Problem- Given the vector field F= <yx, 0, x>, calculate both F•ds and curl(F)•dS over the portion of the plane + + z = 1 where x, y, z > 0, oriented with an upward normal vector.
I took x F to get <0, y-1, z> which I dotted with n_{G}(x,y)= < , ,1> of my parameterization of the plane G(x,y) = (x, y, 1-( + ) to get +z. I then computed +z dy dz to get which I believe is correct.
My issue lies in finding F•ds. I tried defining S as the sum of a set of three parameterized line segments c_{1} c_{2 }c_{3} in the xy-plane, however none have a z-component, so when I took F(c_{1})•c'_{1 }, F(c_{2})•c'_{2}, and F(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, so F•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