Different books seem to define the "patches" of a smooth manifold differently.
One example is defining an $n-$ dimensional smooth manifold $M$ by charts which are homeomorphic maps $\phi_a : U_a \to R^n$ where $U_a$ are open sets in $M$ where the change of basis $\phi_b \circ \phi_a^{-1}$ to be smooth.
On the other hand, some books define injective maps $f_a: V_a \to M$ where $V_a$ is open in $\mathbb{R}^n$ with $p \in M$ and $p \in f_a(V_a)$ to be a parametrization.
Are these maps inverses? Because $f_a$ is not necessarily continuous and not bijective. The vocabulary here just confuses me sometimes.