One of the simplest examples would be to calculate the connection coefficients for the 3D Euclidean space using spherical polar coordinates. Here the line element is of the form [tex]ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2[/tex]. Could you try this one?
As an aside, have you studied and Lagrangian mechanics? If so, there is a method of obtaining the connection coefficients from the Euler-Lagrange equations which is sometimes less time consuming than using the definition involving the metric tensor.
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