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Let $(M,S)$ be a $C^\infty$ $n$-dimensional manifold, where $S$ is atlas.

Let $p\in M, (U,\phi)\in S$ s.t. $p\in U.$

Let $\epsilon>0$ and $\gamma : (-\epsilon, \epsilon)\to M$ be $C^\infty$ curve s.t. $\gamma(0)=p, \gamma((-\epsilon, \epsilon))\subset U$.

And let $f$ be $C^\infty$ function defined on the neighborhood of $p$.

Then, prove $$\dfrac{d}{dt}\bigg|_{t=0}(f\circ \gamma)(t)=\sum_{i=1}^n \frac{\partial(f\circ \phi^{-1})}{\partial x_i}(\phi(p))\overset{\cdot}{x_i}(0),$$ where each $x_i$ is the component of $\phi$, i.e., $\phi=(x_1, \cdots, x_n)$.

First, I have to calculate $\dfrac{d}{dt}(f\circ \gamma)(t)$.

Let $F:=f\circ \phi^{-1}, G:=\phi\circ \gamma$, then $f\circ \gamma=F\circ G.$

$\dfrac{d}{dt}(f\circ \gamma)(t)=\dfrac{d}{dt}(F\circ G)(t)=(DF)(G(t))\cdot G'(t)$.

$(DF)(G(t))=\bigg(\frac{\partial F}{\partial x_1}(G(t)) \ \cdots \ \frac{\partial F}{\partial x_n}(G(t))\bigg)$.

So, letting $t=0$, I get $(DF)(G(0))=\bigg(\frac{\partial (f\circ \phi^{-1})}{\partial x_1}(\phi(p)) \ \cdots \ \frac{\partial (f\circ \phi^{-1})}{\partial x_n}(\phi(p))\bigg)$.

But I don't know how I should calculate $G'(0)$.

$G(t)=(\phi \circ \gamma) (t)$, but can I use chain rule ?

If I do so, I get $G'(t)=(D\phi)(\gamma(t))\cdot \gamma'(t)$, but how can I write this ?

$\phi$ is the function on $U$, and $U$ is not a subset of Euclid space so I cannot write $D\phi$ as $D\phi=(\frac{\partial \phi}{\partial x_1}\ \cdots \ \frac{\partial \phi}{\partial x_n})$.

Arctic Char
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    $G'(0)=DG_0(1)$, the value of $DG_0:\Bbb{R}\to\Bbb{R}^n$ on the real number $1$ (this is essentially the isomorphism between $\text{Hom}(\Bbb{F},W)\cong W$ for any vector space $W$ over a field $F$). So, $G'(0)=(G_1'(0),\dots, G_n'(0))$, and $G_i'(0)$ is abbreviated to $\dot{x}^i(0)$, by slight abuse of notation. Anyway, this question seems to be a duplicate of Clarifying the chain rule terminology in differential geometry calculuations. – peek-a-boo Jun 26 '22 at 03:53
  • If you have access to it, I would very strongly suggest taking a look at Michael Spivak's A Comprehensive Introduction to Differential Geometry, Volume I (pages 35-40 specifically), because Spivak really clarifies how the chain rule computations work, and how the notation fits together in the manifold setting. – peek-a-boo Jun 26 '22 at 04:02

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