Flux integral of a curl = zero

[nonequilibrium]

Does Stokes's theorem imply that the flux integral of a curl of a vector field over a closed surface is always zero? (because then there is no boundary curve and thus the line integral over the boundary curve is zero)

Is there an insightful way to see why this is always true? Maybe a connection with physics or so.

Thank you,
mr vodka

 

[LCKurtz]

By the divergence theorem

$$\int\int_S \nabla \times \vec F\cdot \hat n\, dS = \int\int\int_V \nabla \cdot \nabla \times \vec F\, dV= 0$$

because under appropriate continuity conditions the divergence of a curl is zero.