On covarient derivative

[Tsakanikas Nickos]

Bearing in mind this exercise , prove the following assertion :

Let \( \displaystyle V \) and \( \displaystyle W \) be two differential vector fields along the parametrized curve \( \displaystyle \alpha : I \longrightarrow S \) such that \( \displaystyle \Big| V(t) \Big| = \Big| W(t) \Big| = 1 , \, \forall t \in I \), where \( \displaystyle I \subset \mathbb{R} \) is an interval and \( \displaystyle S \) is a regular surface. Then \[ \displaystyle \Big[ \frac{DW}{dt} \Big] - \Big[ \frac{DV}{dt} \Big] = \frac{d \phi}{dt} \]where \( \displaystyle \phi \) is one of the differentiable determinations of the angle from \( \displaystyle V \) to \( \displaystyle W \).