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A textbook gives the following definition of tangent curves on a smooth manifold.

enter image description here p23

It then asks:

Show that the definition does not depend on the choice of local coordinates in a neighborhood of P.

My suggestion for a proof is as follows:

We have the already given chart map $x: M \to R^d$. Pick any arbitrary different chart map $y:M\to R^d$. Then the chart transition map $y\circ x^{-1}:R^d \to R^d$ is a smooth map (since both chart maps are smooth by assumption).

Then simply take the second equation in the definition of tangent curves and apply the chart transition map (which is possible since both sides of the equation are an instance of $R^d$ since they are derivatives in the charts, not on the manifold):

$$(y\circ x^{-1})^i\left(\frac{d}{dt}|_0x^i(\gamma _1(t))\right )=(y\circ x^{-1})^i\left(\frac{d}{dt}|_0x^i(\gamma_2(t))\right )$$ Edit: I see now that the above equation doesn't make sense, since it is a map $R^d\to R$, while its arguments are $R$.

Now here is the step I am most uncertain about:

Since the transition maps are smooth maps, they preserve the differential structure of the two charts. Hence we can shift the transition maps inside the differential operator:

$$\frac{d}{dt}|_0\left(((y\circ x^{-1})^i \circ x^i)(\gamma _1(t))\right )=\frac{d}{dt}|_0\left(((y\circ x^{-1})^i \circ x^i)(\gamma _2(t))\right )$$

Hence

$$\frac{d}{dt}|_0y^i(\gamma _1(t) )=\frac{d}{dt}|_0y^i(\gamma _2(t) )$$

This concludes the "proof". However, I feel like I have done something wrong, but I don't know what.

So is this "proof" correct, and if not, why, and how would I fix it?

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    It looks like you're mostly pushing symbols around without thinking about it... $\frac{d}{dt}|_0 x^i(\gamma_1(t))$ is a real number and $(y\circ x^{-1})^i$ is a function $\Bbb R^n\to\Bbb R$ so your first equation doesn't even make sense – 2017-02-06
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    Well I was pushing symbols around and **trying** to think about it, but failing. – 2017-02-06
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    But to respond to your point: $\frac{d}{dt}|_0x^i(\gamma_2(t))$ is indeed a number: the derivative w.r.t. $t$ of the $i$'th component of the chart representation of the curve $\gamma_1$ through the point $\gamma_1(t)$, correct? I misunderstood $(y\circ x^{-1})$ to be a map $R \to R^d$, but I see now that that doesn't make sense. – 2017-02-06

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