I am following Bott & Tu, Differential forms in Algebraic Topology, section 6 about Euler class and Thom class. However, I am stucked at a technical detail. To be more specific, let $\pi: E \to M$ be an oriented vector bundle of rank $2$ of smooth manifolds and $E^0$ the complement of the zero section. Fixing a Riemannian structure on $E$ and choose an oriented trivializations $(U_{\alpha})$. The authors observe:
... over each $U_{\alpha}$ we can choose an orthogonal frame. This defines on $\pi^{-1}(U_{\alpha})=E_{\mid U_{\alpha}}$ polar coordinates $r_{\alpha},\theta_{\alpha}$; if $x_1,...,x_n$ are coordinates on $U_{\alpha}$, then $\pi^* x_1,...,\pi^* x_n, r_{\alpha},\theta_{\alpha}$ are coordinates on $E^0_{\mid U_{\alpha}}$. On $U_{\alpha} \cap U_{\beta}$ the angular $\theta_{\alpha},\theta_{\beta}$ differ by a rotation so due to the orientability of $E$ we can write $\theta_{\beta}=\theta_{\alpha}+\pi^*\phi_{\alpha \beta} (0 \leq \phi_{\alpha \beta} < 2\pi)$.
My intuition, at least in some trivial cases, I could figure out what the authors mean here but can not write down everything explicitly. Could someone help me to write the formulas of $r_{\alpha},\theta_{\alpha},\phi_{\alpha \beta}$ and explain why $\pi^* x_1,...,\pi^* x_n, r_{\alpha},\theta_{\alpha}$ are coordinates on $E^0_{\mid U_{\alpha}}$?
If the $\phi_{\alpha \beta}$'s are well-defined then there exists $1$-forms $\zeta_{\alpha}$ on $U_{\alpha}$ such that $d\phi_{\alpha \beta} = 2\pi(\zeta_{\alpha}-\zeta_{\beta})$ and hence the $\zeta_{\alpha}$'s can be glued to a $2$-form on $M$, denote $e \in H^2(M)$ which is called the Euler class of this vector bundle.