Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation.
Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), \cos(v))$$ and $$\alpha(u)=(R \cos(u) \cos(v_0), R \sin(u) \cos(v_0), R \sin(v_0)),$$ where $v_0=v(0)$ and is therefore a constant. Why is it that $$\nabla_{a^{'}}^{\mathbb{R}^{3}} E_1 = (\sin(u) \sin(v_0), - \cos(u) \sin(v_0), 0)?$$
Do you take the derivative of $E_1$ with respect to $u$ because $\alpha(u)$ is just a function of $u$? In addition, let's say that you switch $v_0$ with $v$ so that $\alpha(u)$ becomes $\alpha(u,v)$ does that imply that when taking the covariant derivatie you have to take the derivative of $E_1$ with respect to $u$ and $v$?