show that $\displaystyle (\phi\circ\gamma)^{\prime}(t)=\phi_{*}(\gamma^{\pr ime}(t))$

where if $\displaystyle v\in T_pM$ is a tangent vector and $\displaystyle \phi:M\rightarrow N$ is a smooth function and we define

$\displaystyle (\phi_{*}v)(f)=v(f\circ\phi)$

$\displaystyle \gamma$ is a curve on $\displaystyle M$

$\displaystyle f:N\rightarrow \mathbb{R}$

(Pushing forward)