Identity proof using Stoke's Theorem

[pretzsp]

Homework Statement

Show using Stoke's Theorem that

S is an open surface with boundary C (a space curve). f(\vec r) is a scalar field.

Homework Equations

Stoke's theorem \iint_S (\nabla\times \vec F) \cdot d\vec S = \int_C F \cdot d\vec r

The Attempt at a Solution

Thus far I've tried evaluating (\nabla f)\times d\vec S by taking components in various ways and then trying to force it to become something that I can take the curl of... with no success at all. I'm completely stumped now.

 

[gabbagabbahey]

Try applying Stokes' theorem to the vector function \textbf{F}\equiv f\textbf{c}, where \textbf{c} is any constant vector.

 

[pretzsp]

Well, what I get here is

\iint_S \vec c\cdot\nabla f \times d\vec S = -\oint \vec c \cdot fd\vec r

And I'm not entirely sure whether I can just "cancel" the c dot product...

(should be a vector c there, by the way)

 

[gabbagabbahey]

Well, since \textbf{c} is a constant vector, you can certainly pull it out of the integrals and say

\vec c\cdot\left(\iint_S \nabla f \times d\vec S\right) = \vec c\cdot\left(-\oint fd\vec r\right)

\implies \vec c\cdot \left(\iint_S \nabla f \times d\vec S+\oint fd\vec r\right)=0

And since this is true for arbitrary, constant \vec{c}, what must you conclude?

 

[pretzsp]

Yeah, those last steps were fine for me, I just wasn't sure if that even with c a constant vector that it was "allowed" to just pull it out like so. Thanks for the help.