i'm having trouble showing the smoothness condition for $p \in \partial B^n$.
my charts for such $p$ are wlog of the form $(U_i^+,\varphi_i)$, where $U_i^+ = \{ x \in \overline{B}^n : x^i \geq 0 \}$, and
$$ \varphi_i (x) = (\pi_i \circ \sigma^{-1} )(x)= \frac{1}{|x|^2 + 1} (2x^1, \ldots, 2x^{i-1}, 2x^{i+1}, \ldots, |x|^2 - 1) $$
(where $\pi_i$ is projection $R^{n+1} \to R^n$ that leaves out the $x^i$ coordinate; $\sigma$ is the stereographic projection.)
so i just need to show $$ I_{R^n} \circ i \circ \varphi_i^{-1}=\varphi_i^{-1} = \sigma \circ \pi_i^{-1} $$ is smooth. my issue is i'm having trouble computing $\sigma \circ \pi_i^{-1}$, which makes me think i'm approaching this question in the wrong way.