Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions)
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!
Best Answer
On your first question, we want that differentiable manifolds serve as a generalization of surfaces of Euclidean space and of the Euclidean space itself. The diffeomorphism $\phi$ is a differentiable function from $U\subset M$ to $\mathbb{R}^n$ which possesses a differentiable inverse. Roughly speaking, it means that $M$ looks locally the same that $\mathbb{R}^n$. This way, we can define objects on manifolds throughout the Euclidean definition. In this sense, when we make the composition of the diffeomorphism $\phi$ with the curve on the manifold $\gamma$, we obtain a curve on the Euclidean space $\phi\circ\gamma$. Hence, it's natural to define $\gamma$ to be smooth when its associate Euclidean curve is smooth.
About your second question, I'll let you think a little more. Hint: pay attention on which functions you're using when you're applying the chain rule.