The point here is to use the Dirac's Delta function
so if i have the vector field F = R/r^3 where R is (x,y,z) and r is the magnitude of the vector R, then div(F) = 0 except for at the origin where F is not defined.
i want to show that the volume integral of div(F) over a sphere centered at the origin is 4pi. so using the divergence theorem i have the surface integral of F over the surface of the sphere = the volume integral of div(F) over the solid sphere. it is easy to show that the flux of F through the surface of a sphere is 4pi. but the problem is that F is not continuously differentiable at the origin and the statement of the divergence theorem (at least the one i know of) requires that the vector field F be defined on a neighborhood containing the solid sphere. but that is not true because F is not defined at the origin so i shouldn't be able to apply the divergence theorem right?
however it seems my physics book has overlooked this and came to the conclusion that the volume integral of div(F) is indeed equal to 4pi by the divergence theorem which i am confused about.
so is it that we define the divergence of R/r^3 as 4 pi delta?
or that we somehow derive it using the divergence theorem? this does not seem possible however since you cannot apply the theorem in the first place. so i'm guessing my physics book is just "fudging" when they derive this result using the divergence theorem.
In fact, we discuss this topic in my Electromagnetism course, very naive and physics style. We followed Wagness "Electromagnetic Fields" (by the way the book with most typos I have ever seen, at least the spanish edition), but Dirac's Delta was discussed in the classroom, I think that the book does not mention it.
I'm afraid I don't know the mathematical subtelties.