I am currently trying to define the Covariant Derivative, the question reads:
If $(u_1, u_2)$ are coords on a surface $S$, define the Covariant Derivative $$\frac{DV}{\partial u_i}$$ Of the Vector Field $V$ in the $u_i$-direction, in terms of ordinary Derivatives in $\mathbb{R}^3$.
Now, we have that: $$V_p = \sum_j \omega_j(p)\ U_j(p)$$ Thus, I write: $$\nabla_{u_i} V(p)=\sum_{j}u^i[\omega_j(p)] \ U_j(p)$$ How do I write the previous statement in terms of ordinary derivatives in $\mathbb{R}^3$?
What I told you in a previous reply is that,
$$ \nabla_{u_i} V(p) = \sum_j u_i[\omega_j(p)] \ U_j(p)$$
where $U_j$ is the standard frame of $\mathbb{R}^3$ and $\omega_j = V(p) \cdot U_j(p)$ by orthonormal expansion. I also told you that we've defined,
$$v[f] = \sum_j \frac{\partial f}{\partial x_j} \cdot v_j = \nabla f \cdot v$$
where $f$ is a smooth real-valued function and so you can rewrite the above formula for the covariant derivative as,
$$\nabla_{u_i} V(p) = \sum_j \nabla \omega_j \cdot u_i \ U_j(p)$$
As a standard vector in $\mathbb{R}^3$, we take $U_j = \textbf{e}_j = (0,0,...,x^j = 1,...,0)$. Hence, the coordinate representation of the covariant derivative is given by,
$$ v=\begin{pmatrix} \nabla\omega_1 \cdot u_i \\ \nabla\omega_2 \cdot u_i \\ \nabla\omega_3 \cdot u_i \end{pmatrix}$$
Thus, if you pick a coordinate system $(x_1,x_2,x_3)$ for $\mathbb{R}^3$, then for a smooth function $f$ we have the following,
$$\nabla f(p) = \begin{pmatrix} \frac{\partial f}{\partial x_1} &\frac{\partial f}{\partial x_2} &\frac{\partial f}{\partial x_3}\end{pmatrix}_p$$
and so you can simply $\nabla \omega_j \cdot u_i$ even more by differentiating $(V \cdot U_j)$. We now express $v$ in its compact and expanded form.
$$v = \begin{pmatrix} \frac{\partial \omega_1}{\partial x_1} & \frac{\partial \omega_1}{\partial x_2}& \frac{\partial \omega_1}{\partial x_3} \\ \frac{\partial \omega_2}{\partial x_1} & \frac{\partial \omega_2}{\partial x_2}& \frac{\partial \omega_2}{\partial x_3} \\ \frac{\partial \omega_3}{\partial x_1} & \frac{\partial \omega_3}{\partial x_2}& \frac{\partial \omega_3}{\partial x_3} \end{pmatrix} \cdot u_i = \textbf{A} u_i$$