Application of Stoke's Theorem

Question

$\newcommand{\unit}[1]{\hat{\mathbf{#1}}}$ Verify the truth of Stokes' theorem in the case when $\mathbf{v} = y\unit{i} + 2x\unit{j} + z\unit{k}$, if the path $C$ is a circle $x^2 + y^2 = 1$ in the $xy$ plane and the surface is the plane area bounded by $C$.

I am having trouble determine what the surface is. Is it cylinder cut by the plane formed by $\mathbf{v}$?

Answer 1

No. The surface in this case is the unit disk in the $xy$-plane (i.e. the part of the plane $z=0$ that satisfies $x^2+y^2\leq 1$).

Answer 2

Anything that is protruding out of closed surface. Stokes theorem guarantees that flux through each surface will be constant depending only on the path of the closed curve.
For your particular case, unit disc, i.e. $x^2+y^2 \le 1 , z=0$, would be the best choice because it's easiest.