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Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in GR, the christoffel levi civita connection allows you to take derivatives across a manifold of two different vectors that are separated by some interval. What is being differentiated here? A little confused ...please help.