1. The problem statement, all variables and given/known data Verfify Stokes' theorem for the given surface S and boundary ∂S, and vector fields F. S = [(x,y,z): x2+y2+z2=1, z≥0 ∂S = [(x,y): x2+y2=1 F=<x,y,z> I did this problem and checked the answer - I chose the wrong integral bounds and I am wondering why they are wrong. 2. Relevant equations Stokes' theorem: ∫∫(∇×F)dS = ∫F⋅ds 3. The attempt at a solution I was able to do the left hand of the solution easily. I knew that I should get zero for the cross product, so that's all I had to do. For the right hand side: ∫F⋅dS = ∫xdx+ydy x= cosΘ y=sinΘ dx=-sinΘ dy=cosΘ 0≤Θ≤2π Apparently this is the correct range for theta: 0≤θ≤π Why is it just π?