This is related to my question http://math.stackexchange.com/questions/252742/transforming-the-dirac-operator-on-s1 on stack exchange which has not yet received an answer. For the purposes of this question, fix the spin structure over $S^1$ to be given by the connected double cover.

Sections of the bundle of spinors on $S^1$ can be thought of in two ways. First, by trivializing the bundle over $U_1 = S^1 \setminus \{i\}$ and $U_2 = S^1 \setminus \{-i\},$ sections can be considered as pairs of real valued functions $f_1, f_2$ defined on $\mathbb{R}$ such that $f_1(\frac{1}{x}) = f_2(x)$ when $x> 0,$ and $f_1(\frac{1}{x}) = -f_2(x)$ when $x< 0.$ Following Lawson/Michelson's "Spin Geometry," we can compute the Dirac operator locally. For example, $ f_1 \rightarrow i\frac{df_1}{dx}.$

We can also identify the sections as function $f: S^1 \rightarrow \mathbb{C}$ such that $f(-\theta) = -f(\theta)$ by the procedure described here http://math.stackexchange.com/questions/250835/expressing-the-sections-of-the-mobius-bundle-on-s1-as-globally-defined-real-v. When we identify sections this way, I have seen the Dirac operator expressed as $f \rightarrow -i\frac{df}{d\theta}.$ However I have never seen a derivation of this fact from the local expression given above. My attempt at the computation is contained in the stack exchange link at the top of the page.

Can anyone tell me how to translate between these 2 points of view?