> [!definition] > > Let $X$ and $Y$ be Hausdorff [[Manifold|manifolds]] of class $C^p$, $F: X \to Y$ be a $C^p$-[[Diffeomorphism|diffeomorphism]], and $\xi \in \vf(X)$, then there exists a unique [[Vector Field|vector field]] $F_*\xi \in \vf(Y)$ such that $\xi$ and $F_*\xi$ are $F$-related, known as the **pushforward** of $\xi$ by $F$. > > *Proof*. Let > $ > F_*\xi(F(p)) = dF_{p}(\xi(p)) > $ > then $F_*\xi$ is the unique (rough) vector field that is $F$-related to $F_*\xi$. Since we can express $\xi$ as the composition > $ > \begin{CD} > Y @>{F^{-1}}>> X @>{\xi}>> TX @>{dF}>> TY > \end{CD} > $ > we have that $\xi$ is $C^{p - 1}$ as well.