We had a lecture a few weeks back, looking at Gauss' divergence theorem, and in the definition of the theorem, it specified that the boundary of the volume under consideration, S, had to be a 'piecewise smooth, orientable, closed surface'.
What bothers/intrigues me is that I cannot understand how a closed surface in 3D space CANNOT be orientable. Surely every closed surface is orientable!
My highly non-rigorous, intuitive argument runs as follows:
1) As the surface is closed, we can define two regions, one inside the surface, and one outside
2) We can construct a normal to the surface at any point P that is pointing towards the inside region. Thus the direction of the normal is defined for every point.
3) As the surface is piecewise continuous, this normal will vary continuously.
4) Coupling (2) (defined direction of normal) with (3) (continuously changing normal) gives us an orientation for the closed surface.
5) Therefore every closed surface is orientable.
But of course, the precise wording of the statement for Gauss' Law strongly suggests that people far smarter than me have discovered some exotic non-orientable, closed surface. Is this true?
When I asked my lecturer about this, he just smiled and said he didn't know any examples, but that they do exist, and then said something even more tantalising about 'reflections of higher dimensional objects'
I would love it if anyone could shed some light on my situation.