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The first is about directional derivatives in general. If one has a function [itex]f[/itex] defined in some region and one wishes to know the rate of change of that function (i.e. its derivative) along a particular direction in that region, is the reason why one specifies a curve along the direction one wishes to consider because the curve specifies the direction (in a sense)? That is, if we choose a curve [itex]\gamma[/itex] (parametrised by [itex]t[/itex]) along some direction in the region in which [itex]f[/itex] is defined, then we can evaluate the function along that curve by composing [itex]f[/itex] with [itex]\gamma[/itex], i.e. [itex]f\circ\gamma[/itex]. Then for each value of [itex]\gamma[/itex] we can evaluate [itex]f[/itex] at that point and as such, the rate of change of the function along the direction defined by the curve (at a particular point) [itex]\gamma[/itex] is given by [tex]\frac{d}{dt}(f\circ\gamma)[/tex] Would this be correct?

When it comes to defining tangent vectors on manifolds, is the point that we define a curve [itex]\gamma : (-\varepsilon, \varepsilon)\rightarrow M[/itex] such that a particular direction along the manifold, at a given point [itex]p\in M[/itex], is specified. Then we can consider function [itex]f:M\rightarrow\mathbb{R}[/itex] and evaluate this function along the curve [itex]\gamma[/itex] at the point [itex]p\in M[/itex]. This involves composing the function with the curve [itex]\gamma[/itex] and noting that [itex]\gamma (0)=p[/itex]. Then, [tex]\frac{d}{dt}(f\circ\gamma)\bigg\vert_{p}[/tex] which is the derivative of [itex]f[/itex] along a particular direction (specified by [itex]\gamma[/itex]) on the manifold at a given point [itex]p\in M[/itex]. We note that, in general, there will be more than one curve that will have the same tangent at a given point, and so we identify a tangent vector at the point [itex]p\in M[/itex] as an equivalence class of curves passing through [itex]p\in M[/itex] and satisfying [itex](\phi\circ\gamma_{1})'(0)=(\phi\circ\gamma_{2})'(0)[/itex] (where [itex]\phi[/itex] is some coordinate chart)?!

Adding to this, I was asked a question as to way we don't consider the parameter [itex]t[/itex] to be a one-dimensional coordinate system for the curve [itex]\gamma[/itex]?! My response was that [itex]t[/itex] simply parametrises a curve in [itex]M[/itex], each value of [itex]t\in(-\varepsilon, \varepsilon)\subset\mathbb{R}[/itex] is mapped to a specific point on the manifold, i.e. [itex]t\mapsto \gamma (t)=p\in M[/itex], however, this doesn't specify the actual location of the point on the manifold and therefore [itex]t[/itex] is not a coordinate; one requires a mapping from [itex]M[/itex] to [itex]\mathbb{R}^{n}[/itex] in order to specify the actual location of the point in terms of an [itex]n[/itex]-tuple of

*coordinate values*. I'm unsure whether this is a valid argument though?!

My second question is, given a function [itex]F: \mathbb{R}^{n}\rightarrow\mathbb{R}[/itex] is it valid to consider a curve [itex]\gamma :[0,1]\rightarrow\mathbb{R}^{n}[/itex] defined such that [itex]\gamma (0)=a\in\mathbb{R}^{n} [/itex] and [itex]\gamma (1)=x\in\mathbb{R}^{n} [/itex], and express it as [tex]\gamma(t)=(x^{1}(t),\ldots,x^{n}(t))=\gamma (0)+t\left(\gamma (1)-\gamma (0)\right)=a+t(x-a)[/tex] where the [itex]x^{i}:[0,1]\rightarrow\mathbb{R}[/itex] are coordinate functions defined by [tex]x^{i}(t)=a^{i}+t\left(x^{i}-a^{i}\right)[/tex] Then one can write [tex]\frac{d}{dt}\left((F\circ\gamma)(t)\right)= \frac{d}{dt}\left(F(\gamma(t))\right)=\frac{d}{dt}\left(F((x^{1}(t),\ldots,x^{n}(t)))\right) \\ \qquad\qquad\qquad\qquad\qquad\qquad\;\;\;=\sum_{i=1}^{n}\frac{\partial F(a+t(x-a))}{\partial x^{i}}\frac{dx^{i}}{dt} \\ \qquad\qquad\qquad\qquad\qquad\qquad\;\;\;=\sum_{i=1}^{n}\frac{\partial F(a+t(x-a))}{\partial x^{i}}\left(x^{i}-a^{i}\right)[/tex] and as [itex]x\in\mathbb{R}^{n} [/itex] was chosen arbitrarily, this result holds [itex]\forall x\in\mathbb{R}^{n} [/itex]. Would this be valid?