Verifying Stokes' Theorem
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 nG(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 c1 c2 c3 in the xy-plane, however none have a z-component, so when I took F(c1)•c'1 , F(c2)•c'2, and F(c3)•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.
Re: Verifying Stokes' Theorem
Three parametrized line segments are indeed how you would break up the boundary of S. But one each lies in the xy-, xz-, and yz-planes. Try drawing a picture like the one I've added below.