We can compute the divergence of a vector filed $X$ on $\mathbb{R}^2$ expressed in polar coordinates $(r,\theta)$ in two ways: the first one is the classical formula
$$\text{div}(X)=\frac{1}{r}\frac{\partial(rX^r)}{\partial r}+\frac{1}{r}\frac{\partial X^\theta}{\partial\theta},$$
and the second one is the formula given by the Riemannian definition of the divergence (here $(x^1,x^2)=(r,\theta)$)
$$\text{div}(X)=\frac{\partial X^i}{\partial x^i}+\Gamma_{ij}^iX^j=\frac{\partial X^r}{\partial r}+\frac{\partial X^\theta}{\partial\theta}+\frac{1}{r}X^r.$$
The two expression are not the same: the term $\frac{\partial X^\theta}{\partial\theta}$ is rescaled differently. Why? I suppose that it has something to do with some kind of renormalisation.
For the Christoffel symbols I looked here.
Best Answer
The issue is that the first formula comes from taking components of $X$ with respect to the orthonormal basis $e_r,e_\theta$, whereas the differential geometry formula is in terms of components with respect to the coordinate basis $\partial/\partial r, \partial/\partial\theta$. In particular, the unit vector $e_\theta = \frac1r\partial/\partial\theta$. So the $X^\theta$ appearing in the two formulas are different.