I am having a hard time understanding covariant derivatives. My main problem is working with concrete example. So I was wondering if anybody could help me with explaining it by using simple example.
Let us say we have the curve
\begin{equation*} \gamma(x):=(x,x^2) \end{equation*}
in $\mathbb{R}^2$ with $\mathbb{x} \in \mathbb{R}$. Now suppose I have a vector field $X$ in $\mathbb{R}^2$ how do I write down the covariant derivate of $X$ along $\gamma$.