How do flux, line integrals, and Stokes' theorem relate?

[SpY]]

What is the relation between the flux through a given surface by a vector field? And how does stokes theorem relate to the line integral around a surface in that field

 

[HallsofIvy]

The flux of a vector field $\vec{F}$ through surface S is, by definition,
$$\int\int_S \vec{F}\cdot d\vec{S}$$

And Stokes' theorem say, specifically, that that is equal to
$$\int_{c} \nabla\times\vec{F}\cdot d\vec{r}$$
Where c is the closed path bounding surface S.