theta\end{array}\right]$$ (16)

$$\hat \boldsymbol{\theta} \equiv {{d\boldsymbol{\theta}\over d\theta}\over \left\vert{d\boldsymbol... ...ght\vert} = \left[\begin{array}{c}-\sin \theta\\ \cos \theta\end{array}\right]$$ (17)

$$\dot{\hat {\bf r}} = \left[\begin{array}{c}-\sin\theta\dot\theta\\ \cos\theta\dot\theta\end{array}\right] = \dot\theta\,\hat {\boldsymbol{\theta}}$$ (18)

$$\dot{\hat\boldsymbol{\theta}} = \left[\begin{array}{c}-\cos\theta\dot\theta\\ -\sin\theta\dot\theta\end{array}\right] = -\dot\theta\hat {\bf r}$$ (19)

$$\dot {\bf r} = \left[\begin{array}{c}-r\sin\theta\dot\theta+\cos\theta\dot r\\ ... ...nd{array}\right] = r\dot\theta \,\hat\boldsymbol{\theta}+ \dot r \,\hat {\bf r}$$

$$\ddot{\bf r} = \dot r\dot\theta\,\hat\boldsymbol{\theta}+ r\ddot\theta\hat\bolds... ...a\dot{\hat\boldsymbol{\theta}} + \ddot r\hat {\bf r} + \dot r \dot{\hat{\bf r}}$$

$$= \dot r\dot\theta\hat\boldsymbol{\theta}+\dot r\ddot\theta\hat\bol... ...ta\hat{\bf r}) + \ddot r\hat {\bf r} + \dot r\dot\theta \hat\boldsymbol{\theta}$$

$$= (\ddot r-r{\dot\theta}^2)\hat {\bf r} + (2\dot r\dot\theta+r\ddot\theta)\hat{\boldsymbol{\theta}}$$

$$= (\ddot r-r{\dot\theta}^2)\hat {\bf r} + {1\over r} {d\over dt} (r^2\dot\theta) \hat{\boldsymbol{\theta}}.$$ (20)

See also Cardioid, Circle, Cissoid, Conchoid, Curvilinear Coordinates, Cylindrical Coordinates, Equiangular Spiral, Lemniscate, Limaçon, Rose