Let $\displaystyle F = (y,-x,zx^3y^2)$ Evaluate $\displaystyle \int \int_{S} (curlF)$ dS, where S is the surface $\displaystyle x^2 + y^2 + z^2 = 1, z\geq 0$
I need help with the setup.
Since you titled this "Stoke's theorem" can we assume you know that theorem? (By the way, it should be "Stokes' " theorem. His name was "Stokes", not "Stoke".)
Stokes theorem says $\displaystyle \int_C \vec{F}\cdot d\vec{r}= \int\int_S curl \vec{F}\cdot d\vec{S}$ where r is the closed curve bounding surface S.
Since you are asked to find $\displaystyle \int\int_S (curl F)dS$, Stokes' theorem says that you can instead integrate $\displaystyle \int_C \vec{F}d\vec{r}$ where C is the boundary of that hemi-sphere. When z= 0, $\displaystyle x^2+ y^2= 1$ so the boundary is the unit circle in the xy-plane. Of course, when z= 0, $\displaystyle F= (y, -x, 0)$. I think I might be inclined to integrate around the unit circle by using the angle, $\displaystyle \theta$, as parameter: $\displaystyle x= cos(\theta)$, $\displaystyle y= sin(\theta)$. For the unit circle, $\displaystyle dr= d\theta$.