This is an exercise from Spivak's Calculus on Manifolds, problem 4-27.
Define the singular 1-cube $c_{R,n}:[0,1]\rightarrow \mathbb{R}^2 - \{0\}$ to be $c_{R,n}=(R\cos(2\pi nt), R\sin(2\pi nt))$. Geometrically, this is a circle that winds $n$ times around the origin. A prior problem was to show that if $c$ is a singular 1-cube in $\mathbb{R}^2 - \{0\}$ with $c(0)=c(1)$ (a closed curve), then there exists some $n$ with $c-c_{1,n}=\partial c^2$, for some 2-chain $c^2$. I have solved this problem.
Spivak defines this $n$ to be the winding number of the curve.
My question is: how do I use Stokes theorem to show this $n$ is unique? I think if you assume you have $n_1$ and $n_2$ that both work, then integrating the angle form over a difference of chains involving $n_1$ and $n_2$ works, but I can't quite work out the details.