TITLE: Motivation behind the definition of tangent vectors
QUESTION [17 upvotes]: I've been reading the book Gauge, Fields, Knots and Gravity by Baez.
A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties:

$V(f+g)=V(f) + V(g)$.
$V(\alpha f)= \alpha V(F)$.
$V(fg) = V(f)g(p) + V(g)f(p)$.

Can someone explain me what is the physical interpretation of tangent vectors and the above definition?

REPLY [5 votes]: There are a lot of good answers to this already, but since I just went through this material recently, I'll offer my perspective. The key fact that made viewing "tangent vectors" as derivations make sense to me is that there exists an isomorphism between all derivations at a point $p$ and all vectors based at $p$. Specifically, if

$X$ is an open subset of $\mathbb{R}^n$ and $p \in X$
$\mathcal{D}_p(X)$ represents the vector space of all derivations at $p$
$T_p(X)$ represents the vector space of all vectors $v$ based at $p$

we can define a linear bijection
$$
\phi:T_p(X) \rightarrow \mathcal{D}_p(X)
$$
by
$$
\phi(v) \rightarrow D_v
$$
where $D_v$ indicates the directional derivative operator in the direction $v$. Note that this definition makes sense because $D_v$ is itself a derivation.
Showing that $\phi$ is linear and injective is straightforward; proving surjectivity takes a little more work and requires some tools from calculus; specifically Taylor's theorem with remainder. Tu's Introduction to Manifolds works all of this out in detail on pages 13-14.