Simple Stokes' Theorem/surface integral problem

**limddavid** · January 20th 2011, 05:50 PM

I totally forgot how to use the Stokes' Theorem... I know this problem is really simple, but I'm stuck

"Evaluate the surface integral of r dot dr by Stokes' Theorem."

Thank you!

**Kalter Tod** · January 20th 2011, 08:06 PM

Stoke's Theorem states that the circulation around any closed path is equal to the volume integral of the curl created by an arbitrary surface bounded by the path.

Therefore, you can take the curl of the function (which in this case is simply $r$ in spherical coordinates.)

The formula for which, by the way, can be found at the following link:
http://www.mas.ncl.ac.uk/~nas13/mas251/handout5.pdf

I didn't take the time to work out the curl (you should do this, anyway) because I already know what the final answer will be.

Nonetheless, once you have your answer, you are ready to the do the volume integral.

The outward surface normal for a sphere is the $\hat{r}$ direction.

Therefore, the integral officially becomes

$\int^0_{2\pi} \int^\frac{\pi}{2}_\pi (\nabla \times r)r^2\sin\theta d\theta d\phi$

You'll see I've integrated over a half sphere. I chose these limits because the half sphere is the appropriate volume enclosed by the full circulation around the path $dr$

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