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I read here, Curvilinear page 11, that $$\frac{\partial}{\partial x^i}e_j=\Gamma^k_{ij}e_k$$ where the $e_i$'s are basis vectors. There seems to be some connection, but when I calculate it, for example in polar coordinates, I don't get this. For example, $$\frac{\partial}{\partial\theta} \hat{r}=\hat{\theta}$$ but when I use the formula I wrote above, I get $\frac{1}{r}\hat{\theta}$, close, but no cigar. Is the page wrong, or have I messed up?

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As a hint, try re-writing the time-dependent basis vectors $\hat{r}$ and $\hat{\theta}$ in terms of the time-independent basis vectors $\hat{i}$ and $\hat{j}$. Compute the derivative $\frac{\partial}{\partial\theta}$ and see if the resulting expression can be written in terms of the original basic vectors $\hat{r}$ and $\hat{\theta}$. A visual aid is included below ($\hat{e}_r=\hat{r}$ and $\hat{e}_\theta=\hat{\theta}$). enter image description here

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