Given the surface $S=S_1\cup S_2$ where $S_1=\{ (x,y,z) \in \mathbb{R}^3 / x^2+y^2=1 ; \space 0 \leq z\leq1 \}$ and $S_2=\{ (x,y,z) \in \mathbb{R}^3 / x^2+y^2+(z-1)^2=1 ; \space z\geq1 \}$, orientated with the normal pointing outside the cylinder (with cap) and the sphere, respectively.
Let $F(x,y,z)=(zx + z^2y+x, z^3yx+y,z^4x^2)$. Calculate $\int_S (\nabla \times F)\space dS$
Then, because Stoke's theorem we have that:
$\int_S (\nabla \times F)\space dS= \int_{C_{1}} F \space dS + \int_{C_{2}} F \space dS$
Where, $C_1$ is the simple-closed curve of the cylinder, and $C_2$ of the half of the sphere.
I'm having troubles about defining the appropriate orientation of the curve to make the normal vector pointing outside both surfaces, and also I wonder if I should or not parametrize them with polar coordinates. Can someone help me with this?
Notice that $\mathbf F(x, y, 0) = (x, y, 0)$ and the boundary of the surface is $(\cos t, \sin t, 0)$, therefore $\mathbf F \cdot d\mathbf s = 0$.