Stokes theorem, intersection between cylinder and plane

Question

$\mathcal C$ is the intersection curve between the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$. I tried parameterizing the curve by expanding the cylinder equation $x^2 + (y-1)^2 = 1$. I think I can write the parameterization as follows: $(r \cos(\theta), r \sin(\theta) + 1, 0)$? How do I proceed from here, I want to use Stokes theorem. $\mathcal C$ is oriented counterclockwise.

Answer 1

HINT

We have

• $F=(y^2,xy,xz)\implies \operatorname{rot}(F)=(0,-z,-y)$

then

$$\int_C y^2\,dx+xy\,dy+xz\,dz =\int_S \operatorname{rot}(F)\cdot n \,dS$$

and

$$\operatorname{rot}(F)\cdot n=(0,z,-y)\cdot (0,-1/\sqrt 2,1/\sqrt 2)=\frac z {\sqrt 2}-\frac y {\sqrt 2}$$

Answer 2

Take $S$ to be the region in the plane $y = z$ enclosed by $\mathcal C$. Since $\nabla \times \mathbf F = (0, -z, -y)$ is parallel to the plane $y = z$, $(\nabla \times \mathbf F) \cdot d\mathbf S = 0$.