Consider the following definition: (##M## denotes a manifold structure, ##U## are subsets of the manifold and ##\phi## the transition functions)(adsbygoogle = window.adsbygoogle || []).push({});

Def: A smooth curve in ##M## is a map ##\gamma: I \rightarrow M,## where ##I \subset \mathbb{R}## is an open interval, such that for any chart ##(U,\phi)##, the map ##\phi \circ \gamma : I \rightarrow \mathbb{R}^n## is smooth.

My first question is, why do we define a smooth curve in this way? In particular, why is the map ##\phi \circ \gamma## a good object to consider? The only thing that comes to mind is that now we have a function defined from ##\mathbb{R} \rightarrow \mathbb{R}^n## so differentiation is well defined and thus one may introduce the concept of a tangent vector (as below).

Now let ##f: M \rightarrow \mathbb{R}## be a smooth function on ##M## and ##\gamma: I \rightarrow M##, smooth curve as before. Then ##f \circ \gamma : I \rightarrow \mathbb{R}## is smooth. Hence we take a derivative to find the rate of change of ##f## along the curve ##\gamma##: $$\frac{d}{dt}f(\gamma(t)) = [(f \circ \phi^{-1}) \circ (\phi \circ \gamma)]'(t) = \sum_{i=1}^n \left(\frac{\partial (f \circ \phi^{-1})}{\partial x^i}\right)_{\phi(\gamma(t))} \frac{d}{dt} x^i(\gamma(t))$$

My next question is to simply understand how this equation comes about. I can see it is some application of the chain rule but I am struggling with the precise details of the equation, mostly in how the final equality comes about and the subscript on the ##\partial (f \circ \phi^{-1})/\partial x^i## term. Many thanks!

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# Curves and tangent vectors in a manifold setting

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