Problem:
Show that the value of $\Omega=\iint_{\Sigma}\dfrac{(a-x)dydz+(b-y)dzdx+(c-z)dxdy}{[(a-x)^2+(b-y)^2+(c-z)^2]^{3/2}}$ is independent of the choice of the surface $\Sigma$, provided its boundary $\Gamma$ is kept fixed.
First off, I have no idea how could two different surfaces have the same boundary. Any explanations, preferably with examples, are welcome.
Then, if I want to solve this problem using Stokes's Theorem, I should find a suitable normal vector and the curl of a vector function $F=(f,g,h)$. But all I've been doing are blindly taking stuff from the original integral to make up curl(F), but then I couldn't figure out what the regional function should be.
Using the divergence theorem, I was able to see that if $\Sigma$ is closed and we let F lie in $\Sigma$, the integral is 0. Other than that there's nothing I could think of. How do I prove this?