# Applications of Stokes' Theorem

Let $S$ be a surface and $C$ be a closed curve which is the boundary of $S$, and  $f,g$ are $C^2$ functions. Show that

(i) $\int_C f \nabla g \cdot ds=\iint_S (\nabla f \times \nabla g)\cdot ds$
(ii) $\int_C (f \nabla g+g \nabla f)\cdot ds=0$