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 $\Gamma:[a,b]\to TM$ , 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 $\dot\gamma(f)=\frac{d}{dt}(f\circ\gamma)|_{t=t_0}$ .