Frankly, I have never understood why this is such a roadblock. You don't want edges because the integrals don't behave well there. You prefer not to have complicated shapes because you have to worry about insides and outsides. So why make it complicated?
It's just a surface that surrounds a point, which is intuitively obvious enough for physics. Take further questions over to the math department.
It's okay to say "the flux of the electric field across a closed surface depends only on the total charge contained within it". The problem is when you try to make a mathematical statement out of that. tens of millions of people stand to see their lives ruined because the bureaucrats at the ECB don't understand introductory economics -- Dean Baker
The thing that always tripped me up was that nobody ever explained that the curl is just the direction a little submerged ball would spin...did not find that out until a couple of decades after taking E&M...
Or, you take it in engineering, then a couple of semesters later the physics department comes along and muddies the waters, then the math department later.
And I suppose the philosophy department would want to be in the discussion eventually, as well...
