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## Homework Statement

Hi,

I am reading Ray d'Inverno's book, 'Introducing Einstein's Relativity' and there is a particular derivation of the geodesic equation that I get stumped on (chapter 7). It is a variational method and the final equation is

df/dx_alpha-d/du{df/dx_alpha_dot}=0

where f is the Lagrangian, x_alpha, beta etc are the coordinates

and _dot denotes differenation with respect to an affine parameter.

Now you can carry this computation and derive the equivalent equation but using christoffel symbols. But in this derivation, one line is

f=g[beta,gamma]*dx_beta/du*dx_gamma/du

This equation is a sum over all indices (alpha, beta etc).

df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du

My question is shouldn't this last term be differentiated with the product rule, so that

df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du +

g[beta,gamma]*d/dx_alpha{dx_beta/du*dx_gamma/du}

f is a summation over all indices (ALPHA INCLUDED), so do we not need extra terms to determine these if we are differentiating with respect to alpha??

Why can we jump to just

df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du

???

Please help. I feel I'm missing a point here.