Vector Fields as Derivations

Statements

Proposition. Let $X\in {\mathcal{C}}^{\infty }$ be a vector field on an open subset $U\subset {\mathbb{R}}^n$ and $f\in {\mathcal{C}}^{\infty }(U)$. Define the function $Xf$ on $U$ given, for every $p\in U$, as $(Xf)(p)=X_{p}f$. In terms of the basis of the vector field, $(Xf)(p)$ is given by $(Xf)(p)={\sum }_{i=1}^n{a}^i(p)\frac{\partial f}{\partial x_i}(p).$ By letting, $a\in {\mathcal{C}}^{\infty }(U,\mathbb{R})$ be a function, the vector field $X$ gives rise to the derivation

\mathcal{C}^\infty(U)&\to&\mathcal{C}^\infty\\ f&\mapsto&Xf\;. \end{array}$$ ## References [[Tu, L. W. - An introduction to manifolds|@Tu11 Section 2.5]] and [[Boothby, W. M. - An introduction to differentiable manifolds and riemannian geometry|@Boo75 Theorem 4.2]]