How to evaluate $\int_C {\bf{F}}\cdot d\bf{r}$ for the following surface?
Suppose ${\bf F}(x,y,z)=xy \ {\bf i}+yz\ {\bf j}+zx\ {\bf k}$ and the $C$ is the boundary of the part of the parabloid $z=1-x^2-y^2$ in the first octant.
This question can be easily solved by applying Stoke's theorem. But what if we do not use Stoke's theorem? Is the boundary of the surface the curve that go through all $zy$, $xy$ and $xz$ planes or the one that only lies on $xy$ plane? If we have found the curve, how do we parametrize it?
