velocity vector of curve

Velocity vectors

For a curve $\gamma :J\to M$ on a smooth manifold, the velocity of $\gamma$ at $t_0\in J$ is the differential ${\gamma }^{\prime }(t_0)=d\gamma \left(\frac{d}{dt}{|}_{t_0}\right)\in T_{\gamma (t_0)}M$

Coordinate version

Given a chart $(U,\varphi )$ with coordinate functions $x^i$, we can define

$$\gamma (t)=({\gamma }^1(t),\dots ,{\gamma }^n(t))$$

using these coordinates (for $t\in U$ sufficiently close to $t_0\in U$).

Using this coordinatized version, we can see that by the chain rule

$${\gamma }^{\prime }(t_0)=\frac{d{\gamma }^i}{dt}(t_0)\frac{\partial }{\partial x_i}{|}_{\gamma (t_o)}$$

Note, this is exactly what we would think it is. It is very similar to what happens with a curve in Euclidean space, which is just the derivative of each of the components.