Problem- Given the vector fieldF= <yx, 0, x>, calculate bothF•dsand curl(F)•dSover the portion of the plane + + z = 1 where x, y, z>0, oriented with an upward normal vector.

I took xFto get <0, y-1, z> which I dotted withn_{G}(x,y)= < , ,1> of my parameterization of the plane G(x,y) = (x, y, 1-( + ) to get +z. I then computed +zdy dzto get which I believe is correct.

My issue lies in findingF•ds. I tried defining S as the sum of a set of three parameterized line segmentsc_{1}c_{2 }c_{3}in the xy-plane, however none have a z-component, so when I tookF(c_{1})•c'_{1 },F(c_{2})•c'_{2}, andF(c_{3})•c'_{3 }they all ended up as <0, 0, 0>. I tried defining S in the yz-plane, but I met the same <0, 0, 0> dot products, soF•ds and all of the three scalar line integrals composing it equated to be 0. I feel like I'm doing something wrong.

Any help?

Thanks

Anthony