christoffel symbols of the second kind: how exactly to calculate?

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From: sage/src/sage/calculus/test.py

def christoffel(i,j,k,vars,g):
     s = 0
     ginv = g^(-1)
     for l in range(g.nrows()):
         s = s + (1/2)*ginv[k,l]*(g[j,l].diff(vars[i])+g[i,l].diff(vars[j])-\
                                  g[i,j].diff(vars[l]))
     return s

var('th ph')
g = matrix(SR, [[1,0],[0,sin(th)^2]])
for i in [0,1]:
    for j in [0,1]:
        for k in[0,1]:
            print((i,j,k),christoffel(i,j,k, [th,ph], g))

(0, 0, 0) 0
(0, 0, 1) 0
(0, 1, 0) 0
(0, 1, 1) cos(th)/sin(th)
(1, 0, 0) 0
(1, 0, 1) cos(th)/sin(th)
(1, 1, 0) -cos(th)*sin(th)
(1, 1, 1) 0

SageManifolds version:

S2=manifolds.Sphere(2)     
Phi=S2.embedding()                      
sph.<th,ph>=S2.spherical_coordinates()  
E=S2.ambient()                          
g=E.metric() 
g1 = S2.metric('g1')
g1.set( Phi.pullback(g))
nab = g1.connection()
nab.display(coordinate_labels=False,only_nonzero=False)