- #1

- 601

- 7

Is this the case because the far left-hand side the derivative of [itex]f[\itex] with respect to the parameter [itex]t[\itex] is coordinate independent (as it depends on an equivalence class of curves [itex][c][\itex] on [itex]M[\itex] parameterised by some real parameter [itex]t\in (a,b)\subset\mathbb{R}[\itex], where [itex]a<0<b[\itex] for convenience, defined by [itex]c:(a,b) \rightarrow M, t\mapsto c(t)[\itex], with the equivalence relation [itex]c\sim \tilde{c}[\itex] defined such that [itex]c(0)=p=\tilde{c}(0)[\itex] and [itex]\frac{dx^{\mu}(c(t))}{dt}\Biggr\vert_{t=0}=\frac{dx^{\mu}(\tilde{c}(t))}{dt}\Biggr\vert_{t=0}[\itex] which are themselves coordinate independent)?