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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 $\gamma$.

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In $\mathbb R^2$ it is easy, if say $X=a(x,y)\mathrm dx+ b(x,y)\mathrm dy$, then you have to consider the derivatives of both $a$ and $b$ in the $\gamma'(t)$ direction (and speed) at each $t\in \mathbb R$ moment, i.e. $\nabla_\gamma X(t) = (\partial_{\gamma'(t)}a)\mathrm dx + (\partial_{\gamma'(t)}b)\mathrm dy$ For more sophisticated spaces, a Riemmann metric (local scalar products) is needed on the manifold.

The point is, that for a manifold $M$, and $z\in M$, for any individual vector $v\in T_zM$ one can define $\nabla_v X$ for any vector field defined in a neighborhood of $z$.

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    @BerciPecsi One final stupid question. Is it correct that we are profiting now from the fact that the Christoffel symbols are identically zero for Euclidean co-ordinates? If I had written down the problem in polar co-ordinates you would have to do more work.2012-09-19