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) $\int_C (f \nabla g+g \nabla f)\cdot ds=0$


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Erdos Erdos
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