polar coordinates

Let x,y be Cartesian coordinatesMathworldPlanetmath for 2.

Then r0, θ[0,2π) related to (x,y) by

x(r,θ) = rcosθ,
y(r,θ) = rsinθ,

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

The polar coordinates of Cartesian coordinates (x,y)2{0} are

r(x,y) = x2+y2,
θ(x,y) = arctan(x,y),

where arctan is defined here (http://planetmath.org/OperatornamearcTanWithTwoArguments).

Polar basis. Polar coordinates are equipped with an orthonormal base {𝐞𝐫,𝐞θ}, which can be defined in terms of the standard cartesian base {𝐢,𝐣} in 2 as follows.


where 𝐞𝐫,𝐞θ are so-called radial and traverse or angular vector, respectively. Since these vectors are variable in direction, they are differentiableMathworldPlanetmathPlanetmath. In fact,


The geometrical action of the derivativePlanetmathPlanetmath operator d/dθ is like a rotation operator that rotates each base vector a counter-clockwise angle equals to π/2.

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


RelationsMathworldPlanetmathPlanetmath with complex numbers. When the Euclidean planeMathworldPlanetmath 2 is identified with by


it is possible to define multiplications on 2.  Via polar coordinates, the formulaMathworldPlanetmathPlanetmath for this multiplication becomes very simple, thanks to Euler’s formula (http://planetmath.org/EulerRelation)


Thus, we have the following identification:


If P=(r1,θ1) and Q=(r2,θ2), the productPlanetmathPlanetmath of P and Q works out to be (r1r2,θ1+θ2). (Even if one is not familiar with the complex exponentialPlanetmathPlanetmath, this assertion may be checked directly using the angle sum identities for cos and sin.)

Multiplications of polar coordinates have some simple geometric interpretationsMathworldPlanetmathPlanetmath. For example, if R=(1,α) and Q=(r,β), then QRQ given by (1,α)(r,β)=(r,α+β) is the rotationMathworldPlanetmath of Q by angle α. If S=(t,0), then (t,0)(r,β)=(tr,β) can be viewed as the scalingMathworldPlanetmath of Q along the ray 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 extensionsPlanetmathPlanetmath on domain of polar coordinates r and θ, see here (http://planetmath.org/ConstructionOfPolarCoordinates).

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


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


Partial derivativeMathworldPlanetmath operators transform as follows:

x =cosθr-1rsinθθ
y =sinθr+1rcosθθ
r =cosθx+sinθy
θ =-rsinθx+rcosθy
Title polar coordinates
Canonical name PolarCoordinates
Date of creation 2013-03-22 15:12:16
Last modified on 2013-03-22 15:12:16
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 51-01
Related topic DerivationOfRotationMatrixUsingPolarCoordinates
Related topic CylindricalCoordinates
Related topic ArgumentOfProductAndQuotient