I am trying to evaluate a $\int_\gamma \vec{F}.d\vec{r}$ integral using Stokes' theorem, but I can't visualize the limits I should use in order to integrate properly. The curve $\gamma$ is the intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$, and I am using the sphere as the surface to apply Stokes' theorem. Then:
$(x,y,z)=(a\sin{\phi}\cos{\theta},a\sin{\phi}\sin{\theta},a\cos{\phi})$
But what are the limits? Working with this parametrization and the plane's equation, I got $\tan{\phi}=-\frac{1}{\sin{\theta}+\cos{\theta}}$, what should I do with this relation?
Thanks
We shall take advantage of the fact that ${\rm curl}(\vec F)=(-1,-1,-1)$ is constant and parallel to the normal $n={1\over\sqrt{3}}(1,1,1)$ of the plane $P:\>x+y+z=0$.
The curve $\gamma$ bounds a circular disk $D$ of radius $a$ in $P$. Using Stokes' theorem and writing ${\rm d}\omega$ for the surface element on $D$ we then obtain $$\int_\gamma\vec F\cdot d\vec r=\pm \int_D{\rm curl}(\vec F)\cdot n\>{\rm d}\omega=\pm\sqrt{3}\int_D{\rm d}\omega=\pm\sqrt{3}\pi a^2\ ,$$ depending on the orientation of $\gamma$, which was not specified in the givens.