I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at $p\in M$ to be a derivation on $C_p^{\infty}(M)$. He says that although it may not look it, this in fact makes rigorous the notion of an infinitesimal displacement. The way it's made rigorous is with vector fields. For a global flow $\phi:\mathbb{R}\times M\to M$, fix $q\in M$. Then $\phi_t(p)$ is an integral curve (starting at $p$) of some vector field $X$ (i.e. $\phi_t(p)'$=$X_{\phi_t(p)}$), called the infinitesimal generator of of the one-parameter group $\phi$. Conversely, starting with a vector field $X$, we can find an integral curve for $X$ which amounts to, at least locally, solving a system of ODEs.
I don't really see how this makes rigorous the notion of a displacement of "infinitesimal length", especially since we haven't given any notion of length on an arbitrary smooth manifold. This is the picture I have in my head: Instead of an abstract manifold, let $M=\mathbb{R}^2$ and $C$ be some smooth curve embedded in $\mathbb{R}^2$. In undergraduate math/physics, you might want to compute the work along $C$ is some force field $F$. You would take $\int_C F\cdot dl$, where $dl_p\in T_p\mathbb{R}^2$ for each $p\in C$. $dl$ is a vector field on $\mathbb{R}^2$ and in undergraduate courses, we imagine $dl_p$ has infinitesimally small length. But I just don't see where this infinitesimally small length notion comes from. $dl_p$ is an element of the tangent space at $p$ (so just a vector in $\mathbb{R}^2$ with base point $p$), so is it that we first start with some arbitrary tangent vector field $v$ by assigng to $p\in C$ some element of $T_p\mathbb{R}^2$, called $v_p$, which is tangent to $C$, and then construct the special vector field $dl$ by scaling $v_p$ by something "close to $0$", say $\alpha$, and then set $dl_p:=\alpha v_p$? This isn't very clear to me.
