I have to solve this one:
"Given a smooth curve $\gamma$ on a smooth manifold $M$, prove that $\gamma'$ is a smooth curve on the tangent bundle $TM$".
I have this idea:
let $(U,\varphi)$ a local chart on $M$ with local coordinates $(x_1,\ldots,x_n)$. If $\gamma(t_0)\in U$ then for $t$ close enough to $t_0$ we have $\gamma(t)=(\gamma_1(t),\ldots,\gamma_n(t))$ where $\gamma_i=x^i\circ\gamma$.
\begin{equation} \begin{array}{rcl} & \gamma & \\ \mathbb{R} &\longrightarrow & M \\ \gamma' & \searrow & \uparrow\pi \\ & & TM \end{array} \end{equation} So \begin{eqnarray*} \gamma'(t_0)&=&\left(\gamma(t_0),\gamma_{*t_0}\left(\left.\frac{d}{dt}\right|_{t=t_0}\right)\right)\\ &=&\left(\gamma(t_0),\sum_{i=1}^n\gamma'_i(t_0)\left.\frac{\partial}{\partial x^i}\right|_{t=t_0}\right)\\ &=&\left( \gamma_1(t_0),\ldots,\gamma_n(t_0),\gamma'_1(t_0),\ldots,\gamma'_n(t_0) \right) \end{eqnarray*} is a smooth curve on $TM$.
Is it all right?
Thanks a lot