I stumbled upon a problem on surface integrals which required a parameterization of the cylinder (upper half-cylinder). My surface is $z = \sqrt{1-x^2}$. I found it odd that the normal vector $\vec n = \left< \frac{x^2}{1-x^2}, 0 , 1 \right>$ of the parameterization $r(x,y)=\left< x, y, \sqrt{1-x^2}\right>$ is not defined at the border of the upper half-cylinder $\ e.g.\ P_0=\left(1,0,0\right)$. I would guess that this would just be a vector with the direction of the position vector $\vec{OP_0}$
- Why is it not and why does a more classic parameterization $\ e.g. \ r(θ,z)=\left< cosθ, sinθ, z\right>, \ \vec n_o = \left< sinθ, cosθ , 0 \right>$ not have this problem at that point?
- What does this generally mean for picking a parameterization