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$^1$.
Say that $(I,\gamma)$ is a $C^1$-curve on $M$ through $x\in M$ if $I\subseteq\mathbb R$ is a nontrivial interval with $0\in I$ and $\gamma:I\to M$ is $C^1$-differentiable with $\gamma(0)=x$. Let $$T_x\:M:=\{\gamma'(0):\gamma\text{ is a }C^1\text{-curve on }M\text{ through }x\}$$ denote the tangent space of $M$ at $x\in M$.
If $E$ is a $\mathbb R$-Banach space and $f:M\to E$ is $C^1$-differentiable at $x\in M$, let $$T_x(f)v:=(f\circ\gamma)'(0)$$ denote the pushforward of $v\in T_x\:M$ by $f$, where $\gamma$ is an arbitrary $C^1$-curve on $M$ through $x$.
I've found the following result and would like to know how we can prove it rigorously: Let $x\in\partial M$. Then $v\in T_x\:M\setminus T_x\partial M$ is called
- inward pointing if there is a $C^1$-curve $([0,\varepsilon,\gamma)$ on $M$ through $x$ with $\gamma'(0)=v$;
- outward pointing if $-v$ is inward pointing.
Now let $(\Omega,\phi)$ be a chart$^1$ of $M$ around $x$, $v\in T_x\:M$ and $w:=T_x(\phi)v$.
How can we show that $v$
- $v$ is inward pointing if and only if $w\in(\mathbb H^k)^\circ$;
- $v$ is outward pointing if and only if $w\not\in\mathbb H^k$;
- $v\in T_x\:\partial M$ if and only if $w\in\partial\mathbb H^k$.
Moreover, in any case, $v$ is either inward pointing or outward pointing or $v\in T_x\:\partial M$.
What's clear to me is that by definition of $T_x\:M$, $$v=\gamma'(0)\tag1$$ for some $C^1$-curve $(I,\gamma)$ on $M$ through $x$. Moreover, by definition of the pushforward, $$w=T_x(\phi)v=(\phi\circ\gamma)'(0)\tag2.$$
Now what's the idea of proving (4.), (5.) and (6.)? Is the idea that we can sufficiently "shrink" the interval $I$ if necessary? And why is it important the interval in the definition of "inward pointing" is of the form $[0,\varepsilon)$? Is this crucial?
$^1$ i.e. for each $x\in M$, there is an open subset $\Omega$ of $M$ with $x\in\Omega$ and a $C^1$-diffeomorphism of $\Omega$ onto $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Such a $(\Omega,\phi)$ is called a chart of $M$ around $x$. Denote the manifold boundary of $M$ by $\partial M$. Then $\partial M$ is a $(k-1)$-dimensional embedded submanifold of $\mathbb R^d$ (without boundary). Note that $\phi(\Omega\cap\partial M)=\phi(\Omega)\cap\partial\mathbb H^k$ and if $\pi$ denotes the canonical projection of $\mathbb R^k$ onto $\mathbb R^{k-1}$ with $\pi(\partial\mathbb H^k)=\mathbb R^{k-1}$, then $\left.\pi\circ\phi\right|_{\Omega\cap\partial M}$ is a $C^1$-diffeomorphism onto $\pi(\phi(\Omega)\cap\partial\mathbb H^k)$, which is an open subset of $\mathbb R^{k-1}$, and is called a chart of $\partial M$ around $x$.