Let $x,y$ be Cartesian coordinates for $\mathbbmss{R}^{2}$.

Then $r\geq 0$, $\theta\in[0,2\pi)$ related to $(x,y)$ by

are the polar coordinates for $(x,y)$. It is simply written $(r,\theta)$.

The polar coordinates of Cartesian coordinates $(x,y)\in\mathbbmss{R}^{2}\setminus\{0\}$ are

where $\arctan$ is defined here.

Polar basis. Polar coordinates are equipped with an orthonormal base $\{\mathbf{e_{r},e_{\theta}}\}$, which can be defined in terms of the standard cartesian base $\{\mathbf{i,j}\}$ in $\mathbbmss{R}^{2}$ as follows.

where $\mathbf{e_{r},e_{\theta}}$ are so-called radial and traverse or angular vector, respectively. Since these vectors are variable in direction, they are differentiable. In fact,

The geometrical action of the derivative operator $d/d\theta$ is like a rotation operator that rotates each base vector a counter-clockwise angle equals to $\pi/2$.

Position vector. For an arbitrary point of polar coordinates $(r,\theta)$, its position vector comes given by the single equation

Relations with complex numbers. When the Euclidean plane $\mathbbmss{R}^{2}$ is identified with $\mathbbmss{C}$ by

it is possible to define multiplications on $\mathbbmss{R}^{2}$.  Via polar coordinates, the formula for this multiplication becomes very simple, thanks to Euler’s formula

Thus, we have the following identification:

If $P=(r_{1},\theta_{1})$ and $Q=(r_{2},\theta_{2})$, the product of $P$ and $Q$ works out to be $(r_{1}r_{2},\theta_{1}+\theta_{2})$. (Even if one is not familiar with the complex exponential, this assertion may be checked directly using the angle sum identities for $\cos$ and $\sin$.)

Multiplications of polar coordinates have some simple geometric interpretations. For example, if $R=(1,\alpha)$ and $Q=(r,\beta)$, then $Q\rightarrow RQ$ given by $(1,\alpha)(r,\beta)=(r,\alpha+\beta)$ is the rotation of $Q$ by angle $\alpha$. If $S=(t,0)$, then $(t,0)(r,\beta)=(tr,\beta)$ can be viewed as the scaling of $Q$ along the ray $\overrightarrow{OQ}$ by $t$. Note also that multiplication by $(t,0)$ has the same effect as multiplication by the scalar $t$.

For more on polar coordinates, including their construction and extensions on domain of polar coordinates $r$ and $\theta$, see here.

Calculus in polar coordiantes. For reference, here are some formulae for computing integrals and derivatives in polar coordinates. The Jacobian for transforming from rectangular to polar cordinates is

so we may compute the integral of a scalar field $f$ as

Partial derivative operators transform as follows: