A circle C is cut on the surface of the hemisphere $x^2 + y^2 + z^2 = 1,z≥0 $ by the cylinder $ x^2 + y^2 = y. $Evaluate $ \int_{C} -y^2\,dx +y^2\,dy+z^2\,dz $ where the direction round C is such that the point (0,0,1) is directed into the first octant.
Attempt: So completing the square gives a cylinder of centre (0,1/2) and radius 1/2. Using Stoke's thm, I identified the vector field F to be $ −y^2 i + y^2j +z^2 k $ and took the curl of it to give $ 2yk $. I believe everything is right up to here.
I am confused about what the surface is here that is bounded by C. I realise that to compute $ d\vec{S} =\frac{r_u \times r_v}{|r_u \times r_v|}dS,$ I have to find a suitable parametrisation of some surface. I found where the cylinder and sphere intersected :$ y+z^2 = 1=>z=\sqrt{1−y} $ since $ z≥0 $.So then my parametrisation would be $ r(x,y)=xi+yj+\sqrt{1-y} k $ from which I could then compute two tangent vectors and a normal.
I am not sure if my parametrisation is correct. Can anyone offer any advice? Many thanks