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 .