About the covariant derivative - is this correct?

Question

Let $S \subset \mathbb{R}^3$ be a regular surface, the image of a parametrization $X: U \subset \mathbb{R}^2 \to S$ and $\alpha:I \to W \subset S$ be a curve in $S$. We can write $\alpha(t) = X(u(t), v(t))$, then the covariant derivative of $\alpha'$ at some $t \in I$ is given by:

$$\begin{align} \dfrac{\sf D\alpha'}{{\rm{d}}t} &= {\rm proj}_{T_{\large\alpha(t)}S} \ \alpha''(t) \\ &= \alpha''(t) - \langle\alpha''(t), N(u(t), v(t)) \rangle N(u(t), v(t)) \end{align}$$

Is this correct? If not, what am I getting wrong?

Answer 1

The expression I wrote is correct.