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The covariant differentiation or the levi civita connection represented by the Christoffel symbol, $$\Gamma_{\alpha,\beta}^{\gamma} \frac{dx^k}{dy^{\gamma}} =?= \Gamma_{i,j} \frac{dx^idx^j}{dy^\alpha dy^\beta} + \frac{(dx^2)^k}{dy^\alpha dy^\beta}. $$ how are these two equal using g_(alpha)(beta)=g^(i,j)dx^(i)dx^(j)/dy^(alpha)dy^(beta).

In these frameworks of local coordinate system, {x^i}, {y^alpha}, how one defines them in terms of a multilinear tensor Phi?

Thanks

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 I suggest that you try to rewrite your question more clearly. I don't understand what's being asked. – Deane Yang Mar 7 2012 at 1:42
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 In addition to not understanding the question, the LHS of the equation has $k, \alpha, \beta$ as free indexes, the first term in the RHS does not have k and the second does not make sense (perhaps 2 <-> k ?) – Reimundo Heluani Mar 7 2012 at 9:38
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 Are you trying to ask about the change of variables formula for Christoffel symbols? en.wikipedia.org/wiki/… If that is the case, perhaps this forum is not the most suitable place for this question. Consider asking your question at math.stackexchange.com instead. (And if you do ask it there, please try to write the questions more clearly, right now it is very hard to understand what you are actually asking about.) – Willie Wong Mar 7 2012 at 9:54

closed as not a real question by Deane Yang, Willie Wong, David Roberts, Andy Putman, Andres Caicedo Mar 8 2012 at 4:16

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