content_image/317873.jpg content_image/317872.jpg content_image/317874.jpg
Lemma 4.4.1. Using the notation of figure 4.6, we have ∂ $_{∂r}$g ( s, r ) = ( θ$_{2}$ − θ$_{1}$ ) r . That is, the derivative exists and equals the length of the curve C ( γ ( s ) , r ) ∩ Ω . 

Proof. We have (see figure 4.8) 

<img src='content_image/317873.jpg'>

This difference of areas can be modeled by the difference in the circular sectors of D ( γ ( s ) , r +Δ r ) and D ( γ ( s ) , r ) with angle θ$_{1}$ − θ$_{2}$ . The actual area depends on the image of γ outside of D ( γ ( s ) , r ), but this correction will be a subset of the circular segment of D ( γ ( s ) , r + Δ r ) which is tangent to D ( γ ( s ) , r ) at the point γ exits. This has area O (Δ r $^{2}$) by lemma 4.2.18. 

Thus we have 

<img src='content_image/317872.jpg'>

<img src='content_image/317874.jpg'>

