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

Using the chain rule we find
∂∂x =∂ρ∂x∂∂ρ +∂ϕ∂x∂∂ϕ =xρ∂∂ρ −yρ2∂∂ϕ =cosϕ∂∂ρ −sinϕρ∂∂ϕ ,∂∂y =∂ρ∂y∂∂ρ +∂ϕ∂y∂∂ϕ =yρ∂∂ρ +xρ2∂∂ϕ =sinϕ∂∂ρ +cosϕρ∂∂ϕ ,
where the unit vectors
ˆeρ=(cosϕ,sinϕ),ˆeϕ=(−sinϕ,cosϕ),
We can now use this to evaluate ∇2,
∇2=cos2ϕ∂2∂ρ2 +sinϕcosϕρ2∂∂ϕ +sin2ϕρ∂∂ρ +sin2ϕρ2∂2∂ϕ2 +sinϕcosϕρ2∂∂ϕ +sin2ϕ∂2∂ρ2 −sinϕcosϕρ2∂∂ϕ +cos2ϕρ∂∂ρ +cos2ϕρ2∂2∂ϕ2 −sinϕcosϕρ2∂∂ϕ =∂2∂ρ2 +1ρ∂∂ρ +1ρ2∂2∂ϕ2 =1ρ∂∂ρ (ρ∂∂ρ )+1ρ2∂2∂ϕ2 .
A final useful relation is the integration over these coordinates.

As indicated schematically in Fig. 7.1.2, the surface related to a change ρ→ρ+δρ, ϕ→ϕ+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