Let $M$ be a smooth Hausdorff second-countable $n$-manifold, and identify each [tangent space](/page/Tangent%20Space) $T_pM$ with the space of $\mathbb{R}$-linear derivations $C^\infty(M)\to\mathbb{R}$ at $p$. For every smooth vector field $X\in\mathfrak{X}(M)$, define $D_X:C^\infty(M)\to C^\infty(M)$ by $D_X(f)(p)=X_p(f)$ for every $f\in C^\infty(M)$ and every $p\in M$. Then $D_X$ is $\mathbb{R}$-linear and satisfies the Leibniz rule $D_X(fg)=fD_X(g)+gD_X(f)$ for all $f,g\in C^\infty(M)$. Conversely, if $D:C^\infty(M)\to C^\infty(M)$ is an $\mathbb{R}$-[linear map](/page/Linear%20Map) satisfying $D(fg)=fD(g)+gD(f)$ for all $f,g\in C^\infty(M)$, then there exists a unique smooth vector field $X\in\mathfrak{X}(M)$ such that $D=D_X$.