1. The problem statement, all variables and given/known data If the basic equation for the Christoffel symbol is [tex] \Gamma^l_{ki} = \frac{1}{2} g^{lj} (\partial_k g_{ij} + \partial_i g_{jk} - \partial_j g_{ki}) [/tex] so if you bring multiply the first metric into that equation, won't that turn the first two derivatives into derivatives of a mixed metric [tex] \Gamma^l_{ki} = \frac{1}{2} (\partial_k g^{l}_{i} + \partial_i g^{l}_{k} - g^{lj} \partial_j g_{ki}) [/tex] and then, wouldn't the first two terms go to zero, since they're just the derivative of the kronecker delta, which is constant? If that's correct, why not express the symbol instead as [tex] \Gamma^l_{ki} = \frac{1}{2} ( - g^{lj} \partial_j g_{ki}) [/tex]
Because g^{lj}*partial_k(g_{ij}) is not generally equal to partial_k(g^{lj}*g_{ij})=0. The metric components are not generally constants. Use the product rule on the second expression.