Polar coordinates

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

For an extension to three dimensions, see spherical polar coordinates.
CC Image
Two dimensional polar coordinates r and θ of vector 𝐫→

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 ℝ2 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

OP->≡𝐫→.

Hence we obtain the figure on the right where 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

r=x2+y2x=rcos⁡θy=rsin⁡θ,

so that for r ≠ 0,

θ={arccos⁡(x/r) if y≥03600−arccos⁡(x/r) if y<0.

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

dA=Jdrdθ.

The Jacobian J is the determinant

J=∂(x,y)∂(r,θ)=|cos⁡θ−rsin⁡θsin⁡θrcos⁡θ|=rcos2θ+rsin2θ=r.

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

A=∫02π∫0Rrdrdθ=πR2.