Let be the solid cylinder in defined by the equations
and let be its boundary with the usual orientation; thus consists of a cylindrical surface segment together with the top disc and the bottom disc , each with the appropriate orientation. In addition consider the vector field
and the differential form
1. Calculate , where is the outward unit normal vector to the boundary and is the element of surface area, by using Stokes's Theorem.
2. Calculate .
3. Calculate .
Note especially that the normal in part (2) is not , but .