Let $\displaystyle S_G$ be the surface of the cap (see figure). Let $\displaystyle C$ be the differentiable curve of the cap in the $\displaystyle xy$ plane. Let $\displaystyle S_A$ be the portion of the plane xy that C contains. And let $\displaystyle A$ be the value of the area of $\displaystyle S_A$. Consider $\displaystyle S_G$ oriented outward. Determine with arrows the sense of the curve $\displaystyle C$ and the value of $\displaystyle \iint _{S_G} curl F dS$, where $\displaystyle F=((\ln (z+1))^2xy, (\ln (z+1))^2 \frac{x^2}{2}+x(z+1), \cos z )$.

-------------------------------------------------------

My attempt :

The arrow is easy to draw : counter clockwise.

I've calculated the curl of $\displaystyle F$, but now I realize it is likely useless. Anyway here it comes : $\displaystyle curl F =\left ( \frac{x^2 \ln (z+1)}{2z+2} \right ) \vec i + \left( \frac{xy \ln (z+1)}{z+1} \right) \vec j +(z+1) \vec k$.

I'm sure I have to calculate $\displaystyle \int _C FdS$ (which is worth $\displaystyle \iint _{S_G} curl F dS$ by Stokes' theorem) but how to proceed? I have almost no information about $\displaystyle C$, I mean I don't know its parametrization. The only way I think I might solve the problem is to realize something obvious that imply that $\displaystyle \int _C FdS=0$, but I don't see anything.