Calculating Flux Across a Simple Closed Curve lying on an (x,z)-cylinder

Question

I am having some difficulties with the problem above. The approach I'm using is the following:

Use Stokes Theorem to instead show that $$\iint_{S} \text{curl $(\vec{G})$} \cdot \hat{n} \text{ }ds =0,$$ where $S$ is the region enclosed by $C$. I've computed curl $(\vec{G})=z \hat{j}$. From here I'm not sure where to go as I don't see how to compute $\hat n$ in this case.

Answer 1

The statement is not true. Take the "$(x,z)$" cylinder $x^2+(z-1)^2=1$ for instance. Computing the surface integral of the curl over a slice of constant $y$ gives us

$$\iint\limits_{x^2+(z-1)^2=1}z\hat{j}\cdot \hat{j}dS = \int_0^\pi \int_0^{2\sin\theta}r^2\sin\theta\:dr\:d\theta = \frac{8}{3} \int_0^\pi \sin^4\theta \:d\theta \neq 0$$

Answer 2

HINT: If the cylinder contains lines parallel to the $y$-axis, what do you know about $\vec n$? I believe the hypothesis should be that the curve bounds a region in the cylinder.