Use of the phrase "tangent vector of a curve" Let us understand a curve as a differentiable map $f : J \to \mathbb R^n$ defined on an open interval $J \subset \mathbb R$. The derivative $f'(t_0)$ of $f$ at $t_0 \in J$ is given as the vector $(f'_1(t_0),\ldots,f'_n(t_0)) \in \mathbb R^n$ where the $f_i : J \to \mathbb R$ are the coordinate functions of $f$.
I think for $n > 1$ it is usual to say that $f'(t_0)$ is the tangent vector to the curve $f$ at the point $t_0$ or  the velocity vector of the curve $f$ at the point $t_0$.
For $n = 1$ one can find the wording velocity vector, but I have never seen that $f'(t_0)$ is called the tangent vector of $f  :J \to \mathbb R$ at $t_0$. The definition of the derivative $f'(t_0)$ as the limit $\lim_{t \to t_0}\dfrac{f(t)-f(t_0)}{t-t_0}$ is nevertheless motivated by the concept of tangent by saying that $f'(t_0)$ is the slope of the tangent of the graph $G(f) = \{(t,f(t)) \mid t \in J \} \subset \mathbb R^2$ at the point $(t_0,f(t_0))$. There is also a notational relation to the case $n > 1$: If we consider the curve $\bar f : J \to \mathbb R^2, \bar f(t) = (t,f(t))$, then we get $f'(t_0)$ as the second coordinate of the tangent vector $\bar f'(t_0) \in \mathbb R^2$.
Finally, if we consider a smooth ($C^\infty$) curve $f$ and a point $t_0 \in J$ such that $f'(t_0) \ne 0$, then $M  = f(J)$ is locally around $p_0 = f(t_0)$ a smooth one-dimensional submanifold of $\mathbb R^n$. It has a tangent space $T_{p_0}M$ at $p_0$ which we may regard as a one-dimensional linear subspace of $\mathbb R^n$ and all $v \in T_{p_0}M$ are called tangent vectors at $M$ at $p_0$. This suggests that all scalar multiples of the tangent vector $f'(t_0)$ can also be regarded as tangent vectors (which appears reasonable to me).
I find this notationally confusing. The word "tangent vector" seems to have various different interpretations, but in the most elementary case $n = 1$ it is not used. Would it be better to avoid using the name "tangent vector" for $f'(t_0)$, but to use the unambiguous "velocity vector"? Perhaps somebody can help me to clarify my disorientation.
 A: I think this comes from differential geometry. Let's say you have a smooth manifold $M$ embedded in $R^n$, for example the sphere. Then one defines the tangent space at as the vector space of directions tangent to $M$. For example if
$$
M = F^{-1}(0)
$$
For some nice function $F : \mathbb R^n \to \mathbb R^m$ then the tangent space will be the kernel of the map
$$
v \mapsto \nabla_vF
$$
Where $(\nabla_v F)_i = \sum\frac{\partial F_i}{\partial x_j}v_j$. These are the directions you can move in while staying on the manifold, that is, the tangent space.
A curve is a one dimensional manifold, and so the tangent space at a point in the curve is just a one dimensional vector space (assuming nonzero velocity) and hence the tangent vector is the vector which spans this tangent space.
A: I think your confusion comes from mixing two different situations where the phrase "tangent vector" occurs. These are

*

*Tangent vectors of curves $f : J \to \mathbb R^n$.


*Tangent vectors at points of smooth submanifolds $M \subset \mathbb R^n$.
Perhaps the concept of curve is a bit ambiguous, sometimes it is used in the sense of a (differentiable) map $f : J \to \mathbb R^n$ and sometimes in the sense of a certain subset of some Euclidean space. The latter is often given as the image of some $f : J \to \mathbb R^n$, but it can also have an implicit description as the set of solutions of an algebraic equation (for example, $x^2(x+a) + y^2(x-a) = 0$ yields the strophoid in the plane). Curves in the subset sense need not be one-dimensional submanifolds of $\mathbb R^n$, they may e.g. have "self-intersections". But locally they are one-dimensional submanifolds of $\mathbb R^n$ and therefore it suffices to discuss 1. and 2.
Let us look at 1. As you say

For  $n > 1$ it is usual to say that $f'(t_0)$ is the tangent vector to the curve $f$ at the point $t_0$ or  the velocity vector of the curve $f$ at the point $t_0$.

The concept of velocity vector is related to the physical interpretation via a moving particle. The concept of tangent vector is more geometrical, and in fact one might argue that calling $f'(t_0)$ the tangent vector to $f$ at $t_0$ is a bit too possessive because it seems to claim that $f'(t_0)$ is the only tangent vector. But of course we get a tangent line to $f$ at $p_0 = f(t_0)$ and all non-zero scalar multiples of $f′(t_0)$ can also be regarded as tangent vectors. Clearly $f'(t_0)$ spans the one-dimensional tangent line, but does it have a more convincing unique feature? Probably not from the geometric point of view, but clearly from the functional point of view. Recall that $f$ encodes much more information than its image $f(J)$. The parameterized affine line $\tau(t) = f(t_0) + (t-t_0)f'(t_0)$ is the best approximation of $f$ at $t_0$.
This is not visible when looking globally at the tangent line and $f(J)$, but in terms of functions it is obvious. And that is the difference between 1. and 2.: For one-dimensional submanifolds it does not make sense to pick out a single tangent vector, all are on a par.
I suggest that you accept the phrase "the tangent vector" of a curve as a standard notation used only in context with parameterization. Doing so, each tangent vector $v$ at a point $p$ of a smooth submanifold $M \subset \mathbb R^n$ occurs as the tangent vector $f'(0)$ of a curve $f : J \to \mathbb R^n$ with $0 \in J, f(0) = p$ and $f(J) \subset M$. Of course $f$ is not uniquely determined by $v$, but you certainly know that this relation between tangent vectors $v$ and curves $f$ can be used to define the tangent space $T_pM$ of an abstract manifold $M$ via equivalence classes of curves through $p$.
Concerning the case $n = 1$:
The concept of "tangent" was originally only used for plane curves. Heuristically

the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

This is why the derivative of a real-valued function $f$ was introduced by considering its graph which is a special plane curve.  Anyway, the concept of tangent easily generalizes to curves $f$ in $\mathbb R^n$ with arbitrary $n > 1$. The tangent to $f$ at  $t_0$ is the parameterized affine line which is the best approximation of $f$ at $t_0$. As the curve itself this affine line is a one-dimensional object embedded in a higher-dimensional Euclidean space which makes it non-trivial to associate tangents to curves.
For $n = 1$ it is fairly pointless to consider tangent lines since the real line $\mathbb R$ itself is the only line contained in $\mathbb R$. Thus each curve in $\mathbb R$ has the real line as its tangent line in all of its points, which is a triviality. This is certainly the reason why one usually does not call $f'(t_0) \in \mathbb R$ tangent vector. But formally we can of course do it.
