# Linking surface integral of a gradient field to a contour integral [duplicate]

I have a vector field $F$ deriving from a scalar potential $f$, i.e. $F=\text{grad}(f)$. I want to compute the integral of $F$ over a surface (To evaluate the flux of $F$). I think there exists a theorem linking this integral over a surface to the integral of $f$ over the edge of the surface but I can't find it.

What I find is :

-Green theorem linking integration of $\text{div}(F)$ over a surface to $F$ over a line

-Stokes theorem linking integration of $\text{curl}(F)$ to integration of $F$.

But no theorem linking integration of $F$ over a surface to $f$ over a contour.

Can someone help me ?

Thanks