Surface patch - cylinder, sphere I want to show that the unit cylinder can be covered by a single surface patch,
but that the unit sphere cannot. 
Any hints? I don't have any ideas... 
The definition of the book of a surface patch is: 


Your question isn't clear, but I assume you mean that for a cylinder you can cut a piece of paper into a single shape and fold (roll) it into a cylinder, but that you cannot do that for a sphere.  [For the cylinder, the flat shape is a rectangle with two disks touching opposite sides of length equal to the circumference of each disk.]

Here's is the case for a cone:

Here's the case for a cube:

The reason for this is that the Gaussian curvature of the cylinder is zero everywhere, whereas for a sphere it is zero nowhere.  
A surface of zero Gaussian curvature (such as a cone, cube, etc.) can be "cut and flattened out onto a plane" (and the converse).  A surface with non-zero Gaussian curvature (such as a sphere, or ellipsoid) cannot.
