Let $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $(\Omega,\phi)$ be a $k$-dimensional $C^1$-chart$^1$ of $M$ and $$U:=\phi(\Omega\cap\partial M)=\phi(\Omega)\cap\partial\mathbb H^k\tag1.$$
Let $x\in\Omega\cap\partial M$, $v\in T_x\:\partial M$ and $^2$ $$w:=(T_x\phi)v.$$ Can we show that $w_k=0$?
Maybe it's obvious, but I guess it follows by a suitable characterization of $T_{\phi(x)}\:\partial\mathbb H^k$. The desired claim seems to be used in equation (iv) of this answer and I absolutely don't get why it holds.
$^1$ i.e. $\Omega$ is an open subset of $M$ and $\phi$ is a $C^1$-diffeomorphism of $\Omega$ onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$.
$^2$ If $M_i$ is an embedded submanifold of $\mathbb R^{d_i}$ with boundary and $f:M_1\to M_2$ is $C^1$-differentiable at $x\in M_1$, let $$T_xf:T_x\:M_1\to T_{f(x)}\:M_2$$ denote the pushforward.