it is clear (geometrically) that is parallelizable because we can consider a vector field of unit tangent vectors at each point (say, in anti-clockwise direction).

I am having trouble understanding how to write down the above argument mathematically (Doh) ....we need to come up with a smooth vector field , so that .

what is the choice for a smooth function to have the vector field as above??

can we consider the following (coordinate) vector field ??