Calculate Being S the external surface of understood between and
is a cylinder, with radius a, in three dimensions. It can be written in parametric equations using cylindrical coordinates but taking r= a: , , .
The "position vector" of any point on the cylinder can be written as . The two derivatives:
and
are tangent vectors to the cylinder. Their cross product is
and the "scalar differential of surface area" is given by the length of that: .
(For this simple geometry, you could have just noted that the differential of length around the cylinder is so the differential of area is .)
Since this integral is
.
(In fact, just thinking about the "symmetry" (or, more correctly "asymmetry") makes this problem trivial.)