Definition:Vector Field along Smooth Curve

Definition
Let $M$ be a smooth manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $I \subseteq \R$ be a real inverval.

Let $\gamma : I \to M$ be a smooth curve.

Let $V : I \to TM$ be a continuous map such that:


 * $\forall t \in I : \map V t \in T_{\map \gamma t} M$

Then $V$ is called the vector field along (smooth) curve $\gamma$.