Surface integral computation
Okay, so this is a step in the proof to Liouville's theorem, and it comes from Fritz John's Partial Differential Equations 4th ed (p109, sec. 4.3). The problem is, it requires computation of a surface integral (I think), and I don't know how to do that.
Here's the setup:
Let be harmonic in with in the ball of radius about the origin. Let be the surface area of the unit sphere (i.e. , , and so on.)
We are also given the following identity:
Frankly, I'm not sure what is supposed to denote. I assume it's the "surface integral" with respect to x, but then I'm not sure what the definition of a surface integral is, much less how to evaluate them routinely.
Anyway, we have to use the above information to prove the following identity:
Any ideas on how to do this? Any help would be much appreciated!
Re: Surface integral computation