Problem- Given the vector fieldF= <yx, 0, x>, calculate both $\displaystyle \oint_{\partial S}$F•dsand $\displaystyle \iint_S$curl(F)•dSover the portion of the plane $\displaystyle \frac{x}{2}$+$\displaystyle \frac{y}{3}$+ z = 1 where x, y, z>0, oriented with an upward normal vector.

I took $\displaystyle \nabla$ xFto get <0, y-1, z> which I dotted withn_{G}(x,y)= <$\displaystyle \frac{1}{2}$,$\displaystyle \frac{1}{3}$,1> of my parameterization of the plane G(x,y) = (x, y, 1-($\displaystyle \frac{x}{2}$+$\displaystyle \frac{y}{3}$ ) to get $\displaystyle \frac{y-1}{3}$ +z. I then computed $\displaystyle \int_0^1$$\displaystyle \int_0^{3-3z}$$\displaystyle \frac{y-1}{3}$ +zdy dzto get $\displaystyle \frac{1}{2}$ which I believe is correct.

My issue lies in finding $\displaystyle \oint_{\partial S}$F•ds. I tried defining $\displaystyle \partial$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 $\displaystyle \partial$S in the yz-plane, but I met the same <0, 0, 0> dot products, so $\displaystyle \oint_{\partial S}$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