Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, i.e. for every $x\in M$, there is an open subset $\Omega$ of $M$ with $x\in\Omega$ and a $C^1$-diffeomorphism from $\Omega$ onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. $(\Omega,\phi)$ is called a chart for $M$ and we define $$M^\circ:=\{x\in M:\exists\text{chart }(\Omega,\phi):\phi(\Omega)\text{ is }\mathbb R^k\text{-open}\}$$ and $$\partial M:=\{x\in M:\exists\text{chart }(\Omega,\phi):\phi(x)\in\partial\mathbb H^k\}.$$ $M^\circ$ and $\partial M$ are disjoint.
Now assume $k=d$ let $(\Omega,\phi)$ be any chart for $M$ and $U:=\phi(\Omega)$. We know that $$\phi(\Omega\cap\partial M)=U\cap\partial\mathbb H^d.\tag3$$
I was wondering whether a flow with velocity $v$ started in $\partial M$ remains in $\partial M$ as long as $v(x)\in T_x\:\partial M$ for all $x\in\partial M$ and asked a question how we can prove this intuitively obvious result (which I'm still not able to prove).
I've received an answer to that question, but I don't get how we obtain equation (iv) in that answer. It seems like it's been used that the pushforward$^1$ ${\rm D}_{v(x)}\phi(x)$ in an element of $U\cap\partial\mathbb H^d$.
I don't get why this should hold. So, my question is: How can we show (iv) and is there something we can generally say about ${\rm D}_v\phi(x)$ for $x\in\partial\Omega$ and $v\in T_x\:\partial\Omega$? I guess, it's an element of $T_{\phi(x)}\partial\mathbb H^k$, is that true? Can we consider $\mathbb H^k$ as a submanifold itself?
$^1$ I'm vaguely familiar with the notion of the pushforward: If $N$ is another embedded $C^1$-submanifold of $\mathbb R^d$ with boundary and $f:M\to\mathbb N$ is $C^1$-differentiable at $x\in M$, then we can show that if $v\in T_x\:M$ and $\gamma$ is a $C^1$-curve on $M$ with $\gamma(0)=x$ and $\gamma'(0)=v$, then $${\rm D}_vf(x):=(f\circ\gamma)'(0)\in T_{f(x)}\:N\tag1$$ is independent of the choice of $\gamma$. The map $$T_x\:M\to T_{f(x)}\:N\;,\;\;\;v\mapsto{\rm D}_vf(x)\tag2$$ is called the pushforward.