A Nice Representation of the Laplacian

Nice theorem and write up. Already the one dimensional case is something interesting called “The second symmetric derivative”. And I think it might be used to prove the general case directly: If you add up that result for n orthogonal directions then the left hand side is the Laplacian. The right hand side is a sum of a n limits that at first glance seems to depend heavily on the picked direction, but they can’t as the left hand side is independent of the picked directions. We are free to pick arbitrary many different directions and take the average. In the limit it becomes the average over a sphere but the order of limits is wrong, some standard theorem might apply to resolve that?

Nice. My immediate instinct is to do this with the divergence theorem. The Laplacian is the divergence of the gradient, so as soon as you say "average value on a shell" I'm immediately getting flashbacks to Physics 260 and thinking "well the integral of the Laplacian inside the shell is equal to the integral of the gradient through the shell - is there some way to get the normal component of the gradient out of the difference between the value at the center minus at the surface?"

You made a mistake in the third from bottom line of page four: there is one $\Gamma \left(n/2\right)$ too many in a denominator. One should go into a numerator so it can cancel on the next line. 

Thanks for this by the way, it is quite cool. Sort of reminds me of the interpretation of the Ricci Tensor interpretation, so I'm guessing this generalises to other geometeries without too much trouble.