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I am reading a book that defines tangent vectors and vector fields on a manifold $M$ as derivations:

A vector field is defined as a linear function and derivation $C^{\infty}(M)$ to $C^{\infty}(M)$.
A tangent vector at point $p$ is a linear function from $C^{\infty}(M)$ to $\mathbb R$ that in addition has the property $$v_p(fg)=v_p(f)g(p)+f(p)v_p(g).$$

Now, clearly every vector field $v$ gives rise to tangents vectors via the functions $v_p(f)=v(f)(p)$.

But how can I tell that every tangent vector defined in this abstract way comes from a vector field?

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You use a partition of unity (really, all we need is one bump function) to define a smooth vector field $X$ with $X(p)=v_p$. Namely, choose a coordinate chart $U$ centered at $p$ in which $v_p = \dfrac{\partial}{\partial x_1}(p)$, say. Let $\phi$ be a bump function with $\phi(p) = 1$ and $\operatorname{supp}(\phi)\subset U$. Set $X = \phi\dfrac{\partial}{\partial x_1}$ as a vector field on your manifold.

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