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I am reading the first chapter on Manifolds from the book Warner, in which I have the following doubt.

Let $M$ be a differentiable manifold. A $C^{\infty}$ mapping $\sigma:(a,b)\longrightarrow M$ is called a smooth curve in $M$. Warner says, if $v\in M_m$ (the tangent space to $M$ at $m$), then $v$ is the tangent vector to a smooth curve in $M$. He does this by saying, simply choose a coordinate system $(U,\phi)$ centered at $m$ for which $v=d\phi^{-1}\Big(\frac{\partial}{\partial r_1}|_0\Big)$. How does one do this?

[Here $d\phi^{-1}$ is the induced map on the tangent spaces by $\phi^{-1}:\mathbb{R^d} \longrightarrow M$, i.e. $d\phi^{-1}:\mathbb{R}^d_{\phi(m)}\longrightarrow M_m$, defined by $d\phi^{-1}(v)(g)=v(g\circ\phi^{-1})$, where $v\in\mathbb{R}^d_{\phi(m)}$ and $g$ is a $C^{\infty}$ function in the neighbourhood of $m$. And $\frac{\partial}{\partial r_1}|_0$ represents the normal partial derivative in $\mathbb{R}^d$.]

I tried to do the following. Let $(U,\phi)$ be any coordinate system about $m$, then tangent spaces $\mathbb{R}^d_{\phi(m)}$ and $M_m$ are isomorphic (since $\phi$ is a homeomorphism). Therefore, then there exists some $w\in\mathbb{R}^d_{\phi(m)}$, such that $v=d\phi^{-1}(w)$. Now I am guessing that I should find a homeomorphism of $\phi(U)$ with another subset of $\mathbb{R}^d$ such that $w$ corresponds to $\frac{\partial}{\partial r_1}|_0$ under that homeomorphism. But I am not sure how to do that? Any help will be appreciated. Thanks in advance!

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You are trying too hard. Just use straight-line segments in $R^d$ instead of general curves. –  studiosus Jan 31 at 7:07
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