1. ## Stoke's Theorem

Let $F = (y,-x,zx^3y^2)$ Evaluate $\int \int_{S} (curlF)$ dS, where S is the surface $x^2 + y^2 + z^2 = 1, z\geq 0$

I need help with the setup.

2. 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 $\int_C \vec{F}\cdot d\vec{r}= \int\int_S curl \vec{F}\cdot d\vec{S}$ where r is the closed curve bounding surface S.

Since you are asked to find $\int\int_S (curl F)dS$, Stokes' theorem says that you can instead integrate $\int_C \vec{F}d\vec{r}$ where C is the boundary of that hemi-sphere. When z= 0, $x^2+ y^2= 1$ so the boundary is the unit circle in the xy-plane. Of course, when z= 0, $F= (y, -x, 0)$. I think I might be inclined to integrate around the unit circle by using the angle, $\theta$, as parameter: $x= cos(\theta)$, $y= sin(\theta)$. For the unit circle, $dr= d\theta$.

3. Thank You that was great help