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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.

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    $\begingroup$ It sounds as though you have a good grasp of that is going on. I think perhaps what might have said that the machinery of tangent bundles, vector fields as sections thereof and then flows induced by the vector fields is what stands as a rigorous alternative to the Leibnitz conception. The implicit, seldom stated but always present, link between vector field and flow is the existence and uniqueness theorem (Picard-Lindelöf for $C^2$ manifold) in one direction, and the operation of differentiation (derivation) in the other. You don't need to imagine summing up "infinitessimal lengths" .... $\endgroup$ Sep 10, 2015 at 1:08
  • $\begingroup$ .... Want to ride to $\exp_p(X(p)\,1)$ from $p$? $X$ is your ticket and Picard-Lindelöf is the unfailing locomotive for your train. You know you'll get there, it is well and uniquely defined(although often messy to compute). Actually, as you point out, infinitessimal length is a problem in a general manifold and can't be given a meaning unless you think of the manifold as an embedding in a higher dimensional Euclidean space (there is no notion of "chord" approximating tangent), so we must instead look at how derivations act on scalar fields on the manifold to get to a notion of a vector field. $\endgroup$ Sep 10, 2015 at 1:15

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