Let $S$ be an oriented surface with a normal unit vector $\vec{n}$ which is included in an equation of a vector field $\vec{F}$ where $rot\vec{F}=k\vec{n}$ ($k$ is a positive constant) everywhere in $S$. If $C$ is a closed simple curve located on $S$, what is this equation calculating: $$ \oint\limits_C \vec{F} \cdot d\vec{r} $$
There is a couple of concepts in this question that I don't get the deep understanding of. I know that $\vec{F}$ is a vector field and that $rot\vec{F}$ is a key concept that is used to prove if a vector field is conservative or not (if $rot(\vec{F})=\vec{0}$ the vector field is conservative). I am not sure if $rot$ has another meaning. We have that $rot$ is proportional in a positive direction with $\vec{n}$. So, we have that the $rot$ of $\vec{F}$ is perpendicular and in a positive direction with $S$.
I have difficulties when is the time to associate this information with the equation
$$ \oint\limits_C \vec{F} \cdot d\vec{r} $$
I know that this equation is the line integral of $\vec{F}$ in the direction of $C$. This concept can be generalized to compute Work of a vector field $\vec{F}$ and other physics notions, but is it calculating something in particular?
Am I understanding the question correctly?
Thank you.