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.
Thanks
Best Answer
You're right that if the surface has to fit into 3D, and if it is non-self-intersecting, then it has to be orientable.
But if you allow the 2D surface to self-intersect or go to the 4th dimension, there are many counterexamples. Klein Bottle is the simplest one.
Take a bottle, heat it up, and push the throat through the body, to connect it with a hole that you prepared at the bottom of the bottle. This manifold is equivalent to a $Z_2$ orbifold of a torus - where the $Z_2$ generator shifts by half a period in one direction of the torus and reflects the other direction of the torus.
Your rule (1) fails because there is no well-defined interior and exterior. Indeed, if you make trips around the Klein bottle that reverse the orientation, it inevitably exchanges the interior with the exterior as well. Note that if you enter the hole at the bottom of the bottle (see the picture), you're still outside, but as you travel through the throat, it becomes clear that you have gotten inside the "object" as well, so there's no distinction between the exterior and interior.
The Klein bottle is self-intersecting if embedded into three dimensions. Alternatively, you may avoid the intersections if one of the pieces of the surface that would intersect each other are shifted in a new, fourth dimension of space.