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

Answer

Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
The answer is accepted.