vector field along a curve

Let $M$ be a differentiable manifold and $\gamma :[a,b]\to M$ be a differentiable curve in $M$. Then a vector field along $\gamma$ is a differentiable map $\mathrm{\Gamma }:[a,b]\to TM$, the tangent bundle of $M$, which projects to $\gamma$ under the natural projection $\pi :TM\to M$. That is, it assigns to each point $t_0\in [a,b]$ a vector tangent to $M$ at the point $\gamma (t)$, in a continuous manner. A good example of a vector field along a curve is the speed vector $\dot{\gamma }$. This is the pushforward of the constant vector field $\frac{d}{dt}$ by $\gamma$, i.e., at $t_0$, it is the derivation $\dot{\gamma }(f)={\frac{d}{dt}(f\circ \gamma )|}_{t=t_0}$.