Disjointness of derivations at different points in $\mathbb{R}^{n}$

Question

I know that in $\mathbb{R}^{n}$ a derivation at a point $a$ is a linear map $w \, \colon C^{\infty}(\mathbb{R}^{n}) \to \mathbb{R}$ satisfying the product rule $$w(fg) = f(a)wg + g(a)wf.$$ One then defines the tangent space to $\mathbb{R}^{n}$ at $a$ to be the set $T_{a}\mathbb{R}^{n}$ of all derivations at $a$. It turns out that such set is a vector space isomorphic to the so-called geometric tangent space $\mathbb{R}^{n}_{a} = \{a\} \times \mathbb{R}^{n}$. Here is where my confusion starts: with the above definition, I am not able to spot, given a geometric tangent vector $v_{a}$ at $a$, a tangent vector at $b \neq a$ which corresponds to the same derivation as $v_{a}$. Are $T_{a}\mathbb{R}^{n}$ and $T_{b}\mathbb{R}^{n}$ disjoint?

Answer 1

Yes, the tangent spaces at different points are disjoint. Using the linear structure of the vector space we identify them when convenient. Moreover when the base manifold is Euclidean space itself, such an identification can be described as natural or canonical. However when the base manifold if a more general space there is no natural way of identifying tangent spaces at distinct points. This is where the theory of connections starts.