Someone knows where can I find the proof of the following theorem?
Let be $S$ a surface in $\mathbb{R}^3$ and $R \subset S$ a simply connected region. Let be $(e_1, e_2, e_3)$ an adapted frame defined on R (which changes as $P \in R$ changes). Let be $\phi$ the 1-form which measures the "infinitesimal angle" (or the change of angle) between $e_1$ and the unit tangent vector of $\partial ^+R$. Then
$\int _{\partial ^+R}\phi = 2\pi$
I know that it is similar to the theorem for plane curves, but there you have a fixed frame of $\mathbb{R}^2$, here you have a moving frame on the surface.
Thanks in advance