By divergence Theorem, integral of div (F) dV (in region O) = integral of F‧n dS (surface of O) when the whole Omega region is differentiable.
n= unit vector at any point inside region O.
r= (x,y,z) in O.
Now, for a region O, all the points inside O are differentiable except (0,0,0). Suppose B is a sphere with radius epsilon > 0 and centre at (0,0,0)
And let P be the region that we obtain by removing B from O.
Thus Integral of F‧n dS (in surface of O = Integral of F‧n dS(in surface of P) + Integral of F‧n dS( in surface of B)
as B and O are not differentiable at (0,0,0) , we can't apply divergence theorem on these two regions. By applying the theorem to region P, we get
Integral of F‧n dS (region O)= Integral of divF dV (region P) + Integral of F‧n dS (region B)
thus, prove that
Integral r/(abs(r))^3 ‧n dS (region O) = 4pi
put F= r/(abs(r))^3
I calculated Integral of F‧n dS(region B) = 4 pi epsilon ^2 / (epsilon ^2) = 4 pi
But I have tried so hard but I can't prove the integral in region P to be zero.
Am I on the wrong track?