What do I do with these christoffel symbols?

[Sparkyboy]

Hey guys I'm a bit new to GR and stuck on this question? :/. So we are given that:

d<sup>2xi/dλ2+$\Gamma$i_jk dxi/dλ dxj/dλ = 0

and asked to show that d/dλ(g_ijdxi/dλdxj/dλ) = 0

So I expanded using the product rule to get:

$\Gamma$i_jkd2xi/dλ2 dxj/dλ +$\Gamma$i_jk dxi/dλd2 xj/dλ2

Then rearranged the first equation to get:

d2xi/dλ2 = - $\Gamma$i_jkdxi/dλ dxj/dλ

and substituted for the second order differential equations. That's where I get stuck as I don't know how to get rid of the christoffel symbols. I read somewhere that they have a high degree of symmetry - so maybe I can change the dummy indices to get a symmetric form and they cancel? Very confused - any help would be appreciated.</sup>

 

[Matterwave]

I believe you are missing a term d(g_ij)/dλ*...?

I guess I don't follow exactly your "expand" term. If you skipped a lot of steps, I can't do them in my head haha.

There's something wrong with the indices though, you have two of the same indices "upstairs" (implied sum or not?), which almost never happens.