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I have the following question:

Prove that $$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =0$$ for any closed surface $S$ and twice differentiable vector field $\vec F:\mathbb{R^3} \to \mathbb{R^3} $ .

I need to prove this using Stokes' theorem.

The only thing I want to verify is whether or not for every closed surface $S$, we have: $$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =\int_C \vec F \cdot d\vec r$$ and the last term is trivially zero, because $C=\emptyset $ ($S$ is a closed surface).

Is this correct?

Thanks in advance

Asaf Karagila
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czash
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2 Answers2

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Yes you can use Stokes theorem but as well you can use Gauss(divergence) theorem

$$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =\iiint_{\text{Interior}(S)} \nabla \cdot ( \nabla \times \vec{F}) dV$$

But divergence of curl is identically zero ie

$$ \nabla \cdot ( \nabla \times \vec{F}) = 0$$

tom
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Yes. Correct. Here's an alternative proof:

Choose $C$ to be a band across the surface, like an equator. So now you have two surfaces. So divide the surface integral into two: $$\iint\limits_{S_1}\left(\nabla\times\vec f\right)\cdot \hat n\mbox{ d}S+\iint\limits_{S_2}\left(\nabla\times\vec f\right)\cdot \hat n\mbox{ d}S$$

Since the two surfaces have opposite orientations, they cancel out (alternatively, you may use Stoke's theorem to convert them to line integrals with opposite directions). enter image description here

Abhimanyu Pallavi Sudhir
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  • Wait, what is wrong with my formulation of the solution? Isn't a closed surface is one that has no boundary ? :O – czash Jun 22 '13 at 19:45
  • Can you please explain in a little bit more detail what do you mean by "now you have two surfaces $S_1,S_2$ ? Thanks – czash Jun 22 '13 at 19:52
  • @czash: Nothing is wrong with your solution. I was just giving another (perhaps more rigorous) solution. Bye the "2 surfaces", I mean, that when you are letting $C$, be a band across the closed surface, it cannot be a boundary to the closed surface. Instead, it works as a boundary to two different surfaces. I'll add a picture to my post to make it more clear. – Abhimanyu Pallavi Sudhir Jun 23 '13 at 02:35
  • Got it, Thanks ! – czash Jun 23 '13 at 16:19