Suppose $M$ is a manifold and $f:M\to \mathbb{R}$ a smooth function. Also let $\alpha :I\to \mathbb{R}^n$be a representation of a curve and $(\phi,U)$ be a chart. Now why it holds that $$\frac{\partial f}{\partial x^\alpha}=\frac{d(f\circ \gamma)}{dt}$$ ?

Here, $\gamma(t)=\phi^{-1}\circ\alpha(t).$

  • $\begingroup$ Whats the expression in the LHS ? I think its incomplete. $\endgroup$ – Sou Dec 5 '17 at 15:31
  • $\begingroup$ @Sou燈馬想 It is the $\alpha$-th coordinate curve where $x^{\alpha}(\gamma(t))=t+const$ $\endgroup$ – user122424 Dec 5 '17 at 15:33
  • $\begingroup$ I'm sorry but i cant understand what you're asking. $\endgroup$ – Sou Dec 5 '17 at 15:36
  • $\begingroup$ @Sou燈馬想 Oh, sorry. Fixed now.$\partial$ was missing there. $\endgroup$ – user122424 Dec 5 '17 at 15:38
  • $\begingroup$ @Sou燈馬想 Is it OK now?I've added one line below the question. $\endgroup$ – user122424 Dec 5 '17 at 16:09

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