Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

7.1: Polar Coordinates

( \newcommand{\kernel}{\mathrm{null}\,}\)

Polar coordinates in two dimensions are defined by x=ρcosϕ,y=ρsinϕ,ρ=x2+y2,ϕ=arctan(y/x),

as indicated schematically in Fig. 7.1.1.

polar.png
Figure 7.1.1: Polar coordinates

Using the chain rule we find

x =ρxρ +ϕxϕ =xρρ yρ2ϕ =cosϕρ sinϕρϕ ,y =ρyρ +ϕyϕ =yρρ +xρ2ϕ =sinϕρ +cosϕρϕ ,

We can write =ˆeρρ +ˆeϕ1ρϕ 

where the unit vectors

ˆeρ=(cosϕ,sinϕ),ˆeϕ=(sinϕ,cosϕ),

are an orthonormal set. We say that circular coordinates are orthogonal.

We can now use this to evaluate 2,

2=cos2ϕ2ρ2 +sinϕcosϕρ2ϕ +sin2ϕρρ +sin2ϕρ22ϕ2 +sinϕcosϕρ2ϕ    +sin2ϕ2ρ2 sinϕcosϕρ2ϕ +cos2ϕρρ +cos2ϕρ22ϕ2 sinϕcosϕρ2ϕ =2ρ2 +1ρρ +1ρ22ϕ2 =1ρρ (ρρ )+1ρ22ϕ2 .

A final useful relation is the integration over these coordinates.

imageedit_2_5805175308.png
Figure 7.1.2: Integration in polar coordinates

As indicated schematically in Fig. 7.1.2, the surface related to a change ρρ+δρ, ϕϕ+δϕ is ρδρδϕ. This leads us to the conclusion that an integral over x,y can be rewritten as Vf(x,y)dxdy=Vf(ρcosϕ,ρsinϕ)ρdρdϕ


This page titled 7.1: Polar Coordinates is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?