Let $S$ be a surface in $R^3$ with induced Riemannian metric. let $\gamma : I\to S$ be a smooth curve on $S$ and $V$ a vector field tangent to $S$ along $\gamma$ . $V$ can be thought of as a smooth function $V : I \to R^3$ Show that $V$ is parallel if and only if $V'(t)$ is orthogonal to $T_{\gamma(t)}M$, where $V'(t)$ is the usual time derivative of $V$ .
I know that a vector field $V$ along a curve $\gamma$ is parallel if $ \frac{DV}{dt}=0$ i.e the co variant derivative of $V$ is zero. Then I am trying to solve one direction by using the definition of $\frac{DV }{dt}$ but I got stuck there.I need some help.