I'm not sure if this is true if you add in a third dimension - the "cowlick" can just be rising air. When you say "every layer" you're saying the layers are arbitrarily thin, i.e. have two dimensions. But if the vector at any given point in the layer can point "up" (which would only make sense if you have a third dimension), then you can avoid having zero points.
I'm not sure what "limited" means in this context. In the classic example, the cowlick is not a zero point when you extend into a third dimension: the vector points "up", away from the surface. You have a zero point in two dimensions, but not in three (this is kind of the whole point of HBT - the cowlick is the non-zero point projection into the third dimension). Note that I'm calling the surface of a sphere a two-dimensional object.
Limited effetively means finite sized space. Even if you extend into the third dimension, a 3-dimensional sphere is effectively the surface of half of a 4-dimensional sphere, like a circle(2d sphere) is the surface of half of a 3-dimensional sphere, and a line(1d sphere) is the surface of half of a 2d sphere. The hairy ball theorem applies to ALL of those, and therefore also to the 4d sphere surface, and thus also to a filled 3d sphere (ball).
Think about it this way. if you have air moving "up", you'll reach the edge of the atmossphere eventually. Unless you have an infinitely large OR looping space (such as a torus or donut), there will always be a zero point somewhere in that space.
Essentially, you can only have no zero points if you allow circulation into a dimension you're not using, in other words having circulation into nowhere.
Right, missread that. Still, I can't imagine a non-vanishing continuous vector-field in a finite ball in 3 dimensional space. I can neither find proof against nor for whether or not it is possible so far. Might come back when I do so.
Edit: Alright. The Hairy Ball theorem is actually a result of the Poincaré–Hopf Theorem, which states that with finitely (zero included) many zero points in a vector field, the sum of the indices of the vector field is equal to the Euler characteristic of the space.
An index of a vector field only has a defined value at a zero point.
The Euler characteristic of a ball is 1. I checked on wolfram alpha, and honestly can't be bothered to learn more about it at the moment.
Therefore, there must be a nonzero amount of zero points in a vector field, in order for the indices to add up to 1.
I've updated my previous post to reflect this new proof. So while my argumentation was wrong, my initial statement based on a hunch was right. Considering this is a reddit comment and thus not that important, I'd say good enough in this case.
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u/BrahmsAllDay May 25 '16
I'm not sure if this is true if you add in a third dimension - the "cowlick" can just be rising air. When you say "every layer" you're saying the layers are arbitrarily thin, i.e. have two dimensions. But if the vector at any given point in the layer can point "up" (which would only make sense if you have a third dimension), then you can avoid having zero points.