Polar coordinates

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For an extension to three dimensions, see spherical polar coordinates.

In mathematics and physics, polar coordinates are two numbers—a distance and an angle—that specify the position of a point on a plane.

In their classical ("pre-vector") definition, polar coordinates give the position of a point P with respect to a given point O (the pole) and a given line (the polar axis) through O. One real number (r ) gives the distance of P to O and another number (θ) gives the angle of the line O—P with the polar axis. Given r and θ, one determines P by constructing a circle of radius r with O as origin, and a line with angle θ measured counterclockwise from the polar axis. The point P is on the intersection of the circle and the line.

In modern vector language one identifies the plane with a real Euclidean space ${\displaystyle {\mathbb{ℝ}}^2}$ that has a Cartesian coordinate system. The crossing of the Cartesian axes is on the pole, that is, O is the origin of the Cartesian system and the polar axis is identified with the x-axis of the Cartesian system. The line O—P is generated by the vector

${\displaystyle \overrightarrow{OP}\equiv \overrightarrow{\mathbf{𝐫}}.}$

Hence we obtain the figure on the right where ${\displaystyle \overrightarrow{\mathbf{𝐫}}}$ is the position vector of the point P.

Algebraic definition[edit]

The polar coordinates r and θ are related to the Cartesian coordinates x and y through

${\displaystyle \begin{array}{rl}r& =\sqrt{x^2+y^2}\\ x& =r\cos \theta \\ y& =r\sin \theta ,\\ \end{array}}$

so that for r ≠ 0,

${\displaystyle \theta =\{\begin{array}{ll}\arccos (x/r)& \text{ if }y\ge 0\\ 360^0-\arccos (x/r)& \text{ if }y<0.\\ \end{array}}$

Bounds on the coordinates are: r ≥ 0 and 0 ≤ θ < 360⁰. Coordinate lines are: the circle (fixed r, all θ) and a half-line from the origin (fixed direction θ all r). The slope of the half-line is tanθ = y/x.

Surface element[edit]

The infinitesimal surface element in polar coordinates is

${\displaystyle dA=Jdrd\theta .}$

The Jacobian J is the determinant

${\displaystyle J=\frac{\partial (x,y)}{\partial (r,\theta )}=\left|\begin{array}{cc}\cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \\ \end{array}\right|=r{\cos }^2\theta +r{\sin }^2\theta =r.}$

Example: the area A of a circle of radius R is given by

${\displaystyle A={\displaystyle \underset{0}{\overset{2\pi }{\int }}}{\displaystyle \underset{0}{\overset{R}{\int }}}rdrd\theta =\pi R^2.}$