Answer to Question #138182 in Differential Geometry | Topology for anjali g

Question

(a) Show that the circular cylinder S = {(x,y,z) ∈R^3 : y^2+z^2 = 1}can be covered by a single regular surface patch, and hence is a surface.
(b) Draw a picture describing this surface patch.

Answer 1

We give the patch as $(x,y)\mapsto (\frac{x}{r},\frac{y}{r}, tan (r-\pi/2)).$ Here $\{x,y|\sqrt {x^2+y^2} \in (0,\pi)\}.$ Also, $r=+\sqrt{x^2+y^2}$ . Clearly, the map is smooth since $r\neq 0.$ Also $r<\pi.$ Hence $tan(r-\pi/2)$ tan is well defined. For surjection

, let $(a,b,c)$ be a given point on the cylinder. Since tan is continuous and injective on the range $(-\pi/2,\pi/2)$ and unbounded its surjective, and hence we get $r$ such that $tan(r-\pi/2)=c.$ So take $x=ar, y=br.$ Now $\sqrt{a^2+b^2}=1\Rightarrow a,b\leq 1$ . Hence $x,y\leq r<\pi$. Hence the pre-image lies in our given domain.

b) We append the diagram below.