Answer to Question #138178 in Differential Geometry | Topology for anjali g

Question

(a) Show that the circular cylinder S = {(x,y,z) ∈R3 : y2+z2 = 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

(a) A patch is called regular, if the respective Jacobian has rank 2. It holds, since for the function $f:(y,z)\rightarrow\mathbb{R}^2$ we have: $f_{yy}=1,f_{yz}=f_{zy}=0,f_{zz}=1$ .

(b) The surface has the following form: