Evaluate the surface integral of $\vec F=(-y+z,-z+x,-x+y)$ over the curve $\alpha (t,s)$

Question

Given $\vec F=(-y+z,-z+x,-x+y)$ I am asked to find the integral over the surface $\alpha$ $$\alpha(t,s)=(t\cos(s),t\sin(s),s)$$ for $\quad 0\le s\le 2\pi$ and $\quad0\le t\le 1$

This after explaining the divergence theorem, but the surface is not closed and I have not found how to close it. How else can I integrate it?

Answer 1

Use that $\int \boldsymbol{F}\cdot d\boldsymbol{S}=\int \boldsymbol{F\cdot \hat{n}}dS$, where $\hat{n}$ is the unit vector pointing in the outward direction from the surface and given by the cross-product of two vectors that span up the surface.