TITLE: help in understanding tangent vectors
QUESTION [9 upvotes]: In Aaron's answer here...
"Given a manifold $M$, and a point $p\in M$, we have a vector space $T_pM$ of the tangent vectors to $M$ at $p$.  For example, if you take the hollow sphere sitting inside $R^3$, you can look at the plane that sits tangent to a point, and turn it into a vector space.  These tangent vectors act on functions by taking the directional derivative of a function at a point."
....this is from Aaron's answer in the link mentioned in the question. Here he says that any vector in the tangent plane to the sphere at a given point belongs to the tangent space at that point. fine.. but again he says that "These tangent vectors act on functions by taking the directional derivative of a function at a point." I do not understand the last statement and I'd appreciate some help. Thank you

REPLY [3 votes]: The following may help you (though it is specifically for $\mathbb{R}^n$):
The linear map $\phi$ defined by
$\phi : T_p (\mathbb{R}^n) \to D_p(\mathbb{R}^n)$
Given explicitly by
$v \rightarrow D_v = \sum v^i \frac{\partial}{\partial x^i} \Bigg|_p$
is an isomorphism between vector spaces, where $D_p$ is the vector spaces of all directional derivatives at a point $p$ in $\mathbb{R}^n$. The symbol $D_v$ means the directional derivative in the direction of a tangent vector at $p$.