# Christoffel Symbols for Spherical Polar Coordinates

If we are given a line element;

$$ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2$$

We can easily then see that the metric and the inverse metric are;

$$g=\begin{pmatrix}1&0&0\\0&r^2&0\\0&0&r^2sin^2\theta\\\end{pmatrix}$$

$$g^{-1}=\begin{pmatrix}1&0&0\\0&r^{-2}&0\\0&0&r^{-2}(\sin\theta)^{-2}\\\end{pmatrix}$$

I have used the formula;

$$\Gamma^m_{ij}=\frac12 g^{ml}(\partial_jg_{il}+\partial_ig_{lj}-\partial_lg_{ji} )$$

Where upper indices indictate the inverse matrix to calculate my Christoffel symbols. My results are as follows:

$$\Gamma^1=\begin{pmatrix}0&0&0\\0&-r&0\\0&0&-rsin^2\theta\\\end{pmatrix}$$ $$\Gamma^2=\begin{pmatrix}0&\frac1r&0\\\frac1r&0&0\\0&0&-sin\theta cos\theta\\\end{pmatrix}$$ $$\Gamma^3=\begin{pmatrix}0&0&\frac1r\\0&0&cot\theta\\\frac1r&cot\theta&0\\\end{pmatrix}$$

However the results quoted by Wolfram Mathworld (http://mathworld.wolfram.com/SphericalCoordinates.html) are different, but only in that some entries have been swapped or moved about - all the values in both are the same.

I notice that the article uses the labelling convention $$\{r,\varphi,\theta\}$$, whereas I have used $$\{r,\theta,\varphi\}$$ (which is annoying, since the article uses my preferred notation which I have made a conscious effort to move away from due to none of my professors using it), and was wondering whether or not I made a fundamental mistake, or if the different labeling systems could account for the discrepancy.

The Christoffel symbols you dervied are indeed the correct ones for a spherical coordinate system $(r, \theta, \varphi)$.
If you do the same procedure for a system $(r, \varphi, \theta)$ (in the metric tensor, the entries $(22)$ and $(33)$ are now swapped) you will get the Christoffel symbols as stated on Wolfram Mathworld.
It is simply due to the order of $\theta$ and $\varphi$.