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Just started self teaching myself differential geometry and tried to find the christoffel symbols of the second kind for 2d polar coordinates. I am checking to see if I did everything correctly.

With a line element of:

therefore the metric should be:

The christoffel symbols of the second kind can be found by:

And the non-zero christoffel symbols I found:

I noticed since
is symmetric it is non-zero when m=k so summing over k is not needed, i do not know if I missed anything by doing this.

With a line element of:

I noticed since

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