A detail in the proof orientable manifold admits exactly two orientations

Question

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, $\alpha\in I$, $\beta \in J$ $$ v(p):=\frac{|J(\phi_\alpha\circ\psi^{-1}_\beta)_{\psi_\beta(p)}|}{J(\phi_\alpha\circ\psi^{-1}_\beta)_{\psi_\beta(p)}} $$ I have to notice that $v$ is a smooth function...