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Let $γ : I→R^3$ be a regular parametrization of a curve C. If asked what a vector field on C is I would perhaps answer like this:
1) "a smooth function $v$ associating to any point $γ(t)$ of C an element of the tangent space of $R^3$ at $γ(t)$".
An alternative answer: 2) "a smooth function $f$ associating to any t $∈$ I an element of the tangent space of $R^3$ at $γ(t)$".
My question: is the second answer technically wrong? Thanks.
In fact, I'd prefer the second definition - it's what Lee uses in his Riemannian Manifolds book:
This allows a curve to intersect itself at a point in different directions while being able to record this fact in the velocity vector. I think if one takes seriously the idea that "curve" refers to the map $\gamma:I\to M$, and not its image $\gamma(I)\subset M$, then this is the right definition.
However, I'd say your first definition is appropriate if you're talking about vector fields on the subset $\gamma(I)$, having forgotten about $\gamma$. From Lee's Smooth Manifolds book:
