More on the usual spherical coordinates:
Using the definition of Christoffel symbols: \partial_i e_j =\Gamma^k_ij e_k
find the 6 distinct Christoffel symbols:
\Gamma^1_22, \Gamma^1_33, \Gamma^2_12, \Gamma^2_33, \Gamma^3_13, \Gamma^3_23
where 1=r, 2=\theta, 3=\phi
SEE spherical_coordinates.m

Consider the distance: ds^2=(da^2+\sin(a)^2 db^2)
with coordinates (a,b).  Find the metric tensor g_ij.  Find the Christoffel
symbols from the metric tensor.
Answer check: \Gamma^2_12 = \cot a
Write down the geodesic diff equation and use Mathematica to NDSolve it; use ParametricPlot
to display an example that does not look like a straight line
SEE geodesic_sphere.m

The Mathematica code handout details the 2D parabolic coordinate case.
In particular near the top right find the Christoffel symbols.
Using those results show how equ and eqv (equations for a geodesic)
come about.  Select some initial conditions and NDSolve.
ParametricPlot the results on the (u,v) plane; convert to
show the result is a straight line in (x,y)
SEE parabolic_christoffel.m