Clarifying the definition of the directional derivative of a curve I am reading Kuhnel's Differential Geometry - Curves, Surfaces, Manifolds and I feel like I am missing something about the definition of directional derivative (4.1). Kuhnel defines it as: Let $Y$ be a differentiable vector field defined on an open set of $\mathbb{R}^{n+1}$, and let $(p, X) \in T_p\mathbb{R}^{n+1}$. Then the directional derivative of $Y$ in the direction $X$ at $p$ is (the vector) $D_XY|_p := DY|_p(X)$, where $DY|_p$ is the Jacobian matrix of $Y$ at $p$.
A few pages later, Kuhnel begins talking about geodesics and mentions the following: "If $c(t)$ is the motion of the mass of a particle, then $D_{\dot{c}}\dot{c} = \ddot{c}$ is just the acceleration vector in Euclidean space." But $\dot{c}$ is a curve, not a vector field, so how is the expression $D_{\dot{c}}\dot{c}$ even defined? Even if we consider $\dot{c}$ as a map $c(t) \mapsto \dot{c}(t)$ (i.e. vectors based at points on the curve), it is not defined on an open set of $\mathbb{R}^{n+1}$, so I don't really know how to calculate $D_{\dot{c}}\dot{c}$.
 A: Let $X,Y:\Bbb{R}^n\to\Bbb{R}^n$ be vector fields. Given $p\in \Bbb{R}^n$, we have
\begin{align}
(D_XY)(p)&:=DY_p[X(p)]
\end{align}
where $DY_p:\Bbb{R}^n\to\Bbb{R}^n$ is the (Frechet) derivative of $Y$ at $p$ and $X(p)$ is the vector at $p$ due to the field $X$. Now, using the chain rule, we can rewrite this in another way: if $I\subset \Bbb{R}$ is any open interval containing the origin and $\gamma:I\to \Bbb{R}^n$ is a smooth curve such that $\gamma(0)=p$ and $\gamma'(0)=X(p)$, then the above equation can be written as
\begin{align}
(D_XY)(p)&:= DY_p[X(p)]\equiv DY_{\gamma(0)}[\gamma'(0)]=(Y\circ \gamma)'(0)
\end{align}
(first equality is by definition, second is a rewriting due to our assumptions, last is due to chain rule). The significance is this equation is that on the LHS, the very definition of the symbol $D_XY$ requires vector fields $X,Y$ defined on open subsets of our manifold (in this case $\Bbb{R}^n$). But, the RHS shows that actually we don't really need so much information.
All we need is a smooth curve $\gamma$ and a "smooth" vector field $Y$ which is defined along $\gamma$ in order to calculate directional derivatives. For more details, and also more generality, I would suggest you take a look at John Lee's Riemannian Manifolds particularly chapter 4.
So, in the expression $D_{\dot{c}}\dot{c}$, it is a-priori strictly speaking an abuse of notation and is illogical because it doesn't conform to the definition given. However, after making these observations, we realize that this is a justified (and convenient) abuse because we don't need the information of the vector fields on the whole open set. So, once you "extend you definitions", this should make sense again.
