$Applications of Stokes' Theorem$

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

Multivariable Calculus Stokes' Theorem Integrals

Rosemary Green

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Answer 1

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(i) By Stokes' Theorem,
\[\int_{C} f\nabla g \cdot ds=\iint_{S} \nabla \times (f \nabla g)\cdot dS=\iint_S \left( \nabla f \times \nabla g+f (\nabla \times \nabla g) \right)\cdot dS\]
\[=\iint_S \left( \nabla f \times \nabla g\right)\cdot dS,\]
because $\nabla \times \nabla g=0.$

(ii) Again using Stokes' Theorem we have

\[\iint_C(f\nabla g+g\nabla f)\cdot ds=\int_C\nabla (fg)\cdot ds=\iint_S\nabla \times (\nabla (fg)) \cdot dS=0,\]
since $\nabla \times \nabla \phi=0$, for any function $\phi$.

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$Applications of Stokes' Theorem$

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