For those who don't like to open pdf files (not as prone to viruses as word files, but still)
The problem is to find where and S is the circle in the plane z= 1 using Stokes theorem or Gauss's divergence theorem.
Stoke's theorem says
Since that has " ", that is the obvious one to use.
Taking , ,
(heatly, you don't include " ", which I consider an important part of the parameterization. Of course, here, because z is constant, it doesn't matter).
You then say "r(t)= cos^2t+ sin^2t" which is non-sense! That is the same as saying that r(t)= 1 so of course dr= 0. In order that " ",where F is a vector, make sense, ds must be a vector, not a scalar.
What you want is that r(t) is the position vector xi+ yj+ zk= cos(t)i+ sin(t)j+ k. Then dr= dx i+ dy j+ dz k= -sin(t)dt i+ cos(t)dt j. (dz= 0 so there is no k component.)
That is not 0 because of the .
You then proceed to "use" Gauss's divergence theorem which doesn't apply here because there is no " " in it and because it requires integration over a solid region and there is no solid region given in this problem.