Definition:Tangential Acceleration of Smooth Curve

Definition
Let $I \subseteq \R$ be a real interval.

Let $M \subseteq \R^n$ be an embedded submanifold.

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

Let $\map {\gamma''} t$ be the acceleration of $\gamma$ at $t \in I$.

Let $T_{\map \gamma t} M$ be the tangent space at $\map \gamma t \in M$ for some $t \in I$.

Let $\pi^\top : T_{\map \gamma t} \R^n \to T_{\map \gamma t} M$ be the tangential projection.

The tangential acceleration of $\gamma$ at $t \in I$ is defined as:


 * $\map {\gamma} t^\top := \map {\pi^\top} {\map {\gamma} t}$