> [!definition] > > Let $X$ and $Y$ be [[Hausdorff Space|Hausdorff]] [[Manifold|manifolds]] of class $C^p$, $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], and $\xi \in \vf(X)$, $\xi' \in \vf(Y)$ be $C^{p - 1}$ [[Vector Field|vector fields]]. The vector fields $\xi$ and $\xi'$ are $f$-**related** if the following diagram commutes: > $ > \begin{CD} > X @>f>> Y \\ > @V{\xi}VV @VV{\xi'}V \\ > TX @>>df> TY > \end{CD} > $ > [!theorem] > > Let $X$ and $Y$ be smooth [[n-Manifold|n-manifolds]], $F: X \to Y$ be a smooth map, $\xi \in \vf(Y)$, and $\eta \in \vf(X)$. Then $\xi$ and $\eta$ are $F$-related if and only for any $U \subset Y$ and $f \in C^\infty(U)$, > $ > \xi(f \circ F) = (\eta f) \circ F > $ > In other words, $\xi$ and $\eta$ are $F$-related if and only if they are equal [[Vector Fields as Derivations|as derivations]]. > > *Proof*. Let $p \in X$, $(U, \varphi)$ be a chart at $p$ and $(V, \psi)$ be a chart at $f(p)$, then by expanding through the chain rule, > $ > \begin{align*} > \xi (f \circ F)(p) &= \xi(p) \cdot (f \circ F) \\ > &= D(f \circ F \circ \varphi^{-1})_{\hat p} \cdot \hat \xi(\hat p) \\ > &= D(f \circ \psi^{-1})_{\widehat{F(p)}} \cdot D(F_{U, V}) \cdot \hat \xi(\hat p) > \end{align*} > $ > and > $ > \begin{align*} > \eta f \circ F(p) &= D(f \circ \psi^{-1})_{\widehat{F(p)}} \cdot \hat \xi(\hat p) > \end{align*} > $ > If $\eta(p) = dF_p(\xi(p))$, then $\xi(f \circ F) = \eta f \circ F$. If $\xi(f \circ F) = \eta f \circ F$ for every $f \in C^\infty(U)$, then $\eta(p) = dF_p(\xi(p))$ and they are related.