Consider the vector field: $$\vec{F} = \left(2x-y,-yz^2,-y^2z\right)$$ We are to explicitly show stokes theorem: $$\oint_\Gamma \vec{F}\cdot d\vec{x} = \int\int_{\partial V}\nabla \times \vec{F}\cdot d\vec{A}$$ Where $\partial V$ is the open surface defined by the tetrahedron with vertices at $(0,0,0), (1,0,0), (0,1,0), (0,0,1)$ and the face on $z=0$ being open such that $\Gamma$ is this perimeter.
Now I can't seem to calculate the double integral. I get $\nabla \times \vec{F} = (0,0,1)$ and so the faces on the $xz$ and $yz$ planes give zero, but I cannot get the contribution from the face which is the plane $x+y+z=1$, in other words, I can't seem to compute the surface integral over this plane. Help is much appreciated.
One result of stokes theorem is that you can change your surface of integration as long as its boundary (lip/lid whatever you call it) is not changed. So we will surface integrate the open face.
$$\int _0 ^1 \int _ 0 ^{1-x}1 \,dx\,dy $$
$$\frac12$$