I was stuck on the following question on a practice exam. For the vector field $\vec F(x,y,z) = (\frac{-y}{x^2+2y^2}, \frac{x}{x^2+2y^2}, 1) $
Evaluate $$\int_K\vec F \vec dr,$$
with K the simple closed curve given by $z = x^2+y^2$ and $x+y+z = 2$.
Since K is a simple closed curve i thought i had to use Stokes theorem and i computed $curl(\vec F) = 0$, but i didnt really know how to go from there.
Unfortunately Stokes' theorem cannot be applied naively to the disk because the vector field is defined on a subset of $\Bbb{R}^3$ that is not simply connected (the $z$ axis is taken out). What you can do is define a surface with two boundaries, one being the curve in the problem, other other being the ellipse $x^2+2y^2=1$ in the plane $z = 100$ which gives us
$$\int_CF\cdot dr - \int_{x^2+2y^2=1\cap z=100}F\cdot dr = \iint_S \operatorname{curl}F\cdot dS = 0$$
$$\implies \int_C F\cdot dr = \frac{1}{(1)}\int_{x^2+2y^2=1\cap z=100} -ydx+xdy+dz$$
$$ = \frac{1}{\sqrt{2}}\int_0^{2\pi}\sin^2t+\cos^2t\:dt = \pi\sqrt{2}$$
from parametrizing appropriately.