Proving Borsuk-Ulam with Stokes

Question

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem?

So I want to prove the following: Given a map $f\colon S^2\rightarrow \mathbb{R}^2$, show that there is $x$, such thath $f(x)=f(-x)$.

My idea is: Suppose there is no such $x$ define $$g\colon S^2\rightarrow S^1,x\mapsto\frac{f(x)-f(-x)}{|f(x)-f(-x)|}.$$ Now $g$ preserves antipodal points. I think one can get a contradiction somehow by looking at the 2-Form $dg_1\wedge dg_2$ and invoking Stokes theorem?

Answer 1

You can find an approach here: http://arxiv.org/abs/1205.4540