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## Main Question or Discussion Point

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator [itex] X[\itex] acting on some function [itex]f:M\rightarrow\mathbb{R}[\itex] at a point [itex]p\in M[\itex] (where [itex]M[\itex] is an [itex]m[\itex]-dimensional manifold), [tex]\frac{df(c(t))}{dt}\Biggr\vert_{t=0}=X^{\mu}\left(\frac{\partial f}{\partial x^{\mu}}\right)\equiv X[f][\tex] (with [itex]c(0)=p[\itex]) that a vector [itex]X[\itex] exists without specifying the coordinate (i.e. it is coordinate independent).

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)?

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)?