velocity vector of curve

Velocity vectors

For a curve on a smooth manifold, the velocity of at is the differential

Coordinate version

Given a chart with coordinate functions , we can define

using these coordinates (for sufficiently close to ).

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

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.

Computing differential with curve (locally)

Suppose is a smooth map between smooth manifolds, and . Then

for any smooth curve such that , , and .

Proof

For any , the velocity at of the composition (which is still a curve)

This follows from the chain rule (for the differential):

The lemma then follows directly.