Solving Vector Field Flux with Gauss & Stokes

[galipop]

Hi folks,

I'm working on the following problem...

Show that the flux of the vector field \nabla \times A through a closed surface is zero. Use both Gauss and Stokes.

Where can I begin?

Thanks...

 

[matt grime]

From the definitions. What is the flux over a closed surface, and what do the mentioned theorems allow you to do to integrals etc, and what identities do you know that might be useful. write them ot and use them.

 

[galipop]

Ok I started using this...

\nabla\times A = (\frac{\partial r}{\partial y}-\frac{\partial q}{\partial z})i + (\frac{\partial p}{\partial z}-\frac{\partial r}{\partial x})j+(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y})k

Then...Gauss theorem

\int_V (\nabla \bullet A) dV

So do I basically substitute the top eqn into Gauss's theorem?

Thanks

 

[speeding electron]

Yes - and I don't think you need Stokes' theorem in this.

 

[DarkEternal]

the easiest way i can think of is taking a closed surface and splitting it into two surfaces with a single curve, then applying what you know about Stokes's theorem to the necessary values of the flux on the two surfaces. for example, consider a spherical surface, and then choose the equator as the defining curve. almost no equations needed, just plain logic. of course you could do the substitution of the derivatives and then show that the expression in the integral is zero using Gauss's theorem, but that's more writing (depending on if your teacher makes you prove the vector identity). i think that the first explanation is more "physically" satisfying.